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--- /dev/null +++ b/0dFPT4oBgHgl3EQfTjRk/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1e28a1e22c3830f23ea4c16425c1fefd1148b16fd962b089b2b8d0f603f48cd5 +size 3080237 diff --git a/0tAzT4oBgHgl3EQfevwc/content/tmp_files/2301.01440v1.pdf.txt b/0tAzT4oBgHgl3EQfevwc/content/tmp_files/2301.01440v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c9700ecd27415d5c09a0e8b95e0c1cd142118761 --- /dev/null +++ b/0tAzT4oBgHgl3EQfevwc/content/tmp_files/2301.01440v1.pdf.txt @@ -0,0 +1,867 @@ +1 +Scalable Optimal Design of Incremental Volt/VAR +Control using Deep Neural Networks +Sarthak Gupta, Graduate Student Member, IEEE, Ali Mehrizi-Sani, Senior Member, IEEE, +Spyros Chatzivasileiadis, Senior Member, IEEE, and Vassilis Kekatos, Senior Member, IEEE +Abstract—Volt/VAR control rules facilitate the autonomous +operation of distributed energy resources (DER) to regulate +voltage in power distribution grids. According to non-incremental +control rules, such as the one mandated by the IEEE Standard +1547, the reactive power setpoint of each DER is computed as a +piecewise-linear curve of the local voltage. However, the slopes +of such curves are upper-bounded to ensure stability. On the +other hand, incremental rules add a memory term into the +setpoint update, rendering them universally stable. They can +thus attain enhanced steady-state voltage profiles. Optimal rule +design (ORD) for incremental rules can be formulated as a bilevel +program. We put forth a scalable solution by reformulating ORD +as training a deep neural network (DNN). This DNN emulates the +Volt/VAR dynamics for incremental rules derived as iterations of +proximal gradient descent (PGD). The rule parameters appear as +DNN weights. To reduce the DNN depth, we leverage Nesterov’s +accelerated PGD iterations. Analytical findings and numerical +tests corroborate that the proposed ORD solution can be neatly +adapted to single/multi-phase feeders. +Index Terms—IEEE Standard 1547.8, incremental control +rules, multiphase feeders, proximal gradients, gradient backprop- +agation, deep neural networks. +I. INTRODUCTION +Local Volt/VAR (Volt-Ampere Reactive) control facilitates +voltage regulation on distribution grids by providing reactive +power compensation from DERs equipped with smart invert- +ers. Different from centralized control schemes which incur +large computational and communication burden, local rules +decide DER setpoints based on local measurements. Volt/VAR +control rules can be categorized into non-incremental and +incremental ones. The former compute DER reactive power +setpoints based on local voltage readings. The IEEE Stan- +dard 1547.8 prescribes such non-incremental control rules +as piecewise-linear functions of voltage [1]. On the other +hand, incremental Volt/VAR rules compute the change in VAR +setpoints as a function of voltage [2]–[6]. +The existing literature on designing Volt/VAR control rules +can be classified into stability- and optimality-centric works. +Stability-centric works study the effect of Volt/VAR rules as a +closed-loop dynamical system, which may be rendered unsta- +ble under steep slopes of non-incremental rules [7], [8]. In fact, +to ensure stability, non-incremental rules may have to compro- +mise on the quality of their steady-state voltage profile [5], +[8]. Incremental rules however do not experience stability +limitations and can thus achieve improved voltage profiles +compared to their non-incremental counterparts. Nonetheless, +such improvements may come at the expense of longer settling +times of the associated Volt/VAR dynamics [8]. +Optimality-centric works focus on designing stable control +rules to minimize a voltage regulation objective. To this end, +optimization-based strategies have been employed to design +affine non-incremental rules using heuristics [9]–[11]. Two of +our recent works in [12] and [13] have addressed the problem +of optimally designing the slope, deadband, saturation, and +reference voltage. Reference [12] performs ORD via a bilevel +optimization applicable to single-phase feeders. Reference [13] +proposes DNN-based digital twins that emulate Volt/VAR +dynamics, and reformulates ORD as a DNN training task for +single-/multi-phase feeders. +This letter deals with optimally selecting the shape of +incremental Volt/VAR control rules, with contributions on +three fronts: c1) Although this optimal rule design (ORD) +task can be posed as a mixed-integer nonlinear optimization +program, it does not scale well with the numbers of DERs, +nodes, and grid loading scenarios. To address this challenge, +the genuine idea here is to reformulate ORD as a deep- +learning task and judiciously adapt the fast software modules +widely available for training deep neural networks (DNNs). +We have put forth a similar approach for designing non- +incremental control rules in [13]. However, migrating from +non-incremental to incremental rules is non-trivial due to the +different curve shapes, stability, and settling time properties. +c2) To further expedite ORD for incremental rules, we suggest +implementing accelerated Nesterov-type variants of the rules +to yield a shallower DNN emulator. c3) We also establish the +convergence of incremental rules on multiphase feeders. +Recently, reference [14] deals with the optimal design of +incremental rules. It uses DNNs with a single hidden layer +to model piecewise-linear functions and formulates ORD as +a reinforcement learning task. While [14] also utilizes DNNs +to design incremental rules, we delineate from it in several +ways. Reference [14] focuses on voltage control during tran- +sient dynamics, whereas this work aims at ORD to drive +steady-state voltages closer to unity and over different grid +loading scenarios. Reference [14] utilizes a DNN to model +the piecewise-linear mapping of the rule. In contrast, this +work develops a DNN-based digital twin that emulates end- +to-end Volt/VAR dynamics. Lastly, we provide stability and +convergence analysis for single- and multiphase feeders alike, +whereas [14] applies only to single-phase feeders. +The rest of this letter is organized as follows. Section II +models the feeder and discusses non-incremental and incre- +mental Volt/VAR control rules. Section III formulates DNN- +based digital twins for Volt/VAR dynamics of incremental +rules, and their accelerated version. It also presents ORD +arXiv:2301.01440v1 [math.OC] 4 Jan 2023 + +2 +Fig. 1. Non-incremental Volt/VAR control rule provisioned by the IEEE Std. +1547 for the interconnection of DERs [1]. +for single-phase feeders as a deep learning task. Section IV +extends the ORD process to multiphase feeders. The incre- +mental rules are then benchmarked against non-incremental +rules from [13] using tests on real-world data, in Section V. +The letter is concluded in Section VI. +II. VOLT/VAR CONTROL RULES +Consider a radial feeder serving N buses equipped with +DERs, indexed by n. Let (qℓ, q) collect reactive loads and +generations at all nodes. Vectors (p, v) collect the net active +power injections and voltage magnitudes at all nodes. The +impact of q on v can be approximately captured using the +linearized grid model [13] +v ≃ Xq + ˜v +(1) +where ˜v := Rp − Xqℓ + v01 models the underlying grid +conditions, and v0 is the substation voltage. Vector ˜v rep- +resents the impact of non-controlled quantities (p, qℓ) on +voltages. Matrices (R, X) depend on the feeder topology. For +single-phase feeders, they are symmetric positive definite with +positive entries [15]. For multiphase feeders, they are non- +symmetric and have positive and negative entries [5], [13]. +Vector q in (1) carries the reactive injections by DERs +we would like to control. Per the non-incremental rules of +the IEEE Std. 1547 [1], DER setpoints are decided based +on the Volt/VAR curve of Fig. 1, which is parameterized by +(¯v, δ, σ, ¯q). The standard further constrains these parameters +within a polytopic feasible set [1], [12]. The negative slope of +the linear segment of the curve in Fig. 1 can be expressed as +α := +¯q +σ − δ . +The interaction of Volt/VAR rules with the feeder gives +rise to nonlinear dynamics. These dynamics are stable if +∥ dg(α)X∥2 < 1, where dg(α) is a diagonal matrix carry- +ing the rule slopes over all buses on its diagonal [7]. The +equilibrium setpoints for DERs cannot be expressed in closed +form. However, they coincide with the minimizer of the convex +optimization problem [7] +min +−¯q≤q≤¯q +1 +2q⊤Xq+q⊤(˜v−¯v)+ 1 +2q⊤ dg−1(α)q+δ⊤|q| (2) +where |q| applies the absolute value on q entrywise. Prob- +lem (2) depends on rule parameters (¯v, δ, α, ¯q) across all buses, +collected in the 4N-long vector z := (¯v, δ, α, ¯q). We denote +by qz(˜v) the equilibrium setpoints, and by +vz(˜v) = Xqz(˜v) + ˜v +(3) +the related equilibrium voltages reached by Volt/VAR rules +parameterized by z under grid conditions ˜v. +Optimal rule design (ORD) can be stated as the task of +selecting z to bring equilibrium voltages vz(˜v) close to unity. +To cater to diverse conditions, the utility may sample loading +scenarios {˜vs}S +s=1 for the next hour, and find z as +z∗ ∈ arg min +z +F(z) := 1 +S +S +� +s=1 +∥vz(˜vs) − 1∥2 +2 +(ORD) +subject to (3) and z ∈ Z. +Once found, the customized rules z∗ are sent to DERs to +operate autonomously over the next hour. Note that vz(˜vs) +depends on z because the equilibrium setpoints qz(vs) in (3) +are the minimizers of problem (2), which is parameterized by +z. When solving (ORD) for non-incremental rules, the feasible +set Z consists of the polytopic constraints imposed on z by +the IEEE Std. 1547 as well as additional constraints on α +to ensure ∥ dg(α)X∥2 < 1; see [12]. Therefore, the feasible +set Z can be quite confined. This can lead to less desirable +voltage profiles; that is, higher objective values F(z∗). +The aforesaid issue can be addressed by replacing the non- +incremental Volt/VAR rules of IEEE Std. 1547 by incremental +ones as suggested in [2]–[6]. Incremental rules express the +change rather than the actual value in setpoints as a function +of voltage. One option for incremental rules is to implement +a proximal gradient descent (PGD) algorithm solving (2) +as proposed in [5]. In this case, the control rule coincides +with the PGD iterations, which are implemented by DERs +in a decentralized fashion. Using incremental rules, set Z is +enlarged as now we only need to ensure +z ≥ 0 +0.95 · 1 ≤ ¯v ≤ 1.05 · 1 +and that ¯q are within the reactive power ratings of the DERs. +The PGD algorithm is an extension of gradient descent to +handle constraints and non-differentiable costs [5]. At iteration +t, PGD proceeds with two steps: s1) It first computes the gradi- +ent of the first two terms of F(z), that is Xqt+˜v−¯v = vt−¯v. +Here qt is the latest estimate of the minimizer of (2); s2) PGD +then updates qt+1 as the minimizer of +min +−¯q≤q≤¯q +1 +2q⊤ dg−1(α)q + δ⊤|q| + 1 +2µ∥q − (vt − ¯v)∥2 +2 (4) +for a step size µ > 0. The last problem involves the last two +terms in the cost of (2) regularized by the Euclidean distance +of q to the gradient (vt − ¯v) computed in step s1). +Converting PGD to control rules, step s1) is performed by +the physics of the feeder when injecting qt and measuring +the local voltage deviations. Step s2) is run by each DER +independently as (4) is separable across buses. Using the +subdifferential, solving (4) provides the update [5] +yt +n = ˜αn · +� +qt +n − µ(vt +n − ¯vn) +� +(5a) + +fq +UA3 +qt+1 +n += gn +� +yt +n +� +(5b) +where gn(yn) is the proximal operator +gn(yn) := +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� ++¯qn +, yn > qn + µ˜δn +yn − µ˜δn +, µ˜δn < yn ≤ qn + µ˜δn +0 +, − µ˜δn ≤ yn ≤ µ˜δn +yn + µ˜δn +, − qn − µ˜δn ≤ yn < −µ˜δn +−qn +, yn < −qn − µ˜δn. +(6) +and the new parameters (˜αn, ˜δn) are defined as +˜αn := +1 +1 + µ/αn +and +˜δn := +δn +1 + µ/αn +. +The proximal operator is plotted in the top panel of Figure 2. +Note that in (5), rule parameters are transformed from repre- +sentation z = (¯v, δ, α, ¯q) to representation ˜z := (¯v, ˜δ, ˜α, ¯q). +This is without loss of generality as the transformation is a +bijection, and so one can work exclusively with ˜z. The feasible +set ˜Z for ˜z is similar to Z with the addition that ˜α ≤ 1. As +with non-incremental rules, the rules in (6) are driven by local +data, but now qt+1 +n +depends on (vt +n, qt +n), and not vt +n alone. Both +types of rules solve (2). Hence, they both converge to the same +equilibrium. The advantage of incremental rules is that they are +stable for all α as long as µ < 2/λmax(X); see [5]. It is worth +stressing that z does not have the same physical interpretation +as in non-incremental rules (slopes, deadband, or saturation), +though z parameterizes (2) for both rules. +Accelerated incremental rules. Although PGD rules en- +large Z, their settling times can be long. They reach an +ε-optimal cost of (2) within − 2 log ε +log 2 κ (X) iterations. Here +κ(X) := λmax(X)/λmin(X) is the condition number of +X. References [5], [16] put forth accelerated incremental +rules based on accelerated PGD (APGD). These rules need +− 2 log ε +log 2 +� +κ (X) iterations to attain an ε-optimal cost, and take +the form +˜yt +n := (1 + βt) yt +n − βtyt−1 +n +(7a) +qt+1 +n +:= gn +� +˜yt +n +� +(7b) +where βt := t−1 +t+2, while yt +n and gn(yn) are as defined in (5a) +and (6). Updates (7) remain local, but introduce additional +memory as qt+1 +n +depends on (vt +n, qt +n) and (vt−1 +n +, qt−1 +n +). +III. DEEP LEARNING FOR OPTIMAL RULE DESIGN (ORD) +IN SINGLE-PHASE FEEDERS +Solving (ORD) is challenging as it is a nonconvex bilevel +program. Although it can be modeled as a mixed-integer +nonlinear program, such an approach does not scale well with +the number of DERs and/or scenarios for non-incremental +rules [13]. Seeking a more scalable solution, we reformulate +(ORD) as a deep learning task. The key idea is to design +a DNN that emulates Volt/VAR dynamics under the control +rule of (5). To this end, note that gn(yn) is a piecewise- +linear function with four breakpoints [5]. Interestingly, this +operator can be expressed as the superposition of four rectified +linear units (ReLUs) as illustrated in Fig. 2, where ReLUs are +denoted by ρ(·). The intercepts of the ReLUs depend linearly +on (˜δn, ¯qn). +Fig. 2. Proximal operator g(y) expressed as a sum of four shifted rectified +linear units (ReLUs). +Fig. 3. +A DNN emulating the accelerated incremental rules of (7). Plain +incremental rules can be modeled by dropping the second layer (setting βt = +0) and ignoring output yt +n. +Building on this, one APGD iteration for DER n can be +implemented by the 4-layer DNN in Fig. 3, whose weights +depend affinely on (¯vn, ˜δn, ˜αn, ¯qn). This DNN takes (qt +n, vt +n) +as its input, and computes (qt+1 +n +, yt +n) at its output. It is +termed ICn and will be used as a building block to emulate +Volt/VAR dynamics. This is accomplished by the recursive +neural network (RNN) shown in Fig. 4. Here blocks ICn are +arranged vertically to model the parallel operation of DERs. +Their outputs qt+1 are multiplied by X, and the new voltage +is computed as vt+1 = Xqt+1 + ˜v. This is repeated T times. +Thanks to the RNN structure, there is weight sharing, so the +number of DNN weights is 4N rather than 4NT. +The RNN takes a grid loading vector ˜vs as its input, the rule +parameters ˜z as weights, and computes the voltages vT +˜z (˜vs) +at time T at its output. For the output vT +˜z (˜vs) to approximate +well equilibrium voltages, the depth T can be chosen by the +convergence rate of PGD as follows. +Proposition 1. For the DNN of Fig. 4 to ensure ∥Φ (˜v; z) − +v∗(z)∥2 ≤ ϵ1 ∀ ˜v, its depth T should satisfy +T ≥ +�κ − 1 +2 +� +log +�2∥X∥2∥ˆq∥2 +ϵ1 +� +. +(8) + ++qH +b+ +qp +-uon +一 +qn +t +p +t +p +-uon +p4 +Fig. 4. +Recurrent neural network (RNN) implementation for accelerated +incremental Volt/VAR control rules. +Proof: From the control rule of (5b), it follows that +∥qt − q∗∥2 = ∥g +� +yt� +− g (y∗) ∥2 +≤ ∥yt − y∗∥2 += ∥ dg(˜α)(I − µX) +� +qt−1 − q∗� +∥2 +≤ ∥ dg(˜α)∥2 · ∥I − µX∥2 · ∥qt−1 − q∗∥2 +≤ ∥I − µX∥2 · ∥qt−1 − q∗∥2. +(9) +The first inequality stems from the non-expansive property of +the proximal operator g. The next equality follows from (5a). +The second inequality from the sub-multiplicative property of +the spectral norm. The last inequality follows by the definition +of spectral norm and because ˜αn ≤ 1 for all n. +If ∥I − µX∥2 < 1, inequality (9) implies that the dynamics +in (5) are a non-expansive mapping, and thus, are stable and +converge to q∗. Condition ∥I − µX∥2 < 1 holds when µ < +2/λmax(X). The norm ∥I − µX∥2 achieves its minimum of +� +1 − +2 +κ+1 +� +when +µ0 := +2 +λmax(X) + λmin(X). +Plugging µ0 into (9) and unfolding the dynamics over t +provides +∥qt − q∗∥2 ≤ +� +1 − +2 +κ+1 +�t +∥q0 − q∗∥2 +≤ 2 +� +1 − +2 +κ+1 +�t +∥ˆq∥2. +For +the +voltage +approximation +error +∥vT − v∗∥2 += +∥X +� +qT − q∗� +∥2 at time T to be smaller than ϵ1, we need +∥vT − v∗∥2 ≤ 2∥X∥2 · ∥ˆq∥2 · +� +1 − +2 +κ + 1 +�T +≤ ϵ1. +This can be achieved by selecting T such that +T ≥ +log +� +2∥X∥2∥ˆq∥2 +ϵ1 +� +log +� +1 + +2 +κ−1 +� +≥ +�κ − 1 +2 +� +log 2∥X∥2∥ˆq∥2 +ϵ1 +. +where the last inequality follows from log(1 + x) ≤ x. +Plugging the values ∥X∥2 = 0.463 and κ = 848 for the +IEEE 37-bus feeder, ∥ˆq∥2 = 0.1, and ϵ1 = 10−5 in (8), yields +T ≥ 2, 892 layers, which is relatively large. A key contributor +to this large T is the κ term in (8). This promulgates the +adoption of accelerated rules (7), which are known to have +O(√κ) dependence. Interestingly, during implementation, one +does not need to fix T to the above worst-case bounds. +Leveraging dynamic computation graphs offered by Python +libraries such as Pytorch, one may determine T ‘on the +fly’ depending on the convergence of vt between pairs of +successive layers. +Since the RNN emulates Volt/VAR dynamics, it can surro- +gate vz(˜vs) in (ORD). Then (ORD) can be posed as training +a DNN over its weights ˜z ∈ +˜Z or z ∈ Z. Grid loading +scenarios {˜vs}S +s=1 are treated as features and equilibrium +voltages vz(˜vs) as predictions that should be brought close to +the target value of 1 for scenarios s. The DNN can be trained +using stochastic projected gradient descent (SPGD) as [13] +˜zi+1 = +� +˜zi − λ +2B ∇˜zi +� � +s∈Bi +∥Φ(˜vs; ˜z) − 1∥2 +2 +�� +˜ +Z +(10) +where λ > 0 is the learning rate; set Bi is a batch of +B scenarios; and [·] ˜ +Z is the projection onto +˜Z. Since +˜Z +consists of simple box constraints, projection essentially means +clipping the values to the box. Lastly ∇˜zi(·) represents the +gradient with respect to ˜z evaluated at ˜z = ˜zi, and is +calculated efficiently thanks to gradient back-propagation. +Although our DNN-based ORD assumed PGD-based rule, it +may be applicable to other incremental rules too. +A discussion about control rules and their DNN-based +emulators is due. Recall that all three types of Volt/VAR +control rules (non-incremental, incremental, and accelerated +incremental) reach the same equilibrium voltages, if stable. +The emulators aim at computing these equilibrium voltages. +A natural question is whether the DNN emulator could im- +plement a rule of a type different from the rule actually +implemented on the feeder. This may be desirable to leverage +the advantages of two types. Some caution is needed here. If +the feeder implements non-incremental rules, but incremental +rules converge faster to equilibrium voltages, it makes sense +for the emulator to implement incremental rules. Of course, +in this case, stability constraints on the non-incremental rules +have to be enforced during DNN training. The reverse is not +recommended: If the emulator implements non-incremental +rules, its parameters z should be constrained to be stable and +that would be a restriction of the actual ORD problem. Finally, +given the convergence advantage of accelerated incremental +rules, they are always preferable over plain incremental rules +for the DNN implementation. This showcases the utility of +accelerated control rules even if they are not actually imple- +mented on the feeder. +IV. DEEP LEARNING FOR OPTIMAL RULE DESIGN (ORD) +IN MULTIPHASE FEEDERS +In multiphase feeders, matrix X is non-symmetric and has +both positive and negative entries. Therefore, the rule analysis +and design of Section III has to be revisited. For example, +equilibrium setpoints cannot be found as the minimizers of + +IC1 +七 +q +t+1 +q +IC2 +t +- +t +t-1 +IC N5 +an optimization problem as with (2). Moreover, increasing q +does not mean that all voltages increase. +In multiphase feeders, the non-incremental rules of IEEE +Std. 1547 remain stable as long as ∥ dg(α)X∥2 < 1. This is +the same condition as in the single-phase setup. How about +the stability and equilibrium of incremental rules in multiphase +feeders? Recall that for single-phase rules, incremental rules +were obtained as the PGD iterations solving (2). Lacking an +equivalent inner optimization for multiphase feeders precludes +a similar approach here. Despite the incremental rules of (5) +do not correspond to PGD iterates anymore, they can still be +shown to be stable for multiphase feeders. +Proposition 2. Let UΛU⊤ be the eigen-decomposition of +matrix XX⊤. The incremental rules of (5) are stable for +multiphase feeders if their step size is selected as µ < +λmin +� +Λ−1/2U⊤ � +X + X⊤� +UΛ−1/2� +. +The claim follows readily by adopting the proof of Proposi- +tion 1: If µ is selected as above, then ∥I − µX∥2 < 1 follows +from [5, Prop. 6]. Similar to the single-phase case, incremental +rules in multiphase feeders allow us to enlarge the feasible +set Z of rule parameters z. It is worth stressing that different +from the single-phase setting, incremental and non-incremental +rules do not converge to the same equilibrium on multiphase +feeders. +The ORD task for multiphase feeders can also be formulated +as a deep-learning task, with some modifications. Firstly, +matrices R and X need to be altered. Secondly, the DNNs +for multiphase feeders have 12N trainable parameters, since +each layer consists of 3N building modules corresponding +to bus/phase (node) combinations. Lastly, the step size has +to be selected per Proposition 2. Adopting the proof of +Proposition 1, we next find the minimum DNN depth in +multiphase feeders. +Proposition 3. Let the DNN of Fig. 4 implement the incre- +mental rules of (5) on multiphase feeders with µ selected per +Proposition 2. The DNN depth T ensuring voltage approxi- +mation error ∥Φ (˜v; z) − v∗(z)∥2 ≤ ϵ1 is +T ≥ +log +ϵ1 +2∥X∥2∥ˆq∥2 +log ∥I − µX∥2 +. +We next numerically evaluate the proposed DNN-based +ORD approach in single- and multiphase feeders, and contrast +the performance of incremental control rules with that of non- +incremental rules. +V. NUMERICAL TESTS +We benchmark the performance of DNN-based incremental +rules against non-incremental rules from [13] on single- and +multiphase feeders. Real-world data were sourced from the +Smart* project on April 2, 2011 [17], as explained in [13]. +The DNNs were implemented and trained using Pytorch. +We first compare (non)-incremental rules, both designed +via DNN training for the single-phase IEEE 37-bus feeder +of Figure 5. Homes with IDs 20–369 were averaged 10 +at a time and successively added as active loads to buses +2–26 as shown in Fig. 6. Active generation from solar +Fig. 5. +The IEEE 37-bus feeder converted to single-phase. Node num- +bering follows the format node number {panel ID}. DERs at buses +{6, 9, 11, 12, 15, 16, 20, 22, 24, 25} provide reactive power control; the rest +operate at unit power factor. +TABLE I +INCREMENTAL VS. NON-INCREMENTAL VOLT/VAR CONTROL RULES ON +THE SINGLE-PHASE IEEE 37-BUS FEEDER +Time +q = 0 +Non-incremental +Incremental +Obj. (p.u.) +Time (s) +Obj. (p.u.) +Time (s) +Obj. (p.u.) +1 pm +3.01 · 10−3 +37.98 +3.68 · 10−4 +39.39 +3.66 · 10−4 +2 pm +3.13 · 10−3 +42.93 +4.26 · 10−4 +37.91 +4.25 · 10−4 +3 pm +4.24 · 10−3 +45.02 +8.59 · 10−4 +34.97 +8.50 · 10−4 +4 pm +2.12 · 10−3 +48.30 +1.47 · 10−4 +38.52 +1.48 · 10−4 +5 pm +8.53 · 10−4 +47.37 +9.70 · 10−5 +374.01 +6.90 · 10−5 +panels was also added, as per the mapping in Fig. 6. +Buses {6, 9, 11, 12, 15, 16, 20, 22, 24, 25} were assumed to +host DERs with Volt/VAR control customized per bus. +Incremental rules were simulated in their accelerated ren- +dition. Both sets of rules were trained over S = 80 scenarios +and 200 epochs with a learning rate of 0.001, using the +Adam optimizer, and setting µ = 1 for incremental rules. To +ensure repeatability, the results were repeated across several +time periods between 1–6 PM, and are compiled in Table V. +Incremental rules obtained marginally lower objectives than +non-incremental rules across all periods, with a somewhat +significant difference for the 5 PM period. This behavior is +explained because incremental rules allow for a larger set Z. +DNN-based incremental control rules were also contrasted +with their non-incremental ones on the multiphase IEEE 13- +bus feeder, using the testing setup from [13]. Active loads were +sampled 10 at a time from homes with IDs 20-379 and added +to all three phases for the buses 1-12. Figure 6 also shows the +solar panel assignments shown in Fig 6 for solar generation. +Lastly, nine DERs with inverters were added across phases +and bus indices as shown in Fig 6. +The learning rates for non-incremental and incremental +DNNs were set as 0.1 and 0.001, respectively, with the design + +Z +Z +Z +Z +Z +Z6 +Fig. 6. Multiphase IEEE 13-bus distribution feeder. +TABLE II +INCREMENTAL VS. NON-INCREMENTAL VOLT/VAR CONTROL RULES ON +THE MULTIPHASE IEEE 13-BUS FEEDER +Time +q = 0 +Non-incremental +Incremental +Obj. (p.u.) +Time (s) +Obj. (p.u.) +Time (s) +Obj. (p.u.) +1 pm +2.51 · 10−3 +64.65 +1.15 · 10−3 +199.24 +4.11 · 10−4 +2 pm +1.48 · 10−3 +66.60 +6.89 · 10−4 +209.92 +3.03 · 10−4 +3 pm +6.89 · 10−4 +74.68 +4.94 · 10−4 +263.37 +2.16 · 10−4 +4 pm +8.03 · 10−4 +68.32 +5.26 · 10−4 +126.81 +2.47 · 10−4 +5 pm +5.51 · 10−4 +62.58 +4.11 · 10−4 +129.71 +1.95 · 10−4 +parameters z := (¯v, δ, σ, α) initialized to feasible values +(0.95, 0.01, 0.3, 1.5). Table V compares the performance of +the two rule categories over multiple periods for S = 80. +While incremental rules took longer times to train, they were +successful in lowering the cost F(z) by more than 50%, thus +yielding improved voltage profiles across all periods. +VI. CONCLUSIONS +We have devised a DNN approach to optimally design +incremental Volt/VAR control rules for single- and multi-phase +feeders. The key idea is to construct a DNN that emulates +end-to-end the associated Volt/VAR dynamics. The DNN takes +grid conditions as the input, the rule parameters as weights, +and outputs the associated equilibrium voltages. Leveraging +the convergence rates of the related optimization algorithms, +we have provided bounds on the minimum depth of the +DNN emulator to approximate equilibrium voltages within the +desired accuracy. We have also established the stability of +incremental control rules for multiphase feeders. Numerical +tests have demonstrated that the designed control rules attain +improved voltage profiles compared to their non-incremental +alternatives. The improvement was found to be starker for +mutiphase feeders, wherein (non)-incremental rules do not +reach the same equilibrium. Our findings motivate further +research to possibly characterize the equilibria of control +rules for multiphase feeders; the convergence of accelerated +incremental rules for multiphase feeders; and to deal with +chance-constrained formulations or ORD problems targeting +phase imbalances. +REFERENCES +[1] IEEE Standard for Interconnection and Interoperability of DERs with +Associated Electric Power Systems Interfaces, IEEE Std., 2018. +[2] N. Li, G. Qu, and M. Dahleh, “Real-time decentralized voltage control in +distribution networks,” in Proc. Allerton Conf., Allerton, IL, Oct. 2014. +[3] M. Farivar, X. Zhou, and L. Chen, “Local voltage control in distribution +systems: An incremental control algorithm,” in Proc. IEEE Intl. Conf. +on Smart Grid Commun., Miami, FL, Nov. 2015. +[4] H. Zhu and H. J. Liu, “Fast local voltage control under limited reactive +power: Optimality and stability analysis,” IEEE Trans. Power Syst., +vol. 31, no. 5, pp. 3794–3803, 2016. +[5] V. Kekatos, L. 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Available: +http://arxiv.org/abs/2211.09557 +[14] W. Cui, J. Li, and B. Zhang, “Decentralized safe reinforcement learning +for inverter-based voltage control,” Electric Power Systems Research, +vol. 211, p. 108609, 2022. +[15] S. Taheri, M. Jalali, V. Kekatos, and L. Tong, “Fast probabilistic hosting +capacity analysis for active distribution systems,” IEEE Trans. Smart +Grid, vol. 12, no. 3, pp. 2000–2012, May 2021. +[16] V. Kekatos, L. Zhang, G. B. Giannakis, and R. Baldick, “Accelerated +localized voltage regulation in single-phase distribution grids,” in Proc. +IEEE Intl. Conf. on Smart Grid Commun., Miami, FL, Nov. 2015. +[17] D. Chen, S. Iyengar, D. Irwin, and P. Shenoy, “Sunspot: Exposing the +location of anonymous solar-powered homes,” in ACM Intl. Conf. on +Systems for Energy-Efficient Built Environ., Palo Alto, CA, Nov. 2016. + +2280 + 2340 2344 +106 116 119 +1574 +1577 1619 +Z +296 372 650 +734 +841 933 +3155 +3156 3188 +3877 +3888 3912 +1632 +1856 2150 +2844 + 2940 2968 +2408 +2495 2500 +3552 +3741 3752 +2500 +2567 2764 \ No newline at end of file diff --git a/0tAzT4oBgHgl3EQfevwc/content/tmp_files/load_file.txt b/0tAzT4oBgHgl3EQfevwc/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cab8ebe00c1b99a6dec272108430d9bcbd0c0a9a --- /dev/null +++ b/0tAzT4oBgHgl3EQfevwc/content/tmp_files/load_file.txt @@ -0,0 +1,504 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf,len=503 +page_content='1 Scalable Optimal Design of Incremental Volt/VAR Control using Deep Neural Networks Sarthak Gupta, Graduate Student Member, IEEE, Ali Mehrizi-Sani, Senior Member, IEEE, Spyros Chatzivasileiadis, Senior Member, IEEE, and Vassilis Kekatos, Senior Member, IEEE Abstract—Volt/VAR control rules facilitate the autonomous operation of distributed energy resources (DER) to regulate voltage in power distribution grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' According to non-incremental control rules, such as the one mandated by the IEEE Standard 1547, the reactive power setpoint of each DER is computed as a piecewise-linear curve of the local voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' However, the slopes of such curves are upper-bounded to ensure stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' On the other hand, incremental rules add a memory term into the setpoint update, rendering them universally stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' They can thus attain enhanced steady-state voltage profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Optimal rule design (ORD) for incremental rules can be formulated as a bilevel program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' We put forth a scalable solution by reformulating ORD as training a deep neural network (DNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' This DNN emulates the Volt/VAR dynamics for incremental rules derived as iterations of proximal gradient descent (PGD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The rule parameters appear as DNN weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' To reduce the DNN depth, we leverage Nesterov’s accelerated PGD iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Analytical findings and numerical tests corroborate that the proposed ORD solution can be neatly adapted to single/multi-phase feeders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Index Terms—IEEE Standard 1547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='8, incremental control rules, multiphase feeders, proximal gradients, gradient backprop- agation, deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' INTRODUCTION Local Volt/VAR (Volt-Ampere Reactive) control facilitates voltage regulation on distribution grids by providing reactive power compensation from DERs equipped with smart invert- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Different from centralized control schemes which incur large computational and communication burden, local rules decide DER setpoints based on local measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Volt/VAR control rules can be categorized into non-incremental and incremental ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The former compute DER reactive power setpoints based on local voltage readings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The IEEE Stan- dard 1547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='8 prescribes such non-incremental control rules as piecewise-linear functions of voltage [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' On the other hand, incremental Volt/VAR rules compute the change in VAR setpoints as a function of voltage [2]–[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The existing literature on designing Volt/VAR control rules can be classified into stability- and optimality-centric works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Stability-centric works study the effect of Volt/VAR rules as a closed-loop dynamical system, which may be rendered unsta- ble under steep slopes of non-incremental rules [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' In fact, to ensure stability, non-incremental rules may have to compro- mise on the quality of their steady-state voltage profile [5], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Incremental rules however do not experience stability limitations and can thus achieve improved voltage profiles compared to their non-incremental counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Nonetheless, such improvements may come at the expense of longer settling times of the associated Volt/VAR dynamics [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Optimality-centric works focus on designing stable control rules to minimize a voltage regulation objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' To this end, optimization-based strategies have been employed to design affine non-incremental rules using heuristics [9]–[11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Two of our recent works in [12] and [13] have addressed the problem of optimally designing the slope, deadband, saturation, and reference voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Reference [12] performs ORD via a bilevel optimization applicable to single-phase feeders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Reference [13] proposes DNN-based digital twins that emulate Volt/VAR dynamics, and reformulates ORD as a DNN training task for single-/multi-phase feeders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' This letter deals with optimally selecting the shape of incremental Volt/VAR control rules, with contributions on three fronts: c1) Although this optimal rule design (ORD) task can be posed as a mixed-integer nonlinear optimization program, it does not scale well with the numbers of DERs, nodes, and grid loading scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' To address this challenge, the genuine idea here is to reformulate ORD as a deep- learning task and judiciously adapt the fast software modules widely available for training deep neural networks (DNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' We have put forth a similar approach for designing non- incremental control rules in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' However, migrating from non-incremental to incremental rules is non-trivial due to the different curve shapes, stability, and settling time properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' c2) To further expedite ORD for incremental rules, we suggest implementing accelerated Nesterov-type variants of the rules to yield a shallower DNN emulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' c3) We also establish the convergence of incremental rules on multiphase feeders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Recently, reference [14] deals with the optimal design of incremental rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' It uses DNNs with a single hidden layer to model piecewise-linear functions and formulates ORD as a reinforcement learning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' While [14] also utilizes DNNs to design incremental rules, we delineate from it in several ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Reference [14] focuses on voltage control during tran- sient dynamics, whereas this work aims at ORD to drive steady-state voltages closer to unity and over different grid loading scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Reference [14] utilizes a DNN to model the piecewise-linear mapping of the rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' In contrast, this work develops a DNN-based digital twin that emulates end- to-end Volt/VAR dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Lastly, we provide stability and convergence analysis for single- and multiphase feeders alike, whereas [14] applies only to single-phase feeders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The rest of this letter is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Section II models the feeder and discusses non-incremental and incre- mental Volt/VAR control rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Section III formulates DNN- based digital twins for Volt/VAR dynamics of incremental rules, and their accelerated version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' It also presents ORD arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='01440v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='OC] 4 Jan 2023 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Non-incremental Volt/VAR control rule provisioned by the IEEE Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' 1547 for the interconnection of DERs [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' for single-phase feeders as a deep learning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Section IV extends the ORD process to multiphase feeders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The incre- mental rules are then benchmarked against non-incremental rules from [13] using tests on real-world data, in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The letter is concluded in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' VOLT/VAR CONTROL RULES Consider a radial feeder serving N buses equipped with DERs, indexed by n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Let (qℓ, q) collect reactive loads and generations at all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Vectors (p, v) collect the net active power injections and voltage magnitudes at all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The impact of q on v can be approximately captured using the linearized grid model [13] v ≃ Xq + ˜v (1) where ˜v := Rp − Xqℓ + v01 models the underlying grid conditions, and v0 is the substation voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Vector ˜v rep- resents the impact of non-controlled quantities (p, qℓ) on voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Matrices (R, X) depend on the feeder topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' For single-phase feeders, they are symmetric positive definite with positive entries [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' For multiphase feeders, they are non- symmetric and have positive and negative entries [5], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Vector q in (1) carries the reactive injections by DERs we would like to control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Per the non-incremental rules of the IEEE Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' 1547 [1], DER setpoints are decided based on the Volt/VAR curve of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' 1, which is parameterized by (¯v, δ, σ, ¯q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The standard further constrains these parameters within a polytopic feasible set [1], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The negative slope of the linear segment of the curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' 1 can be expressed as α := ¯q σ − δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The interaction of Volt/VAR rules with the feeder gives rise to nonlinear dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' These dynamics are stable if ∥ dg(α)X∥2 < 1, where dg(α) is a diagonal matrix carry- ing the rule slopes over all buses on its diagonal [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The equilibrium setpoints for DERs cannot be expressed in closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' However, they coincide with the minimizer of the convex optimization problem [7] min −¯q≤q≤¯q 1 2q⊤Xq+q⊤(˜v−¯v)+ 1 2q⊤ dg−1(α)q+δ⊤|q| (2) where |q| applies the absolute value on q entrywise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Prob- lem (2) depends on rule parameters (¯v, δ, α, ¯q) across all buses, collected in the 4N-long vector z := (¯v, δ, α, ¯q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' We denote by qz(˜v) the equilibrium setpoints, and by vz(˜v) = Xqz(˜v) + ˜v (3) the related equilibrium voltages reached by Volt/VAR rules parameterized by z under grid conditions ˜v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Optimal rule design (ORD) can be stated as the task of selecting z to bring equilibrium voltages vz(˜v) close to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' To cater to diverse conditions, the utility may sample loading scenarios {˜vs}S s=1 for the next hour, and find z as z∗ ∈ arg min z F(z) := 1 S S � s=1 ∥vz(˜vs) − 1∥2 2 (ORD) subject to (3) and z ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Once found, the customized rules z∗ are sent to DERs to operate autonomously over the next hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Note that vz(˜vs) depends on z because the equilibrium setpoints qz(vs) in (3) are the minimizers of problem (2), which is parameterized by z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' When solving (ORD) for non-incremental rules, the feasible set Z consists of the polytopic constraints imposed on z by the IEEE Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' 1547 as well as additional constraints on α to ensure ∥ dg(α)X∥2 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' see [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Therefore, the feasible set Z can be quite confined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' This can lead to less desirable voltage profiles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' that is, higher objective values F(z∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The aforesaid issue can be addressed by replacing the non- incremental Volt/VAR rules of IEEE Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' 1547 by incremental ones as suggested in [2]–[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Incremental rules express the change rather than the actual value in setpoints as a function of voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' One option for incremental rules is to implement a proximal gradient descent (PGD) algorithm solving (2) as proposed in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' In this case, the control rule coincides with the PGD iterations, which are implemented by DERs in a decentralized fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Using incremental rules, set Z is enlarged as now we only need to ensure z ≥ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='95 · 1 ≤ ¯v ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='05 · 1 and that ¯q are within the reactive power ratings of the DERs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The PGD algorithm is an extension of gradient descent to handle constraints and non-differentiable costs [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' At iteration t, PGD proceeds with two steps: s1) It first computes the gradi- ent of the first two terms of F(z), that is Xqt+˜v−¯v = vt−¯v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Here qt is the latest estimate of the minimizer of (2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' s2) PGD then updates qt+1 as the minimizer of min −¯q≤q≤¯q 1 2q⊤ dg−1(α)q + δ⊤|q| + 1 2µ∥q − (vt − ¯v)∥2 2 (4) for a step size µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The last problem involves the last two terms in the cost of (2) regularized by the Euclidean distance of q to the gradient (vt − ¯v) computed in step s1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Converting PGD to control rules, step s1) is performed by the physics of the feeder when injecting qt and measuring the local voltage deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Step s2) is run by each DER independently as (4) is separable across buses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Using the subdifferential, solving (4) provides the update [5] yt n = ˜αn · � qt n − µ(vt n − ¯vn) � (5a) fq UA3 qt+1 n = gn � yt n � (5b) where gn(yn) is the proximal operator gn(yn) := � � � � � � � � � � � � � � � +¯qn , yn > qn + µ˜δn yn − µ˜δn , µ˜δn < yn ≤ qn + µ˜δn 0 , − µ˜δn ≤ yn ≤ µ˜δn yn + µ˜δn , − qn − µ˜δn ≤ yn < −µ˜δn −qn , yn < −qn − µ˜δn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' (6) and the new parameters (˜αn, ˜δn) are defined as ˜αn := 1 1 + µ/αn and ˜δn := δn 1 + µ/αn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The proximal operator is plotted in the top panel of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Note that in (5), rule parameters are transformed from repre- sentation z = (¯v, δ, α, ¯q) to representation ˜z := (¯v, ˜δ, ˜α, ¯q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' This is without loss of generality as the transformation is a bijection, and so one can work exclusively with ˜z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The feasible set ˜Z for ˜z is similar to Z with the addition that ˜α ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' As with non-incremental rules, the rules in (6) are driven by local data, but now qt+1 n depends on (vt n, qt n), and not vt n alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Both types of rules solve (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Hence, they both converge to the same equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The advantage of incremental rules is that they are stable for all α as long as µ < 2/λmax(X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' see [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' It is worth stressing that z does not have the same physical interpretation as in non-incremental rules (slopes, deadband, or saturation), though z parameterizes (2) for both rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Accelerated incremental rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Although PGD rules en- large Z, their settling times can be long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' They reach an ε-optimal cost of (2) within − 2 log ε log 2 κ (X) iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Here κ(X) := λmax(X)/λmin(X) is the condition number of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' References [5], [16] put forth accelerated incremental rules based on accelerated PGD (APGD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' These rules need − 2 log ε log 2 � κ (X) iterations to attain an ε-optimal cost, and take the form ˜yt n := (1 + βt) yt n − βtyt−1 n (7a) qt+1 n := gn � ˜yt n � (7b) where βt := t−1 t+2, while yt n and gn(yn) are as defined in (5a) and (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Updates (7) remain local, but introduce additional memory as qt+1 n depends on (vt n, qt n) and (vt−1 n , qt−1 n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' DEEP LEARNING FOR OPTIMAL RULE DESIGN (ORD) IN SINGLE-PHASE FEEDERS Solving (ORD) is challenging as it is a nonconvex bilevel program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Although it can be modeled as a mixed-integer nonlinear program, such an approach does not scale well with the number of DERs and/or scenarios for non-incremental rules [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Seeking a more scalable solution, we reformulate (ORD) as a deep learning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The key idea is to design a DNN that emulates Volt/VAR dynamics under the control rule of (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' To this end, note that gn(yn) is a piecewise- linear function with four breakpoints [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Interestingly, this operator can be expressed as the superposition of four rectified linear units (ReLUs) as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' 2, where ReLUs are denoted by ρ(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The intercepts of the ReLUs depend linearly on (˜δn, ¯qn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Proximal operator g(y) expressed as a sum of four shifted rectified linear units (ReLUs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' A DNN emulating the accelerated incremental rules of (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Plain incremental rules can be modeled by dropping the second layer (setting βt = 0) and ignoring output yt n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Building on this, one APGD iteration for DER n can be implemented by the 4-layer DNN in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' 3, whose weights depend affinely on (¯vn, ˜δn, ˜αn, ¯qn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' This DNN takes (qt n, vt n) as its input, and computes (qt+1 n , yt n) at its output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' It is termed ICn and will be used as a building block to emulate Volt/VAR dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' This is accomplished by the recursive neural network (RNN) shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Here blocks ICn are arranged vertically to model the parallel operation of DERs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Their outputs qt+1 are multiplied by X, and the new voltage is computed as vt+1 = Xqt+1 + ˜v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' This is repeated T times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Thanks to the RNN structure, there is weight sharing, so the number of DNN weights is 4N rather than 4NT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The RNN takes a grid loading vector ˜vs as its input, the rule parameters ˜z as weights, and computes the voltages vT ˜z (˜vs) at time T at its output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' For the output vT ˜z (˜vs) to approximate well equilibrium voltages, the depth T can be chosen by the convergence rate of PGD as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' For the DNN of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' 4 to ensure ∥Φ (˜v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' z) − v∗(z)∥2 ≤ ϵ1 ∀ ˜v, its depth T should satisfy T ≥ �κ − 1 2 � log �2∥X∥2∥ˆq∥2 ϵ1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' (8) +qH b+ qp uon 一 qn t p t p uon p4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Recurrent neural network (RNN) implementation for accelerated incremental Volt/VAR control rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Proof: From the control rule of (5b), it follows that ∥qt − q∗∥2 = ∥g � yt� − g (y∗) ∥2 ≤ ∥yt − y∗∥2 = ∥ dg(˜α)(I − µX) � qt−1 − q∗� ∥2 ≤ ∥ dg(˜α)∥2 · ∥I − µX∥2 · ∥qt−1 − q∗∥2 ≤ ∥I − µX∥2 · ∥qt−1 − q∗∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' (9) The first inequality stems from the non-expansive property of the proximal operator g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The next equality follows from (5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The second inequality from the sub-multiplicative property of the spectral norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The last inequality follows by the definition of spectral norm and because ˜αn ≤ 1 for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' If ∥I − µX∥2 < 1, inequality (9) implies that the dynamics in (5) are a non-expansive mapping, and thus, are stable and converge to q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Condition ∥I − µX∥2 < 1 holds when µ < 2/λmax(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The norm ∥I − µX∥2 achieves its minimum of � 1 − 2 κ+1 � when µ0 := 2 λmax(X) + λmin(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Plugging µ0 into (9) and unfolding the dynamics over t provides ∥qt − q∗∥2 ≤ � 1 − 2 κ+1 �t ∥q0 − q∗∥2 ≤ 2 � 1 − 2 κ+1 �t ∥ˆq∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' For the voltage approximation error ∥vT − v∗∥2 = ∥X � qT − q∗� ∥2 at time T to be smaller than ϵ1, we need ∥vT − v∗∥2 ≤ 2∥X∥2 · ∥ˆq∥2 · � 1 − 2 κ + 1 �T ≤ ϵ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' This can be achieved by selecting T such that T ≥ log � 2∥X∥2∥ˆq∥2 ϵ1 � log � 1 + 2 κ−1 � ≥ �κ − 1 2 � log 2∥X∥2∥ˆq∥2 ϵ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' where the last inequality follows from log(1 + x) ≤ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Plugging the values ∥X∥2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='463 and κ = 848 for the IEEE 37-bus feeder, ∥ˆq∥2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='1, and ϵ1 = 10−5 in (8), yields T ≥ 2, 892 layers, which is relatively large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' A key contributor to this large T is the κ term in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' This promulgates the adoption of accelerated rules (7), which are known to have O(√κ) dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Interestingly, during implementation, one does not need to fix T to the above worst-case bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Leveraging dynamic computation graphs offered by Python libraries such as Pytorch, one may determine T ‘on the fly’ depending on the convergence of vt between pairs of successive layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Since the RNN emulates Volt/VAR dynamics, it can surro- gate vz(˜vs) in (ORD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Then (ORD) can be posed as training a DNN over its weights ˜z ∈ ˜Z or z ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Grid loading scenarios {˜vs}S s=1 are treated as features and equilibrium voltages vz(˜vs) as predictions that should be brought close to the target value of 1 for scenarios s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The DNN can be trained using stochastic projected gradient descent (SPGD) as [13] ˜zi+1 = � ˜zi − λ 2B ∇˜zi � � s∈Bi ∥Φ(˜vs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' ˜z) − 1∥2 2 �� ˜ Z (10) where λ > 0 is the learning rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' set Bi is a batch of B scenarios;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' and [·] ˜ Z is the projection onto ˜Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Since ˜Z consists of simple box constraints, projection essentially means clipping the values to the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Lastly ∇˜zi(·) represents the gradient with respect to ˜z evaluated at ˜z = ˜zi, and is calculated efficiently thanks to gradient back-propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Although our DNN-based ORD assumed PGD-based rule, it may be applicable to other incremental rules too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' A discussion about control rules and their DNN-based emulators is due.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Recall that all three types of Volt/VAR control rules (non-incremental, incremental, and accelerated incremental) reach the same equilibrium voltages, if stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The emulators aim at computing these equilibrium voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' A natural question is whether the DNN emulator could im- plement a rule of a type different from the rule actually implemented on the feeder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' This may be desirable to leverage the advantages of two types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Some caution is needed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' If the feeder implements non-incremental rules, but incremental rules converge faster to equilibrium voltages, it makes sense for the emulator to implement incremental rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Of course, in this case, stability constraints on the non-incremental rules have to be enforced during DNN training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The reverse is not recommended: If the emulator implements non-incremental rules, its parameters z should be constrained to be stable and that would be a restriction of the actual ORD problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Finally, given the convergence advantage of accelerated incremental rules, they are always preferable over plain incremental rules for the DNN implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' This showcases the utility of accelerated control rules even if they are not actually imple- mented on the feeder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' DEEP LEARNING FOR OPTIMAL RULE DESIGN (ORD) IN MULTIPHASE FEEDERS In multiphase feeders, matrix X is non-symmetric and has both positive and negative entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Therefore, the rule analysis and design of Section III has to be revisited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' For example, equilibrium setpoints cannot be found as the minimizers of IC1 七 q t+1 q IC2 t t t-1 IC N5 an optimization problem as with (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Moreover, increasing q does not mean that all voltages increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' In multiphase feeders, the non-incremental rules of IEEE Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' 1547 remain stable as long as ∥ dg(α)X∥2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' This is the same condition as in the single-phase setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' How about the stability and equilibrium of incremental rules in multiphase feeders?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Recall that for single-phase rules, incremental rules were obtained as the PGD iterations solving (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Lacking an equivalent inner optimization for multiphase feeders precludes a similar approach here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Despite the incremental rules of (5) do not correspond to PGD iterates anymore, they can still be shown to be stable for multiphase feeders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Let UΛU⊤ be the eigen-decomposition of matrix XX⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The incremental rules of (5) are stable for multiphase feeders if their step size is selected as µ < λmin � Λ−1/2U⊤ � X + X⊤� UΛ−1/2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The claim follows readily by adopting the proof of Proposi- tion 1: If µ is selected as above, then ∥I − µX∥2 < 1 follows from [5, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Similar to the single-phase case, incremental rules in multiphase feeders allow us to enlarge the feasible set Z of rule parameters z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' It is worth stressing that different from the single-phase setting, incremental and non-incremental rules do not converge to the same equilibrium on multiphase feeders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The ORD task for multiphase feeders can also be formulated as a deep-learning task, with some modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Firstly, matrices R and X need to be altered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Secondly, the DNNs for multiphase feeders have 12N trainable parameters, since each layer consists of 3N building modules corresponding to bus/phase (node) combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Lastly, the step size has to be selected per Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Adopting the proof of Proposition 1, we next find the minimum DNN depth in multiphase feeders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Let the DNN of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' 4 implement the incre- mental rules of (5) on multiphase feeders with µ selected per Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The DNN depth T ensuring voltage approxi- mation error ∥Φ (˜v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' z) − v∗(z)∥2 ≤ ϵ1 is T ≥ log ϵ1 2∥X∥2∥ˆq∥2 log ∥I − µX∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' We next numerically evaluate the proposed DNN-based ORD approach in single- and multiphase feeders, and contrast the performance of incremental control rules with that of non- incremental rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' NUMERICAL TESTS We benchmark the performance of DNN-based incremental rules against non-incremental rules from [13] on single- and multiphase feeders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Real-world data were sourced from the Smart* project on April 2, 2011 [17], as explained in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The DNNs were implemented and trained using Pytorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' We first compare (non)-incremental rules, both designed via DNN training for the single-phase IEEE 37-bus feeder of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Homes with IDs 20–369 were averaged 10 at a time and successively added as active loads to buses 2–26 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Active generation from solar Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The IEEE 37-bus feeder converted to single-phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Node num- bering follows the format node number {panel ID}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' DERs at buses {6, 9, 11, 12, 15, 16, 20, 22, 24, 25} provide reactive power control;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' the rest operate at unit power factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' TABLE I INCREMENTAL VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' NON-INCREMENTAL VOLT/VAR CONTROL RULES ON THE SINGLE-PHASE IEEE 37-BUS FEEDER Time q = 0 Non-incremental Incremental Obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=') Time (s) Obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=') Time (s) Obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=') 1 pm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='01 · 10−3 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='98 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='68 · 10−4 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='39 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='66 · 10−4 2 pm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='13 · 10−3 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='93 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='26 · 10−4 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='91 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='25 · 10−4 3 pm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='24 · 10−3 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='02 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='59 · 10−4 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='97 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='50 · 10−4 4 pm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='12 · 10−3 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='47 · 10−4 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='52 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='48 · 10−4 5 pm 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='53 · 10−4 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='37 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='70 · 10−5 374.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='01 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='90 · 10−5 panels was also added, as per the mapping in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Buses {6, 9, 11, 12, 15, 16, 20, 22, 24, 25} were assumed to host DERs with Volt/VAR control customized per bus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Incremental rules were simulated in their accelerated ren- dition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Both sets of rules were trained over S = 80 scenarios and 200 epochs with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='001, using the Adam optimizer, and setting µ = 1 for incremental rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' To ensure repeatability, the results were repeated across several time periods between 1–6 PM, and are compiled in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Incremental rules obtained marginally lower objectives than non-incremental rules across all periods, with a somewhat significant difference for the 5 PM period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' This behavior is explained because incremental rules allow for a larger set Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' DNN-based incremental control rules were also contrasted with their non-incremental ones on the multiphase IEEE 13- bus feeder, using the testing setup from [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Active loads were sampled 10 at a time from homes with IDs 20-379 and added to all three phases for the buses 1-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Figure 6 also shows the solar panel assignments shown in Fig 6 for solar generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Lastly, nine DERs with inverters were added across phases and bus indices as shown in Fig 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The learning rates for non-incremental and incremental DNNs were set as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='001, respectively, with the design Z Z Z Z Z Z6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Multiphase IEEE 13-bus distribution feeder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' TABLE II INCREMENTAL VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' NON-INCREMENTAL VOLT/VAR CONTROL RULES ON THE MULTIPHASE IEEE 13-BUS FEEDER Time q = 0 Non-incremental Incremental Obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=') Time (s) Obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=') Time (s) Obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=') 1 pm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='51 · 10−3 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='15 · 10−3 199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='11 · 10−4 2 pm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='48 · 10−3 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='60 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='89 · 10−4 209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='92 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='03 · 10−4 3 pm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='89 · 10−4 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='68 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='94 · 10−4 263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='37 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='16 · 10−4 4 pm 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='03 · 10−4 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='32 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='26 · 10−4 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='81 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='47 · 10−4 5 pm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='51 · 10−4 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='58 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='11 · 10−4 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='71 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='95 · 10−4 parameters z := (¯v, δ, σ, α) initialized to feasible values (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='95, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Table V compares the performance of the two rule categories over multiple periods for S = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' While incremental rules took longer times to train, they were successful in lowering the cost F(z) by more than 50%, thus yielding improved voltage profiles across all periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' CONCLUSIONS We have devised a DNN approach to optimally design incremental Volt/VAR control rules for single- and multi-phase feeders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The key idea is to construct a DNN that emulates end-to-end the associated Volt/VAR dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The DNN takes grid conditions as the input, the rule parameters as weights, and outputs the associated equilibrium voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Leveraging the convergence rates of the related optimization algorithms, we have provided bounds on the minimum depth of the DNN emulator to approximate equilibrium voltages within the desired accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' We have also established the stability of incremental control rules for multiphase feeders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Numerical tests have demonstrated that the designed control rules attain improved voltage profiles compared to their non-incremental alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' The improvement was found to be starker for mutiphase feeders, wherein (non)-incremental rules do not reach the same equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' Our findings motivate further research to possibly characterize the equilibria of control rules for multiphase feeders;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' the convergence of accelerated incremental rules for multiphase feeders;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' and to deal with chance-constrained formulations or ORD problems targeting phase imbalances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'} +page_content=' REFERENCES [1] IEEE Standard for Interconnection and Interoperability of DERs with Associated Electric Power Systems Interfaces, IEEE Std.' metadata={'source': 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Chern number in a supercell framework +Roberta Favata1 and Antimo Marrazzo1, ∗ +1Dipartimento di Fisica, Universit`a di Trieste, Strada Costiera 11, I-34151 Trieste, Italy +(Dated: January 9, 2023) +We present an approach for the calculation of the Z2 topological invariant in non-crystalline +two-dimensional quantum spin Hall insulators. While topological invariants were originally mathe- +matically introduced for crystalline periodic systems, and crucially hinge on tracking the evolution +of occupied states through the Brillouin zone, the introduction of disorder or dynamical effects can +break the translational symmetry and imply the use of larger simulation cells, where the k−point +sampling is typically reduced to the single Γ-point. Here, we introduce a single-point formula for +the spin Chern number that enables to adopt the supercell framework, where a single Hamiltonian +diagonalisation is performed. Inspired by the work of E. Prodan [Phys. Rev. B, 80, 12 (2009)], our +single-point approach allows to calculate the spin Chern number even when the spin operator ˆsz +does not commute with the Hamiltonian, as in the presence of Rashba spin-orbit coupling. We val- +idate our method on the Kane-Mele model, both pristine and in the presence of Anderson disorder. +Finally, we investigate the disorder-driven transition from the trivial phase to the topological state +known as topological Anderson insulator. Beyond disordered systems, our approach is particularly +useful to investigate the role of defects, to study topological alloys and in the context of ab-initio +molecular dynamics simulations at finite temperature. +I. +INTRODUCTION +Two-dimensional (2D) topological insulators (TI) are +materials with an insulating bulk and robust edge states +protected by the non-trivial topology of the bulk elec- +tronic structure [1, 2]. +These systems are discussed +through topological invariants, integer quantities which +characterise the ground-state wavefunction in the bulk. +As long as the topological invariant is non-trivial and, +possibly, the symmetries needed to define that topology +are preserved, the material is said to be in a topolog- +ical phase. These invariants are geometrical properties +of the electronic structure, as they are defined in terms +of quantities such as the Berry phase or the Berry cur- +vature, which involve derivatives of the occupied states +in reciprocal space with respect to the quasi-momentum +k [2]. Standard geometrical formulas are usually discre- +tised on a regular mesh of k-points for numerical imple- +mentation. However, most electronic structure calcula- +tions for non-crystalline systems are normally performed +by diagonalising the Hamiltonian at a single k-point in a +large supercell. Usually the Γ point at the center of the +Brillouin zone (BZ) is considered, although potentially +more efficient choices based on the Baldereschi point [3] +can be employed. The derivation of single-point formu- +las for geometrical and topological properties is not at all +a trivial task, although successful single-point formalism +have been developed for the Berry phase [4], the orbital +magnetisation and the Chern number [5]. +In this work, we target the calculation of the topologi- +cal invariant for non-crystalline 2D insulators with time- +reversal (TR) symmetry. For these systems, encompass- +ing all non-magnetic 2D materials [6], the invariant ν is +∗ antimo.marrazzo@units.it +a Z2 number: if ν = 0 the topology is trivial, otherwise +if ν = 1 we have a quantum spin Hall insulator (QSHI), +where topologically-protected gapless helical edge states +cross the bulk gap [2]. +Over the years, several meth- +ods have been developed to calculate the Z2 invariant +in crystalline systems with periodic boundary conditions +(PBCs). +In the following, we briefly outline some of the most +popular and practical methods in the context of elec- +tronic structure simulations. +If inversion symmetry is +present, there is a particularly simple method introduced +by Fu and Kane [7], which requires the knowledge of the +parity of the occupied states at the four TR-invariant +points in the BZ. In the more general case, the Z2 in- +variant can be obtained by tracking the evolution of +hermaphrodite [8] (a.k.a. hybrid) Wannier charge cen- +tres [9–11], or equivalently the eigenvalues of the Wilson +loop [12–14], over half BZ. More recently, a generalisa- +tion of the Fu-Kane approach based on elementary band +representations [15, 16] has allowed to calculate the in- +variant by using only the knowledge of the irreducible +representations of the occupied states at selected high- +symmetry points in the BZ [16]. The Z2 invariant can +be also computed as an individual Chern number [2] on +half of the Hilbert space [10, 11], where the split is per- +formed by two projectors which are smooth and related +by TR symmetry. Although several formulas to compute +the Z2 invariant have been introduced, all the ones we +mentioned, and most other existing approaches, require +the knowledge of the occupied states at multiple k-points +and become ill-defined for non-crystalline systems; hence +in the supercell framework they are of no avail. +Nonetheless, a number of methods have been proposed +to deal with non-periodic systems. Some of these [17– +19] calculate the Z2 invariant by means of a Pfaffian +with twisted boundary conditions, as firstly advocated +by Kane and Mele in their original discussion of the Z2 +arXiv:2301.02612v1 [cond-mat.mes-hall] 6 Jan 2023 + +2 +invariant in presence of disorder and electron-electron in- +teractions [20]. A different method is based on construct- +ing the Z2 invariant from the scattering matrix of the +system at the Fermi level [21, 22]. Further, there exists +a formulation based on the non-commutative index the- +orem [23, 24], where the Z2 index for disordered topolog- +ical insulators is computed from the discrete spectrum +of a certain compact operator, which is defined as the +difference of a proper pair of projection operators [25– +27]. +An alternative non-commmutative approach was +proposed by Loring and Hastings [28, 29] and relates the +Z2 index to the topological obstruction to approximat- +ing almost commuting matrices by exactly commuting +matrices; its robustness with respect to the introduction +of disorder has been investigated in Ref. [30]. The most +practical approach from the point of electronic structure +simulations has been arguably put forward by Huang +and Liu [31, 32], who addressed the problem of calcu- +lating the Z2 invariant for non-periodic system in the +context of quantum spin Hall quasicrystals, and intro- +duced the spin Bott index, which measures the commu- +tativity of the projected position operators. The connec- +tion between the Bott indices and Chern or Z2 invariants +has been investigated theoretically [28–30, 33], while nu- +merical simulations [32, 34, 35] provided evidence that +Bott indices can be used to study non-periodic topolog- +ical systems. Still, it is conceptually rather unsatisfac- +tory that the calculation of topological invariants in a +supercell framework requires introducing radically differ- +ent formalisms, which call for rather non-trivial equiva- +lence proofs and extensive testing. As a matter of fact, +the use of the primitive cell and k-points is an arbitrary— +although indeed very convenient—choice; there is no con- +ceptual reason preventing bona fide Z2 invariants to be +calculated directly in the supercell by deriving a suitable +single-point limit. In addition, it is important to assess +the convergence with respect to the system size, as dif- +ferent approaches might deliver the same correct answer +at very different computational costs. For instance, re- +cent works [32, 33] claimed that the difference between +the Chern number and Bott index is within a correc- +tion of the order O(1/L), where L is the linear size of +the system. Such slow convergence can hinder the study +of the system close to a topological phase transition; in +fact Huang and Liu empirically added a singular value +decomposition (SVD) to their algorithm to improve an +otherwise slow convergence [32]. +Here, we take a different approach, that essentially +combines the work of Ceresoli and Resta on the single- +point Chern number [5] and the insights from Prodan +on a generalised spin Chern number [36]. Notably, our +single-point invariant is directly derived by its parent for- +mula for crystalline systems, it shows exponential conver- +gence with the supercell size, both in the pristine and dis- +ordered case, it is easy to implement in electronic struc- +ture codes, and it works well also in presence of strong +Rashba spin-orbit coupling (SOC). +II. +METHODS +In absence of spin-mixing spin-orbit interactions, the +spin operator ˆsz commutes with the Hamiltonian and it +is possible to discuss the Z2 invariant in terms of the spin +Chern number. In this case, the occupied states diago- +nalise ˆsz and can be divided in two subsets, either purely +spin-up or spin-down, and the regular Chern number can +be calculated for each spin. As soon as the Hamiltonian +does not commute any more with ˆsz, for instance because +Rashba SOC is present, such simple-minded spin Chern +number cannot be defined any more. Notably, Prodan +has shown [36] that it is possible to generalise this def- +inition by projecting the spin operator on the occupied +states: +Pz = P(k)ˆszP(k), +(1) +where P is the ground-state projector +P(k) = +� +n +|unk⟩ ⟨unk| , +(2) +unk are the periodic part of the Bloch eigenstates and +n labels the occupied state at each k-point in the BZ. +Then, we diagonalise Pz: +Pz |uλ⟩ = sλ |uλ⟩ . +(3) +If only diagonal SOC terms are present, the eigenvalue +spectrum of Pz consists of two values only sλ = ± 1 +2 and +one can select a single spin component by choosing the +eigenstates which correspond to one of the two eigenval- +ues sλ. +The crucial observation made by Prodan [36] +is that, even if Rashba SOC is present, the spectrum of +Pz displays two separate bands of eigenvalues symmet- +ric around the origin and one can still introduce a well- +defined spin Chern number by selecting the eigenvectors +with positive (or negative) eigenvalues. Finally, the spin +Chern number can be computed as: +Cs = C+ − C− +2 +mod 2 +(4) +where C± are calculated on the uλ eigenstates with posi- +tive and negative eigenvalues respectively; in general it is +sufficient to compute either C+ or C− only and consider +its parity. The results are of paramount practical rele- +vance, as it is typically much simpler to deal with a for- +mulation based on generalised Chern numbers, which can +be written as full BZ integrals and do not require taking +into account TR symmetry or complex gauge fixing, as +required instead by more general Z2 formulations [20, 37]. +In principle, if the Rashba interaction is strong enough +then the gap of the Pz spectrum might close, preventing +the spin Chern number to be defined. Remarkably, as +we will discuss in full detail in the Sec. III, this does not +seem to occur in practice. As long as the system is in- +sulating, Eq. 4 is well defined even if the Rashba SOC is +several times larger than the diagonal SOC. Hence, we + +3 +adopt the approach of Prodan [36] and target the deriva- +tion of a single-point formula. In order to obtain the cor- +rect single-point limit, we follow the approach of Ceresoli +and Resta [5] for the derivation of the single-point Chern +number in TR-broken systems (the latter admit a Z topo- +logical invariant). Let us start with the formula for the +generalised spin Chern number in 2D periodic systems: +Cσ = 1 +2π +� +BZ +TrσΩxy(k)dk += − 1 +π +� +sλ=σ +� +BZ +Im ⟨∂kxuλ(k)|∂kyuλ(k)⟩ dkxdky, +(5) +where uλ are the eigenvectors of Pz (see Eq. 3) and σ = ± +corresponds to one of the sectors of the Pz spectrum. +Now we consider the parallelogram Brillouin zone and +change coordinate system to have a rectangular integra- +tion domain: +Cσ = − 1 +π Im +� +sλ=σ +� b1 +0 +dk1 +� b2 +0 +dk2 ⟨∂k1uλ(k)|∂k2uλ(k)⟩ +≃ −|b1||b2| +π +Im +� +sλ=σ +⟨∂k1uλ(k)|∂k2uλ(k)⟩ |k=Γ, +(6) +where b1,2 are the two reciprocal lattice vectors and the +last step is performed in the limit of a very large supercell. +In the same limit, we can calculate derivatives through +finite differences: +∂kj |uλ(k)⟩ |k=Γ = lim +η→0 +|uλ(ηbj)⟩ − |uλ(Γ)⟩ +η|bj| +, +(7) +where we can drop the limit for a large supercell and +just consider the difference |uλ(bj)⟩ − |uλ(Γ)⟩. Eq. 7 re- +quires a differentiable function, which is not guaranteed +in numerical diagonalisations. Hence, we fix the gauge +by adopting a discretised version of the covariant deriva- +tive [38, 39] as successfully performed for the Chern num- +ber by Ceresoli and Resta [5]. One replaces the states +with their “duals”: +|˜un(bj)⟩ = +� +m +S−1 +mn(bj) |um(bj)⟩ +(8) +where +we +define +the +overlap +matrix +Snm(bj) += +⟨un(Γ)|um(bj)⟩ +and +the +dual +states +satisfy +⟨un(Γ)|˜um(bj)⟩ += +δnm. +Next, +we construct the +states un(bj) by imposing the periodic gauge, which +allows us to perform a single diagonalisation at Γ: +|uλ(bj)⟩ = e−ibj·r |uλ(Γ)⟩ . +(9) +The states in Eq. 9 are Hamiltonian eigenstates, but they +might correspond to a different eigenvalue with respect +to the one at Γ; the ordering is anyway fixed by the +covariant derivative. We note in passing, that while a +non-trivial Chern number would prevent the adoption of +a periodic gauge for the wavefunction, here the periodic +gauge is only temporarily imposed to build each |un(bj)⟩ +from the knowledge of the |un(Γ)⟩, but it is effectively +replaced by the parallel transport gauge enforced by the +covariant derivative. The final single-point formula for +the spin Chern number is +C(asym) +σ += −|b1||b2| +π +Im +� +sλ=σ +⟨˜uλ(b1)|˜uλ(b2)⟩ . +(10) +In Eq. 10, we emphasise with the superscript “asym” the +implicit choice made in Eq. 7, which corresponds to the +right-hand derivative. In fact, an alternative choice is the +symmetric derivative +∂kj |uλ(k)⟩ |k=Γ ≃ |uλ(bj)⟩ − |uλ(−bj)⟩ +2|bj| +, +(11) +which can also be computed with a single Γ-only diago- +nalisation and leads to the following formula for the spin +Chern number: +C(sym) +σ += −|b1||b2| +4π +Im +� +sλ=σ +(⟨˜uλ(b1)| − ⟨˜uλ(−b1)|) (|˜uλ(b2)⟩ − |˜uλ(−b2)⟩) . +(12) +In Sec. III, we will show how the symmetric formula con- +verges much faster than the asymmetric version, at es- +sentially the same computational cost. +We +have +implemented +the +single-point +formulas +in a dedicated Python package, freely available on +GitHub [40]. +The code provides user-friendly inter- +faces to two popular tight-binding packages such as +PythTB [41] and TBmodels [42], and it can be easily +interfaced to other codes. +III. +NUMERICAL RESULTS AND DISCUSSION +We validate our approach on the paradigmatic Kane- +Mele (KM) model [20, 43] on the honeycomb lattice, both +pristine and in presence of Anderson disorder (see Fig. 1). + +4 +FIG. 1. The Kane-Mele model in the supercell approach. Left panel: pristine Kane-Mele model, the primitive cell is shown +in orange while a 3 × 3 supercell is marked in blue. Right panel: random realisation of a disordered Kane-Mele model in a +3 × 3 supercell (green) with periodic boundary conditions, where different colours are used to represent the on-site terms. In +the following, supercells are labelled by their integer size L × L (in units of the pristine primitive cell) and the corresponding +number of sites N = 2L2. +The tight-binding Hamiltonian reads +HKM = t +� +⟨i,j⟩ +c† +icj + ∆ +� +i +ξic† +ici ++ iλSO +� +⟨⟨i,j⟩⟩ +νijc† +iσzcj +(13) ++ iλR +� +⟨i,j⟩ +c† +i(σ × ˆdij)zcj, +where i and j run over all sites in the lattice and the +creation and annihilation operators are expressed in the +contracted form c† +i = (c† +i↑, c† +i↓). The first term is a real +nearest-neighbour hopping (denoted by ⟨ , ⟩), if taken +alone that would yield four (pair-degenerate) bands with +gapless Dirac cones centred on the high-symmetry points +K and K +′ in the Brillouin zone. +The second term +is a staggered on-site potential (ξi = ±1 is the sub- +lattice index of the i−th site) while the third term is +the KM SOC [20, 43] which involves a complex next- +nearest neighbour hopping (denoted by ⟨⟨ , ⟩⟩) with a +spin-dependent amplitude proportional to the Pauli ma- +trix σz. +The factor νij = sign(d1 × d2)z depends on +the orientation of the vectors d1 and d2 along the two +bonds connecting i to the next-nearest neighbour site j. +The fourth term is the Rashba SOC and is a complex +nearest-neighbour hopping with off-diagonal spin com- +ponents, where σ = (σx, σy, σz) is the vector of Pauli +matrices and ˆdij is the unit vector between sites j and +i. In the following, we consider a KM Hamiltonian at +fixed parameters t = 1 and λSO = 0.03 t, which ensure +that the energy gap is insulating all over the entire phase +diagram [20, 43]. +A. +Validation and convergence tests for crystalline +systems +In the single-point approach, the topological invari- +ants become exact integer numbers only in the thermody- +namic limit of an infinite supercell. First, we test the con- +vergence properties of the single-point spin Chern num- +ber (SPSCN) on the pristine KM model, in both asym- +metric (Eq. 10) and symmetric (Eq. 12) formulation. We +inspect the SPSCN as a function of the supercell size L, +here defined as the number of primitive cells along each +lattice vector that makes the supercell L×L (see Fig. 1); +the number of sites inside the supercell is N = 2L2. A +representation of a supercell 3 × 3 is given in the left- +hand panel of Fig. 1. In our calculations only values of L +which are multiple of 3 are considered, to always include +the special points K and K +′ folded at Γ. +We bench- +mark the accuracy of the formulas inside the Z2-even +and Z2-odd domains in Fig. 2. +The symmetric for- +mula converges faster than the asymmetric one in both +trivial and topological phases. Remarkably, the quantity +∆Cσ = |Cσ(L)−Cσ(∞)|, which is the difference between +the spin Chern number given by the single-point formulas +at finite sizes and the exact value obtained in the ther- +modynamic limit, decreases exponentially in both formu- +lations. However, the global prefactor in the symmetric +case is an order of magnitude smaller than the one of the +asymmetric formula, leading to more accurate results at +significantly smaller sizes L. Hence, in the following we +adopt the symmetric formula only and study the topo- +logical phase transition as a function of the on-site ∆, +results are reported in Fig. 3. +Our SPSCN is able to +reproduce the sharp topological transition already at rel- +atively small supercell sizes, as shown in the left-hand +panel of Fig. 3. The band gap vanishes on the boundary +of the phase transition and in the corresponding neigh- +bourhood of parameters convergence is slower and larger + +5 +6 +12 +18 +24 +30 +36 +42 +48 +54 +60 +L +−0.100 +−0.075 +−0.050 +−0.025 +0.000 +0.025 +0.050 +0.075 +0.100 +Single-point Cσ +Asymmetric +Symmetric +Exact value +0 +20 +40 +60 +L +10−4 +10−3 +10−2 +10−1 +∆Cσ +−5 +0 +5 +∆/λSO +−5 +0 +5 +λR/λSO +6 +12 +18 +24 +30 +36 +42 +48 +54 +60 +L +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +1.05 +1.10 +1.15 +Single-point Cσ +Asymmetric +Symmetric +Exact value +0 +20 +40 +60 +L +10−4 +10−3 +10−2 +10−1 +∆Cσ +−5 +0 +5 +∆/λSO +−5 +0 +5 +λR/λSO +FIG. 2. Convergence of the single-point spin Chern number, in its symmetric and asymmetric implementation, with respect +to supercell size for the Kane-Mele model, where the Hamiltonian is diagonalised at the Γ-point only. In the uppest insets, a +sketch of the corresponding point in the pristine phase diagram. The lowest insets show the difference between the single-point +calculations of the spin Chern number and the thermodynamic limit. Left panel: the spin Chern number converges to zero in +the trivial phase (∆/λSO = 5.5, λR/λSO = 3). Right panel: in the topological phase (∆/λSO = 0.8 , λR/λSO = 2) the spin +Chern number converges to one. In all cases, the asymptotic convergence is exponential, but the symmetric formula converges +visibly faster than its asymmetric counterpart. +0 +1 +2 +3 +4 +5 +6 +∆/λSO +−0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Single-point Cσ +L = 9 +L = 24 +L = 51 +Exact value +−5 +0 +5 +∆/λSO +−5 +0 +5 +λR/λSO +0 +1 +2 +3 +4 +5 +6 +∆/λSO +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +˜Eg +L = 9 +L = 24 +L = 51 +FIG. 3. Left panel: the single-point spin Chern number (symmetric formula) versus the on-site term ∆ at fixed λR/λSO = 2 +for the Kane-Mele model. Different supercell sizes are considered (L = 9, 21, 51, and corresponding number of sites N = +162, 882, 5202). As the supercell size increases, the transition becomes sharper and approaches the analytical solution. Right +panel: gap ˜Eg of the P ˆszP operator versus the on-site term ∆ for the same supercell sizes as on the left-hand panel. +A +non-vanishing ˜Eg guarantees that the spin Chern number is well defined. +supercell sizes must be employed. In the right-hand panel +of Fig. 3, we show how the gap ˜Eg of the Pz operator +varies across the topological phase transition, but always +remains finite, ensuring that our single-point invariant is +everywhere well defined. Then, we validate the SPSCN +by calculating the entire topological phase diagram of the +KM model, which is reported in the upper panel of Fig. 4. +Notably, the method can distinguish topological and triv- +ial phases even for small, but still finite, values of both +the gap of the Hamiltonian and the gap of Pz (lower left- +hand panel in Fig. 4). Larger differences between the SP- +SCN and the exact value (zero), which are visibile in the +upper-left side of the topological phase diagram (marked +in blue), are finite size effects and are reduced for large +supercells, as highlighted in the lower right-hand panel in +Fig. 4: in that region both to Hamiltonian and Pz oper- +ators gap are indeed very small. Therefore, our formulas +works well also in presence of very strong Rashba SOC +and small band gaps. + +6 +0 +1 +2 +3 +4 +5 +6 +∆/λSO +0 +1 +2 +3 +4 +λR/λSO +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +Single-point Cσ +0 +1 +2 +3 +4 +5 +6 +∆/λSO +0 +1 +2 +3 +4 +λR/λSO +1e-14 +0.2 +0.4 +0.6 +0.8 +1 +˜Eg +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +5.0 +λR/λSO +−0.4 +−0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Single-point Cσ +L = 21 +L = 36 +L = 48 +Exact value +−5 +0 +5 +∆/λSO +−5 +0 +5 +λR/λSO +FIG. 4. Upper panel: topological phase diagram of the Kane-Mele model calculated with the single-point spin Chern number +(symmetric formula), for a supercell size L = 36 containing N = 2592 sites. Black dashed line marks the analytical solution +for the semi-metallic state separating the topological and trivial phases. Lower-left panel: gap ˜Eg of the P ˆszP operator for +the same calculations performed in the upper panel. Notably, ˜Eg is non-vanishing over all the phase diagram and guarantees +that the spin Chern number is well defined everywhere. Lower-right panel: the single-point spin Chern number versus the +Rashba coupling λR, at fixed ∆/λSO = 0.3, for different supercell size L = 21, 36, 48 and corresponding number of sites +N = 882, 2592, 4608. In that region of the phase diagram, band gaps are very small and finite size effects intensify; still the +single-point approach can distinguish the two phases. +B. +Disorder-driven topological phase transitions +The presence of disorder is often modelled by means +of an ensemble of large supercells, each representing a +specific random realisation as schematically represented +in the right-hand panel of Fig. 1. In electronic structure +simulations, defect calculations are performed by consid- +ering large supercells, to suppress the spurious interac- +tions due to the periodic replicas. Alloys are often sim- +ulated through the so-called special quasi-random struc- +tures [44]. In addition, a non-perturbative treatment of +temperature effects always require working with super- +cells, being a single structure with special atomic dis- +placements [45] or a collection of snapshots obtained from +ab initio molecular dynamics. +The SPSCN particularly suits this framework, and we +now assess the accuracy and convergence properties of +our formula on the KM model supplemented by an An- +derson disorder term [46], where we highlight its capa- +bility to detect disorder-driven topological transitions. +We emphasise that the simple KM model in presence +of rather strong Anderson disorder is used as a proto- +type and a proxy for testing, although our approach +targets the more general scenario mentioned above, of +supercell calculations, either for model Hamiltonians or +first-principles simulations. +The Hamiltonian of the disordered KM model reads +Hdis = HKM + +� +i +wic† +ici, +(14) +where wi ∈ +� +− W +2 , W +2 +� +is a randomly distributed on-site +potential and W is the disorder strength which, in the fol- +lowing, is reported in units of the nearest-neighbour hop- +ping amplitude t. In Fig. 5 we test the convergence of the +single-point formulas (Eqs. 10 and 12) with increasing su- +percell size L for the disorder strength W/t = 1, which is +weak enough not to destroy the topological phases of the +corresponding pristine KM model. The SPSCN is eval- + +7 +5 +10 +15 +20 +25 +30 +35 +40 +45 +L +−0.15 +−0.10 +−0.05 +0.00 +0.05 +0.10 +0.15 +Single-point Cσ +Asymmetric +Symmetric +0 +20 +40 +L +10−4 +10−3 +10−2 +10−1 +∆Cσ +−5 +0 +5 +∆/λSO +−5 +0 +5 +λR/λSO +5 +10 +15 +20 +25 +30 +35 +40 +45 +L +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +1.05 +1.10 +1.15 +Single-point Cσ +Asymmetric +Symmetric +0 +20 +40 +L +10−4 +10−3 +10−2 +10−1 +∆Cσ +−5 +0 +5 +∆/λSO +−5 +0 +5 +λR/λSO +FIG. 5. Convergence of the single-point spin Chern number, in its symmetric and asymmetric implementation, with respect to +supercell size L for the disordered Kane-Mele model. We report the average and standard deviation of the single-point invariant +calculated on M = 100 realisations with disorder strength W/t = 1. In the upper insets, the point in the corresponding pristine +phase diagram is shown. In the lowest insets, we report the difference between the mean value and the thermodynamic limit as +a function of L. Left panel: the spin Chern number converges to zero for ∆/λSO = 5.5 and λR/λSO = 3. Right panel: the spin +Chern number converges to one for ∆/λSO = 0.8 and λR/λSO = 2. Also in presence of disorder, the asymptotic convergence +is exponential and the symmetric formula converges visibly faster than its asymmetric counterpart. Statistical fluctuations are +very small and negligible at almost any supercell size. +uated as the mean value over M realisations of random +disorder with supercells of size L × L. Also in presence +of disorder, the convergence of the formulas is exponen- +tial and the symmetric version converges faster than the +asymmetric one. In addition, we consider increasing dis- +order strengths and study the robustness of the topolog- +ical phase, results are reported in Fig. 6. For sufficiently +strong disorder, the topological phase is destroyed and +the SPSCN becomes trivial. As expected, the width of +the phase transition becomes smaller with increasing su- +percell sizes. As investigated in Ref. [47], for a certain +range of parameters, the disordered KM model given by +Eq. 14 displays a topological state called topological An- +derson insulator (TAI). It is a phase of quantized con- +ductance which is obtained adding Anderson disorder to +a trivial insulator or metal which are relatively close to +a topological phase transition [48–50]. The mechanism +for this disorder-induced transition has been discussed in +terms of a renormalization of the model parameters such +as the on-site term [49]. +The weak-disorder boundary +of a TAI can be studied within an effective-medium the- +ory and the self-consistent Born approximation [47, 49], +but these perturbative approach might fail in the strong- +disorder regime, where the TAI phase is destroyed in +favour of a trivial insulating phase, as we show next. +In Fig. 7 we use the SPSCN to inspect these topological +phase transitions driven by disorder. In order to compare +with previous work on the disordered KM model [47] and +for the sake of clarity, we consider a value of λSO = 0.3 t +which is an order of magnitude greater than the one used +for the previous examples. First, we fix λR = 0 (left- +hand panel) and observe that the TAI appears at about +W/t = 2, in agreement with the conductance calcula- +tion in [47] and the spin Bott index results in [32] (note +the factor of two with respect to the W defined therein). +Then, we consider finite Rashba SOC and show the re- +sults in the right-hand panel of Fig. 7, where we note +that the TAI region has become narrower, in agreement +with Ref. [47]. A check on the gap ˜Eg of operator Pz +is performed for every SPSCN calculation in presence of +disorder: Anderson disorder never fully closes the gap +and the invariant can always be computed. +IV. +CONCLUSIONS +In this work, we have introduced a robust and effi- +cient single-point formula to calculate the Z2 topological +invariant in non-crystalline 2D materials. We have val- +idated our method with supercell numerical simulations +on the KM model, both pristine and disordered. +Our +approach can reproduce the entire phase diagram of the +KM model, where each calculation requires only a single- +point diagonalisation in the supercell framework, even in +presence of strong Rashba SOC. In addition, we have +extensively tested our method in presence of Anderson +disorder, and we have shown how the single-point for- +mula can correctly describe disorder-driven topological +phase transitions. In particular, we have discussed both +the process where disorder destroys the topological phase +and where disorder actually promotes it, as for the TAI +phase; that is in agreement with calculations of the con- +ductance [22, 47] and spin Bott index [32] reported in +the literature. Our single-point approach converges ex- +ponentially with size, so it is typically sufficient to work +with relatively small supercells, which is critical for ap- + +8 +0 +2 +4 +6 +8 +10 +W/t +−0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +Single-point Cσ +L = 15 +L = 42 +−5 +0 +5 +∆/λSO +−5 +0 +5 +λR/λSO +0 +2 +4 +6 +8 +10 +W/t +10−1 +100 +min( ˜Eg) +FIG. 6. +Robustness of the topological phase with respect to disorder. +The symmetric single-point spin Chern number is +calculated as function of disorder strength W/t, starting from the system in the topological phase (∆/λSO = 3, λR/λSO = 1). +For each W, we report the mean and standard deviation over M = 50 realisations of Anderson disorder for supercells of sizes +L = 15, 42 and number of sites N = 450, 3528 respectively. Upper inset: a sketch of the point where the calculations are +computed reported on the pristine phase diagram (W/t = 0). Lower inset: minimum value, over the disorder realizations, of +the gap ˜Eg of the P ˆszP operator as a function of W/t. With increasing supercell size L, the transition becomes sharper. ˜Eg +does not vanish with Anderson disorder and the approach performs well also in the strong-disorder regime. +0 +2 +4 +6 +8 +10 +12 +W/t +−0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +Single-point Cσ +L = 15 +L = 42 +−5 +0 +5 +∆/λSO +−5 +0 +5 +λR/λSO +0 +2 +4 +6 +8 +10 +12 +W/t +−0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +Single-point Cσ +L = 15 +L = 42 +−5 +0 +5 +∆/λSO +−5 +0 +5 +λR/λSO +0 +2 +4 +6 +8 +10 +12 +W/t +10−3 +10−2 +10−1 +100 +min( ˜Eg) +FIG. 7. Topological Anderson insulator (TAI). The symmetric single-point spin Chern number is calculated as function of +disorder strength W/t, starting from the system in a trivial state close to the phase transition. We report the mean value and +the standard deviation of single-point invariant over M = 50 realisations of Anderson disorder for supercells of sizes L = 15, 42 +and corresponding numbers of sites N = 450, 3528 respectively. For 5 ≤ W/t ≤ 10 the number of random realisations is +purposely increased to M = 100 to reduce the standard deviation. +Left panel: TAI state in absence of Rashba coupling +(∆/λSO = 5.5, λR = 0). Right panel: TAI at finite Rashba coupling (∆/λSO = 5.3, λR/λSO = 1). Here, the minimum value +of the gap ˜Eg (over M disorder realizations) is reported versus W/t in the lower inset (the same plot is not present in the +right-hand left panel since ˜Eg is constantly equal to one for λR = 0). + +9 +plications in ab initio modelling. One of the side benefits +of adopting Prodan’s approach is that the formula can, +at least in principle, be meaningful also in presence of +weak TR-breaking perturbations [36]. 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Smogunov, I. Timrov, T. Thonhauser, P. Umari, +N. Vast, X. Wu, and S. Baroni, Journal of Physics: Con- +densed Matter 29, 465901 (2017). + diff --git a/3tE0T4oBgHgl3EQfvAFe/content/tmp_files/load_file.txt b/3tE0T4oBgHgl3EQfvAFe/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b4111edf717358c07a02048e0050d846f2f19bde --- /dev/null +++ b/3tE0T4oBgHgl3EQfvAFe/content/tmp_files/load_file.txt @@ -0,0 +1,824 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf,len=823 +page_content='Single-point spin Chern number in a supercell framework Roberta Favata1 and Antimo Marrazzo1, ∗ 1Dipartimento di Fisica, Universit`a di Trieste, Strada Costiera 11, I-34151 Trieste, Italy (Dated: January 9, 2023) We present an approach for the calculation of the Z2 topological invariant in non-crystalline two-dimensional quantum spin Hall insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' While topological invariants were originally mathe- matically introduced for crystalline periodic systems, and crucially hinge on tracking the evolution of occupied states through the Brillouin zone, the introduction of disorder or dynamical effects can break the translational symmetry and imply the use of larger simulation cells, where the k−point sampling is typically reduced to the single Γ-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Here, we introduce a single-point formula for the spin Chern number that enables to adopt the supercell framework, where a single Hamiltonian diagonalisation is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Inspired by the work of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Prodan [Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' B, 80, 12 (2009)], our single-point approach allows to calculate the spin Chern number even when the spin operator ˆsz does not commute with the Hamiltonian, as in the presence of Rashba spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' We val- idate our method on the Kane-Mele model, both pristine and in the presence of Anderson disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Finally, we investigate the disorder-driven transition from the trivial phase to the topological state known as topological Anderson insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Beyond disordered systems, our approach is particularly useful to investigate the role of defects, to study topological alloys and in the context of ab-initio molecular dynamics simulations at finite temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' INTRODUCTION Two-dimensional (2D) topological insulators (TI) are materials with an insulating bulk and robust edge states protected by the non-trivial topology of the bulk elec- tronic structure [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' These systems are discussed through topological invariants, integer quantities which characterise the ground-state wavefunction in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' As long as the topological invariant is non-trivial and, possibly, the symmetries needed to define that topology are preserved, the material is said to be in a topolog- ical phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' These invariants are geometrical properties of the electronic structure, as they are defined in terms of quantities such as the Berry phase or the Berry cur- vature, which involve derivatives of the occupied states in reciprocal space with respect to the quasi-momentum k [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Standard geometrical formulas are usually discre- tised on a regular mesh of k-points for numerical imple- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' However, most electronic structure calcula- tions for non-crystalline systems are normally performed by diagonalising the Hamiltonian at a single k-point in a large supercell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Usually the Γ point at the center of the Brillouin zone (BZ) is considered, although potentially more efficient choices based on the Baldereschi point [3] can be employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' The derivation of single-point formu- las for geometrical and topological properties is not at all a trivial task, although successful single-point formalism have been developed for the Berry phase [4], the orbital magnetisation and the Chern number [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' In this work, we target the calculation of the topologi- cal invariant for non-crystalline 2D insulators with time- reversal (TR) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' For these systems, encompass- ing all non-magnetic 2D materials [6], the invariant ν is ∗ antimo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='marrazzo@units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='it a Z2 number: if ν = 0 the topology is trivial, otherwise if ν = 1 we have a quantum spin Hall insulator (QSHI), where topologically-protected gapless helical edge states cross the bulk gap [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Over the years, several meth- ods have been developed to calculate the Z2 invariant in crystalline systems with periodic boundary conditions (PBCs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' In the following, we briefly outline some of the most popular and practical methods in the context of elec- tronic structure simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' If inversion symmetry is present, there is a particularly simple method introduced by Fu and Kane [7], which requires the knowledge of the parity of the occupied states at the four TR-invariant points in the BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' In the more general case, the Z2 in- variant can be obtained by tracking the evolution of hermaphrodite [8] (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' hybrid) Wannier charge cen- tres [9–11], or equivalently the eigenvalues of the Wilson loop [12–14], over half BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' More recently, a generalisa- tion of the Fu-Kane approach based on elementary band representations [15, 16] has allowed to calculate the in- variant by using only the knowledge of the irreducible representations of the occupied states at selected high- symmetry points in the BZ [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' The Z2 invariant can be also computed as an individual Chern number [2] on half of the Hilbert space [10, 11], where the split is per- formed by two projectors which are smooth and related by TR symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Although several formulas to compute the Z2 invariant have been introduced, all the ones we mentioned, and most other existing approaches, require the knowledge of the occupied states at multiple k-points and become ill-defined for non-crystalline systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' hence in the supercell framework they are of no avail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Nonetheless, a number of methods have been proposed to deal with non-periodic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Some of these [17– 19] calculate the Z2 invariant by means of a Pfaffian with twisted boundary conditions, as firstly advocated by Kane and Mele in their original discussion of the Z2 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='02612v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='mes-hall] 6 Jan 2023 2 invariant in presence of disorder and electron-electron in- teractions [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' A different method is based on construct- ing the Z2 invariant from the scattering matrix of the system at the Fermi level [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Further, there exists a formulation based on the non-commutative index the- orem [23, 24], where the Z2 index for disordered topolog- ical insulators is computed from the discrete spectrum of a certain compact operator, which is defined as the difference of a proper pair of projection operators [25– 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' An alternative non-commmutative approach was proposed by Loring and Hastings [28, 29] and relates the Z2 index to the topological obstruction to approximat- ing almost commuting matrices by exactly commuting matrices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' its robustness with respect to the introduction of disorder has been investigated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' The most practical approach from the point of electronic structure simulations has been arguably put forward by Huang and Liu [31, 32], who addressed the problem of calcu- lating the Z2 invariant for non-periodic system in the context of quantum spin Hall quasicrystals, and intro- duced the spin Bott index, which measures the commu- tativity of the projected position operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' The connec- tion between the Bott indices and Chern or Z2 invariants has been investigated theoretically [28–30, 33], while nu- merical simulations [32, 34, 35] provided evidence that Bott indices can be used to study non-periodic topolog- ical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Still, it is conceptually rather unsatisfac- tory that the calculation of topological invariants in a supercell framework requires introducing radically differ- ent formalisms, which call for rather non-trivial equiva- lence proofs and extensive testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' As a matter of fact, the use of the primitive cell and k-points is an arbitrary— although indeed very convenient—choice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' there is no con- ceptual reason preventing bona fide Z2 invariants to be calculated directly in the supercell by deriving a suitable single-point limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' In addition, it is important to assess the convergence with respect to the system size, as dif- ferent approaches might deliver the same correct answer at very different computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' For instance, re- cent works [32, 33] claimed that the difference between the Chern number and Bott index is within a correc- tion of the order O(1/L), where L is the linear size of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Such slow convergence can hinder the study of the system close to a topological phase transition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' in fact Huang and Liu empirically added a singular value decomposition (SVD) to their algorithm to improve an otherwise slow convergence [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Here, we take a different approach, that essentially combines the work of Ceresoli and Resta on the single- point Chern number [5] and the insights from Prodan on a generalised spin Chern number [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Notably, our single-point invariant is directly derived by its parent for- mula for crystalline systems, it shows exponential conver- gence with the supercell size, both in the pristine and dis- ordered case, it is easy to implement in electronic struc- ture codes, and it works well also in presence of strong Rashba spin-orbit coupling (SOC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' METHODS In absence of spin-mixing spin-orbit interactions, the spin operator ˆsz commutes with the Hamiltonian and it is possible to discuss the Z2 invariant in terms of the spin Chern number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' In this case, the occupied states diago- nalise ˆsz and can be divided in two subsets, either purely spin-up or spin-down, and the regular Chern number can be calculated for each spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' As soon as the Hamiltonian does not commute any more with ˆsz, for instance because Rashba SOC is present, such simple-minded spin Chern number cannot be defined any more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Notably, Prodan has shown [36] that it is possible to generalise this def- inition by projecting the spin operator on the occupied states: Pz = P(k)ˆszP(k), (1) where P is the ground-state projector P(k) = � n |unk⟩ ⟨unk| , (2) unk are the periodic part of the Bloch eigenstates and n labels the occupied state at each k-point in the BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Then, we diagonalise Pz: Pz |uλ⟩ = sλ |uλ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' (3) If only diagonal SOC terms are present, the eigenvalue spectrum of Pz consists of two values only sλ = ± 1 2 and one can select a single spin component by choosing the eigenstates which correspond to one of the two eigenval- ues sλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' The crucial observation made by Prodan [36] is that, even if Rashba SOC is present, the spectrum of Pz displays two separate bands of eigenvalues symmet- ric around the origin and one can still introduce a well- defined spin Chern number by selecting the eigenvectors with positive (or negative) eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Finally, the spin Chern number can be computed as: Cs = C+ − C− 2 mod 2 (4) where C± are calculated on the uλ eigenstates with posi- tive and negative eigenvalues respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' in general it is sufficient to compute either C+ or C− only and consider its parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' The results are of paramount practical rele- vance, as it is typically much simpler to deal with a for- mulation based on generalised Chern numbers, which can be written as full BZ integrals and do not require taking into account TR symmetry or complex gauge fixing, as required instead by more general Z2 formulations [20, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' In principle, if the Rashba interaction is strong enough then the gap of the Pz spectrum might close, preventing the spin Chern number to be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Remarkably, as we will discuss in full detail in the Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' III, this does not seem to occur in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' As long as the system is in- sulating, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 4 is well defined even if the Rashba SOC is several times larger than the diagonal SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Hence, we 3 adopt the approach of Prodan [36] and target the deriva- tion of a single-point formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' In order to obtain the cor- rect single-point limit, we follow the approach of Ceresoli and Resta [5] for the derivation of the single-point Chern number in TR-broken systems (the latter admit a Z topo- logical invariant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Let us start with the formula for the generalised spin Chern number in 2D periodic systems: Cσ = 1 2π � BZ TrσΩxy(k)dk = − 1 π � sλ=σ � BZ Im ⟨∂kxuλ(k)|∂kyuλ(k)⟩ dkxdky, (5) where uλ are the eigenvectors of Pz (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 3) and σ = ± corresponds to one of the sectors of the Pz spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Now we consider the parallelogram Brillouin zone and change coordinate system to have a rectangular integra- tion domain: Cσ = − 1 π Im � sλ=σ � b1 0 dk1 � b2 0 dk2 ⟨∂k1uλ(k)|∂k2uλ(k)⟩ ≃ −|b1||b2| π Im � sλ=σ ⟨∂k1uλ(k)|∂k2uλ(k)⟩ |k=Γ, (6) where b1,2 are the two reciprocal lattice vectors and the last step is performed in the limit of a very large supercell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' In the same limit, we can calculate derivatives through finite differences: ∂kj |uλ(k)⟩ |k=Γ = lim η→0 |uλ(ηbj)⟩ − |uλ(Γ)⟩ η|bj| , (7) where we can drop the limit for a large supercell and just consider the difference |uλ(bj)⟩ − |uλ(Γ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 7 re- quires a differentiable function, which is not guaranteed in numerical diagonalisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Hence, we fix the gauge by adopting a discretised version of the covariant deriva- tive [38, 39] as successfully performed for the Chern num- ber by Ceresoli and Resta [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' One replaces the states with their “duals”: |˜un(bj)⟩ = � m S−1 mn(bj) |um(bj)⟩ (8) where we define the overlap matrix Snm(bj) = ⟨un(Γ)|um(bj)⟩ and the dual states satisfy ⟨un(Γ)|˜um(bj)⟩ = δnm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Next, we construct the states un(bj) by imposing the periodic gauge, which allows us to perform a single diagonalisation at Γ: |uλ(bj)⟩ = e−ibj·r |uλ(Γ)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' (9) The states in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 9 are Hamiltonian eigenstates, but they might correspond to a different eigenvalue with respect to the one at Γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' the ordering is anyway fixed by the covariant derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' We note in passing, that while a non-trivial Chern number would prevent the adoption of a periodic gauge for the wavefunction, here the periodic gauge is only temporarily imposed to build each |un(bj)⟩ from the knowledge of the |un(Γ)⟩, but it is effectively replaced by the parallel transport gauge enforced by the covariant derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' The final single-point formula for the spin Chern number is C(asym) σ = −|b1||b2| π Im � sλ=σ ⟨˜uλ(b1)|˜uλ(b2)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' (10) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 10, we emphasise with the superscript “asym” the implicit choice made in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 7, which corresponds to the right-hand derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' In fact, an alternative choice is the symmetric derivative ∂kj |uλ(k)⟩ |k=Γ ≃ |uλ(bj)⟩ − |uλ(−bj)⟩ 2|bj| , (11) which can also be computed with a single Γ-only diago- nalisation and leads to the following formula for the spin Chern number: C(sym) σ = −|b1||b2| 4π Im � sλ=σ (⟨˜uλ(b1)| − ⟨˜uλ(−b1)|) (|˜uλ(b2)⟩ − |˜uλ(−b2)⟩) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' (12) In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' III, we will show how the symmetric formula con- verges much faster than the asymmetric version, at es- sentially the same computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' We have implemented the single-point formulas in a dedicated Python package, freely available on GitHub [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' The code provides user-friendly inter- faces to two popular tight-binding packages such as PythTB [41] and TBmodels [42], and it can be easily interfaced to other codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' NUMERICAL RESULTS AND DISCUSSION We validate our approach on the paradigmatic Kane- Mele (KM) model [20, 43] on the honeycomb lattice, both pristine and in presence of Anderson disorder (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' The Kane-Mele model in the supercell approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Left panel: pristine Kane-Mele model, the primitive cell is shown in orange while a 3 × 3 supercell is marked in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Right panel: random realisation of a disordered Kane-Mele model in a 3 × 3 supercell (green) with periodic boundary conditions, where different colours are used to represent the on-site terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' In the following, supercells are labelled by their integer size L × L (in units of the pristine primitive cell) and the corresponding number of sites N = 2L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' The tight-binding Hamiltonian reads HKM = t � ⟨i,j⟩ c† icj + ∆ � i ξic† ici + iλSO � ⟨⟨i,j⟩⟩ νijc† iσzcj (13) + iλR � ⟨i,j⟩ c† i(σ × ˆdij)zcj, where i and j run over all sites in the lattice and the creation and annihilation operators are expressed in the contracted form c† i = (c† i↑, c† i↓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' The first term is a real nearest-neighbour hopping (denoted by ⟨ , ⟩), if taken alone that would yield four (pair-degenerate) bands with gapless Dirac cones centred on the high-symmetry points K and K ′ in the Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' The second term is a staggered on-site potential (ξi = ±1 is the sub- lattice index of the i−th site) while the third term is the KM SOC [20, 43] which involves a complex next- nearest neighbour hopping (denoted by ⟨⟨ , ⟩⟩) with a spin-dependent amplitude proportional to the Pauli ma- trix σz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' The factor νij = sign(d1 × d2)z depends on the orientation of the vectors d1 and d2 along the two bonds connecting i to the next-nearest neighbour site j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' The fourth term is the Rashba SOC and is a complex nearest-neighbour hopping with off-diagonal spin com- ponents, where σ = (σx, σy, σz) is the vector of Pauli matrices and ˆdij is the unit vector between sites j and i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' In the following, we consider a KM Hamiltonian at fixed parameters t = 1 and λSO = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='03 t, which ensure that the energy gap is insulating all over the entire phase diagram [20, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Validation and convergence tests for crystalline systems In the single-point approach, the topological invari- ants become exact integer numbers only in the thermody- namic limit of an infinite supercell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' First, we test the con- vergence properties of the single-point spin Chern num- ber (SPSCN) on the pristine KM model, in both asym- metric (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 10) and symmetric (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 12) formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' We inspect the SPSCN as a function of the supercell size L, here defined as the number of primitive cells along each lattice vector that makes the supercell L×L (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' the number of sites inside the supercell is N = 2L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' A representation of a supercell 3 × 3 is given in the left- hand panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' In our calculations only values of L which are multiple of 3 are considered, to always include the special points K and K ′ folded at Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' We bench- mark the accuracy of the formulas inside the Z2-even and Z2-odd domains in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' The symmetric for- mula converges faster than the asymmetric one in both trivial and topological phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Remarkably, the quantity ∆Cσ = |Cσ(L)−Cσ(∞)|, which is the difference between the spin Chern number given by the single-point formulas at finite sizes and the exact value obtained in the ther- modynamic limit, decreases exponentially in both formu- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' However, the global prefactor in the symmetric case is an order of magnitude smaller than the one of the asymmetric formula, leading to more accurate results at significantly smaller sizes L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Hence, in the following we adopt the symmetric formula only and study the topo- logical phase transition as a function of the on-site ∆, results are reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Our SPSCN is able to reproduce the sharp topological transition already at rel- atively small supercell sizes, as shown in the left-hand panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' The band gap vanishes on the boundary of the phase transition and in the corresponding neigh- bourhood of parameters convergence is slower and larger 5 6 12 18 24 30 36 42 48 54 60 L −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='100 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='075 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='050 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='100 Single-point Cσ Asymmetric Symmetric Exact value 0 20 40 60 L 10−4 10−3 10−2 10−1 ∆Cσ −5 0 5 ∆/λSO −5 0 5 λR/λSO 6 12 18 24 30 36 42 48 54 60 L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='15 Single-point Cσ Asymmetric Symmetric Exact value 0 20 40 60 L 10−4 10−3 10−2 10−1 ∆Cσ −5 0 5 ∆/λSO −5 0 5 λR/λSO FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Convergence of the single-point spin Chern number, in its symmetric and asymmetric implementation, with respect to supercell size for the Kane-Mele model, where the Hamiltonian is diagonalised at the Γ-point only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' In the uppest insets, a sketch of the corresponding point in the pristine phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' The lowest insets show the difference between the single-point calculations of the spin Chern number and the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Left panel: the spin Chern number converges to zero in the trivial phase (∆/λSO = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='5, λR/λSO = 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Right panel: in the topological phase (∆/λSO = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='8 , λR/λSO = 2) the spin Chern number converges to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' In all cases, the asymptotic convergence is exponential, but the symmetric formula converges visibly faster than its asymmetric counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 0 1 2 3 4 5 6 ∆/λSO −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='0 Single-point Cσ L = 9 L = 24 L = 51 Exact value −5 0 5 ∆/λSO −5 0 5 λR/λSO 0 1 2 3 4 5 6 ∆/λSO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='0 ˜Eg L = 9 L = 24 L = 51 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Left panel: the single-point spin Chern number (symmetric formula) versus the on-site term ∆ at fixed λR/λSO = 2 for the Kane-Mele model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Different supercell sizes are considered (L = 9, 21, 51, and corresponding number of sites N = 162, 882, 5202).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' As the supercell size increases, the transition becomes sharper and approaches the analytical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Right panel: gap ˜Eg of the P ˆszP operator versus the on-site term ∆ for the same supercell sizes as on the left-hand panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' A non-vanishing ˜Eg guarantees that the spin Chern number is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' supercell sizes must be employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' In the right-hand panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 3, we show how the gap ˜Eg of the Pz operator varies across the topological phase transition, but always remains finite, ensuring that our single-point invariant is everywhere well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Then, we validate the SPSCN by calculating the entire topological phase diagram of the KM model, which is reported in the upper panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Notably, the method can distinguish topological and triv- ial phases even for small, but still finite, values of both the gap of the Hamiltonian and the gap of Pz (lower left- hand panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Larger differences between the SP- SCN and the exact value (zero), which are visibile in the upper-left side of the topological phase diagram (marked in blue), are finite size effects and are reduced for large supercells, as highlighted in the lower right-hand panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 4: in that region both to Hamiltonian and Pz oper- ators gap are indeed very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Therefore, our formulas works well also in presence of very strong Rashba SOC and small band gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 6 0 1 2 3 4 5 6 ∆/λSO 0 1 2 3 4 λR/λSO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='8 1 Single-point Cσ 0 1 2 3 4 5 6 ∆/λSO 0 1 2 3 4 λR/λSO 1e-14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='8 1 ˜Eg 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='0 λR/λSO −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='0 Single-point Cσ L = 21 L = 36 L = 48 Exact value −5 0 5 ∆/λSO −5 0 5 λR/λSO FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Upper panel: topological phase diagram of the Kane-Mele model calculated with the single-point spin Chern number (symmetric formula), for a supercell size L = 36 containing N = 2592 sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Black dashed line marks the analytical solution for the semi-metallic state separating the topological and trivial phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Lower-left panel: gap ˜Eg of the P ˆszP operator for the same calculations performed in the upper panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Notably, ˜Eg is non-vanishing over all the phase diagram and guarantees that the spin Chern number is well defined everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Lower-right panel: the single-point spin Chern number versus the Rashba coupling λR, at fixed ∆/λSO = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='3, for different supercell size L = 21, 36, 48 and corresponding number of sites N = 882, 2592, 4608.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' In that region of the phase diagram, band gaps are very small and finite size effects intensify;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' still the single-point approach can distinguish the two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Disorder-driven topological phase transitions The presence of disorder is often modelled by means of an ensemble of large supercells, each representing a specific random realisation as schematically represented in the right-hand panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' In electronic structure simulations, defect calculations are performed by consid- ering large supercells, to suppress the spurious interac- tions due to the periodic replicas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Alloys are often sim- ulated through the so-called special quasi-random struc- tures [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' In addition, a non-perturbative treatment of temperature effects always require working with super- cells, being a single structure with special atomic dis- placements [45] or a collection of snapshots obtained from ab initio molecular dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' The SPSCN particularly suits this framework, and we now assess the accuracy and convergence properties of our formula on the KM model supplemented by an An- derson disorder term [46], where we highlight its capa- bility to detect disorder-driven topological transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' We emphasise that the simple KM model in presence of rather strong Anderson disorder is used as a proto- type and a proxy for testing, although our approach targets the more general scenario mentioned above, of supercell calculations, either for model Hamiltonians or first-principles simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' The Hamiltonian of the disordered KM model reads Hdis = HKM + � i wic† ici, (14) where wi ∈ � − W 2 , W 2 � is a randomly distributed on-site potential and W is the disorder strength which, in the fol- lowing, is reported in units of the nearest-neighbour hop- ping amplitude t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 5 we test the convergence of the single-point formulas (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 10 and 12) with increasing su- percell size L for the disorder strength W/t = 1, which is weak enough not to destroy the topological phases of the corresponding pristine KM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' The SPSCN is eval- 7 5 10 15 20 25 30 35 40 45 L −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='15 Single-point Cσ Asymmetric Symmetric 0 20 40 L 10−4 10−3 10−2 10−1 ∆Cσ −5 0 5 ∆/λSO −5 0 5 λR/λSO 5 10 15 20 25 30 35 40 45 L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='15 Single-point Cσ Asymmetric Symmetric 0 20 40 L 10−4 10−3 10−2 10−1 ∆Cσ −5 0 5 ∆/λSO −5 0 5 λR/λSO FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Convergence of the single-point spin Chern number, in its symmetric and asymmetric implementation, with respect to supercell size L for the disordered Kane-Mele model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' We report the average and standard deviation of the single-point invariant calculated on M = 100 realisations with disorder strength W/t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' In the upper insets, the point in the corresponding pristine phase diagram is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' In the lowest insets, we report the difference between the mean value and the thermodynamic limit as a function of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Left panel: the spin Chern number converges to zero for ∆/λSO = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='5 and λR/λSO = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Right panel: the spin Chern number converges to one for ∆/λSO = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='8 and λR/λSO = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Also in presence of disorder, the asymptotic convergence is exponential and the symmetric formula converges visibly faster than its asymmetric counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Statistical fluctuations are very small and negligible at almost any supercell size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' uated as the mean value over M realisations of random disorder with supercells of size L × L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Also in presence of disorder, the convergence of the formulas is exponen- tial and the symmetric version converges faster than the asymmetric one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' In addition, we consider increasing dis- order strengths and study the robustness of the topolog- ical phase, results are reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' For sufficiently strong disorder, the topological phase is destroyed and the SPSCN becomes trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' As expected, the width of the phase transition becomes smaller with increasing su- percell sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' As investigated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' [47], for a certain range of parameters, the disordered KM model given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 14 displays a topological state called topological An- derson insulator (TAI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' It is a phase of quantized con- ductance which is obtained adding Anderson disorder to a trivial insulator or metal which are relatively close to a topological phase transition [48–50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' The mechanism for this disorder-induced transition has been discussed in terms of a renormalization of the model parameters such as the on-site term [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' The weak-disorder boundary of a TAI can be studied within an effective-medium the- ory and the self-consistent Born approximation [47, 49], but these perturbative approach might fail in the strong- disorder regime, where the TAI phase is destroyed in favour of a trivial insulating phase, as we show next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 7 we use the SPSCN to inspect these topological phase transitions driven by disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' In order to compare with previous work on the disordered KM model [47] and for the sake of clarity, we consider a value of λSO = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='3 t which is an order of magnitude greater than the one used for the previous examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' First, we fix λR = 0 (left- hand panel) and observe that the TAI appears at about W/t = 2, in agreement with the conductance calcula- tion in [47] and the spin Bott index results in [32] (note the factor of two with respect to the W defined therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Then, we consider finite Rashba SOC and show the re- sults in the right-hand panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 7, where we note that the TAI region has become narrower, in agreement with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' A check on the gap ˜Eg of operator Pz is performed for every SPSCN calculation in presence of disorder: Anderson disorder never fully closes the gap and the invariant can always be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' CONCLUSIONS In this work, we have introduced a robust and effi- cient single-point formula to calculate the Z2 topological invariant in non-crystalline 2D materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' We have val- idated our method with supercell numerical simulations on the KM model, both pristine and disordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Our approach can reproduce the entire phase diagram of the KM model, where each calculation requires only a single- point diagonalisation in the supercell framework, even in presence of strong Rashba SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' In addition, we have extensively tested our method in presence of Anderson disorder, and we have shown how the single-point for- mula can correctly describe disorder-driven topological phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' In particular, we have discussed both the process where disorder destroys the topological phase and where disorder actually promotes it, as for the TAI phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' that is in agreement with calculations of the con- ductance [22, 47] and spin Bott index [32] reported in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Our single-point approach converges ex- ponentially with size, so it is typically sufficient to work with relatively small supercells, which is critical for ap- 8 0 2 4 6 8 10 W/t −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='50 Single-point Cσ L = 15 L = 42 −5 0 5 ∆/λSO −5 0 5 λR/λSO 0 2 4 6 8 10 W/t 10−1 100 min( ˜Eg) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Robustness of the topological phase with respect to disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' The symmetric single-point spin Chern number is calculated as function of disorder strength W/t, starting from the system in the topological phase (∆/λSO = 3, λR/λSO = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' For each W, we report the mean and standard deviation over M = 50 realisations of Anderson disorder for supercells of sizes L = 15, 42 and number of sites N = 450, 3528 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Upper inset: a sketch of the point where the calculations are computed reported on the pristine phase diagram (W/t = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Lower inset: minimum value, over the disorder realizations, of the gap ˜Eg of the P ˆszP operator as a function of W/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' With increasing supercell size L, the transition becomes sharper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' ˜Eg does not vanish with Anderson disorder and the approach performs well also in the strong-disorder regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 0 2 4 6 8 10 12 W/t −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='50 Single-point Cσ L = 15 L = 42 −5 0 5 ∆/λSO −5 0 5 λR/λSO 0 2 4 6 8 10 12 W/t −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='50 Single-point Cσ L = 15 L = 42 −5 0 5 ∆/λSO −5 0 5 λR/λSO 0 2 4 6 8 10 12 W/t 10−3 10−2 10−1 100 min( ˜Eg) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Topological Anderson insulator (TAI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' The symmetric single-point spin Chern number is calculated as function of disorder strength W/t, starting from the system in a trivial state close to the phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' We report the mean value and the standard deviation of single-point invariant over M = 50 realisations of Anderson disorder for supercells of sizes L = 15, 42 and corresponding numbers of sites N = 450, 3528 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' For 5 ≤ W/t ≤ 10 the number of random realisations is purposely increased to M = 100 to reduce the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Left panel: TAI state in absence of Rashba coupling (∆/λSO = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='5, λR = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Right panel: TAI at finite Rashba coupling (∆/λSO = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content='3, λR/λSO = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Here, the minimum value of the gap ˜Eg (over M disorder realizations) is reported versus W/t in the lower inset (the same plot is not present in the right-hand left panel since ˜Eg is constantly equal to one for λR = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 9 plications in ab initio modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' One of the side benefits of adopting Prodan’s approach is that the formula can, at least in principle, be meaningful also in presence of weak TR-breaking perturbations [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' This feature could be useful to study how the bulk topology is affected by the presence of magnetic impurities, or of a magnetic substrate through the proximity effect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' even though the absence of TR symmetry would allow backscattering be- tween the two helical edge states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' To encourage the use of our approach, we release a dedicated Python package that allows to seamlessly calculate the single-point Chern (Z) and spin-Chern (Z2) invariants of any TB model thanks to dedicated interfaces to PythTB and TBmodels, two very popular TB codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Notably, these two packages also allow working with Wannier Hamiltonians, which are read in the format produced by Wannier90 [51, 52];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' that provides a simple way to apply our work in the con- text of first-principles calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Then, it would be interesting to explore the effect of the TB approximation where the real-space position operator is taken to be diag- onal, versus considering all off-diagonal elements, essen- tially taking into account the overlap between Wannier functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Nonetheless, the formalism is rather simple and it could be implemented with limited effort directly into plane-wave first-principles codes, such as Quantum ESPRESSO [53, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' In short, our approach allows study- ing 2D topological insulators in a supercell framework, which is crucial to investigate very relevant phenomena such as disorder, defects, alloying, and to study dynam- ical and temperature effects through ab initio molecular dynamics simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Bernevig and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Hughes, Topological Insulators and Topological Superconductors (Princeton University Press, 2013).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Ce- pellotti, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Pizzi, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Marzari, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Nanotechnol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' 13, 246 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' [7] L.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Peressi, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Resta, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' B 64, 115202 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE0T4oBgHgl3EQfvAFe/content/2301.02612v1.pdf'} 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0000000000000000000000000000000000000000..7a115dacdfcb02ef03c30ef947443f4bcd093e62 --- /dev/null +++ b/3tFRT4oBgHgl3EQfojeI/content/tmp_files/2301.13609v1.pdf.txt @@ -0,0 +1,4507 @@ + +Physarum Inspired Bicycle Lane Network Design +in a Congested Mega City + + +By +Md. Ahsan Habib +Roll: 1507082 + + + + + + + + +Department of Computer Science and Engineering +Khulna University of Engineering & Technology +Khulna 9203, Bangladesh + + +March 2020 + + + +ii + +Certification + + +The thesis titled “Physarum Inspired Bicycle Lane Network Design in a Congested +Mega City” submitted by Md. Ahsan Habib, Roll No: 1507082, Academic Year: 2018- +19, for partial fulfillment of the requirements for the degree of “Bachelor of Science in +Computer Science and Engineering”. + +Supervisor + + + + Dr. Muhammad Aminul Haque Akhand + +Professor + +Dept. of Computer Science and Engineering + +Khulna University of Engineering & Technology + +Khulna, Bangladesh. + + + + + + + + + + + + + + + +iii + + +Acknowledgements + +First and foremost, I must sense grateful to and wish to acknowledge my insightful +indebtedness to Dr. Muhammad Aminul Haque Akhand, Professor of Department of +Computer Science and Engineering and the supervisor of the thesis. His unfathomable +knowledge in this field influenced me to carry out this thesis up to this point. His endless +endurance, scholarly guidance, continual encouragement, +constant and lively +supervision, constructive criticism, priceless suggestion made it possible to come up to +this phase. Without his inspiring, enthusiasm and encouragement, this work could not +be completed. +Last, but by no means least, I thank Allah for the talents and abilities I was given that +made it possible to undertake this thesis. + + + + + + + + + + + + + + + + +iv + + +Abstract + +Mobility is a key factor in urban life and transport network plays a vital role in mobility. +Worse transport network having less mobility is one of the key reasons to decline the +living standard in any unplanned mega city. Transport mobility enhancement in an +unplanned mega city is always challenging due to various constraints including +complex design and high cost involvement. The aim of this thesis is to enhance +transport mobility in a megacity introducing a bicycle lane. To design the bicycle lane +natural Physarum, brainless single celled multi-nucleated protist, is studied and +modified for better optimization. Recently Physarum inspired techniques are drawn +significant attention to the construction of effective networks. Exiting Physarum +inspired models effectively and efficiently solves different problems including +transport network design and modification and implication for bicycle lane is the unique +contribution of this study. Central area of Dhaka, the capital city of Bangladesh, is +considered to analyze and design the bicycle lane network bypassing primary roads. + + + + +v + + +Contents + + Title Page + + + + + + + i + Certification + + + + + + + ii + Acknowledgements + + + + + + iii + Abstract + + + + + + + + iv + Contents + + + + + + + + v + List of Tables + + + + + + + vii + List of Figures + + + + + + viii +CHAPTER 1 Introduction +1 + + + 1.1 Overview of Transport Network in a Mega City + + 1 + + + 1.2 Motivation + + + + + + + 1 + 1.3 Objectives of the Thesis +2 + 1.4 Organization of the Thesis +3 +CHAPTER 2 Physarum Inspired Network Design +4 + 2.1 Physarum and its Properties +4 + 2.2 Network Design Inspired on Physarum +5 + 2.3 Review of Existing Physarum Inspired Works +6 + + + +2.3.1 Transpiration Network Design 6 + + + +2.3.1 Other Optimization Task + + + + 7 +CHAPTER 3 Physarum Inspired Bicycle Lane Design in an Unplanned + + + Mega City 10 + 3.1 Mobility Problem in an Unplanned Mega City: Dhaka as a Case +Study 9 + + + +3.1.1 History and Overview of Dhaka City + + + 9 + + + +3.1.2 Transportation Crisis in Dhaka City + + + 9 + + + +3.1.3 Effect of Transportation Crisis to Other Problems + 13 + 3.2 Importance of Bicycle Lane in an Mega City 14 + 3.3 Challenges to Increase Mobility in Dhaka City +15 + 3.4 Bicycle Lane Design in an Unplanned Mega City +16 + 3.5 Significance of Study +23 +CHAPTER 4 Experimental Studies +24 + 4.1 Experimental Settings +24 + + + +vi + + 4.2 Bicycle Lane Network Design in a Prominent Area +Error! +Bookmark not defined. + 4.2.1 Network Design +25 + 4.2.2 Effectiveness Analysis 32 + 4.2.2.1 Time Saving +33 + 4.2.2.2 Fuel and Cost Saving +33 + 4.2.2.3 CO2 Emission Reduction +36 + 4.3 Bicycle Lane Network Design for Entire Dhaka City +36 + 4.3.1 Network Design 36 + 4.3.2 Effectiveness Analysis 38 + 4.3.2.1 Time Saving +39 + 4.3.2.2 Fuel and Cost Saving +42 + 4.3.2.3 CO2 emission reduction +44 +CHAPTER 5 Conclusions +46 + + + 5.1 Achievements + + + + + + 46 + + + 5.2 Future Study + + + + + + + 46 +References +47 + + + + + + + + +vii + + +List of Tables + + + + +Table No +Description +Page +2.1 +Network construction using Physarum. +7 +4.1 +The locations of prominent area of Dhaka city including +traffic pressure. +27 +4.2 +Routes from node point 1. +32 +4.3 +Time comparison in car, bus and bicycle considering +from node point 1. +34 +4.4 +Time saving per day. +34 +4.5 +Cost comparison in car, bus and bicycle considering +from node point 1. +35 +4.6 +Cost saving per day. +35 +4.7 +CO2 emission reduction per day. +36 +4.8 +The general data on locations in Dhaka city, including +population, area, and traffic pressure. +38 +4.9 +Routes from node point 7. +39 +4.10 +Time comparison in car, bus and bicycle considering +from node point 7. +41 +4.11 +Time saving per day. +41 +4.12 +Cost comparison in car, bus and bicycle considering from +node point 1. +43 +4.13 +Cost saving per day. +43 +4.14 +CO2 emission reduction per day. +44 + + + +viii + + +List of Figures + + + + + + + + + + + + + + + + + + + + +Figure No +Description +Page +2.1 +Physarum polycephalum. +4 +3.1 +Farmgate to University of Dhaka routes and time +needed in driving mode. +11 +3.2 +Physarum inspired network design of 11 nodes. +16 +3.3 +Real traffic network. +17 +4.1. +Selected Dhaka city Map. +19 +4.2 +Network design using modified Physarum inspired +technique. +21 +4.3 +Dhaka city Map. +28 +4.4 +Constructed network 29 points of Dhaka city. + +29 + + + +2 + +CHAPTER 1 +Introduction +A transport network can be described as a collection of linear features that permits +either vehicular movement or flow of some commodity. The characteristics of +Physarum can be used to design a transportation network. This chapter discusses the +background study of network design, objectives, and organization of the thesis. +1.1 Overview of Transport Network in a Mega City +Mobility, a key factor in planning and designing urban transport, is a fundamental part +of human beings [1]. For mobility purposes, people can use both motorized and non- +motorized vehicles within a city. Motorized vehicles like buses, cars, motorbikes, +cycles, etc. are hazardous in many kinds [2]. Non-motorized transports involve walking +and cycling as well as variants such as small-wheeled transportation like skates, +skateboards, push scooters and hand carts [3]. Nowadays, non-motorized mobility is +trendy [4]. + +The idea of mega city emerged to characterize the world's largest metropolitan +agglomerations at the end of the 20th century. In the 1970s, only two mega cities had +over ten million residents. Currently, 9.9% of the urban population globally resides in +23 megacities. It is projected that in 2025, the number will increase to 37 if 13.6% of +global urban population are to be accommodated [5]. + +A transportation network is the formation of a spatial network that enables vehicle +movement or the flow of some commodities. A vast network of rail, subways and bus +lines passes through well-organized mega cities. There are also exist special footways +and networks for cycling routes. +1.2 Motivation +In a well-planned and well-organized mega city, both footways and roads are available +for mobility purpose but in case of an unplanned and unorganized mega city, both +footpaths and roads are hardly available. In some cases, footpaths are snatched by + + + +3 + +hawkers and street vendors. The public transport system is defined by far a lack of +people's desired travel needs in terms of mobility, reliability, convenience, pace and +safety. In fact, some transports like buses are considered unreliable and time consuming +to reach their destinations. The Texas Transportation Institute reported a delay of 3.6 +billion vehicle-hours in the 75 biggest metropolitan regions in 2000, culminating in 5.7 +billion U.S. gallons (21.6 billion liters) of waste fuel and a loss of productivity of $67.5 +billion or around 0.7 percent of GDP. + +Bicycling or walking activity may help increase blood flow, release endorphins, and +decrease overall stress. It can even help to improve mental health and energy by +tracking 30 minutes of bicycling or walking a day[6]–[9]. Efficient and effective +bicycle lane network design in an unplanned and unorganized mega-city can minimize +total travel time, fuel usage, costs, carbon dioxide (CO2) emission, etc. Physarum +polycephalum is multi-headed, brainless, a giant multi-nucleated, single-celled protist +that can solve different complex problems. Physarum networks are believed to have +achieved a good balance between cost, efficiency, and resilience. +1.3 Objectives of the Thesis +In the case of an unorganized and unplanned mega city, where the transportation +network is congested and unplanned and there are hardly any footpaths available and +no further transport facilities can be expanded. So there are some huge problems in +those cities like traffic jam, noise pollution, air pollution, CO2 emission, etc. With a +planned lane network with non-motorized vehicles nearly all of this problem can be +solved. Since it is not feasible to completely rebuild the transportation network and +infrastructure of a mega city but possible to transform mega city towards a green city. +The objective of this study is given below: + Study of Physarum + Physarum related paper study + Network design + Bicycle lane network in mega city + + + +4 + +1.4 Organization of the Thesis +The main attraction of this thesis is to present a modified Physarum inspired technique +to construct bicycle lane network design. The thesis has five chapters. An introduction +to network design and Physarum has been given in Chapter 1. Chapter-wise overviews +of the rest of the thesis are as follows: +Chapter 2: Describes the literature review that includes a brief description of Physarum +with its properties and previous related work to Physarum inspired network design. +Chapter 3: Explains the proposed modified Physarum Inspired Bicycle Lane Design +in an Unplanned Mega City in detail. +Chapter 4: Reports the experimental result of modified Physarum Inspired Bicycle +Lane Design. Also, in this chapter, a case study of Dhaka is demonstrated. Finally, +Chapter 5: This chapter is for the conclusions of this thesis together with the outline +of future directions of research opened by this work. + + + + +CHAPTER 2 +Physarum Inspired Network Design +Physarum polycephalum is a brainless amoeboid organism. Physarum-inspired network +design model has demonstrated extraordinary skill in designing effective networks. In +this chapter firstly, we discuss the Physarum polycephalum, secondly the Physarum- +based network design and lastly, the existing network design inspired by Physarum. +2.1 Physarum and its Properties +Physarum polycephalum, accurately the 'many-headed' slime mold, is a gigantic multi- +nucleated but single-celled protist [10]. The slime mold Physarum polycephalum +creates a form of spatial memory by avoiding areas it has previously explored to +navigate in a complex environment[11]. Recently, Physarum polycephalum (true slime +mold) has arisen as a fascinating illustration of biological computation through +morphogenesis[12]. Although it is a single-cell organism, studies have shown that the +Physarum can overcome different minimum cost flow problems through its growth +process[12]. In the following Fig. 2.1, an example of the Physarum polycephalum is +shown. + +Figure 2.1: Physarum polycephalum. [61]. + + +FS +FS +FS +Source +FS +FS + +5 + +Here Physarum polycephalum is shown to grow up the network towards the FSs from +the source. (FS = Food Source) +2.2 Network Design Inspired on Physarum +The intelligent behavior of slime mold was first observed by Nakagaki et al. in +2000[13]. In previous biological experiments, Physarum-inspired network model has +exhibited an extraordinary intelligence to build efficient networks to connect multiple +food sources. Physarum networks are believed to have achieved a good balance +between cost, efficiency, and resilience. For instance, Physarum constructed networks +with comparable qualities to those of the Tokyo rail system in a renowned experiment +performed by Tero et al. in 2010[14]. +They developed a mathematical model for adaptive network construction to emulate +the behavior of Physarum which is based on feedback loops between the thickness of +each tube and internal protoplasmic flow in which high rates of streaming stimulate an +increase in tube diameter, whereas tubes tend to decline at low flow rates. The edges +represent plasmodial tubes in which protoplasm flows, and nodes are junctions between +tubes. They consider the pressure at nodes 𝑖 and 𝑗 are 𝑃𝑖 and 𝑃𝑗, respectively, and the +two nodes are connected by a cylinder of length 𝐿𝑖𝑗 and radius 𝑟𝑖𝑗. They assume that +the flow is laminar and follows the Hagen-Poiseuille equation, the flux through the tube +is, +𝑄𝑖𝑗 = 𝜋𝑟𝑖𝑗 +4(𝑃𝑖 − 𝑃𝑗) +8𝜀𝐿𝑖𝑗 + = 𝐷𝑖𝑗(𝑃𝑖 − 𝑃𝑗) +𝐿𝑖𝑗 + , (2.1) +here 𝜀 is the viscosity of the fluid, and 𝐷𝑖𝑗 = +𝜋𝑟𝑖𝑗 +4 +8𝜀 is a measure of the conductivity of +the tube. As the length 𝐿𝑖𝑗 is a constant, the behavior of the network is described by the +conductivities of the edges. +The constrains must be maintained, + + ∑ 𝑄1𝑗 = 𝐼0 +𝑗 +, +For source node-1 + + ∑ 𝑄2𝑗 = −𝐼0 +𝑗 +, +For sink node-2 + + ∑ 𝑄𝑖𝑗 = 0 +𝑗 +, +Inflow and outflow must be conserved + + + +6 + +To accommodate the adaptive behavior of the plasmodium, the conductivity of each +tube evolves according to +𝑑𝐷𝑖𝑗 +𝑑𝑡 = 𝑓(|𝑄𝑖𝑗|) − 𝐷𝑖𝑗, where 𝑓(|𝑄𝑖𝑗|) describes the +expansion of tubes in response to the flux and 𝐷𝑖𝑗 represents the rate of tube +constriction, so the tubes will gradually disappear. +The functional 𝑓(|𝑄|) = +|𝑄|𝛾 +(1+ |𝑄|𝛾) which describes a sigmoidal response where 𝛾 is a +parameter that controls the nonlinearity of feedback (𝛾 > 0). +2.3 Review of Existing Physarum Inspired Works +Physarum can successfully overcome many problems in real life even more +complicated problems. In this section initially, we discuss and summarize about the +Physarum inspired network design techniques, and then discuss about other +optimization problems solved using Physarum inspired methods. +2.3.1 Transpiration Network Design +To link several food points Physarum can build high-quality networks. A mathematical +model of the adaptive dynamics of a transport network of the true slime mold that shows +path-finding behavior in a maze is developed in 2007 by Tero et al. [15]. In 2010, +Physarum developed networks shows similar qualities to the Tokyo rail system in a +famous experiment conducted by Tero et al. [14]. Since then, Physarum inspired other +real-world transport networks, such as Iberian motorways [16] and Mexican Federal +highways [17] have also been constructed. Adamatzky et al. [18] develops a model to +construct networks on major urban areas of China. Becker et al. [19] developed in 2011 +a fault tolerant connection networks for the Tokyo rail system using an agent based +simulation of Physarum polycephalum. Physarum-inspired cellular automaton (CA)- +based network designing model was developed by Tsompanas et al. [20] inspired by +Slime Mould. Zhang et al. [21] recently proposed a method to solve the problem of +network design in supply chain for multiple source nodes and multiple sink nodes. +Physarum is excellent at doing other network design [22].Here we summarize various +works of network construction using Physarum inspired technique in the following +Table 2.1. + + + + +7 + + +2.3.2 Other Optimization Task +Nakagaki et al. in 2000 [13] observed that Physarum productively found the shortest +path between two selected points in a maze. In addition, the Physarum can solve many +other famous problems like the shortest paths [23]–[25], towers of Hanoi problem [26] +and minimum risk problem [27]. Physarum can effectively solve many other complex +problems in the real world like traveling salesman problem [28]–[30], population +migration [31], etc. Logic gates design and boolean operations can be performed by a +slime mold network [32], [33]. Chaining these logic gates together can enable a slime +mold computer to perform binary computation operations. Physarum works very well +in logical computing as well[34]–[38]. Identifying critical components [39], [40] and +many other problems [41], [42] are effectively and efficiently solved through Physarum +bio-inspired technique. Most interestingly, many other studies have shown that +Table 2.1: Network construction using Physarum. +Authors & Year +Title of Paper +Contribution +Tero et al., 2007 +[15] +A mathematical model for +adaptive transport network in +path finding by true slime mold +Model for adaptive transport +network in Path-finding in a maze +Tero et al., 2010 +[14] +Rules for Biologically Inspired +Adaptive Network Design +Tokyo Rail Network construction +Adamatzky et al., +2011 [16] +Rebuilding Iberian motorways +with slime mould +Iberian motorway network +construction +Adamatzky et al., +2011 [17] +Approximating Mexican +highways with slime mould +Mexican Federal highway +network construction +Adamatzky et al., +2013 [18] +Slime mould imitates transport +networks in China +Slime mould protoplasmic +networks on major urban areas of +China +Becker et al., 2011 +[19] +Design of fault tolerant +networks with agent-based +simulation of Physarum +polycephalum +Construction of fault tolerant +connection networks for the +Tokyo rail system using an agent +based simulation of Physarum +polycephalum +Tsompanas et al., +2015 [20] +Evolving Transport Networks +With Cellular Automata Models +Inspired by Slime Mould +Physarum-inspired cellular +automaton (CA)-based network +designing model +Zhang et al. 2016 +[21] +A Physarum-inspired approach +to supply chain network design +Supply chain network design + + + + + +8 + +Physarum's tubular topologies often mimic those of complex mathematical networks +[43], [44] like the Steiner tree problems [45]–[49]. + +Studies with Physarum related works showed that the organism can solve many +complex real-life problems efficiently and effectively, particularly in the sense of +network design. This can be applied with some changes for designing the bicycle lane +network. In this work bicycle lane network is planned using local lanes in congested +mega city. + + + + + + + + + + + +10 + +CHAPTER 3 +Physarum Inspired Bicycle Lane Design in an Unplanned +Mega City +An unplanned mega city suffers various problems including transporation and mobility. +Transport mobility enhancement in an unplanned mega city is always challenging due +to various constraints including complex design and high cost involvement. In this +thesis, we try to increase the mobility in an unplanned mega city Dhaka. In this chapter, +problems of an unplanned mega city Dhaka are addressed firstly, then challenges in +transformation an unplanned megacity to green city, and finally, the importance of +bicycle lane in an unplanned mega city and bicycle lane network design in an unplanned +mega city are discussed. +3.1 Mobility Problem in an Unplanned Mega City: Dhaka as a Case +Study +This thesis aim is to enhance transport mobility in an unplanned mega city introducing +a bicycle lane. In this section initially, we discuss the history and overview of Dhaka +city, then the transportation crisis in Dhaka city and finally, the effect of transportation +crisis on other problems. +3.1.1 History and Overview of Dhaka City +It is mentioned that the concept of a Mega City originated at the end of the 20th century +to describe the largest city in the world. Although, literature has little disagreement +about the population threshold used as a megacity concept, the UN (2003) defines most +precisely: a conurbation of ten million or more inhabitants is a megacity which has now +been widely accepted [5]. + +Since 1971, Dhaka has experienced incredible growth and rapid growth. It is one of the +world's only seven cities with a population of over 2.4 percent between 1975 and 2005 +(UN 2006). In 2011, it was one of the world's top ten mega cities. The developments +have unfortunately happened unplanned, especially since the 1990s [5]. The word + + + +11 + +Dhaka is nowadays mentioned regularly in the most unlivable cities. Dhaka was the +world's fastest-growing town between 1950 and 2000 [50]. While population growth +has declined recently, it is still the second-largest growth mega city in the world [50]. +3.1.2 Transportation Crisis in Dhaka City +The mega city has neither efficient public transport nor mass transit [50]. It is probably +the world's only mega-city without efficient public transit and public transit [50]. Dhaka +has a poorly developed transport system with 200 km of main roads and about 260 km +(too few) secondary and collector roads, in addition to 250 km of narrow roads +(approximately) [50]. There are many incomplete critical connections in the road +network and several regions have insufficient connectivity to the network [50]. Separate +bicycling lanes and footpaths are barely available in the city, which enhance the +mobility crisis. + +There was a time when traffic congestion was only suffered by commuters on the main +streets of the city, but now it starts right from the door. Traffic jam has turned into +nightmares for daily trips. According to a World Bank report, the average traffic speed +in Dhaka has dropped from 21 kilometers per hour (kmph) to 7 kilometers per hour in +the last 10 years, and by 2035 the speed could drop to 4 kilometers per hour, which is +slower than the walking speed [55]. Another study commissioned by the BRAC +Institute of Government and Development indicates that traffic congestion in Dhaka +consumes about 5 million working hours a day and costs the country $11.4 billion a +year [55]. The financial loss is a measure of the time lost in traffic congestion and the +extra hours expended on cars. + +It should be noted that there is no adequate and proper routing of our public transport +system. In 2016, According to the BRTA, 20,304 new cars were introduced to Dhaka's +traffic, which means more than 55 new cars hit the streets every day [55]. As the number +of cars increases, there is also an increasing demand for parking space. Unfortunately, +however, the parking space in our city is quite inadequate. Many vehicles on the streets +are stored. Many buses and trucks are parked on the streets on a regular basis [55]. + + + + + +12 + +According to the Dhaka Metropolitan Police (DMP) Traffic Department, traffic jams +have become intolerable in some urban areas over the past few days, including Mirpur- +12 to Mirpur-10 crossing, Rokeya Sarani, Gulshan, Banani, Badda, Moghbazar, +Eskaton, Tejgaon, Airport Road, and Uttara, for a number of reasons, including the +ongoing Dhaka International Trade Fair, the building of underground trains and the +increase in private transport[56]. Urban analyst and former chairman of UGC Prof +Nazrul Islam said traffic jams are gradually deteriorating due to an increase in urban +population and the number of small vehicles and lack of effective control measures +[56]. “We have built over half dozens of flyovers, but it is not a solution to solve the +problem. We will not be able to reduce traffic jams without increasing public transport +and ensuring better traffic management", he observed [56]. Transport and urban experts +believe that the government should take practical steps to ensure effective mass +transportation, restore transportation efficiency, decrease the use of private and small +cars, replace micro-buses and mini-busses with single-decker, double-decker, and +articulated buses, and extend the city to dramatically alleviate traffic jams without +spending huge money [56]. The experts also said that railways and waterways can also +be used effectively to relieve road traffic pressure and facilitate trouble-free transport +services for the commuters [56]. + +Figure 3.1 illustrates the traffic jam in the city. Here, three routes are available from +Farmgate to the University of Dhaka. During driving mode, it takes around 15 minutes +at 06:00 a.m., 18-35 minutes at 10:00 a.m., and 18-40 minutes at 5:00 p.m. on average. +On the other side, it takes an average of 45-50 minutes in walking mode. We note that +the speed of driving is slightly higher than that of walking. Not only in some areas, but +throughout the city, it's the case. + +The number of automobiles has been increasing in Dhaka city at the rate of at least 10 +percent annually, which has been contributing to environmental pollution on the one +hand and traffic congestion on the other. This transportation problem enhances other +problems like air pollution, noise pollution, fuel consumption, CO2 emission, worst in +road conditions, etc. + + + + + + +13 + + + + + +(A) At 06.00AM +(B) At 10.00AM + + +(C) At 5.00PM +(D) Walking mode +Figure 3.1: Farmgate to University of Dhaka routes and time needed in driving mode. (A) At 06.00am. (B) At +10.00am. (C) At 5.00pm. (D) Time needed in walking mode. + +回 +of1 +ManikMiaAve +IndiraRd +CG +OCK +W.Raza +回 +回 +园 +OFarmgate +LKISAREA +Insaf Bal +Bashundhara City +&Gen +spital +ShoppingComplex +nthapath +园 +aHatirheel +hani Playground +KALABAGAN +NEWESK +12A +FreeSchool St, +Kalabegan.1st Ln +16min +IANMONDI +3.9 km +15min +OLDE +5.5 km +RdNo.8 +回 +17 min +3.7 km +Rd8/A +TA +RdNo.6 +Off +回 +H +HATIRPOOL +Rd No.5 +ATOLA +MintoRd +DhakaCityCollege +SHAHBAGH +回 +Kazi Food Industries Ltd. +KATABON +Bangladesh +Dhaka CollegeDhaka +National Museum +Shreshtha +Noor +hammad +Dhaka New Market +UniversityofDhaka +icCollege +RAMNA +et-pilkhana Rd +回 +Bangla Acad0 +OFarmgate +IKISAREA +GA +KawranBazar +InsafBarakah +Bashundhara City +ShoppingComplex +&GeneralHos +Ranthapath@ +Hatirjheel +上 +ind +KALABAGAN +NEWESKATON +F +Free SohoolSt +Kalabagan 1etLn +DI +KATHALBAGAN +OLD ESKATON +回 +3 +18-35mins +3.9km +PARIBAG +RdNo.6 +HATIRPOOL +H +=18-40mins +Rd No.5 +3.7km +Dhaka CityCollege +Baily Rd +SHAHBAGH +18-35mins +Rd +4.1km +Bangladesh +ziFoodIndustriesLtd +DhakaCollege +KATABON +Ant +HANA +Dhaka NewMarket +UniversityofDhakaO +SaniRd +RAMNA +t-Pilkhana Rd +Google +BanglaAcademy +Eden Mohila日 +OFarmgate +TallabapRd +IKISAREA +KawranBazar +InsafBaral +Bashundhara City +ShoppingComplex +&General +thap +atn +Hatirjheel +KALABAGAN +sffofaa +NEWESKATON +FroeSchoolSt +6610-日 +Katabagan1stLn +H +KATHALBAGAN +OLD ESKATON +3 +20-40mins +3.9km +PARIBAGH +Rd No-6 +HATIRPOOL +18-40mins +RdNo.5 +H +3.7km +Dhaka City College +Baily Rd +SHAHBAGH +22-40mins +113419 +4.1km +回 +Bangladesh +FoodIndustries Ltd +DhakaCollege +KATABON +11 +ANA +T +DhakaNewMarket +UniversityofDhaka +SaniRd +RAMNA +BanglaAcademy日 +OFarmgate +TallabieRd +IKISAREA +H +KawranBazar +Insaf +Bashundhara City +ShoppingComplex +&Ger +st +白aHatirjheel +H +KALABAGAN +NEW ESKATON +Ftee Schodl St +8010-回 +Kalsbigan1st La +KATHALBAGAN +OLDESKATO +3 +50min +PARIBAGR +专T +3.9km +RdNo.6 +HATIRPOOL +Officers'Clut +H +RdNo.5 +48min + 3.7km +Shaka CityCollege +O +Baily +49min +3.8km +Bangladesh +IndustriesLtd +NationalMuseum +RamnaParl +DhakaCollege +KATABON +DhakaNewMarket +University of Dhaka0o +ani +RAMNA +PilkhanaRd +回 +BanglaAcademy + +14 + +3.1.3 Effect of Transportation Crisis on Other Problems +Transportation crisis affects the environment badly. It may affect the air pollution, noise +pollution, fuel consumption, etc. According to the Department of Environment (DoE), +the standard value of the Air Quality Index (AQI) is 50 represents good air quality with +little potential to affect public health [51]. But, according to AirVisual information, +Dhaka the capital city of Bangladesh has been ranked the worst in the Air Quality Index + + (AQI) valued 309, which is hazardous and would trigger health warnings of emergency +conditions [51]. The entire population is more likely to be affected by the enormous +number of diseases like nausea, asthma, high blood pressure, heart disease, and cancer. +It also impacts the respiratory tract severely and causing irritation. Children's cognitive +faculty will be adversely affected by lead exposure, which can also distress the central +nervous system, causing hypertension and renal injury. In the last two months, the +capitalist has been enjoying just nineteen hours of good air [51]. Diesel-run vehicles +account for more than 80 percent of the air pollution in Dhaka as most of them fail to +comply with the approved emission standard, said a recently published survey report +[52]. + +In Dhaka, the average sound level is between 80dB and 110dB in prime areas such as +Farmgate, Karwan Bazar, Shahbagh, Gabtoli, and Mohakhali Bus Terminal, says the +study report [53]. According to the World Health Organization (WHO), this is almost +twice the maximum noise level that can be tolerated by humans – 60dB – without +suffering a gradual loss of hearing [53]. According to a recent study conducted by WHO +at 45 locations of Dhaka city, most of the traffic points and many of the industrial, +residential, commercial, silent and mixed areas are suffering noises exceeding the +standard limits of Bangladesh [54]. WHO has also identified several areas as severe +red, moderate red, mild red and green zones in terms of noise pollution in Dhaka city +[54]. Around 11.7% of the population in Bangladesh have lost their hearing due to noise +pollution, says the Development of Environment (DoE) study, which was conducted in +2017 [53]. The major sources of noise pollution in urban areas are traffic and loud +horns. The DoE found that in Dhaka, 500-1,000 vehicles honk at the same time when +stuck in traffic[53]. Around 5% of the world population is facing several kinds of health +hazards due to complexities related to noise pollution, According to the WHO [53]. + + + +15 + + + +There is a scarcity of natural gas and petroleum in Bangladesh also. Gas supplies meet +56% of domestic energy demand [57]. Bangladesh has a very limited energy reserve; +small amounts of oil, coal and countable natural gas reserves [58]. The country is a net +importer of crude oil and petroleum products [57]. +3.2 Importance of Bicycle Lane in Mega City +In more ways than one, driving a bicycle has a positive impact on the environment. +They are also less expensive than other forms of transportation and environment- +friendly. Bicycles are considered zero-emission vehicles i.e. they do not release any +carbon emissions. Bicycles, as vehicles with zero emissions, do not contribute to air +pollution. People can have moderate fresh air. They do not contribute to sound +pollution. When bicycles are used as a consistent form of travel by a large percentage +of the population in a particular area especially in an urban area, there is a great relief +on road traffic conditions. Bicycles also have the effect of alleviating parking +difficulties in urban areas, because they simply take up so much less space than cars. +Low physical activity or Physical inactivity is recognized as one of the country's leading +risk factors and the fourth leading cause of deaths due to non-communicable disease +(NCDs) worldwide - cardiovascular diseases, chronic lung diseases, heart disease, +stroke, diabetes and cancers - and each year contributes to over three million +preventable deaths [59]. There may be some physical exercise every day by using bi- +cycle. Bicycles also offer more freedom of movement without time constraints, +crowded and unpleasant conditions and, if desired, the ability to travel alone. So, people +can have eco-friendly Travel. Bicycles are lighter and usually cause less damage to the +roads than others. This will reduce the number of injuries. So the area would be +environment-friendly. Most of the well-organized mega city criteria are met by using +bicycles. +3.3 Challenges to Increase Mobility in Dhaka City +In the case of an unplanned or unorganized city, one of the big issues is that road +conditions are not good enough and the cycling lanes & footways are hardly available + + + +16 + +which is the major cause of worst traffic. On the other hand, noise pollution and traffic +congestion are troubling and there is a huge CO2 gas emission occurs. + +Between the well-organized city and unplanned city, there is a huge gap in road +conditions, footways & cycling lanes, air quality, noise pollution, and traffic +congestion. There may have some parameters to increase transport mobility in the city +like the construction of separate roads, underground roads, railways, etc. But the +construction of those parameters is not a feasible solution because of huge budgets and +spaces. There are two alternative low-cost solutions exist, the first one is to make ready +the footpaths for routing purpose and the next one is to introduce local lanes with +bicycles as vehicles for moving around the city. But the first one is not possible because +most of the time, street vendors and hawkers snatch up footways and in some case +footways not exist. The next solution is feasible and effective as bicycles have several +environmental benefits. + +For this purpose, we have to plan a network and always try to use local lanes for routing +through one place to another place, if it is not feasible to use local roads for some cases, +we will use main roads and will always try to minimize the use of main roads. There +exist some constraints to be handled to plan network that we cannot access all possible +roads like VIP roads, heavy traffic roads, etc. On the other hand, it is almost impossible +to plant more trees to improve air quality and reduce CO2. And most of the noise +pollution and traffic congestion is caused by motor vehicle use. +3.4 Bicycle Lane Design in an Unplanned Mega City +Different approaches have been presented over the past decades to design networks. It +is possible to split the solutions into two categories: exact solutions and heuristic +solutions. Exact approaches can treat Network Design Problem in a rigorous way which +is inefficient when dealing with real-world large-scale networks. And, an approximate +yet efficient approach is provided by heuristic approaches, more popular than exact +approaches, that have emerged in recent decades which can tackle large-scale real- +world problems. Without using an exact and heuristic approach, here we present the +Physarum-inspired technique which takes into account the constraints to construct the +bicycle lane network. Basically, we always try to use local lanes for routing through + + + +17 + +one place to another place, if it is not feasible to use local roads for some cases, we will +use main roads and will always try to minimize the use of main roads. + +Compared to previous studies it is noted that the network has only one direction +between two nodes, so the stream is only flowing from one node to another, but is never +flowing in the opposite direction. However, most roads have the features of double-way +traffic in real traffic networks as demonstrated in Fig. 3.2. There is a clear distinction +between opposite directions, where flows do not interfere in two directions opposite. +Apparently, in the traffic network shown in Fig. 3.2, the initial approach influenced by +the Physarum cannot be applied. +Here in the following, we discuss the modified Physarum inspired lane design +technique. +Given a graph 𝐺 = (𝑁, 𝐸), where + 𝑁 denotes a set of 𝑛 cities, + 𝐸 represents a set of 𝑚 connections or linkages. + +There is a protoplasmic flow in each link of this model. The two terminals of the link +represent two locations of the specified area. One terminal is called the source node, +and the other terminal is called the sink node. Protoplasmic flows from the source node +into the network and from the sink node out of the network. At each city there is +pressure and the amount of flux in each edge is proportional to the difference in pressure +between the two terminals of this edge. Specifically, the flux 𝑄𝑖𝑗 in edge (i,j) is given +by the modified Hagen-Poiseuille equation below. + +Figure 3.2: Real traffic network. + + + + +18 + +𝑄𝑖𝑗 = +𝐷𝑖𝑗 +𝑐𝑖𝑗 +(𝑃𝑖 + 𝑃𝑗) (3.1) + 𝐷𝑖𝑗 = +𝜋𝑟𝑖𝑗 +4 +8𝜀 (3.2) +In the above equation, 𝐷𝑖𝑗 is the conductivity of the linkage, 𝑐𝑖𝑗/𝐿𝑖𝑗 is the length of the +edge, 𝑃𝑖 and 𝑃𝑗 are the pressure of the vertices 𝑖 and 𝑗, 𝑟𝑖𝑗 is the radius of the edge, +𝜀 (𝑒𝑝𝑠𝑖𝑙𝑜𝑛) is the coefficient of viscosity. In the case of conductivity(𝐷𝑖𝑗), which is +linkage specific, we are using a fixed conductivity value for all the linkages for +simplicity. The length(𝑐𝑖𝑗/𝐿𝑖𝑗) is not the direct length from 𝑖 city to 𝑗 city rather we +consider all possible path length with no use or hardly use of main roads and we +calculate pressure of each city based on the amount of population in that city. As we +use initial fixed conductivity value, 𝑟𝑖𝑗 is considered to be the same for all connections. +Eq. (3.2) indicates that the tubular thickness( 𝑟𝑖𝑗) of Physarum increases with the +conductivity of the tube. Therefore, the conductivity update formula can explain the +change in tubular thickness of Physarum as follows, +𝑑 +𝑑𝑡 𝐷𝑖𝑗 = 𝑓(|𝑄𝑖𝑗|) − 𝜇𝐷𝑖𝑗 , (3.3) +here 𝑓(|𝑄𝑖𝑗|) is an increasing function, 𝜇 is a positive constant. In our case, we +considered 𝑓(|𝑄𝑖𝑗|) = |𝑄𝑖𝑗| for simplicity. The equation of conductivity update +suggests that conductivity tends to increase with large flux edges. Consequently, the +equation of conductivity update reflects the above physiological mechanism. We must +first calculate the pressures to calculate the flux and update the edge conductivities. The +pressures can be determined using the Poisson equation network below by considering +the flux conservation law at each vertex, +∑ 𝐷𝑖𝑗 +𝑐𝑖𝑗 +(𝑃𝑖 + 𝑃𝑗) = { + −𝐼0, +𝑗 = 𝑠𝑜𝑢𝑟𝑐𝑒 + +𝐼0, 𝑗 = 𝑠𝑖𝑛𝑘 +0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 +𝑖 ∈𝑉(𝑗) + , (3.4) +here 𝑉(𝑗) is the set of vertices linked to vertex j, I0 is the amount of flux flowing into +and out of the node of the source. +Let the pressure at the sink node be 0, and give an initial value to each edge +conductivity, then use Eq. (3.4) to measure the other pressures. After that, we can +calculate the amount of flux in each edge using Eq. (3.1), and we can change the +conductivity of each edge using Eq. (3.2). According to an edge conductivity threshold +value, edges with conductivity lower than this value are cut off from the network. + + + +19 + +Let’s consider a simple small network consist of only eleven points picked from Dhaka +City. Physarum always finds the optimal route network among the eleven nodes and is +believed to have achieved a good balance between cost, efficiency, and resilience. Here +are the illustrations in the following which are the generalized design using modified +Physarum inspired technique. This technique can be applied any real-life traffic +network design. In this work, the technique is applied both prominent area centered +with Motijheel and entire Dhaka city. +In Fig. 3.2, (A) there are 11 node points between Farmgate (node point 2) and the +University of Dhaka (node point 10) existing network with both main and local roads. +Here, (B) – (E) shows that the network growing process with time (t = iteration) i.e. (B) +Network at 10th iteration; (C) Network at 20th iteration; (D) Network at 50th iteration; +(E) Final network for 11 nodes. For those cases, we always try to avoid main roads to +grow up the networks. And the Fig. 3.3 depicts the possible road network design if all +main roads are available. + + +Figure 3.3: Sample Region with 11 points. + +日. +日 +Government +FAR +2 +GATE +Science +College +CRd +IKI'SAREA +mited +H +BashundharaCity +Insaf BarakahKi +Rd +ShoppingComnlex +&GeneralHos +apath +日 +Panthapath +Hatiriheel +fosfa +KAL +4 +AGAN +NEW ESKATON +F +专T +KATHAI +AGAN +OLD +ESKAT +RdNo.8 +LinkRdD +35 +P +JAGH +R +Rd No.S. +H +HAT +6 +OOL +MintoRd +y.College +Ba +SHAH +GH +oSquare +NewElephantRd +日 +Ramna +KA +7 +BON +Bangladesh +914 +NationalMuseum +eDhaka +University +NewMarket +ofDhaka +日 +3 +RA +10 +NA +EdenMohila +College +Udayan Higher +Suprem +SecondarySchool +ofBang + +20 + + + + +(A) 11 node points (t = 0) + +(B) At t = 10 +Figure 3.4: Physarum inspired network design of 11 nodes. (A) 11 node points. The network is +expanding with t. (B) At t = 10. + +11 +- +6 +7 +1011 +4 +6 +7 +10 + +21 + + + +(C) At t = 20 + +(D) At t = 50 +Figure 3.4: Physarum inspired network design of 11 nodes. (A) 11 node points. The network is +expanding with t. (C) At t = 20. (D) At t = 50. + +6 +7 +104 +L +7 + +22 + + + + +(E) Final network +Figure 3.4: Physarum inspired network design of 11 nodes. (E) Final network. + +4 +10 + +23 + + + +Figure 3.5: Physarum inspired network design of 11 nodes using main roads (if available). + +4 +6 + +24 + +3.5 Significance of Study +Modified Physarum Polycephalum Inspired Network Design Technique is used to +design any real-life traffic network. In this research, the bicycle lane network is planned +using this strategy in a congested mega city Dhaka. This addresses many complicated +problems, including the problem of mobility. + +Modified Physarum Polycephalum Inspired Network Design Technique holds a +significantly different form from the existing Physarum Polycephalum Inspired +Network Design Technique. Compared to previous studies it is noted that the network +has only one direction between two nodes, so the stream is only flowing from one node +to another, but is never flowing in the opposite direction. However, most roads have +the features of double-way traffic in real traffic. There is a clear distinction between +opposite directions, where flows do not interfere in two directions opposite. Apparently, +in real life traffic network, the initial approach influenced by the Physarum cannot be +applied. Modified Physarum Polycephalum Inspired Network Design Technique +calculates flux and pressure using Eq. (3.1) and Eq. (3.4) respectively as we know that +the flow of traffic is not uni-directional rather bi-directional. + + + + + + + + + + + +CHAPTER 4 +Experimental Studies +This chapter experimentally investigates the efficacy of the proposed modified +Physarum inspired bicycle lane network design technique. For both a certain portion of +Dhaka City and the entire Dhaka City, we are assuming a reduction in the number of +buses, cars, taxicabs, and motorcycles. Time saving, fuel saving, user cost saving, and +CO2 emission reduction are calculated with some standard average measurement. In +this chapter at first, we discuss the prominent points and description of this experiment, +then the experimental setting which includes both parameter setting and machine +description. In the third section of this chapter, we describe the experimental outcomes +and explanation of achievements for both 10km ranged portion and entire Dhaka city +are described. +4.1 Experimental Settings +In the experiment, the number of node points was 29 for both case but 77 linkages/edges +are considered for prominent area of Dhaka city and 89 linkages/edges for entire Dhaka +city; the length of edges(𝑐𝑖𝑗/𝐿𝑖𝑗) which is not linear and tubular thickness( 𝑟𝑖𝑗) are +estimated using the google map; the value of meu(𝜇) was varied from .8 to 1; for certain +cases we simply ignore the tubular thickness( 𝑟𝑖𝑗) variations and considered a fixed +conductivity value(𝐷𝑖𝑗); it is assumed that, the value of pressure(𝑃𝑖) at each node point +is proportional to its population in that area, and a random initial threshold is applied +which is increased with the iteration. + +The modified Physarum Inspired Bicycle Lane Design was implemented on Visual C++ +of Visual Studio 2013. The experiments have been done on a PC (Intel Core i3-5005U +CPU @ 2.00 GHz CPU, 2GB NVIDIA GeForce 940M, 4GB RAM) with Windows 10 +OS. + +According to Bangladesh Road Transport Authority (BRTA), on February 04, 2020, +there are 127398 registered buses (including microbus, minibus), 293268 registered +cars, are 724800 registered motorcycles, and 36600 registered taxicabs currently + + + +25 + +available in Dhaka city [60]. There may have some unregistered cars & buses and a lot +of unregistered motorcycles & taxicabs in the city, we are not considering them. It is +assumed that 40 passengers per bus, 1 person per car, taxicab and motorcycle on +average. And let’s assume that buses take 20km ride, cars use 10km ride, taxicabs use +100km per day and motorcycles ply 15km ride. All those rides are supposed to take +place per day. + +In this work, both prominent area centered with Motijheel and entire Dhaka city are +considered to apply the modified Physarum inspired technique. In this section, the +achievements or environmental effects of the prominent area and the entire city of +Dhaka with bicycles as vehicle are respectively addressed with the proposed network. +4.2 Bicycle Lane Network Design in a Prominent Area +At first, a 10km selected prominent area of Dhaka city centered with Motijheel is +considered to construct the network using Physarum inspired technique. In this area, +we have chosen 29 vital points and numbering those from 1 to 29 arbitrarily. Here, we +are considering a 10 km range because the average speed of cycling is 20kmph, so 10 +km a day can be traveled easily in 30 minutes. And it can also lead to better mental +health and energy by bicycling 30 minutes a day [6], [8]. Fig. 4.1 illustrates the selected +portion of Dhaka city. Here, (A) depicts the portion of Dhaka city marked within the +entire Dhaka city. (B) Shows the 10km range centered with Motijheel. + +For 10km ranged area, it is assumed that overall 2% of users switch from car to bicycle +and 10% of users switch from motorcycle to bicycle. And a 5% bus and 10% taxicab +are being reduced because of using bicycle. Then the atmosphere would change +significantly. This section first explains the designed networks and then calculate the +time saving and then fuel and cost saving is estimated and finally CO2 emission +reduction is calculated. + +The locations of prominent area of Dhaka city including traffic pressure is presented in +Table 4.1. In this case, we assign some random values as node point's traffic pressure +which is proportional to its population. For examples, Taltola, Donia, etc are less and +on the other hand Motijheel, Tejgaon, etc are high traffic traffic-pressured area. + + + +26 + + + + +(A) Bangladesh +(B) Dhaka + + +(C) Prominent area +(D) Node points in Prominent area. +1 Motijheel +2 Mohakhali +3 Gulshan +4 Shahinbag +5 Tejgaon +6 Badda +7 EWU +8 Bashundhara +9 Mogbazar +10 Mirbag +11 Dhanmondi +12 Shahbag +13 DU +14 Kamlapur +15 Ramna +16 Kotwali +17 Lalbagh +18 Khilgaon +19 Taltola +20 Aftabnagar +21 Sadarghat +22 Matuail +23 Wari +24 Golapbag +25 Donia +26 Rajarbagh +27 Sobujbagh +28 Green Model Town +29 Nandipara +Figure 4.1: Selected Dhaka city map. (A) Bangladesh. (B) Dhaka. (C) Prominent area. (D) Node points in Prominent area. + + + +Uzbekistan +Kyrgyzstan +Beijing +北京 +Turkmenistan +Tajikistan +China +Yello +Afghanistan +Iran +Shanghai +上海 +New.Delhi +Pakistan +Ea +Nepal +ianGulf +Bhutan +Taipei +台北 +UnitedArab +Bangladesh +Emirates +Taiwan +India +My +imar +HongKong +Oman +Mumbai +rma) +香港 +Laos +Thailand +South +Luzon +Bengaluru +Vietnam +China Sea +23orfedodo +Bayof Bengal +Bangkok +CUMLHIMUICEU +Arabian Sea +Cambodia +Philipp +AndamanSea +.HoChi +Panay +Gulf of +MinhCity +Thailand +Palawan +Negro +SriLanka +Minda +Laccadive Sea +BasilanIsla +Malaysia +Kuala Lumpur +Celebes SHO1 +H07 +N5 +Batshar +ASSAM +Kishanganj +Guwahati +Nagaon +Purnia +AHIDim +MEGHALAYA +oShillong +Sahibganj +N5 +N2 +Pakur +Silchar +Sylhet +nka +Raishahi +N502 +ICGTG +Hailakandi +MAI +T +N2 +Bangladesh +N6 +gapur +N704 +TRIPURAV +Aizawl +Dhaka +N7 +PT +MIZORAM +AH +Jessore +BENGAL +Madaripur: +Satkhira +Kolkata +Gk +N1 +gpur +Sundarban +Chittagong +Forest, +Bangladesh +Chandanaish +eqey +yangarh +T5(* +H +135153 +N1 +Digha +Cox'sBazar +Whaikhyang +Mrauk-UN302 +UTTARA +9114 +Hazrat +N511 +Shahjalal +N105 +N3 +International +N501 +Airport +N301 +MIRPUR +BASUNDHARA +Dhaka Zoo +RESIDENTIAL + +N3 +AREA +Dhaka +Jalshiri.Abason +raid +GULSHAN +N5 +Gabtoli +Z5069 +5114 +MOHAKHALI +BADDA +R202 +MOHAMMADPUR +TEJGAON +KHILGAON +ROTST +Boshila +DHANMONDI +R +ty +RAMNA +MOTIJHEEL +Chanpara +尼 +b +N2 +LalbaghFort +Hizla +R110 +KOTWALI +N2 +Matuail +Keraniganji +8N +R820 +R820 +N1 +Shiddhirganj +Z5069 +R810 +R111 +Ruhitpur +Bashundhara +Kadamtoli +Riverview +Baghair +Ekuria +R810BARIDHARA +MadaniAve +Banani +Suvastu Nazar Valley +ShoppingComplex +SHER-E-BANGLA +NAGAR +National +Parliament House +LALMATIA +H +DhakaNewMarket +2 +Madina Filling Station +R820 +R110 +Buriganga River +R820 +N1 +Z1102 +Keraniganj +21 +NB +R820 +RanaCNG& +GASStation + +27 + +4.2.1 Network Design +Here in the planned network, the distance is not the linear distance between two node +points rather distance is calculated using google map. And the time is calculated +considering standard speed 20kmph for bicycle in minutes. + +In Fig. 4.2, (A) there are 29 node points around 10km range centered with Motijheel +(node point 1) existing network with both main and local roads. Figure (B) – (E) shows +that the network growing process with time (t = iteration) i.e. (B) Network in 10th +iteration; (C) Network 20th iteration; (D) Network 50th iteration; (E) Finale network. +For those cases, we always try to avoid main roads to grow up the networks. In Fig. 4.3 +Network is planned with main roads (if available). + +Table 4.1: The locations of prominent area of Dhaka city including traffic pressure. +Sl +Location +Traffic Pressure +01 +Motijheel +9 +02 +Mohakhali +8 +03 +Gulshan +5 +04 +Shahinbag +8 +05 +Tejgaon +9 +06 +Badda +4 +07 +EWU +5 +08 +Bashundhara +5 +09 +Mogbazar +5 +10 +Mirbag +5 +11 +Dhanmondi +7 +12 +Shahbag +9 +13 +DU +9 +14 +Kamlapur +8 +15 +Ramna +5 +16 +Kotwali +3 +17 +Lalbagh +4 +18 +Khilgaon +6 +19 +Taltola +3 +20 +Aftabnagar +4 +21 +Sadarghat +5 +22 +Matuail +8 +23 +Wari +7 +24 +Golapbag +5 +25 +Donia +3 +26 +Rajarbagh +5 +27 +Sobujbagh +5 +28 +Green Model Town +4 +29 +Nandipara +6 + + + + + + +28 + + + +(A) 10km range existing road network + +(B) When t=10 + +Figure 4.2: Network design using modified Physarum inspired technique. (A) 10km range existing road +network. (B) When t=10. + +12 +26 +16 +2127 +16 + +29 + + + + +(C) When t=20 + +(D) When t=50 + +Figure 4.2: Network design using modified Physarum inspired technique. (C) When t=20. (D) When t=50. + +27 +16 +219 +12 +16 + +30 + + + + +(E) Final Network +Figure 4.2: Final network design using modified Physarum inspired technique. + +27 +26 + +31 + + + +Figure 4.3: Network design using modified Physarum inspired technique using main roads (if available). + +27 +2 +16 + +32 + +Here the Table 4.2 shows routes of all node points from starting node point 1. For +example, the path 1-9-10-2-3 means that we have to cross node points 9, 10 and 2 in +order to go into destination node point 3 from starting node point 1. The minimum +distance is considered for accessing any node here. For example, the destination node +point 3 can be accessed using the routes 1-9-10-2-3, 1-20-7-6-2-3, 1-12-5-4-3 and so +on but the minimum distanced one is considered for calculation. +4.2.2 Effectiveness Analysis +Driving a paddled-bicycle has a more than one beneficial environmental effect. For +many cases bicycles take less time to ride, no fuel usage, saving user money, CO 2 +emission reduction in unplanned mega city and do not contribute to air pollution, sound +pollution, great relief on road traffic conditions, alleviating parking difficulties in urban +areas, cause less damage to the roads, get relief of some non-communicable disease +Table 4.2: Routes from node point 1. +Des. Node +Point +Routes +Distance +(km) +Estimated Travel +Time (min) +2 +1-9-10-2 +8.55 +25.65 +3 +1-9-10-2-3 +11.25 +33.75 +4 +1-9-4 +7.3 +21.9 +5 +1-5 +7.5 +22.5 +6 +1-9-10-7-6 +8.2 +24.6 +7 +1-9-10-7 +6.9 +20.7 +8 +1-8 +6.85 +20.55 +9 +1-9 +2.9 +8.7 +10 +1-9-10 +4.4 +13.2 +11 +1-12-13-11 +9.6 +28.8 +12 +1-12 +5.3 +15.9 +13 +1-12-13 +6.4 +19.2 +14 +1-14 +5.7 +17.1 +15 +1-15 +2.4 +7.2 +16 +1-23-16 +4.5 +13.5 +17 +1-23-16-17 +8.1 +24.3 +18 +1-20-18 +9 +27 +19 +1-19 +3.7 +11.1 +20 +1-20 +6.2 +18.6 +21 +1-23-21 +4.5 +13.5 +22 +1-23-21-25-22 +10.3 +30.9 +23 +1-23 +2.8 +8.4 +24 +1-14-24 +7.15 +21.45 +25 +1-23-21-25 +7.1 +21.3 +26 +1-14-26 +6.7 +20.1 +27 +1-14-26-27 +9.9 +29.7 +28 +1-14-28 +7.9 +23.7 +29 +1-14-26-29 +8.1 +24.3 + + + + + + +33 + +(NCDs) for have some physical exercise every day. The considering factors of this +paper are only time saving, fuel and user money saving and CO2 emission reduction. +4.2.2.1 Time Saving +As we mentioned that, according to a World Bank report, the average traffic speed in +Dhaka has dropped from 21 kilometers per hour (kmph) to 7 kilometers per hour in the +last 10 years, and by 2035 the speed could drop to 4 kilometers per hour, which is +slower than the walking speed. In our constructed network, we mainly try to avoid main +roads or minimum usage of main roads if requires. So, here traffic jam is hardly +available. For convenience, we assume no traffic jam exists in the following calculation +of time. + +In Table 4.3, the time needed for both cars and buses are listed in three different times +(at 6:00 AM, 10:00 AM, and 4:00 PM) calculated from google map and the time needed +for a bicycle is always constant. Here, we measure the distance and time of all the node +point from center node point 1. + +The time required for a car, taxicab and motorcycle are almost the same that’s why we +don’t mention it differently in the Table 4.3. Here we notice that at the morning 6:00 +AM the travel time is less because of minimal traffic in the roads, at 10:00 AM (peak +hour) when traffic jam occurs severe the need time to travel is huge and at 4:00 PM +there exist traffic jam also but sometimes a bit less than peak hour. This condition is +true for all types of cars, buses, taxicabs, etc. For the prominent area, the gross working +hours saving are described in the following Table 4.4. So, around 152216 working +hours per day can be saved using bicycle in the prominent area. +4.2.2.2 Fuel and Cost Saving +The cost of installation or repair is not listed here rather we considering only running +cost. Because whenever we switch from car to bicycle there would be a significant +reduction in costs. + +Here the mileage of car and taxicab is assumed at 20kmpl and 25kmpl respectively with +diesel and 65tk per liter diesel. On the other hand, cycling has no cost. The bus mileage +considers 5kmpl with diesel and motorcycle has 50kmpl with petrol. + + + +34 + + +In Table 4.4, the needed fuel for cars, taxicabs, motorcycles, and buses is calculated +and there is no running cost & fuel cost for the bicycle. In case of bus, the per km fare +is fixed by BRTA of 1.7tk and we assumed that taxicab fare is 50tk per km. Here, the +distance of all the node points are measured from center node point 1. + +Table 4.3: Time comparison in car, bus and bicycle considering from node point 1. +Node +Point +Dista +nce +(km) +Car & Taxicab & Motor cycle +Bus +Bicycle +6:00am +(min) +10:00am +(min) +4:00pm +(min) +6:00am +(min) +10:00am +(min) +4:00pm +(min) +Dista +nce +(km) +Time +(min) +2 +8.6 +18 +39 +36 +20 +43 +40 +8.55 +25.65 +3 +8.9 +19 +42 +39 +21 +46 +43 +11.25 +33.75 +4 +7.1 +14 +33 +30 +16 +36 +33 +7.3 +21.9 +5 +5.1 +12 +27 +26 +14 +30 +29 +7.5 +22.5 +6 +7.6 +15 +36 +36 +17 +39 +39 +8.2 +24.6 +7 +7.6 +15 +36 +36 +17 +39 +39 +6.9 +20.7 +8 +4.8 +11 +24 +24 +13 +27 +27 +6.85 +20.55 +9 +3.6 +9 +18 +15 +11 +20 +17 +2.9 +8.7 +10 +4 +10 +20 +18 +12 +22 +20 +4.4 +13.2 +11 +6.1 +12 +30 +30 +14 +33 +33 +9.6 +28.8 +12 +3.4 +9 +23 +23 +11 +26 +26 +5.3 +15.9 +13 +3.8 +10 +24 +21 +12 +27 +24 +6.4 +19.2 +14 +1.2 +3 +11 +11 +5 +12 +12 +5.7 +17.1 +15 +3.1 +8 +18 +18 +10 +20 +20 +2.4 +7.2 +16 +3.2 +8 +27 +33 +10 +29 +35 +4.5 +13.5 +17 +6.1 +13 +33 +30 +15 +36 +33 +8.1 +24.3 +18 +7.2 +14 +53 +48 +16 +56 +51 +9 +27 +19 +3 +8 +21 +18 +10 +23 +20 +3.7 +11.1 +20 +10 +24 +42 +38 +26 +47 +43 +6.2 +18.6 +21 +3.5 +9 +36 +36 +11 +38 +38 +4.5 +13.5 +22 +8.3 +18 +33 +30 +20 +37 +34 +10.3 +30.9 +23 +2.5 +6 +21 +21 +8 +23 +23 +2.8 +8.4 +24 +4.2 +10 +27 +27 +12 +30 +30 +7.15 +21.45 +25 +5.7 +13 +29 +27 +15 +32 +30 +7.1 +21.3 +26 +4.7 +11 +29 +29 +13 +32 +32 +6.7 +20.1 +27 +3.1 +8 +15 +12 +10 +17 +14 +9.9 +29.7 +28 +5.8 +12 +42 +39 +14 +45 +42 +7.9 +23.7 +29 +5.4 +12 +33 +33 +14 +36 +36 +8.1 +24.3 + + + +Table 4.4: Time saving per day. +Transit +% of Transit +Reduction +Transit +Reduces +Riding Distance +(km) +Time saves +(min) +Bus +5% +6370 +127400 +709800 +Car +2% +5865 +58650 +326764 +Taxicab +10% +3660 +366000 +2039143 +Motor cycle +10% +72480 +1087200 +6057257 +Total time saving +9132964 + + + + + +35 + +The gross fuel saving and user cost saving for motijheel area are calculated for buses, +cars, taxicabs, and motorcycles and described in the Table 4.6. So, around 64797 liters +fuel and 20.6 million user money costs can be saved per day using bicycle in the +motijheel area. + + +Table 4.5: Cost comparison in car, bus and bicycle considering from node point 1. +Node +Point +Dista +nce +(km) +Car +Taxicabs +Motor cycle +Bus +Bicycle +Fuel +(litre) +User +Cost +(tk) +Fuel +(litre) +User +Cost +(tk) +Fuel +(litre) +User +Cost +(tk) +Fuel +(litre) +User +Cost +(tk) +Dista +nce +(km) +Cost +(tk) +2 +8.6 +0.43 +27.95 +0.34 +430 +0.17 +15.31 + + + + +As we +assume +bicycles +reduces +5% of +the total +bus in +Dhaka +city. + +The fuel +consump +tion is +reduced += +6370×20 +×1/5 +litres + += 25480 +litres +14.62 +8.55 + + + + + + + + + + + + + +N/A +3 +8.9 +0.45 +28.93 +0.36 +445 +0.18 +15.84 +15.13 +11.25 +4 +7.1 +0.36 +23.08 +0.28 +355 +0.14 +12.64 +12.07 +7.3 +5 +5.1 +0.26 +16.58 +0.2 +255 +0.1 +9.08 +8.67 +7.5 +6 +6.8 +0.34 +22.1 +0.27 +340 +0.14 +12.1 +11.56 +8.2 +7 +7.6 +0.38 +24.7 +0.3 +380 +0.15 +13.53 +12.92 +6.9 +8 +4.8 +0.24 +15.6 +0.19 +240 +0.1 +8.54 +8.16 +6.85 +9 +3.6 +0.18 +11.7 +0.14 +180 +0.07 +6.41 +6.12 +2.9 +10 +4 +0.2 +13 +0.16 +200 +0.08 +7.12 +6.8 +4.4 +11 +6.1 +0.31 +19.83 +0.24 +305 +0.12 +10.86 +10.37 +9.6 +12 +3.4 +0.17 +11.05 +0.14 +170 +0.07 +6.05 +5.78 +5.3 +13 +3.8 +0.19 +12.35 +0.15 +190 +0.08 +6.76 +6.46 +6.4 +14 +1.2 +0.06 +3.9 +0.05 +60 +0.02 +2.14 +2.04 +5.7 +15 +3.1 +0.16 +10.08 +0.12 +155 +0.06 +5.52 +5.27 +2.4 +16 +3.2 +0.16 +10.4 +0.13 +160 +0.06 +5.7 +5.44 +4.5 +17 +6.1 +0.31 +19.83 +0.24 +305 +0.12 +10.86 +10.37 +8.1 +18 +7.2 +0.36 +23.4 +0.29 +360 +0.14 +12.82 +12.24 +9 +19 +3 +0.15 +9.75 +0.12 +150 +0.06 +5.34 +5.1 +3.7 +20 +10 +0.5 +32.5 +0.4 +500 +0.2 +17.8 +17 +6.2 +21 +3.5 +0.18 +11.38 +0.14 +175 +0.07 +6.23 +5.95 +4.5 +22 +8.3 +0.42 +26.98 +0.33 +415 +0.17 +14.77 +14.11 +10.3 +23 +2.5 +0.13 +8.13 +0.1 +125 +0.05 +4.45 +4.25 +2.8 +24 +4.2 +0.21 +13.65 +0.17 +210 +0.08 +7.48 +7.14 +7.15 +25 +5.7 +0.29 +18.53 +0.23 +285 +0.11 +10.15 +9.69 +7.1 +26 +4.7 +0.24 +15.28 +0.19 +235 +0.09 +8.37 +7.99 +6.7 +27 +3.1 +0.16 +10.08 +0.12 +155 +0.06 +5.52 +5.27 +9.9 +28 +5.8 +0.29 +18.85 +0.23 +290 +0.12 +10.32 +9.86 +7.9 +29 +5.4 +0.27 +17.55 +0.22 +270 +0.11 +9.61 +9.18 +8.1 + + + +Table 4.6: Cost saving per day. +Transit +% of Transit +Reduction +Transit +Reduces +Riding +Distance (km) +Fuel +(litres) +User Cost +(tk) +Bus +5% +6370 +127400 +25480 +216580 +Car +2% +5865 +58650 +2933 +190613 +Taxicab +10% +3660 +366000 +14640 +18300000 +Motor cycle +10% +72480 +1087200 +21744 +1935216 +Total cost saving +64797 +20642409 + + + + + +36 + +4.2.2.3 CO2 Emission Reduction +In the calculation, 887 g/km, 258 g/km, 237 g/km and 40 g/km are the considered +amount of CO2 emission in 1km ride of bus, car, taxicab and motorcycle respectively. +For motijheel, the gross CO2 emission reduction is described in the following Table 4.7. +In total, around 6.5 × 105 kg CO2 emission is reduced in the motijheel area per day. +4.3 Bicycle Lane Network Design for Entire Dhaka City +Here, 29 important locations of entire Dhaka city are selected and numbering those +from 1 to 29 randomly, which are depicted in following Fig. 4.4. As it is considered +that the paddled-bicycle for routing 10km ranged area, it is not feasible to move through +all over the Dhaka city using paddled-bicycle. But in case of electric bicycle, it is +possible. That’s why electric bicycles are considered as vehicles for entire Dhaka city. + +The general data of Dhaka city including location, population and traffic pressure are +described in the Table 4.8. In this case, the consideration is that the node point's traffic +pressure is proportional to its population. + +For entire Dhaka city, it is assumed that 5% of users switch from car to electrical +motorcycle or bicycle and 50% of users switch from motorcycle to electrical +motorcycle or bicycle. And a 10% bus and 20% taxicab are being reduced because of +using electric motorcycles or bicycles. Then there will be some noteworthy change in +the environment. +4.3.1 Network Design +The constructed network of selected 29 points of Dhaka city is illustrated in Fig. 4.5. +In the Table 4.9, routes are shown to all node points from starting node point 7. For +Table 4.7: CO2 emission reduction per day. +Transit +% of Transit +Reduction +Transit +Reduces +Riding +Distance (km) +CO2 Emission +(g) +Bus +5% +6370 +127400 +1.13 × 108 +Car +2% +5865 +58650 +1.51 × 107 +Taxicab +10% +3660 +366000 +8.67 × 107 +Motor cycle +10% +72480 +1087200 +4.35 × 107 +Total CO2 saving +2.58 × 108 + + + + + +37 + + + + +01 Uttara +02 Mirpur +03 Uttara ABM City +04 Basundhara R/A +05 Khilkhet +06 Cantonment +07 Gulshan +08 Badda +09 Mohakhali +10 Tejgaon +11 Motijheel +12 Khilgoan +13 Gabtoli +14 Mohammadpur +15 Dhanmondi +16 Shahbagh +17 Matuail +18 Kotwali +19 New Market +20 Baridhara +21 Banani +22 Monipur +23 Sher-E-Bangla Nagar +24 Airport +25 Poradia +26 Madarbari +27 Beraid +28 NutanPara +29 Pagla Rail station + +Figure 4.3: Entire Dhaka city. + + + +Asnua +N302 +N501 +R303 +N511 +N302 +N501 +Dhaka Zoo +JalshiriAb +23 +R20 +OL +Boshila +1.5 +28 +Chanpa +LalbaghFort +b +Hizla +R110 +Keraniganj +R820 +N8 +820 +N1 +Shiddhirganj +5069 +RB10 +Bashundhara +R111 +pur +Kada +Riverview +Baghair +Ekuria +N8 +folafe +29 +eshviariRN +Kalakandi +Gode +Zazira + +38 + + +example, the route 7-6-24-5-1 means that we have to pass node points 6, 26 and 5 in +order to arrive destination node point 1 from starting node point 7. We are considering +the minimum distance for accessing any node here. For example, the destination node +point 1 can be accessed using the routes 7-6-24-5-1, 7-6-4-26-5-1, 7-21-22-2-24-5-1 +and so on. But the minimum distanced route is considered for calculation. +4.3.2 Effectiveness Analysis +Using an electric-bicycle has a more than one beneficial environmental effect. For most +of the cases bicycles take less time to ride, no fuel usage, saving user money, CO2 +emission reduction in unplanned mega city and very less contribution to air pollution, +sound pollution, great relief on road traffic conditions, alleviating parking difficulties +in urban areas, cause less damage to the roads, get relief of some non-communicable +disease (NCDs) for have some physical exercise every day. The considering factors of +Table 4.8: The general data on locations in Dhaka city, including population, area, and traffic pressure. +Sl +Location +Population +Area (km²) +Traffic Pressure +01 +Uttara +345,097 +36.91 +10 +02 +Mirpur +274,530 +4.71 +8 +03 +Uttara ABM City +145,097 +27.91 +5 +04 +Bashundhara R/A +274,200 +13.54 +8 +05 +Khilkhet +130,053 +15.88 +5 +06 +Cantonment +117,464 +14.47 +4 +07 +Gulshan +145,969 + 8.85 +5 +08 +Badda +157,924 +16.78 +5 +09 +Mohakhali +145,969 +8.85 +5 +10 +Tejgaon +148,255 +2.46 +5 +11 +Motijheel +225,999 +3.69 +7 +12 +Khilgaon +327,717 +13.8 +9 +13 +Gabtoli +198,723 +4.98 +6 +14 +Mohammadpur +355,843 +11.65 +10 +15 +Dhanmondi +147,643 +2.86 +5 +16 +Shahbag +74,113 +3.49 +3 +17 +Matuail +125,312 +19.36 +4 +18 +Kotwali +210,504 +0.67 +6 +19 +New Market +66,439 +1.67 +2 +20 +Baridhara +105,969 +5.45 +4 +21 +Banani +145,969 +8.85 +5 +22 +Monipur +274,530 +4.71 +8 +23 +Sher-E-Bangla Nagar +248,871 +5.25 +7 +24 +Airport +130,053 +15.88 +5 +25 +Poradia +52,014 +20.09 +2 +26 +Madarbari +93,153 +12.65 +3 +27 +Beraid +157,924 +16.78 +5 +28 +NutanPara +125,312 +19.36 +4 +29 +Pagla Rail station +194,019 +246.21 +6 + + + + + + +39 + +this paper is only time saving, fuel and user money saving and CO2 emission reduction. +4.3.2.1 Time Saving +In the following measurements, 7 kilometers per hour (kmph) for the buses, cars, +taxicabs & motorcycles which is the average speed of traffic in Dhaka city and 40 km/h +for electric bicycles using our constructed network. We are taken into consideration our +constructed network as jam-free. In the Table 4.10, the time necessary for cars and +buses are listed in three different times (at 6:00 AM, 10:00 AM, and 4:00 PM) +calculated from google map and the time needed for a bicycle is always constant. The +required time for taxicabs and motorcycles is considered to be the same as the time +needed for a car. Here, we measure the distance and time of all the node point from +center node point 7. + +Table 4.9: Routes from node point 7. +Des. Node +Point +Routes +Distance +(km) +Estimated Travel +Time(min) +1 +7-6-24-5-1 +15.65 +23.48 +2 +7-21-22-2 +9.89 +14.84 +3 +7-6-24-5-26-3 +19.51 +29.27 +4 +7-6-4 +9.02 +13.53 +5 +7-6-24-5 +11.9 +17.85 +6 +7-6 +3.95 +5.93 +8 +7-20-8 +3.17 +4.76 +9 +7-9 +2.17 +3.26 +10 +7-9-10 +5.98 +8.97 +11 +7-9-10-16-11 +10.61 +15.92 +12 +7-20-12 +9.04 +13.56 +13 +7-21-13 +9.44 +14.16 +14 +7-23-14 +7.31 +10.97 +15 +7-23-14-15 +11.55 +17.33 +16 +7-9-10-16 +8.7 +13.05 +17 +7-9-28-17 +15.61 +23.42 +18 +1-20-18 +14.78 +22.17 +19 +7-9-10-19 +9.87 +14.81 +20 +7-20 +2.12 +3.18 +21 +7-21 +1.76 +2.64 +22 +7-21-22 +7.34 +11.01 +23 +7-23 +4.07 +6.11 +24 +7-6-24 +8.55 +12.83 +25 +7-6-4-25 +18.72 +28.08 +26 +7-6-24-5-26 +15.5 +23.25 +27 +7-20-27 +11.2 +16.8 +28 +7-9-28 +9.41 +14.12 +29 +7-9-28-17-29 +22.02 +33.03 + + + + + + +40 + + + +Figure 4.4: Constructed network 29 points of Dhaka city. + + + +25 + +41 + +In the Table 4.10, we have noticed that at the morning 6:00 AM the travel time is less +because of minimal traffic in the roads, at 10:00 AM (peak hour) when traffic jam +occurs severe amount the need time to travel is huge and at 4:00 PM there exist traffic +jam also but sometimes a bit less than peak hour or sometimes a bit high. This condition +is true for all types of cars, buses, taxicabs, etc. For the entire Dhaka city, the gross +working hours saving are described in the following Table 4.11. So, around .2 million +working hours per day using bicycle in the motijheel area. +Table 4.10: Time comparison in car, bus and bicycle considering from node point 7. +Node +Point +Dista +nce +(km) +Car & Taxicab & Motor cycle +Bus +Bicycle +6:00am +(min) +10:00am +(min) +4:00pm +(min) +6:00am +(min) +10:00am +(min) +4:00pm +(min) +Dista +nce +(km) +Time +(min) +1 +14 +24 +42 +45 +26 +50 +51 +15.65 +23.48 +2 +7.5 +16 +33 +36 +18 +43 +42 +9.89 +14.84 +3 +18.2 +40 +75 +75 +42 +82 +81 +19.51 +29.27 +4 +10.2 +22 +42 +36 +22 +42 +42 +9.02 +13.53 +5 +8.7 +12 +24 +24 +14 +31 +30 +11.9 +17.85 +6 +6 +10 +26 +24 +12 +32 +30 +3.95 +5.93 +8 +4.7 +8 +30 +27 +9 +35 +32 +3.17 +4.76 +9 +3.5 +7 +24 +22 +9 +28 +24 +2.17 +3.26 +10 +5.5 +10 +25 +24 +12 +31 +28 +5.98 +8.97 +11 +9.1 +18 +45 +42 +20 +52 +48 +10.61 +15.92 +12 +12 +40 +60 +57 +42 +68 +65 +9.04 +13.56 +13 +9.7 +22 +44 +45 +24 +51 +51 +9.44 +14.16 +14 +8.3 +14 +36 +33 +16 +44 +39 +7.31 +10.97 +15 +8.7 +16 +33 +36 +17 +41 +42 +11.55 +17.33 +16 +8 +16 +39 +39 +18 +47 +45 +8.7 +13.05 +17 +16.6 +30 +63 +60 +32 +71 +66 +15.61 +23.42 +18 +11.4 +24 +62 +60 +26 +60 +66 +14.78 +22.17 +19 +8.7 +18 +42 +42 +20 +50 +48 +9.87 +14.81 +20 +2.2 +5 +12 +11 +6 +15 +14 +2.12 +3.18 +21 +2.3 +5 +17 +15 +7 +17 +15 +1.76 +2.64 +22 +6.8 +20 +36 +33 +22 +44 +39 +7.34 +11.01 +23 +6.6 +12 +24 +23 +14 +32 +29 +4.07 +6.11 +24 +8.7 +12 +24 +24 +14 +32 +30 +8.55 +12.83 +25 +15.9 +30 +68 +71 +32 +75 +77 +18.72 +28.08 +26 +14.7 +30 +63 +60 +32 +71 +66 +15.5 +23.25 +27 +8.5 +14 +36 +33 +16 +45 +39 +11.2 +16.8 +28 +10.2 +24 +63 +60 +26 +71 +66 +9.41 +14.12 +29 +19.1 +35 +79.5 +75 +37 +87 +81 +22.02 +33.03 + + + +Table 4.11: Time saving per day. +Transit +% of Transit +Reduction +Transit +Reduces +Riding +Distance (km) +Time Saves +(min) +Bus +10% +12740 +254796 +7.2 × 107 + + + + + +Car +5% +14663 +146630 +1.04 × 106 +Taxicab +20% +7320 +732000 +5.2 × 106 +Motor cycle +50% +362400 +5436000 +3.8 × 107 +Total time saving +1.16 × 108 + + + + + +42 + +Time Saving Explanation: +Total Bus ride reduce: 0. 1 × 127398 × 20 km = 254796 km +Time saving for Bus: 254796 × 40 × ( 8.57 – 1.5) mins = 7.2 × 107 mins += 1.2 × 106 hours = 50000 day +Total Car ride reduce: 0.05 × 293268 × 10 km = 146634 km +Time saving for Car: 146634 × (8.57 – 1.5) mins = 1.04 × 106 mins + + + += 1.73 × 104 hours = 720 days +Total Taxicab ride reduce: 0.2 × 36600 × 100 km = 732000 km +Time saving for Taxicab: 732000 × (8.57 – 1.5) mins = 5.2 × 106 mins + + + += 8.66 × 104 hours = 3611 days +Total Motorcycle ride reduce: 0.5 × 724800 × 15 km = 5.4 × 106 km +Time saving for Motorcycle: 5.4 × 106 × (8.57 – 1.5) mins = 3.8 × 107 mins + + + += 6.3 × 105 hours = 26289 days +Total working time saving: 50000 + 720 + 3611 + 26289 days = 80620 days +4.3.2.2 Fuel and Cost Saving +The cost of installation or repair is not listed here rather we considering only running +cost. Because whenever users switch from car to bicycle there would be a significant +reduction in costs. + +Here it is assumed that the mileage is 5km per liter diesel for bus and 20km per liter +diesel for both car and taxicab of 65tk per liter and the mileage is estimated to be 50km +per liter for motorcycle of 89tk per liter. On the other hand, the electric cycle charges +of 10tk with 50km of travel per charge charges. The fuel consumption is calculated per +day basis and user saving is grand saving of considering all users of the Dhaka city. + +In the following Table 4.12, the needed fuel for cars, taxicabs, motorcycles, and buses +is calculated and there is no running cost & fuel cost for the bicycle. In case of bus, the +per km fare is fixed by BRTA of 1.7tk and we assumed that taxicab fare is 50tk per km. +Here the distance of all the node point is measured from center node point 7. + + + + +43 + +The gross fuel saving and user cost saving of the entire Dhaka city is described in the +following Table 4.13. So, it is saved around 195570 liters fuel and 46.72 million user +costs per day for using bicycle in the entire Dhaka area. + +Fuel and Cost Saving Explanation: +Fuel saving for Bus: 254796 km × 1/5 litres = 50959 litres + +Table 4.13: Cost saving per day. +Transit +% of Transit +Reduction +Transit +Reduces +Riding +Distance (km) +Fuel +(litres) +User Cost (tk) +Bus +10% +12740 +254796 +50959 +1.27 million +Car +5% +14663 +146630 +7331 +4.5 lakh +Taxicab +20% +7320 +732000 +29280 +36.5 million +Motor cycle +50% +362400 +5436000 +1.08 × 105 +8.5 million +Total cost saving +195570 +46.72 million + + +Table 4.12: Cost comparison in car, bus and bicycle considering from node point 7. +Node +Point +Dist +ance +(km) +Car +Taxicabs +Motor Cycle +Bus +Electric +Bicycle +Fuel +(litre) +User +Cost +(tk) +Fuel +(litre) +User +Cost +(tk) +Fuel +(litre) +User +Cost +(tk) +Fuel +(litre) +User +Cost +(tk) +Dista +nce +(km) +User +Cost +(tk) +1 +14 +0.7 +45.5 +0.56 +700 +0.28 +15.49 + + + + + + +As we +assume +bicycles +reduces +10% of the +total bus in +Dhaka +city. + +The fuel +consumpti +on is +reduced += +12740×20 +×1/5 litres + += 50959 +litres +23.8 +15.65 +3.13 +2 +7.5 +0.38 +24.38 +0.3 +375 +0.15 +10.68 +12.75 +9.89 +1.98 +3 +18.2 +0.91 +59.15 +0.73 +910 +0.36 +8.37 +30.94 +19.51 +3.9 +4 +10.2 +0.51 +33.15 +0.41 +510 +0.2 +6.23 +17.34 +9.02 +1.8 +5 +8.7 +0.44 +28.28 +0.35 +435 +0.17 +9.79 +14.79 +11.9 +2.38 +6 +6 +0.3 +19.5 +0.24 +300 +0.12 +16.2 +10.2 +3.95 +0.79 +8 +4.7 +0.24 +15.28 +0.19 +235 +0.09 +21.36 +7.99 +3.17 +0.63 +9 +3.5 +0.18 +11.38 +0.14 +175 +0.07 +17.27 +5.95 +2.17 +0.43 +10 +5.5 +0.28 +17.88 +0.22 +275 +0.11 +14.77 +9.35 +5.98 +1.2 +11 +9.1 +0.46 +29.58 +0.36 +455 +0.18 +15.49 +15.47 +10.61 +2.12 +12 +12 +0.6 +39 +0.48 +600 +0.24 +14.24 +20.4 +9.04 +1.81 +13 +9.7 +0.49 +31.53 +0.39 +485 +0.19 +29.55 +16.49 +9.44 +1.89 +14 +8.3 +0.42 +26.98 +0.33 +415 +0.17 +20.29 +14.11 +7.31 +1.46 +15 +8.7 +0.44 +28.28 +0.35 +435 +0.17 +15.49 +14.79 +11.55 +2.31 +16 +8 +0.4 +26 +0.32 +400 +0.16 +3.92 +13.6 +8.7 +1.74 +17 +16.6 +0.83 +53.95 +0.66 +830 +0.33 +4.09 +28.22 +15.61 +3.12 +18 +11.4 +0.57 +37.05 +0.46 +570 +0.23 +12.1 +19.38 +14.78 +2.96 +19 +8.7 +0.44 +28.28 +0.35 +435 +0.17 +11.75 +14.79 +9.87 +1.97 +20 +2.2 +0.11 +7.15 +0.09 +110 +0.04 +15.49 +3.74 +2.12 +0.42 +21 +2.3 +0.12 +7.48 +0.09 +115 +0.05 +28.3 +3.91 +1.76 +0.35 +22 +6.8 +0.34 +22.1 +0.27 +340 +0.14 +26.17 +11.56 +7.34 +1.47 +23 +6.6 +0.33 +21.45 +0.26 +330 +0.13 +15.13 +11.22 +4.07 +0.81 +24 +8.7 +0.44 +28.28 +0.35 +435 +0.17 +18.16 +14.79 +8.55 +1.71 +25 +15.9 +0.8 +51.68 +0.64 +795 +0.32 +34 +27.03 +18.72 +3.74 +26 +14.7 +0.74 +47.78 +0.59 +735 +0.29 +15.49 +24.99 +15.5 +3.1 +27 +8.5 +0.43 +27.63 +0.34 +425 +0.17 +10.68 +14.45 +11.2 +2.24 +28 +10.2 +0.51 +33.15 +0.41 +510 +0.2 +8.37 +17.34 +9.41 +1.88 +29 +19.1 +0.96 +62.08 +0.76 +955 +0.38 +6.23 +32.47 +22.02 +4.4 + + + + + + +44 + +User money saving: 254796 km × ( +65 +5 − 40 × +10 +50) tk = 1.27 million tk +Fuel saving for Car: 146634 km × +1 +20 litres = 7331 litres +User money saving: 146634 km × ( +65 +20 − +10 +50) tk = 4.5 × 105 tk = 4.5 lakh tk +Fuel saving for Taxicab: 732000 km × +1 +25 litres = 29280 litres +User money saving: 732000 km × (50 − +10 +50) tk = 3.65 × 107 tk = 36.5 million tk +Fuel saving for Motor cycle: 5.4 × 106 km × +1 +50 litres = 1.1 × 105 litres +User money saving 5.4 × 106 × ( +89 +50 − +10 +50) tk = 8.5 × 106 tk = 8.5 million tk +Total fuel saving: 50959 + 7331 + 29280 + 1.1 × 105 litres = 197570 litres +Total User money saving: 1.27 + .45 + 36.5 + 8.5 million tk = 46.72 million tk +4.3.2.3 CO2 Emission Reduction + In the following calculation, 887 g/km, 258 g/km, 237 g/km and 40 g/km are the +considered amount of CO2 emission in 1km ride of bus, car, taxicab, and motorcycle +respectively. Then CO2 emission is reduced in a significant amount. For the entire +Dhaka city, the gross CO2 emission reduction is described in the Table 4.14. In total, +around 6.58 × 105 kg CO2 emission is reduced in the entire Dhaka city per day. + +CO2 Emission Reduction Explanation: +CO2 emission reduction for Bus: 0.1 × 127398 × 20 km × 887 gm = 2.3 × 105 kg +CO2 emission reduction for Car: 0.05 × 293268 × 10 km × 258 gm = 3.8 × 104 kg +CO2 emission reduction for Taxicab: 0.2 × 36600 × 100 km × 237 gm = 1.7 × 105 kg +CO2 emission reduction for Motorcycle: 0.5 ×724800 ×15 km × 40 gm = 2.2 × 105 kg +Total CO2 emission reduction: 6.58 × 105 kg + +Table 4.14: CO2 emission reduction per day. +Transit +% of Transit +Reduction +Transit +Reduces +Riding +Distance (km) +CO2 +Emission (g) +Bus +10% +12740 +254796 +2.3 × 108 +Car +5% +14663 +146630 +3.8 × 107 +Taxicab +20% +7320 +732000 +1.7 × 108 +Motor cycle +50% +362400 +5436000 +2.2 × 108 +Total CO2 emission reduction +6.58 × 108 + + + + + +45 + +When CNG is being used as fuel in buses, cars, and taxicabs the CO2 emission is more +significant. Buying a car and motorcycle also increases traffic jams, air carbon dioxide, +pollution of the environment and costly in the current situation in Dhaka. Alternatively, +electric bicycle does not make an enormous traffic jam, CO2 in air, and less expensive. +In order to do this, governments should build parking lots in various places where +necessary. + + + + + + + + + +CHAPTER 5 +Conclusions +A modified Physarum-inspired model is presented in this paper to address the design +of the bicycle lane network. Different approaches, like exact approaches and heuristic +approaches, have been presented over the past decades to design transportation +networks. Recently bio-inspired method had drawn great attraction to network design. +In real two-way traffic networks, the modified technique is more effective and efficient. +This chapter will now give a short summary of the main points described in this thesis. +Also, it discusses possible future works based on the outcome of the present work + 5.1 Achievements +The network design technology inspired by Physarum is believed to have balanced +costs, effectiveness, and resilience. Inside Dhaka city, an unorganized and unplanned +city, we have developed an electric bicycle network system where there's little footway. +To meet this challenge, we primarily use local roads and try to avoid major roads +towards the construction of the electric bicycle network. Since bicycles are non- +motorized vehicles do not produce greenhouse gases so they do not cause air pollution. +They also don't contribute to noise pollution. If a large number of people use bicycle in +the city, traffic jams will be eliminated. The costs will be reduced and people can have +some physical activity also, which is beneficial to health. Since bicycles do not need to +use gasoline, the importation of gasoline will be reduced. That also enriches the +economy and the environment. +5.2 Future Study +In the future, parallel computing and the optimal model for the design of the transport +network are part of our work. Furthermore, our research includes the implementation +of the Physarum polycephalum inspired model for the dynamic traffic network and the +elastic demand traffic network. + + + + +47 + +References +[1] +P. Merriman, “Mobility,” in International Encyclopedia of Human Geography, +Elsevier, 2009, pp. 134–143. +[2] +H. Hao, Y. Geng, H. Wang, and M. Ouyang, “Regional disparity of urban +passenger transport associated GHG (greenhouse gas) emissions in China: A +review,” Energy, vol. 68, pp. 783–793, Apr. 2014. +[3] +M. R. 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Sports Med., vol. 46, no. 10, pp. 709– +712, Aug. 2012. +[60] “Number of registered Vehicles in Dhaka Metro,” Dhaka, 2020. +[61] Moskvitch Katia, “Slime Molds Remember — but Do They Learn?,” +https://www.quantamagazine.org/slime-molds-remember-but-do-they-learn- +20180709/, Jul-2018. + + diff --git a/3tFRT4oBgHgl3EQfojeI/content/tmp_files/load_file.txt b/3tFRT4oBgHgl3EQfojeI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f0a1f840844ff50cbea54fe339668dbaf8aa72f2 --- /dev/null +++ b/3tFRT4oBgHgl3EQfojeI/content/tmp_files/load_file.txt @@ -0,0 +1,2167 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf,len=2166 +page_content='Physarum Inspired Bicycle Lane Network Design in a Congested Mega City By Md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Ahsan Habib Roll: 1507082 Department of Computer Science and Engineering Khulna University of Engineering & Technology Khulna 9203, Bangladesh March 2020 ii Certification The thesis titled “Physarum Inspired Bicycle Lane Network Design in a Congested Mega City” submitted by Md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Ahsan Habib, Roll No: 1507082, Academic Year: 2018- 19, for partial fulfillment of the requirements for the degree of “Bachelor of Science in Computer Science and Engineering”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Supervisor Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Muhammad Aminul Haque Akhand Professor Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' of Computer Science and Engineering Khulna University of Engineering & Technology Khulna, Bangladesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' iii Acknowledgements First and foremost, I must sense grateful to and wish to acknowledge my insightful indebtedness to Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Muhammad Aminul Haque Akhand, Professor of Department of Computer Science and Engineering and the supervisor of the thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' His unfathomable knowledge in this field influenced me to carry out this thesis up to this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' His endless endurance, scholarly guidance, continual encouragement, constant and lively supervision, constructive criticism, priceless suggestion made it possible to come up to this phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Without his inspiring, enthusiasm and encouragement, this work could not be completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Last, but by no means least, I thank Allah for the talents and abilities I was given that made it possible to undertake this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' iv Abstract Mobility is a key factor in urban life and transport network plays a vital role in mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Worse transport network having less mobility is one of the key reasons to decline the living standard in any unplanned mega city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Transport mobility enhancement in an unplanned mega city is always challenging due to various constraints including complex design and high cost involvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The aim of this thesis is to enhance transport mobility in a megacity introducing a bicycle lane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' To design the bicycle lane natural Physarum, brainless single celled multi-nucleated protist, is studied and modified for better optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Recently Physarum inspired techniques are drawn significant attention to the construction of effective networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Exiting Physarum inspired models effectively and efficiently solves different problems including transport network design and modification and implication for bicycle lane is the unique contribution of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Central area of Dhaka, the capital city of Bangladesh, is considered to analyze and design the bicycle lane network bypassing primary roads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' v Contents Title Page i Certification ii Acknowledgements iii Abstract iv Contents v List of Tables vii List of Figures viii CHAPTER 1 Introduction 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 Overview of Transport Network in a Mega City 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 Motivation 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 Objectives of the Thesis 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='4 Organization of the Thesis 3 CHAPTER 2 Physarum Inspired Network Design 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 Physarum and its Properties 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 Network Design Inspired on Physarum 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 Review of Existing Physarum Inspired Works 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 Transpiration Network Design 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 Other Optimization Task 7 CHAPTER 3 Physarum Inspired Bicycle Lane Design in an Unplanned Mega City 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 Mobility Problem in an Unplanned Mega City: Dhaka as a Case Study 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 History and Overview of Dhaka City 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 Transportation Crisis in Dhaka City 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 Effect of Transportation Crisis to Other Problems 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 Importance of Bicycle Lane in an Mega City 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 Challenges to Increase Mobility in Dhaka City 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='4 Bicycle Lane Design in an Unplanned Mega City 16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 Significance of Study 23 CHAPTER 4 Experimental Studies 24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 Experimental Settings 24 vi 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 Bicycle Lane Network Design in a Prominent Area Error!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Bookmark not defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 Network Design 25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 Effectiveness Analysis 32 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 Time Saving 33 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 Fuel and Cost Saving 33 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 CO2 Emission Reduction 36 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 Bicycle Lane Network Design for Entire Dhaka City 36 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 Network Design 36 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 Effectiveness Analysis 38 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 Time Saving 39 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 Fuel and Cost Saving 42 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 CO2 emission reduction 44 CHAPTER 5 Conclusions 46 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 Achievements 46 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 Future Study 46 References 47 vii List of Tables Table No Description Page 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 Network construction using Physarum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 The locations of prominent area of Dhaka city including traffic pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 27 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 Routes from node point 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 32 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 Time comparison in car, bus and bicycle considering from node point 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 34 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='4 Time saving per day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 34 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 Cost comparison in car, bus and bicycle considering from node point 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 35 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='6 Cost saving per day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 35 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7 CO2 emission reduction per day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 36 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='8 The general data on locations in Dhaka city, including population, area, and traffic pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 38 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='9 Routes from node point 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 39 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='10 Time comparison in car, bus and bicycle considering from node point 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 41 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='11 Time saving per day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 41 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='12 Cost comparison in car, bus and bicycle considering from node point 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 43 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='13 Cost saving per day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 43 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='14 CO2 emission reduction per day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 44 viii List of Figures Figure No Description Page 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 Physarum polycephalum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 Farmgate to University of Dhaka routes and time needed in driving mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 Physarum inspired network design of 11 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 Real traffic network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Selected Dhaka city Map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 Network design using modified Physarum inspired technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 Dhaka city Map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 28 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='4 Constructed network 29 points of Dhaka city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 29 2 CHAPTER 1 Introduction A transport network can be described as a collection of linear features that permits either vehicular movement or flow of some commodity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The characteristics of Physarum can be used to design a transportation network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' This chapter discusses the background study of network design, objectives, and organization of the thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 Overview of Transport Network in a Mega City Mobility, a key factor in planning and designing urban transport, is a fundamental part of human beings [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' For mobility purposes, people can use both motorized and non- motorized vehicles within a city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Motorized vehicles like buses, cars, motorbikes, cycles, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' are hazardous in many kinds [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Non-motorized transports involve walking and cycling as well as variants such as small-wheeled transportation like skates, skateboards, push scooters and hand carts [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Nowadays, non-motorized mobility is trendy [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=" The idea of mega city emerged to characterize the world's largest metropolitan agglomerations at the end of the 20th century." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In the 1970s, only two mega cities had over ten million residents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Currently, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='9% of the urban population globally resides in 23 megacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' It is projected that in 2025, the number will increase to 37 if 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='6% of global urban population are to be accommodated [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' A transportation network is the formation of a spatial network that enables vehicle movement or the flow of some commodities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' A vast network of rail, subways and bus lines passes through well-organized mega cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' There are also exist special footways and networks for cycling routes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 Motivation In a well-planned and well-organized mega city, both footways and roads are available for mobility purpose but in case of an unplanned and unorganized mega city, both footpaths and roads are hardly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In some cases, footpaths are snatched by 3 hawkers and street vendors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=" The public transport system is defined by far a lack of people's desired travel needs in terms of mobility, reliability, convenience, pace and safety." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In fact, some transports like buses are considered unreliable and time consuming to reach their destinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The Texas Transportation Institute reported a delay of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='6 billion vehicle-hours in the 75 biggest metropolitan regions in 2000, culminating in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7 billion U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' gallons (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='6 billion liters) of waste fuel and a loss of productivity of $67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 billion or around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7 percent of GDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Bicycling or walking activity may help increase blood flow, release endorphins, and decrease overall stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' It can even help to improve mental health and energy by tracking 30 minutes of bicycling or walking a day[6]–[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Efficient and effective bicycle lane network design in an unplanned and unorganized mega-city can minimize total travel time, fuel usage, costs, carbon dioxide (CO2) emission, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Physarum polycephalum is multi-headed, brainless, a giant multi-nucleated, single-celled protist that can solve different complex problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Physarum networks are believed to have achieved a good balance between cost, efficiency, and resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 Objectives of the Thesis In the case of an unorganized and unplanned mega city, where the transportation network is congested and unplanned and there are hardly any footpaths available and no further transport facilities can be expanded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' So there are some huge problems in those cities like traffic jam, noise pollution, air pollution, CO2 emission, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' With a planned lane network with non-motorized vehicles nearly all of this problem can be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Since it is not feasible to completely rebuild the transportation network and infrastructure of a mega city but possible to transform mega city towards a green city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The objective of this study is given below: \uf0b7 Study of Physarum \uf0b7 Physarum related paper study \uf0b7 Network design \uf0b7 Bicycle lane network in mega city 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='4 Organization of the Thesis The main attraction of this thesis is to present a modified Physarum inspired technique to construct bicycle lane network design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The thesis has five chapters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' An introduction to network design and Physarum has been given in Chapter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Chapter-wise overviews of the rest of the thesis are as follows: Chapter 2: Describes the literature review that includes a brief description of Physarum with its properties and previous related work to Physarum inspired network design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Chapter 3: Explains the proposed modified Physarum Inspired Bicycle Lane Design in an Unplanned Mega City in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Chapter 4: Reports the experimental result of modified Physarum Inspired Bicycle Lane Design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Also, in this chapter, a case study of Dhaka is demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Finally, Chapter 5: This chapter is for the conclusions of this thesis together with the outline of future directions of research opened by this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' CHAPTER 2 Physarum Inspired Network Design Physarum polycephalum is a brainless amoeboid organism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Physarum-inspired network design model has demonstrated extraordinary skill in designing effective networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In this chapter firstly, we discuss the Physarum polycephalum, secondly the Physarum- based network design and lastly, the existing network design inspired by Physarum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content="1 Physarum and its Properties Physarum polycephalum, accurately the 'many-headed' slime mold, is a gigantic multi- nucleated but single-celled protist [10]." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The slime mold Physarum polycephalum creates a form of spatial memory by avoiding areas it has previously explored to navigate in a complex environment[11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Recently, Physarum polycephalum (true slime mold) has arisen as a fascinating illustration of biological computation through morphogenesis[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Although it is a single-cell organism, studies have shown that the Physarum can overcome different minimum cost flow problems through its growth process[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In the following Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1, an example of the Physarum polycephalum is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1: Physarum polycephalum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' FS FS FS Source FS FS 5 Here Physarum polycephalum is shown to grow up the network towards the FSs from the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (FS = Food Source) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 Network Design Inspired on Physarum The intelligent behavior of slime mold was first observed by Nakagaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' in 2000[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In previous biological experiments, Physarum-inspired network model has exhibited an extraordinary intelligence to build efficient networks to connect multiple food sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Physarum networks are believed to have achieved a good balance between cost, efficiency, and resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' For instance, Physarum constructed networks with comparable qualities to those of the Tokyo rail system in a renowned experiment performed by Tero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' in 2010[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' They developed a mathematical model for adaptive network construction to emulate the behavior of Physarum which is based on feedback loops between the thickness of each tube and internal protoplasmic flow in which high rates of streaming stimulate an increase in tube diameter, whereas tubes tend to decline at low flow rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The edges represent plasmodial tubes in which protoplasm flows, and nodes are junctions between tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' They consider the pressure at nodes 𝑖 and 𝑗 are 𝑃𝑖 and 𝑃𝑗, respectively, and the two nodes are connected by a cylinder of length 𝐿𝑖𝑗 and radius 𝑟𝑖𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' They assume that the flow is laminar and follows the Hagen-Poiseuille equation, the flux through the tube is, 𝑄𝑖𝑗 = 𝜋𝑟𝑖𝑗 4(𝑃𝑖 − 𝑃𝑗) 8𝜀𝐿𝑖𝑗 = 𝐷𝑖𝑗(𝑃𝑖 − 𝑃𝑗) 𝐿𝑖𝑗 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1) here 𝜀 is the viscosity of the fluid, and 𝐷𝑖𝑗 = 𝜋𝑟𝑖𝑗 4 8𝜀 is a measure of the conductivity of the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' As the length 𝐿𝑖𝑗 is a constant, the behavior of the network is described by the conductivities of the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The constrains must be maintained, ∑ 𝑄1𝑗 = 𝐼0 𝑗 , For source node 1 ∑ 𝑄2𝑗 = −𝐼0 𝑗 , For sink node 2 ∑ 𝑄𝑖𝑗 = 0 𝑗 , Inflow and outflow must be conserved 6 To accommodate the adaptive behavior of the plasmodium, the conductivity of each tube evolves according to 𝑑𝐷𝑖𝑗 𝑑𝑡 = 𝑓(|𝑄𝑖𝑗|) − 𝐷𝑖𝑗, where 𝑓(|𝑄𝑖𝑗|) describes the expansion of tubes in response to the flux and 𝐷𝑖𝑗 represents the rate of tube constriction, so the tubes will gradually disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The functional 𝑓(|𝑄|) = |𝑄|𝛾 (1+ |𝑄|𝛾) which describes a sigmoidal response where 𝛾 is a parameter that controls the nonlinearity of feedback (𝛾 > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 Review of Existing Physarum Inspired Works Physarum can successfully overcome many problems in real life even more complicated problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In this section initially, we discuss and summarize about the Physarum inspired network design techniques, and then discuss about other optimization problems solved using Physarum inspired methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 Transpiration Network Design To link several food points Physarum can build high-quality networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' A mathematical model of the adaptive dynamics of a transport network of the true slime mold that shows path-finding behavior in a maze is developed in 2007 by Tero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In 2010, Physarum developed networks shows similar qualities to the Tokyo rail system in a famous experiment conducted by Tero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Since then, Physarum inspired other real-world transport networks, such as Iberian motorways [16] and Mexican Federal highways [17] have also been constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Adamatzky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' [18] develops a model to construct networks on major urban areas of China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' [19] developed in 2011 a fault tolerant connection networks for the Tokyo rail system using an agent based simulation of Physarum polycephalum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Physarum-inspired cellular automaton (CA)- based network designing model was developed by Tsompanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' [20] inspired by Slime Mould.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' [21] recently proposed a method to solve the problem of network design in supply chain for multiple source nodes and multiple sink nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Physarum is excellent at doing other network design [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='Here we summarize various works of network construction using Physarum inspired technique in the following Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 Other Optimization Task Nakagaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' in 2000 [13] observed that Physarum productively found the shortest path between two selected points in a maze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In addition, the Physarum can solve many other famous problems like the shortest paths [23]–[25], towers of Hanoi problem [26] and minimum risk problem [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Physarum can effectively solve many other complex problems in the real world like traveling salesman problem [28]–[30], population migration [31], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Logic gates design and boolean operations can be performed by a slime mold network [32], [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Chaining these logic gates together can enable a slime mold computer to perform binary computation operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Physarum works very well in logical computing as well[34]–[38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Identifying critical components [39], [40] and many other problems [41], [42] are effectively and efficiently solved through Physarum bio-inspired technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Most interestingly, many other studies have shown that Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1: Network construction using Physarum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Authors & Year Title of Paper Contribution Tero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=', 2007 [15] A mathematical model for adaptive transport network in path finding by true slime mold Model for adaptive transport network in Path-finding in a maze Tero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=', 2010 [14] Rules for Biologically Inspired Adaptive Network Design Tokyo Rail Network construction Adamatzky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=', 2011 [16] Rebuilding Iberian motorways with slime mould Iberian motorway network construction Adamatzky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=', 2011 [17] Approximating Mexican highways with slime mould Mexican Federal highway network construction Adamatzky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=', 2013 [18] Slime mould imitates transport networks in China Slime mould protoplasmic networks on major urban areas of China Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=', 2011 [19] Design of fault tolerant networks with agent-based simulation of Physarum polycephalum Construction of fault tolerant connection networks for the Tokyo rail system using an agent based simulation of Physarum polycephalum Tsompanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=', 2015 [20] Evolving Transport Networks With Cellular Automata Models Inspired by Slime Mould Physarum-inspired cellular automaton (CA)-based network designing model Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=" 2016 [21] A Physarum-inspired approach to supply chain network design Supply chain network design 8 Physarum's tubular topologies often mimic those of complex mathematical networks [43], [44] like the Steiner tree problems [45]–[49]." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Studies with Physarum related works showed that the organism can solve many complex real-life problems efficiently and effectively, particularly in the sense of network design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' This can be applied with some changes for designing the bicycle lane network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In this work bicycle lane network is planned using local lanes in congested mega city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 10 CHAPTER 3 Physarum Inspired Bicycle Lane Design in an Unplanned Mega City An unplanned mega city suffers various problems including transporation and mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Transport mobility enhancement in an unplanned mega city is always challenging due to various constraints including complex design and high cost involvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In this thesis, we try to increase the mobility in an unplanned mega city Dhaka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In this chapter, problems of an unplanned mega city Dhaka are addressed firstly, then challenges in transformation an unplanned megacity to green city, and finally, the importance of bicycle lane in an unplanned mega city and bicycle lane network design in an unplanned mega city are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 Mobility Problem in an Unplanned Mega City: Dhaka as a Case Study This thesis aim is to enhance transport mobility in an unplanned mega city introducing a bicycle lane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In this section initially, we discuss the history and overview of Dhaka city, then the transportation crisis in Dhaka city and finally, the effect of transportation crisis on other problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 History and Overview of Dhaka City It is mentioned that the concept of a Mega City originated at the end of the 20th century to describe the largest city in the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Although, literature has little disagreement about the population threshold used as a megacity concept, the UN (2003) defines most precisely: a conurbation of ten million or more inhabitants is a megacity which has now been widely accepted [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Since 1971, Dhaka has experienced incredible growth and rapid growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=" It is one of the world's only seven cities with a population of over 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='4 percent between 1975 and 2005 (UN 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=" In 2011, it was one of the world's top ten mega cities." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The developments have unfortunately happened unplanned, especially since the 1990s [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The word 11 Dhaka is nowadays mentioned regularly in the most unlivable cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=" Dhaka was the world's fastest-growing town between 1950 and 2000 [50]." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' While population growth has declined recently, it is still the second-largest growth mega city in the world [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 Transportation Crisis in Dhaka City The mega city has neither efficient public transport nor mass transit [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=" It is probably the world's only mega-city without efficient public transit and public transit [50]." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Dhaka has a poorly developed transport system with 200 km of main roads and about 260 km (too few) secondary and collector roads, in addition to 250 km of narrow roads (approximately) [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' There are many incomplete critical connections in the road network and several regions have insufficient connectivity to the network [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Separate bicycling lanes and footpaths are barely available in the city, which enhance the mobility crisis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' There was a time when traffic congestion was only suffered by commuters on the main streets of the city, but now it starts right from the door.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Traffic jam has turned into nightmares for daily trips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' According to a World Bank report, the average traffic speed in Dhaka has dropped from 21 kilometers per hour (kmph) to 7 kilometers per hour in the last 10 years, and by 2035 the speed could drop to 4 kilometers per hour, which is slower than the walking speed [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Another study commissioned by the BRAC Institute of Government and Development indicates that traffic congestion in Dhaka consumes about 5 million working hours a day and costs the country $11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='4 billion a year [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The financial loss is a measure of the time lost in traffic congestion and the extra hours expended on cars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' It should be noted that there is no adequate and proper routing of our public transport system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=" In 2016, According to the BRTA, 20,304 new cars were introduced to Dhaka's traffic, which means more than 55 new cars hit the streets every day [55]." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' As the number of cars increases, there is also an increasing demand for parking space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Unfortunately, however, the parking space in our city is quite inadequate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Many vehicles on the streets are stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Many buses and trucks are parked on the streets on a regular basis [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 12 According to the Dhaka Metropolitan Police (DMP) Traffic Department, traffic jams have become intolerable in some urban areas over the past few days, including Mirpur- 12 to Mirpur-10 crossing, Rokeya Sarani, Gulshan, Banani, Badda, Moghbazar, Eskaton, Tejgaon, Airport Road, and Uttara, for a number of reasons, including the ongoing Dhaka International Trade Fair, the building of underground trains and the increase in private transport[56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Urban analyst and former chairman of UGC Prof Nazrul Islam said traffic jams are gradually deteriorating due to an increase in urban population and the number of small vehicles and lack of effective control measures [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' “We have built over half dozens of flyovers, but it is not a solution to solve the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' We will not be able to reduce traffic jams without increasing public transport and ensuring better traffic management", he observed [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Transport and urban experts believe that the government should take practical steps to ensure effective mass transportation, restore transportation efficiency, decrease the use of private and small cars, replace micro-buses and mini-busses with single-decker, double-decker, and articulated buses, and extend the city to dramatically alleviate traffic jams without spending huge money [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The experts also said that railways and waterways can also be used effectively to relieve road traffic pressure and facilitate trouble-free transport services for the commuters [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 illustrates the traffic jam in the city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Here, three routes are available from Farmgate to the University of Dhaka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' During driving mode, it takes around 15 minutes at 06:00 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=', 18-35 minutes at 10:00 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=', and 18-40 minutes at 5:00 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' On the other side, it takes an average of 45-50 minutes in walking mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' We note that the speed of driving is slightly higher than that of walking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=" Not only in some areas, but throughout the city, it's the case." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The number of automobiles has been increasing in Dhaka city at the rate of at least 10 percent annually, which has been contributing to environmental pollution on the one hand and traffic congestion on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' This transportation problem enhances other problems like air pollution, noise pollution, fuel consumption, CO2 emission, worst in road conditions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 13 (A) At 06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='00AM (B) At 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='00AM (C) At 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='00PM (D) Walking mode Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1: Farmgate to University of Dhaka routes and time needed in driving mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (A) At 06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='00am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (B) At 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='00am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (C) At 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='00pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (D) Time needed in walking mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 回 of1 ManikMiaAve IndiraRd CG OCK W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='Raza 回 回 园 OFarmgate LKISAREA Insaf Bal Bashundhara City &Gen spital ShoppingComplex nthapath 园 aHatirheel hani Playground KALABAGAN NEWESK 12A FreeSchool St, Kalabegan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1st Ln 16min IANMONDI 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='9 km 15min OLDE 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 km RdNo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='8 回 17 min 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7 km Rd8/A TA RdNo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='6 Off 回 H HATIRPOOL Rd No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 ATOLA MintoRd DhakaCityCollege SHAHBAGH 回 Kazi Food Industries Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' KATABON Bangladesh Dhaka CollegeDhaka National Museum Shreshtha Noor hammad Dhaka New Market UniversityofDhaka icCollege RAMNA et pilkhana Rd 回 Bangla Acad0 OFarmgate IKISAREA GA KawranBazar InsafBarakah Bashundhara City ShoppingComplex &GeneralHos Ranthapath@ Hatirjheel 上 ind KALABAGAN NEWESKATON F Free SohoolSt Kalabagan 1etLn DI KATHALBAGAN OLD ESKATON 回 3 18 35mins 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='9km PARIBAG RdNo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='6 HATIRPOOL H =18 40mins Rd No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7km Dhaka CityCollege Baily Rd SHAHBAGH 18 35mins Rd 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1km Bangladesh ziFoodIndustriesLtd DhakaCollege KATABON Ant HANA Dhaka NewMarket UniversityofDhakaO SaniRd RAMNA t Pilkhana Rd Google BanglaAcademy Eden Mohila日 OFarmgate TallabapRd IKISAREA KawranBazar InsafBaral Bashundhara City ShoppingComplex &General thap atn Hatirjheel KALABAGAN sffofaa NEWESKATON FroeSchoolSt 6610 日 Katabagan1stLn H KATHALBAGAN OLD ESKATON 3 20 40mins 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='9km PARIBAGH Rd No 6 HATIRPOOL 18 40mins RdNo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 H 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7km Dhaka City College Baily Rd SHAHBAGH 22 40mins 113419 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1km 回 Bangladesh FoodIndustries Ltd DhakaCollege KATABON 11 ANA T DhakaNewMarket UniversityofDhaka SaniRd RAMNA BanglaAcademy日 OFarmgate TallabieRd IKISAREA H KawranBazar Insaf Bashundhara City ShoppingComplex &Ger st 白aHatirjheel H KALABAGAN NEW ESKATON Ftee Schodl St 8010 回 Kalsbigan1st La KATHALBAGAN OLDESKATO 3 50min PARIBAGR 专T 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='9km RdNo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content="6 HATIRPOOL Officers'Clut H RdNo." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 48min 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7km Shaka CityCollege O Baily 49min 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='8km Bangladesh IndustriesLtd NationalMuseum RamnaParl DhakaCollege KATABON DhakaNewMarket University of Dhaka0o ani RAMNA PilkhanaRd 回 BanglaAcademy 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 Effect of Transportation Crisis on Other Problems Transportation crisis affects the environment badly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' It may affect the air pollution, noise pollution, fuel consumption, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' According to the Department of Environment (DoE), the standard value of the Air Quality Index (AQI) is 50 represents good air quality with little potential to affect public health [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' But, according to AirVisual information, Dhaka the capital city of Bangladesh has been ranked the worst in the Air Quality Index (AQI) valued 309, which is hazardous and would trigger health warnings of emergency conditions [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The entire population is more likely to be affected by the enormous number of diseases like nausea, asthma, high blood pressure, heart disease, and cancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' It also impacts the respiratory tract severely and causing irritation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=" Children's cognitive faculty will be adversely affected by lead exposure, which can also distress the central nervous system, causing hypertension and renal injury." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In the last two months, the capitalist has been enjoying just nineteen hours of good air [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Diesel-run vehicles account for more than 80 percent of the air pollution in Dhaka as most of them fail to comply with the approved emission standard, said a recently published survey report [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In Dhaka, the average sound level is between 80dB and 110dB in prime areas such as Farmgate, Karwan Bazar, Shahbagh, Gabtoli, and Mohakhali Bus Terminal, says the study report [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' According to the World Health Organization (WHO), this is almost twice the maximum noise level that can be tolerated by humans – 60dB – without suffering a gradual loss of hearing [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' According to a recent study conducted by WHO at 45 locations of Dhaka city, most of the traffic points and many of the industrial, residential, commercial, silent and mixed areas are suffering noises exceeding the standard limits of Bangladesh [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' WHO has also identified several areas as severe red, moderate red, mild red and green zones in terms of noise pollution in Dhaka city [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Around 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7% of the population in Bangladesh have lost their hearing due to noise pollution, says the Development of Environment (DoE) study, which was conducted in 2017 [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The major sources of noise pollution in urban areas are traffic and loud horns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The DoE found that in Dhaka, 500-1,000 vehicles honk at the same time when stuck in traffic[53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Around 5% of the world population is facing several kinds of health hazards due to complexities related to noise pollution, According to the WHO [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 15 There is a scarcity of natural gas and petroleum in Bangladesh also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Gas supplies meet 56% of domestic energy demand [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Bangladesh has a very limited energy reserve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' small amounts of oil, coal and countable natural gas reserves [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The country is a net importer of crude oil and petroleum products [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 Importance of Bicycle Lane in Mega City In more ways than one, driving a bicycle has a positive impact on the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' They are also less expensive than other forms of transportation and environment- friendly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Bicycles are considered zero-emission vehicles i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' they do not release any carbon emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Bicycles, as vehicles with zero emissions, do not contribute to air pollution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' People can have moderate fresh air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' They do not contribute to sound pollution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' When bicycles are used as a consistent form of travel by a large percentage of the population in a particular area especially in an urban area, there is a great relief on road traffic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Bicycles also have the effect of alleviating parking difficulties in urban areas, because they simply take up so much less space than cars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=" Low physical activity or Physical inactivity is recognized as one of the country's leading risk factors and the fourth leading cause of deaths due to non-communicable disease (NCDs) worldwide - cardiovascular diseases, chronic lung diseases, heart disease, stroke, diabetes and cancers - and each year contributes to over three million preventable deaths [59]." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' There may be some physical exercise every day by using bi- cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Bicycles also offer more freedom of movement without time constraints, crowded and unpleasant conditions and, if desired, the ability to travel alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' So, people can have eco-friendly Travel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Bicycles are lighter and usually cause less damage to the roads than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' This will reduce the number of injuries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' So the area would be environment-friendly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Most of the well-organized mega city criteria are met by using bicycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 Challenges to Increase Mobility in Dhaka City In the case of an unplanned or unorganized city, one of the big issues is that road conditions are not good enough and the cycling lanes & footways are hardly available 16 which is the major cause of worst traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' On the other hand, noise pollution and traffic congestion are troubling and there is a huge CO2 gas emission occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Between the well-organized city and unplanned city, there is a huge gap in road conditions, footways & cycling lanes, air quality, noise pollution, and traffic congestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' There may have some parameters to increase transport mobility in the city like the construction of separate roads, underground roads, railways, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' But the construction of those parameters is not a feasible solution because of huge budgets and spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' There are two alternative low-cost solutions exist, the first one is to make ready the footpaths for routing purpose and the next one is to introduce local lanes with bicycles as vehicles for moving around the city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' But the first one is not possible because most of the time, street vendors and hawkers snatch up footways and in some case footways not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The next solution is feasible and effective as bicycles have several environmental benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' For this purpose, we have to plan a network and always try to use local lanes for routing through one place to another place, if it is not feasible to use local roads for some cases, we will use main roads and will always try to minimize the use of main roads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' There exist some constraints to be handled to plan network that we cannot access all possible roads like VIP roads, heavy traffic roads, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' On the other hand, it is almost impossible to plant more trees to improve air quality and reduce CO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' And most of the noise pollution and traffic congestion is caused by motor vehicle use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='4 Bicycle Lane Design in an Unplanned Mega City Different approaches have been presented over the past decades to design networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' It is possible to split the solutions into two categories: exact solutions and heuristic solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Exact approaches can treat Network Design Problem in a rigorous way which is inefficient when dealing with real-world large-scale networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' And, an approximate yet efficient approach is provided by heuristic approaches, more popular than exact approaches, that have emerged in recent decades which can tackle large-scale real- world problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Without using an exact and heuristic approach, here we present the Physarum-inspired technique which takes into account the constraints to construct the bicycle lane network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Basically, we always try to use local lanes for routing through 17 one place to another place, if it is not feasible to use local roads for some cases, we will use main roads and will always try to minimize the use of main roads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Compared to previous studies it is noted that the network has only one direction between two nodes, so the stream is only flowing from one node to another, but is never flowing in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' However, most roads have the features of double-way traffic in real traffic networks as demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' There is a clear distinction between opposite directions, where flows do not interfere in two directions opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Apparently, in the traffic network shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2, the initial approach influenced by the Physarum cannot be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Here in the following, we discuss the modified Physarum inspired lane design technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Given a graph 𝐺 = (𝑁, 𝐸), where 𝑁 denotes a set of 𝑛 cities, 𝐸 represents a set of 𝑚 connections or linkages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' There is a protoplasmic flow in each link of this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The two terminals of the link represent two locations of the specified area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' One terminal is called the source node, and the other terminal is called the sink node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Protoplasmic flows from the source node into the network and from the sink node out of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' At each city there is pressure and the amount of flux in each edge is proportional to the difference in pressure between the two terminals of this edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Specifically, the flux 𝑄𝑖𝑗 in edge (i,j) is given by the modified Hagen-Poiseuille equation below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2: Real traffic network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 18 𝑄𝑖𝑗 = 𝐷𝑖𝑗 𝑐𝑖𝑗 (𝑃𝑖 + 𝑃𝑗) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1) 𝐷𝑖𝑗 = 𝜋𝑟𝑖𝑗 4 8𝜀 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2) In the above equation, 𝐷𝑖𝑗 is the conductivity of the linkage, 𝑐𝑖𝑗/𝐿𝑖𝑗 is the length of the edge, 𝑃𝑖 and 𝑃𝑗 are the pressure of the vertices 𝑖 and 𝑗, 𝑟𝑖𝑗 is the radius of the edge, 𝜀 (𝑒𝑝𝑠𝑖𝑙𝑜𝑛) is the coefficient of viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In the case of conductivity(𝐷𝑖𝑗), which is linkage specific, we are using a fixed conductivity value for all the linkages for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The length(𝑐𝑖𝑗/𝐿𝑖𝑗) is not the direct length from 𝑖 city to 𝑗 city rather we consider all possible path length with no use or hardly use of main roads and we calculate pressure of each city based on the amount of population in that city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' As we use initial fixed conductivity value, 𝑟𝑖𝑗 is considered to be the same for all connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2) indicates that the tubular thickness( 𝑟𝑖𝑗) of Physarum increases with the conductivity of the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Therefore, the conductivity update formula can explain the change in tubular thickness of Physarum as follows, 𝑑 𝑑𝑡 𝐷𝑖𝑗 = 𝑓(|𝑄𝑖𝑗|) − 𝜇𝐷𝑖𝑗 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3) here 𝑓(|𝑄𝑖𝑗|) is an increasing function, 𝜇 is a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In our case, we considered 𝑓(|𝑄𝑖𝑗|) = |𝑄𝑖𝑗| for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The equation of conductivity update suggests that conductivity tends to increase with large flux edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Consequently, the equation of conductivity update reflects the above physiological mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' We must first calculate the pressures to calculate the flux and update the edge conductivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The pressures can be determined using the Poisson equation network below by considering the flux conservation law at each vertex, ∑ 𝐷𝑖𝑗 𝑐𝑖𝑗 (𝑃𝑖 + 𝑃𝑗) = { −𝐼0, 𝑗 = 𝑠𝑜𝑢𝑟𝑐𝑒 +𝐼0, 𝑗 = 𝑠𝑖𝑛𝑘 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝑖 ∈𝑉(𝑗) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='4) here 𝑉(𝑗) is the set of vertices linked to vertex j, I0 is the amount of flux flowing into and out of the node of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Let the pressure at the sink node be 0, and give an initial value to each edge conductivity, then use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='4) to measure the other pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' After that, we can calculate the amount of flux in each edge using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1), and we can change the conductivity of each edge using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' According to an edge conductivity threshold value, edges with conductivity lower than this value are cut off from the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 19 Let’s consider a simple small network consist of only eleven points picked from Dhaka City.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Physarum always finds the optimal route network among the eleven nodes and is believed to have achieved a good balance between cost, efficiency, and resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Here are the illustrations in the following which are the generalized design using modified Physarum inspired technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' This technique can be applied any real-life traffic network design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In this work, the technique is applied both prominent area centered with Motijheel and entire Dhaka city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2, (A) there are 11 node points between Farmgate (node point 2) and the University of Dhaka (node point 10) existing network with both main and local roads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Here, (B) – (E) shows that the network growing process with time (t = iteration) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (B) Network at 10th iteration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (C) Network at 20th iteration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (D) Network at 50th iteration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (E) Final network for 11 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' For those cases, we always try to avoid main roads to grow up the networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' And the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 depicts the possible road network design if all main roads are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3: Sample Region with 11 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 日.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=" 日 Government FAR 2 GATE Science College CRd IKI'SAREA mited H BashundharaCity Insaf BarakahKi Rd ShoppingComnlex &GeneralHos apath 日 Panthapath Hatiriheel fosfa KAL 4 AGAN NEW ESKATON F 专T KATHAI AGAN OLD ESKAT RdNo." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='8 LinkRdD 35 P JAGH R Rd No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' H HAT 6 OOL MintoRd y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='College Ba SHAH GH oSquare NewElephantRd 日 Ramna KA 7 BON Bangladesh 914 NationalMuseum eDhaka University NewMarket ofDhaka 日 3 RA 10 NA EdenMohila College Udayan Higher Suprem SecondarySchool ofBang 20 (A) 11 node points (t = 0) (B) At t = 10 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='4: Physarum inspired network design of 11 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (A) 11 node points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The network is expanding with t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (B) At t = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 11 6 7 1011 4 6 7 10 21 (C) At t = 20 (D) At t = 50 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='4: Physarum inspired network design of 11 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (A) 11 node points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The network is expanding with t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (C) At t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (D) At t = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 6 7 104 L 7 22 (E) Final network Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='4: Physarum inspired network design of 11 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (E) Final network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 4 10 23 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5: Physarum inspired network design of 11 nodes using main roads (if available).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 4 6 24 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 Significance of Study Modified Physarum Polycephalum Inspired Network Design Technique is used to design any real-life traffic network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In this research, the bicycle lane network is planned using this strategy in a congested mega city Dhaka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' This addresses many complicated problems, including the problem of mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Modified Physarum Polycephalum Inspired Network Design Technique holds a significantly different form from the existing Physarum Polycephalum Inspired Network Design Technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Compared to previous studies it is noted that the network has only one direction between two nodes, so the stream is only flowing from one node to another, but is never flowing in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' However, most roads have the features of double-way traffic in real traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' There is a clear distinction between opposite directions, where flows do not interfere in two directions opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Apparently, in real life traffic network, the initial approach influenced by the Physarum cannot be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Modified Physarum Polycephalum Inspired Network Design Technique calculates flux and pressure using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='4) respectively as we know that the flow of traffic is not uni-directional rather bi-directional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' CHAPTER 4 Experimental Studies This chapter experimentally investigates the efficacy of the proposed modified Physarum inspired bicycle lane network design technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' For both a certain portion of Dhaka City and the entire Dhaka City, we are assuming a reduction in the number of buses, cars, taxicabs, and motorcycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Time saving, fuel saving, user cost saving, and CO2 emission reduction are calculated with some standard average measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In this chapter at first, we discuss the prominent points and description of this experiment, then the experimental setting which includes both parameter setting and machine description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In the third section of this chapter, we describe the experimental outcomes and explanation of achievements for both 10km ranged portion and entire Dhaka city are described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 Experimental Settings In the experiment, the number of node points was 29 for both case but 77 linkages/edges are considered for prominent area of Dhaka city and 89 linkages/edges for entire Dhaka city;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' the length of edges(𝑐𝑖𝑗/𝐿𝑖𝑗) which is not linear and tubular thickness( 𝑟𝑖𝑗) are estimated using the google map;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' the value of meu(𝜇) was varied from .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='8 to 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' for certain cases we simply ignore the tubular thickness( 𝑟𝑖𝑗) variations and considered a fixed conductivity value(𝐷𝑖𝑗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' it is assumed that, the value of pressure(𝑃𝑖) at each node point is proportional to its population in that area, and a random initial threshold is applied which is increased with the iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The modified Physarum Inspired Bicycle Lane Design was implemented on Visual C++ of Visual Studio 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The experiments have been done on a PC (Intel Core i3-5005U CPU @ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='00 GHz CPU, 2GB NVIDIA GeForce 940M, 4GB RAM) with Windows 10 OS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' According to Bangladesh Road Transport Authority (BRTA), on February 04, 2020, there are 127398 registered buses (including microbus, minibus), 293268 registered cars, are 724800 registered motorcycles, and 36600 registered taxicabs currently 25 available in Dhaka city [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' There may have some unregistered cars & buses and a lot of unregistered motorcycles & taxicabs in the city, we are not considering them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' It is assumed that 40 passengers per bus, 1 person per car, taxicab and motorcycle on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' And let’s assume that buses take 20km ride, cars use 10km ride, taxicabs use 100km per day and motorcycles ply 15km ride.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' All those rides are supposed to take place per day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In this work, both prominent area centered with Motijheel and entire Dhaka city are considered to apply the modified Physarum inspired technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In this section, the achievements or environmental effects of the prominent area and the entire city of Dhaka with bicycles as vehicle are respectively addressed with the proposed network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 Bicycle Lane Network Design in a Prominent Area At first, a 10km selected prominent area of Dhaka city centered with Motijheel is considered to construct the network using Physarum inspired technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In this area, we have chosen 29 vital points and numbering those from 1 to 29 arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Here, we are considering a 10 km range because the average speed of cycling is 20kmph, so 10 km a day can be traveled easily in 30 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' And it can also lead to better mental health and energy by bicycling 30 minutes a day [6], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 illustrates the selected portion of Dhaka city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Here, (A) depicts the portion of Dhaka city marked within the entire Dhaka city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (B) Shows the 10km range centered with Motijheel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' For 10km ranged area, it is assumed that overall 2% of users switch from car to bicycle and 10% of users switch from motorcycle to bicycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' And a 5% bus and 10% taxicab are being reduced because of using bicycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Then the atmosphere would change significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' This section first explains the designed networks and then calculate the time saving and then fuel and cost saving is estimated and finally CO2 emission reduction is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The locations of prominent area of Dhaka city including traffic pressure is presented in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=" In this case, we assign some random values as node point's traffic pressure which is proportional to its population." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' For examples, Taltola, Donia, etc are less and on the other hand Motijheel, Tejgaon, etc are high traffic traffic-pressured area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 26 (A) Bangladesh (B) Dhaka (C) Prominent area (D) Node points in Prominent area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 1 Motijheel 2 Mohakhali 3 Gulshan 4 Shahinbag 5 Tejgaon 6 Badda 7 EWU 8 Bashundhara 9 Mogbazar 10 Mirbag 11 Dhanmondi 12 Shahbag 13 DU 14 Kamlapur 15 Ramna 16 Kotwali 17 Lalbagh 18 Khilgaon 19 Taltola 20 Aftabnagar 21 Sadarghat 22 Matuail 23 Wari 24 Golapbag 25 Donia 26 Rajarbagh 27 Sobujbagh 28 Green Model Town 29 Nandipara Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1: Selected Dhaka city map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (A) Bangladesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (B) Dhaka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (C) Prominent area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (D) Node points in Prominent area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Uzbekistan Kyrgyzstan Beijing 北京 Turkmenistan Tajikistan China Yello Afghanistan Iran Shanghai 上海 New.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='Delhi Pakistan Ea Nepal ianGulf Bhutan Taipei 台北 UnitedArab Bangladesh Emirates Taiwan India My imar HongKong Oman Mumbai rma) 香港 Laos Thailand South Luzon Bengaluru Vietnam China Sea 23orfedodo Bayof Bengal Bangkok CUMLHIMUICEU Arabian Sea Cambodia Philipp AndamanSea .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='HoChi Panay Gulf of MinhCity Thailand Palawan Negro SriLanka Minda Laccadive Sea BasilanIsla Malaysia Kuala Lumpur Celebes SHO1 H07 N5 Batshar ASSAM Kishanganj Guwahati Nagaon Purnia AHIDim MEGHALAYA oShillong Sahibganj N5 N2 Pakur Silchar Sylhet nka Raishahi N502 ICGTG Hailakandi MAI T N2 Bangladesh N6 gapur N704 TRIPURAV Aizawl Dhaka N7 PT MIZORAM AH Jessore BENGAL Madaripur: Satkhira Kolkata Gk N1 gpur Sundarban Chittagong Forest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=" Bangladesh Chandanaish eqey yangarh T5( H 135153 N1 Digha Cox'sBazar Whaikhyang Mrauk UN302 UTTARA 9114 Hazrat N511 Shahjalal N105 N3 International N501 Airport N301 MIRPUR BASUNDHARA Dhaka Zoo RESIDENTIAL N3 AREA Dhaka Jalshiri." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='Abason ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='raid ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='GULSHAN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='N5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='Gabtoli ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='Z5069 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5114 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='MOHAKHALI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='BADDA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='R202 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='MOHAMMADPUR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='TEJGAON ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='KHILGAON ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='ROTST ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='Boshila ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='DHANMONDI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='ty ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='RAMNA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='MOTIJHEEL ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='R820 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='RanaCNG& ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='GASStation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 Network Design Here in the planned network, the distance is not the linear distance between two node points rather distance is calculated using google map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' And the time is calculated considering standard speed 20kmph for bicycle in minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2, (A) there are 29 node points around 10km range centered with Motijheel (node point 1) existing network with both main and local roads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Figure (B) – (E) shows that the network growing process with time (t = iteration) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (B) Network in 10th iteration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (C) Network 20th iteration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (D) Network 50th iteration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (E) Finale network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' For those cases, we always try to avoid main roads to grow up the networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 Network is planned with main roads (if available).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1: The locations of prominent area of Dhaka city including traffic pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Sl Location Traffic Pressure 01 Motijheel 9 02 Mohakhali 8 03 Gulshan 5 04 Shahinbag 8 05 Tejgaon 9 06 Badda 4 07 EWU 5 08 Bashundhara 5 09 Mogbazar 5 10 Mirbag 5 11 Dhanmondi 7 12 Shahbag 9 13 DU 9 14 Kamlapur 8 15 Ramna 5 16 Kotwali 3 17 Lalbagh 4 18 Khilgaon 6 19 Taltola 3 20 Aftabnagar 4 21 Sadarghat 5 22 Matuail 8 23 Wari 7 24 Golapbag 5 25 Donia 3 26 Rajarbagh 5 27 Sobujbagh 5 28 Green Model Town 4 29 Nandipara 6 28 (A) 10km range existing road network (B) When t=10 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2: Network design using modified Physarum inspired technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (A) 10km range existing road network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (B) When t=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 12 26 16 2127 16 29 (C) When t=20 (D) When t=50 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2: Network design using modified Physarum inspired technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (C) When t=20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' (D) When t=50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 27 16 219 12 16 30 (E) Final Network Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2: Final network design using modified Physarum inspired technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 27 26 31 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3: Network design using modified Physarum inspired technique using main roads (if available).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 27 2 16 32 Here the Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 shows routes of all node points from starting node point 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' For example, the path 1-9-10-2-3 means that we have to cross node points 9, 10 and 2 in order to go into destination node point 3 from starting node point 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The minimum distance is considered for accessing any node here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' For example, the destination node point 3 can be accessed using the routes 1-9-10-2-3, 1-20-7-6-2-3, 1-12-5-4-3 and so on but the minimum distanced one is considered for calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 Effectiveness Analysis Driving a paddled-bicycle has a more than one beneficial environmental effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' For many cases bicycles take less time to ride, no fuel usage, saving user money, CO 2 emission reduction in unplanned mega city and do not contribute to air pollution, sound pollution, great relief on road traffic conditions, alleviating parking difficulties in urban areas, cause less damage to the roads, get relief of some non-communicable disease Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2: Routes from node point 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Des.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Node Point Routes Distance (km) Estimated Travel Time (min) 2 1-9-10-2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='55 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='65 3 1-9-10-2-3 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='25 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='75 4 1-9-4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='9 5 1-5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 6 1-9-10-7-6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='6 7 1-9-10-7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='9 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7 8 1-8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='85 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='55 9 1-9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='9 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7 10 1-9-10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='4 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 11 1-12-13-11 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='6 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='8 12 1-12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='9 13 1-12-13 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='4 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 14 1-14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 15 1-15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 16 1-23-16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 17 1-23-16-17 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 18 1-20-18 9 27 19 1-19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 20 1-20 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='6 21 1-23-21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 22 1-23-21-25-22 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='9 23 1-23 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='4 24 1-14-24 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='15 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='45 25 1-23-21-25 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 26 1-14-26 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 27 1-14-26-27 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='9 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7 28 1-14-28 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='9 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7 29 1-14-26-29 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 33 (NCDs) for have some physical exercise every day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The considering factors of this paper are only time saving, fuel and user money saving and CO2 emission reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 Time Saving As we mentioned that, according to a World Bank report, the average traffic speed in Dhaka has dropped from 21 kilometers per hour (kmph) to 7 kilometers per hour in the last 10 years, and by 2035 the speed could drop to 4 kilometers per hour, which is slower than the walking speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In our constructed network, we mainly try to avoid main roads or minimum usage of main roads if requires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' So, here traffic jam is hardly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' For convenience, we assume no traffic jam exists in the following calculation of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3, the time needed for both cars and buses are listed in three different times (at 6:00 AM, 10:00 AM, and 4:00 PM) calculated from google map and the time needed for a bicycle is always constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Here, we measure the distance and time of all the node point from center node point 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The time required for a car, taxicab and motorcycle are almost the same that’s why we don’t mention it differently in the Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Here we notice that at the morning 6:00 AM the travel time is less because of minimal traffic in the roads, at 10:00 AM (peak hour) when traffic jam occurs severe the need time to travel is huge and at 4:00 PM there exist traffic jam also but sometimes a bit less than peak hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' This condition is true for all types of cars, buses, taxicabs, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' For the prominent area, the gross working hours saving are described in the following Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' So, around 152216 working hours per day can be saved using bicycle in the prominent area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 Fuel and Cost Saving The cost of installation or repair is not listed here rather we considering only running cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Because whenever we switch from car to bicycle there would be a significant reduction in costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Here the mileage of car and taxicab is assumed at 20kmpl and 25kmpl respectively with diesel and 65tk per liter diesel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' On the other hand, cycling has no cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The bus mileage considers 5kmpl with diesel and motorcycle has 50kmpl with petrol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 34 In Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='4, the needed fuel for cars, taxicabs, motorcycles, and buses is calculated and there is no running cost & fuel cost for the bicycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In case of bus, the per km fare is fixed by BRTA of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7tk and we assumed that taxicab fare is 50tk per km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Here, the distance of all the node points are measured from center node point 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3: Time comparison in car, bus and bicycle considering from node point 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Node Point Dista nce (km) Car & Taxicab & Motor cycle Bus Bicycle 6:00am (min) 10:00am (min) 4:00pm (min) 6:00am (min) 10:00am (min) 4:00pm (min) Dista nce (km) Time (min) 2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='6 18 39 36 20 43 40 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='55 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='65 3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='9 19 42 39 21 46 43 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='25 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='75 4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 14 33 30 16 36 33 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='9 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 12 27 26 14 30 29 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='6 15 36 36 17 39 39 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='6 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='6 15 36 36 17 39 39 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='9 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='8 11 24 24 13 27 27 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='85 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='55 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='6 9 18 15 11 20 17 2.' metadata={'source': 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30 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='15 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='45 25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7 13 29 27 15 32 30 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 26 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7 11 29 29 13 32 32 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 27 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 8 15 12 10 17 14 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='9 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7 28 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='8 12 42 39 14 45 42 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='9 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7 29 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='4 12 33 33 14 36 36 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='4: Time saving per day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Transit % of Transit Reduction Transit Reduces Riding Distance (km) Time saves (min) Bus 5% 6370 127400 709800 Car 2% 5865 58650 326764 Taxicab 10% 3660 366000 2039143 Motor cycle 10% 72480 1087200 6057257 Total time saving 9132964 35 The gross fuel saving and user cost saving for motijheel area are calculated for buses, cars, taxicabs, and motorcycles and described in the Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' So, around 64797 liters fuel and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='6 million user money costs can be saved per day using bicycle in the motijheel area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5: Cost comparison in car, bus and bicycle considering from node point 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Node Point Dista nce (km) Car Taxicabs Motor cycle Bus Bicycle Fuel (litre) User Cost (tk) Fuel (litre) User Cost (tk) Fuel (litre) User Cost (tk) Fuel (litre) User Cost (tk) Dista nce (km) Cost (tk) 2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='43 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='34 430 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='17 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='31 As we assume bicycles reduces 5% of the total bus in Dhaka city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The fuel consump tion is reduced = 6370×20 ×1/5 litres = 25480 litres 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='62 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='55 N/A 3 8.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='27 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='22 270 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='11 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='61 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='18 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='6: Cost saving per day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Transit % of Transit Reduction Transit Reduces Riding Distance (km) Fuel (litres) User Cost (tk) Bus 5% 6370 127400 25480 216580 Car 2% 5865 58650 2933 190613 Taxicab 10% 3660 366000 14640 18300000 Motor cycle 10% 72480 1087200 21744 1935216 Total cost saving 64797 20642409 36 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 CO2 Emission Reduction In the calculation, 887 g/km, 258 g/km, 237 g/km and 40 g/km are the considered amount of CO2 emission in 1km ride of bus, car, taxicab and motorcycle respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' For motijheel, the gross CO2 emission reduction is described in the following Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In total, around 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 × 105 kg CO2 emission is reduced in the motijheel area per day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 Bicycle Lane Network Design for Entire Dhaka City Here, 29 important locations of entire Dhaka city are selected and numbering those from 1 to 29 randomly, which are depicted in following Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' As it is considered that the paddled-bicycle for routing 10km ranged area, it is not feasible to move through all over the Dhaka city using paddled-bicycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' But in case of electric bicycle, it is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' That’s why electric bicycles are considered as vehicles for entire Dhaka city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The general data of Dhaka city including location, population and traffic pressure are described in the Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=" In this case, the consideration is that the node point's traffic pressure is proportional to its population." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' For entire Dhaka city, it is assumed that 5% of users switch from car to electrical motorcycle or bicycle and 50% of users switch from motorcycle to electrical motorcycle or bicycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' And a 10% bus and 20% taxicab are being reduced because of using electric motorcycles or bicycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Then there will be some noteworthy change in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 Network Design The constructed network of selected 29 points of Dhaka city is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In the Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='9, routes are shown to all node points from starting node point 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' For Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7: CO2 emission reduction per day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Transit % of Transit Reduction Transit Reduces Riding Distance (km) CO2 Emission (g) Bus 5% 6370 127400 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='13 × 108 Car 2% 5865 58650 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='51 × 107 Taxicab 10% 3660 366000 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='67 × 107 Motor cycle 10% 72480 1087200 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='35 × 107 Total CO2 saving 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='58 × 108 37 01 Uttara 02 Mirpur 03 Uttara ABM City 04 Basundhara R/A 05 Khilkhet 06 Cantonment 07 Gulshan 08 Badda 09 Mohakhali 10 Tejgaon 11 Motijheel 12 Khilgoan 13 Gabtoli 14 Mohammadpur 15 Dhanmondi 16 Shahbagh 17 Matuail 18 Kotwali 19 New Market 20 Baridhara 21 Banani 22 Monipur 23 Sher E Bangla Nagar 24 Airport 25 Poradia 26 Madarbari 27 Beraid 28 NutanPara 29 Pagla Rail station Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3: Entire Dhaka city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Asnua N302 N501 R303 N511 N302 N501 Dhaka Zoo JalshiriAb 23 R20 OL Boshila 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 28 Chanpa LalbaghFort b Hizla R110 Keraniganj R820 N8 820 N1 Shiddhirganj 5069 RB10 Bashundhara R111 pur Kada Riverview Baghair Ekuria N8 folafe 29 eshviariRN Kalakandi Gode Zazira 38 example, the route 7-6-24-5-1 means that we have to pass node points 6, 26 and 5 in order to arrive destination node point 1 from starting node point 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' We are considering the minimum distance for accessing any node here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' For example, the destination node point 1 can be accessed using the routes 7-6-24-5-1, 7-6-4-26-5-1, 7-21-22-2-24-5-1 and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' But the minimum distanced route is considered for calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 Effectiveness Analysis Using an electric-bicycle has a more than one beneficial environmental effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' For most of the cases bicycles take less time to ride, no fuel usage, saving user money, CO2 emission reduction in unplanned mega city and very less contribution to air pollution, sound pollution, great relief on road traffic conditions, alleviating parking difficulties in urban areas, cause less damage to the roads, get relief of some non-communicable disease (NCDs) for have some physical exercise every day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The considering factors of Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='8: The general data on locations in Dhaka city, including population, area, and traffic pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Sl Location Population Area (km²) Traffic Pressure 01 Uttara 345,097 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='91 10 02 Mirpur 274,530 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='71 8 03 Uttara ABM City 145,097 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='91 5 04 Bashundhara R/A 274,200 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='54 8 05 Khilkhet 130,053 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='88 5 06 Cantonment 117,464 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='47 4 07 Gulshan 145,969 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='85 5 08 Badda 157,924 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='78 5 09 Mohakhali 145,969 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='85 5 10 Tejgaon 148,255 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='46 5 11 Motijheel 225,999 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='69 7 12 Khilgaon 327,717 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='8 9 13 Gabtoli 198,723 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='98 6 14 Mohammadpur 355,843 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='65 10 15 Dhanmondi 147,643 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='86 5 16 Shahbag 74,113 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='49 3 17 Matuail 125,312 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='36 4 18 Kotwali 210,504 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='67 6 19 New Market 66,439 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='67 2 20 Baridhara 105,969 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='45 4 21 Banani 145,969 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='85 5 22 Monipur 274,530 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='71 8 23 Sher-E-Bangla Nagar 248,871 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='25 7 24 Airport 130,053 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='88 5 25 Poradia 52,014 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='09 2 26 Madarbari 93,153 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='65 3 27 Beraid 157,924 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='78 5 28 NutanPara 125,312 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='36 4 29 Pagla Rail station 194,019 246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='21 6 39 this paper is only time saving, fuel and user money saving and CO2 emission reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 Time Saving In the following measurements, 7 kilometers per hour (kmph) for the buses, cars, taxicabs & motorcycles which is the average speed of traffic in Dhaka city and 40 km/h for electric bicycles using our constructed network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' We are taken into consideration our constructed network as jam-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In the Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='10, the time necessary for cars and buses are listed in three different times (at 6:00 AM, 10:00 AM, and 4:00 PM) calculated from google map and the time needed for a bicycle is always constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The required time for taxicabs and motorcycles is considered to be the same as the time needed for a car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Here, we measure the distance and time of all the node point from center node point 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='9: Routes from node point 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Des.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Node Point Routes Distance (km) Estimated Travel Time(min) 1 7-6-24-5-1 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='65 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='48 2 7-21-22-2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='89 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='84 3 7-6-24-5-26-3 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='51 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='27 4 7-6-4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='02 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='53 5 7-6-24-5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='9 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='85 6 7-6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='95 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='93 8 7-20-8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='76 9 7-9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='26 10 7-9-10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='98 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='97 11 7-9-10-16-11 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='61 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='92 12 7-20-12 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='04 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='56 13 7-21-13 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='44 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='16 14 7-23-14 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='31 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='97 15 7-23-14-15 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='55 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='33 16 7-9-10-16 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='05 17 7-9-28-17 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='61 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='42 18 1-20-18 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='78 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='17 19 7-9-10-19 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='87 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='81 20 7-20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='18 21 7-21 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='76 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='64 22 7-21-22 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='34 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='01 23 7-23 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='07 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='11 24 7-6-24 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='55 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='83 25 7-6-4-25 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='72 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='08 26 7-6-24-5-26 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='25 27 7-20-27 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='8 28 7-9-28 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='41 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='12 29 7-9-28-17-29 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='02 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='03 40 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='4: Constructed network 29 points of Dhaka city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 25 41 In the Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='10, we have noticed that at the morning 6:00 AM the travel time is less because of minimal traffic in the roads, at 10:00 AM (peak hour) when traffic jam occurs severe amount the need time to travel is huge and at 4:00 PM there exist traffic jam also but sometimes a bit less than peak hour or sometimes a bit high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' This condition is true for all types of cars, buses, taxicabs, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' For the entire Dhaka city, the gross working hours saving are described in the following Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' So, around .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 million working hours per day using bicycle in the motijheel area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='10: Time comparison in car, bus and bicycle considering from node point 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Node Point Dista nce (km) Car & Taxicab & Motor cycle Bus Bicycle 6:00am (min) 10:00am (min) 4:00pm (min) 6:00am (min) 10:00am (min) 4:00pm (min) Dista nce (km) Time (min) 1 14 24 42 45 26 50 51 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='65 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='48 2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 16 33 36 18 43 42 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='89 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='84 3 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 40 75 75 42 82 81 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='51 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='27 4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 22 42 36 22 42 42 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='02 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='53 5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7 12 24 24 14 31 30 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='9 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='85 6 6 10 26 24 12 32 30 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='95 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='93 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7 8 30 27 9 35 32 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='76 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 7 24 22 9 28 24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='26 10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 10 25 24 12 31 28 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='98 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='97 11 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 18 45 42 20 52 48 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='61 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='92 12 12 40 60 57 42 68 65 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='04 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='56 13 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7 22 44 45 24 51 51 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='44 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='16 14 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 14 36 33 16 44 39 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='31 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='97 15 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7 16 33 36 17 41 42 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='55 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='33 16 8 16 39 39 18 47 45 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='05 17 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='6 30 63 60 32 71 66 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='61 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='42 18 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='4 24 62 60 26 60 66 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='78 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='17 19 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7 18 42 42 20 50 48 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='87 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='81 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 5 12 11 6 15 14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='18 21 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 5 17 15 7 17 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='76 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='64 22 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='8 20 36 33 22 44 39 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='34 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='01 23 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='6 12 24 23 14 32 29 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='07 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='11 24 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7 12 24 24 14 32 30 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='55 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='83 25 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='9 30 68 71 32 75 77 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='72 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='08 26 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7 30 63 60 32 71 66 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='25 27 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 14 36 33 16 45 39 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='8 28 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 24 63 60 26 71 66 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='41 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='12 29 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 35 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 75 37 87 81 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='02 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='03 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='11: Time saving per day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Transit % of Transit Reduction Transit Reduces Riding Distance (km) Time Saves (min) Bus 10% 12740 254796 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 × 107 Car 5% 14663 146630 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='04 × 106 Taxicab 20% 7320 732000 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 × 106 Motor cycle 50% 362400 5436000 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='8 × 107 Total time saving 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='16 × 108 42 Time Saving Explanation: Total Bus ride reduce: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 1 × 127398 × 20 km = 254796 km Time saving for Bus: 254796 × 40 × ( 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='57 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5) mins = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 × 107 mins = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 × 106 hours = 50000 day Total Car ride reduce: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='05 × 293268 × 10 km = 146634 km Time saving for Car: 146634 × (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='57 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5) mins = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='04 × 106 mins = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='73 × 104 hours = 720 days Total Taxicab ride reduce: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 × 36600 × 100 km = 732000 km Time saving for Taxicab: 732000 × (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='57 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5) mins = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 × 106 mins = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='66 × 104 hours = 3611 days Total Motorcycle ride reduce: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 × 724800 × 15 km = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='4 × 106 km Time saving for Motorcycle: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='4 × 106 × (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='57 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5) mins = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='8 × 107 mins = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 × 105 hours = 26289 days Total working time saving: 50000 + 720 + 3611 + 26289 days = 80620 days 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 Fuel and Cost Saving The cost of installation or repair is not listed here rather we considering only running cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Because whenever users switch from car to bicycle there would be a significant reduction in costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Here it is assumed that the mileage is 5km per liter diesel for bus and 20km per liter diesel for both car and taxicab of 65tk per liter and the mileage is estimated to be 50km per liter for motorcycle of 89tk per liter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' On the other hand, the electric cycle charges of 10tk with 50km of travel per charge charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The fuel consumption is calculated per day basis and user saving is grand saving of considering all users of the Dhaka city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In the following Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='12, the needed fuel for cars, taxicabs, motorcycles, and buses is calculated and there is no running cost & fuel cost for the bicycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In case of bus, the per km fare is fixed by BRTA of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7tk and we assumed that taxicab fare is 50tk per km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Here the distance of all the node point is measured from center node point 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 43 The gross fuel saving and user cost saving of the entire Dhaka city is described in the following Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' So, it is saved around 195570 liters fuel and 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='72 million user costs per day for using bicycle in the entire Dhaka area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Fuel and Cost Saving Explanation: Fuel saving for Bus: 254796 km × 1/5 litres = 50959 litres Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='13: Cost saving per day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Transit % of Transit Reduction Transit Reduces Riding Distance (km) Fuel (litres) User Cost (tk) Bus 10% 12740 254796 50959 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='27 million Car 5% 14663 146630 7331 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 lakh Taxicab 20% 7320 732000 29280 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 million Motor cycle 50% 362400 5436000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='08 × 105 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 million Total cost saving 195570 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='72 million Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='12: Cost comparison in car, bus and bicycle considering from node point 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Node Point Dist ance (km) Car Taxicabs Motor Cycle Bus Electric Bicycle Fuel (litre) User Cost (tk) Fuel (litre) User Cost (tk) Fuel (litre) User Cost (tk) Fuel (litre) User Cost (tk) Dista nce (km) User Cost (tk) 1 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='56 700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='28 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='49 As we assume bicycles reduces 10% of the total bus in Dhaka city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The fuel consumpti on is reduced = 12740×20 ×1/5 litres = 50959 litres 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='8 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='65 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='13 2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='38 24.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='41 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='88 29 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='96 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='76 955 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='38 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='23 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='47 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='02 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='4 44 User money saving: 254796 km × ( 65 5 − 40 × 10 50) tk = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='27 million tk Fuel saving for Car: 146634 km × 1 20 litres = 7331 litres User money saving: 146634 km × ( 65 20 − 10 50) tk = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 × 105 tk = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 lakh tk Fuel saving for Taxicab: 732000 km × 1 25 litres = 29280 litres User money saving: 732000 km × (50 − 10 50) tk = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='65 × 107 tk = 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 million tk Fuel saving for Motor cycle: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='4 × 106 km × 1 50 litres = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 × 105 litres User money saving 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='4 × 106 × ( 89 50 − 10 50) tk = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 × 106 tk = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 million tk Total fuel saving: 50959 + 7331 + 29280 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 × 105 litres = 197570 litres Total User money saving: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='27 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='45 + 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 + 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 million tk = 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='72 million tk 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 CO2 Emission Reduction In the following calculation, 887 g/km, 258 g/km, 237 g/km and 40 g/km are the considered amount of CO2 emission in 1km ride of bus, car, taxicab, and motorcycle respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Then CO2 emission is reduced in a significant amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' For the entire Dhaka city, the gross CO2 emission reduction is described in the Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In total, around 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='58 × 105 kg CO2 emission is reduced in the entire Dhaka city per day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' CO2 Emission Reduction Explanation: CO2 emission reduction for Bus: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 × 127398 × 20 km × 887 gm = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 × 105 kg CO2 emission reduction for Car: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='05 × 293268 × 10 km × 258 gm = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='8 × 104 kg CO2 emission reduction for Taxicab: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 × 36600 × 100 km × 237 gm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7 × 105 kg CO2 emission reduction for Motorcycle: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='5 ×724800 ×15 km × 40 gm = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 × 105 kg Total CO2 emission reduction: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='58 × 105 kg Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='14: CO2 emission reduction per day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Transit % of Transit Reduction Transit Reduces Riding Distance (km) CO2 Emission (g) Bus 10% 12740 254796 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='3 × 108 Car 5% 14663 146630 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='8 × 107 Taxicab 20% 7320 732000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='7 × 108 Motor cycle 50% 362400 5436000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 × 108 Total CO2 emission reduction 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='58 × 108 45 When CNG is being used as fuel in buses, cars, and taxicabs the CO2 emission is more significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Buying a car and motorcycle also increases traffic jams, air carbon dioxide, pollution of the environment and costly in the current situation in Dhaka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Alternatively, electric bicycle does not make an enormous traffic jam, CO2 in air, and less expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In order to do this, governments should build parking lots in various places where necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' CHAPTER 5 Conclusions A modified Physarum-inspired model is presented in this paper to address the design of the bicycle lane network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Different approaches, like exact approaches and heuristic approaches, have been presented over the past decades to design transportation networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Recently bio-inspired method had drawn great attraction to network design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' In real two-way traffic networks, the modified technique is more effective and efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' This chapter will now give a short summary of the main points described in this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Also, it discusses possible future works based on the outcome of the present work 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='1 Achievements The network design technology inspired by Physarum is believed to have balanced costs, effectiveness, and resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=" Inside Dhaka city, an unorganized and unplanned city, we have developed an electric bicycle network system where there's little footway." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' To meet this challenge, we primarily use local roads and try to avoid major roads towards the construction of the electric bicycle network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Since bicycles are non- motorized vehicles do not produce greenhouse gases so they do not cause air pollution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=" They also don't contribute to noise pollution." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' If a large number of people use bicycle in the city, traffic jams will be eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' The costs will be reduced and people can have some physical activity also, which is beneficial to health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Since bicycles do not need to use gasoline, the importation of gasoline will be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' That also enriches the economy and the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content='2 Future Study In the future, parallel computing and the optimal model for the design of the transport network are part of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Furthermore, our research includes the implementation of the Physarum polycephalum inspired model for the dynamic traffic network and the elastic demand traffic network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 47 References [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' Merriman, “Mobility,” in International Encyclopedia of Human Geography, Elsevier, 2009, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' 134–143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFRT4oBgHgl3EQfojeI/content/2301.13609v1.pdf'} +page_content=' [2] H.' 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+LEVEL-2 LARGE DEVIATION PRINCIPLE FOR +COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES +HIROKI TAKAHASI +Abstract. We consider level-2 large deviations for the one-sided countable full +shift without assuming the existence of Bowen’s Gibbs state. To deal with non- +compact closed sets, we provide a sufficient condition in terms of inducing which +ensures the exponential tightness of a sequence of Borel probability measures +constructed from periodic configurations. Under this condition we establish the +level-2 Large Deviation Principle. We apply our results to the continued fraction +expansion of real numbers in [0, 1) generated by the R´enyi map, and obtain the +level-2 Large Deviation Principle, as well as a weighted equidistribution of a set +of quadratic irrationals to equilibrium states of the R´enyi map. +1. Introduction +Dynamical systems (iterated maps) equipped with finite Markov partitions are +represented as finite Markov shifts, and the construction of relevant invariant mea- +sures and the investigation of their statistical properties are done on the symbolic +level, with adaptations of ideas in statistical mechanics (see e.g., [4, 5, 29, 32, 39]). +This thermodynamic formalism initiated in the 60s has been successfully extended +to maps with infinite Markov partitions and shift spaces with countably infinite +number of states (see e.g., [1, 6, 10, 11, 18, 30, 31, 41]). This paper is concerned +with level-2 large deviations for such countable Markov shifts, and its application +to a dynamical system related to number theory. +The theory of large deviations aims to characterize limit behaviors of probability +measures in terms of rate functions. Let X be a topological space, and let M(X ) +denote the space of Borel probability measures on X endowed with the weak* +topology. We say a sequence {˜µn}∞ +n=1 in M(X ) satisfies the Large Deviation Prin- +ciple (LDP) if there exists a lower semicontinuous function I : X → [0, ∞] such +that +(1.1) +lim inf +n→∞ +1 +n log ˜µn(G) ≥ − inf +G I for any open set G ⊂ X , +and +(1.2) +lim sup +n→∞ +1 +n log ˜µn(C) ≤ − inf +C I for any closed set C ⊂ X . +We call x ∈ X a minimizer if I(x) = 0 holds. The set of minimizers is a closed +set. The LDP means that in the limit n → ∞ the measure ˜µn assigns all but +Date: January 10, 2023. +2020 Mathematics Subject Classification. 37A44, 37A50, 37A60, 60F10. +Keywords: thermodynamic formalism; Gibbs state; Large Deviation Principle; periodic points; +equidistribution. +1 + +2 +HIROKI TAKAHASI +exponentially small mass to the set of minimizers. +The function I is called a +rate function, and called a good rate function if its level set {x ∈ X : I(x) ≤ +α} is compact for any α > 0. If X is a metric space and {˜µn}∞ +n=1 satisfies the +LDP, the rate function is unique. The setup in our mind is that X is the space +of Borel probability measures on a topological space X on which a Borel map +σ: X → X acts, and each ˜µn ∈ M(X ) is given in terms of empirical measures +δn +x = (1/n) �n−1 +k=0 δσkx, where δσkx ∈ X denotes the unit point mass at σkx ∈ X. +We refer to the LDP in this setup as level-2 [8, Chapter 1]. +For topologically mixing finite Markov shifts together with H¨older continuous +potentials, the level-2 LDP for empirical distributions and that for sequences con- +structed from empirical measures on periodic orbits were established in [15, 22, 33] +and [16] respectively. A key ingredient in these classical cases is the existence of +Bowen’s Gibbs states [4]. With the aid of Bowen’s Gibbs states, one can deduce +the lower bound (1.1) by combining Birkhoff’s and Shannon-McMillan-Breiman’s +theorems, and the upper bound (1.2) by modifying the standard proof of the +variational principle [40] (see [33]). For countable Markov shifts, Bowen’s Gibbs +states were constructed under the assumption of a good regularity of potentials +and a strong connectivity of transition matrices defining the shift spaces (see e.g., +[1, 6, 11, 18, 30]). Several level-2 LDPs were established in [34] under the existence +of Bowen’s Gibbs states. +It has been realized that not all dynamically relevant invariant probability mea- +sures correspond to Bowen’s Gibbs states. One of the best known examples is +the absolutely continuous invariant probability measure of an interval map of +Manneville-Pomeau type, with finitely many branches and a neutral fixed point. +Such a measure still retains a weak form of Bowen’s Gibbs state [41], and has +the weak Gibbsian property in statistical mechanics sense [9, 17, 42]. For a ther- +modynamic formalism and level-2 large deviations for a class of this map, see +[12, 28, 41] and [25, 26] respectively. With these historical developments and the +abundance of interesting dynamical systems modeled by countable Markov shifts +without Bowen’s Gibbs states (see e.g., [13, 43]), it is important to establish the +level-2 LDP for countable Markov shifts without assuming the existence of Bowen’s +Gibbs states. +A main new difficulty for countable Markov shifts is a treatment of non-compact +closed sets. We say a sequence {˜µn}∞ +n=1 of Borel probability measures on a non- +compact space X is exponentially tight if for any L > 0 there exists a compact set +K ⊂ X such that +lim sup +n→∞ +1 +n log ˜µn(X \ K) ≤ −L. +If {˜µn}∞ +n=1 is exponentially tight, then the upper bound (1.2) for any compact +closed set implies (1.2) for any closed set which is not necessarily compact, see +e.g., [7] for details. The proof of the level-2 LDPs in [34] relies on the existence of +Bowen’s Gibbs states in order to verify the exponential tightness. +Our strategy for countable Markov shifts without Bowen’s Gibbs states is to use +inducing to verify the exponential tightness. Inducing is a familiar procedure in +ergodic theory originally considered in works by Kakutani, Rohlin and others, and + +LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES +3 +was used in the construction of absolutely continuous invariant measures or Gibbs- +equilibrium states (see e.g., [2, 3, 23, 24]). An inducing scheme we use here is given +by the first return map to an a priori fixed domain. In terms of this inducing, we +will formulate a sufficient condition which ensures the exponential tightness for +the original system. +A key concept is that of local Gibbs states introduced in +Section 2.2. +1.1. Statements of results. Throughout the rest of this paper, let N denote the +discrete set of positive integers. +Let X denote the one-sided infinite Cartesian +product topological space of N, called a countable full shift. The topology of X +has a base that consists of cylinders +[p1 · · · pn] = {x = (xn)∞ +n=1 ∈ X : xk = pk for every k ∈ {1, 2, . . . , n}}, +where n ≥ 1 and p1 · · ·pn ∈ Nn. +This topology is metrizable with the metric +d(x, y) = exp (− inf{n ≥ 1: xn ̸= yn}) where exp(−∞) = 0 by convention. Let σ +denote the left shift acting on X: (σx)n = xn+1 for n ≥ 1. +Let φ: X → R be a function, called a potential. We say φ is acceptable if it is +uniformly continuous and satisfies +sup +p∈N +� +sup +[p] +φ − inf +[p] φ +� +< ∞. +We say φ is locally H¨older continuous if there exist C > 0 and α ∈ (0, 1] such that +for any p ∈ N and all x, y ∈ [p], +|φ(x) − φ(y)| ≤ Cd(x, y)α. +Clearly, if φ is locally H¨older continuous then it is acceptable. For each n ≥ 1 we +write Snφ for the Birkhoff sum �n−1 +k=0 φ ◦ σk, and introduce a pressure +(1.3) +P(φ) = lim +n→∞ +1 +n log +� +p1···pn∈Nn +sup +[p1···pn] +exp Snφ. +This limit exists by the sub-additivity [4, Lemma 1.18], which is never −∞. +Let φ: X → R be acceptable and satisfy P(φ) < ∞. We consider a sequence +{˜µn}∞ +n=1 of Borel probability measures on M(X) given by +(1.4) +˜µn = +1 +Zn(φ) +� +x∈En +exp Snφ(x)δδn +x, +where +En = {x ∈ X : σnx = x}, +and δδn +x denotes the unit point mass at δn +x, and Zn(φ) the normalizing constant. +In dynamical systems terms, En is the set of periodic points of period n. +In +statistical mechanics terms, the measure ˜µn is closely related to the canonical +ensemble subject to a periodic boundary condition. +An inducing scheme consists of a subset X∗ of X of the form +(1.5) +X∗ = X \ +� +p∈N∩[1,p∗−1] +[p], + +4 +HIROKI TAKAHASI +where p∗ ≥ 2, and a function R: X∗ → N ∪ {∞} given by +(1.6) +R(x) = inf{n ≥ 1: σnx ∈ X∗}. +Given an inducing scheme (X∗, R) we define an induced map +(1.7) +τ : X∗ ∩ +∞ +� +k=1 +σ−kX∗ �→ σR(x)x ∈ X∗, +and an inducing domain +(1.8) +Σ = +∞ +� +n=0 +τ −n +� +X∗ ∩ +∞ +� +k=1 +σ−kX∗ +� +. +In other words, τ is the first return map to X∗ and Σ is the domain on which τ +can be iterated infinitely many times. We call (Σ, τ|Σ) an induced system. Given +a potential φ: X → R, we introduce a parametrized family of induced potentials +Φγ : Σ → R (γ ∈ R) by +(1.9) +Φγ(x) = SR(x)φ(x) − γR(x). +As shown in Section 2.1, the induced system has a countably infinite partition that +conjugates the system to the countable full shift. The local H¨older continuity of +the induced potential Φγ and its pressure P(Φγ) are well-defined in terms of this +conjugacy. Our main result is stated as follows. +Theorem A (the level-2 Large Deviation Principle). Let φ: X → R be acceptable +and satisfy P(φ) < ∞. +Assume there exists an induced system for which the +induced potentials Φγ, γ ∈ R are locally H¨older continuous, and there exists γ0 ∈ R +such that P(Φγ0) = 0. Then {˜µn}∞ +n=1 is exponentially tight and satisfies the LDP +with the good rate function. +Let us define the rate function in Theorem A. Let M(X, σ) denote the set of σ- +invariant elements of M(X) and let Mφ(X, σ) = {µ ∈ M(X, σ): +� +φdµ > −∞}. +Define Fφ : M(X) → [−∞, 0] by +Fφ(µ) = +� +h(µ) + +� +φdµ − P(φ) +if µ ∈ Mφ(X, σ); +−∞ +otherwise, +where h(µ) ∈ [0, ∞] denotes the measure-theoretic entropy of µ with respect to σ. +Since φ is acceptable and P(φ) < ∞, sup φ < ∞ is finite. For each µ ∈ Mφ(X, σ), +� +φdµ is finite [18, Theorem 2.1.9] and we have h(µ) + +� +φdµ ≤ P(φ) < ∞, and so +h(µ) < ∞. If φ is acceptable, then we have +P(φ) = sup +� +h(µ) + +� +φdµ: µ ∈ Mφ(X, σ) +� +, +known as the variational principle [18, Theorem 2.1.8]. A measure µ ∈ Mφ(X, σ) +which attains this supremum is called an equilibrium state for the potential φ. The +rate function Iφ : M(X) → [0, ∞] in Theorem A is given by +(1.10) +Iφ(µ) = − inf +G∋µ sup +G +Fφ, + +LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES +5 +where the infimum is taken over all open subsets G of M(X) containing µ. Since +the entropy is not upper semicontinuous on M(X, σ), Iφ may not be equal to −Fφ. +Since the sequence {˜µn}∞ +n=1 in Theorem A is exponentially tight, it is tight. By +Prohorov’s theorem, it has a limit point. Since the rate function Iφ in Theorem A +is the good rate function, there exists at least one minimizer. If the minimizer +is unique, we obtain a “level-2 weighted equidistribution of elements of �∞ +n=1 En +toward minimizers”. +Theorem B (level-2 weighted equidistribution). Let φ: X → R be acceptable and +satisfy P(φ) < ∞. Assume there exists an induced system for which the induced +potentials Φγ, γ ∈ R are locally H¨older continuous, and there exists γ0 ∈ R such +that P(Φγ0) = 0. Assume that the minimizer of the rate function Iφ is unique, +denoted by µmin. For any bounded continuous function ˜ϕ: M(X) → R, +lim +n→∞ +1 +Zn(φ) +� +x∈En +exp Snφ(x) ˜ϕ(δn +x) = ˜ϕ(µmin). +Under the assumption of Theorem A, minimizers are not always unique, and not +always an equilibrium state. A sufficient condition was given in [35] which ensures +that minimizers are equilibrium states. +Taking various continuous functions ˜ϕ in Theorem B, we obtain convergences +of various time averages over the elements of En. Let C(X) denote the set of +real-valued bounded continuous functions on X. +Corollary (Inspired by Olsen [21, Section 1.1]). Under the assumption of Theo- +rem B, assume moreover the minimizer is unique, denoted by µmin. +(a) For all ϕ, ψ ∈ C(X), +lim +n→∞ +1 +Zn(φ) +� +x∈En +exp Snφ(x) 1 +n2Snϕ(x)Snψ(x) = +� +ϕdµmin +� +ψdµmin. +(b) For ϕ, ψ ∈ C(X) with inf ψ > 0, +lim +n→∞ +1 +Zn(φ) +� +x∈En +exp Snφ(x)Snϕ(x) +Snψ(x) = +� +ϕdµmin +� +ψdµmin +. +(c) For π1, π2 ∈ C(X) and a bounded continuous function f : R → R, +lim +n→∞ +1 +Zn(φ) +� +x∈En +exp Snφ(x) 1 +n2 +n−1 +� +k1,k2=0 +f(π1(σk1x) + π2(σk2x)) += +� +fd(µmin ◦ π−1 +1 +⊗ µmin ◦ π−1 +2 ), +where ⊗ denotes the convolution. +Proof. Apply Theorem B to the bounded continuous functions µ ∈ M(X) �→ +� +ϕdµ +� +ψdµ, µ ∈ M(X) �→ +� +ϕdµ/ +� +ψdµ, µ ∈ M(X) �→ +� +fd(µ ◦ π−1 +1 +⊗ µ ◦ π−1 +2 ) +respectively. +□ + +6 +HIROKI TAKAHASI + 1 +0 +1/2 2/3 +1 +Figure 1. The graph of the R´enyi map T. +1.2. Applications. Our results can be applied to dynamical systems modeled by +the countable full shift without Bowen’s Gibbs state. The assumption in Theo- +rem A can be verified, for example, for the infinite Manneville-Pomeau map [13, +Section 2.2], and the two-dimensional conformal maps in [43, Section 5] related to +number theory. Minimizers of the associated rate functions are not unique, and so +Theorem B does not apply. Further applications of different taste will be given in +our forthcoming paper. +A prime example to which our results apply is the R´enyi map T : [0, 1) → [0, 1) +given by +(1.11) +T(ξ) = +1 +1 − ξ − +� +1 +1 − ξ +� +, +where ⌊·⌋ denotes the floor function. The graph of T is obtained by reversing the +graph of the well-known Gauss map ξ ∈ (0, 1] → 1/ξ − ⌊1/ξ⌋ ∈ [0, 1) around the +axis {ξ = 1/2}, as shown in FIGURE 1. The map T leaves invariant the absolutely +continuous infinite measure dx/x, and x = 0 is its neutral fixed point:T(0) = 0, +T ′(0) = 1. The asymptotic distribution of typical orbits, in the Lebesgue measure +sense, are concentrated on this neutral fixed point. +The iteration of T generates an infinite continued fraction expansion of each +number ξ ∈ [0, 1) of the form +(1.12) +ξ = 1 − +1 +d1(ξ) − +1 +d2(ξ) − ... +, +where dn(ξ) = ⌊1/(1 − T n−1(ξ))⌋ + 1 ≥ 2 for n ≥ 1. Using the infinite Markov +partition {Jp}p∈N, Jp = [1 − 1/p, 1 − 1/(p + 1)) of [0, 1), one can represent T as +the left shift acting on X [13, 37]. The map +(1.13) +π: (xn)∞ +n=1 ∈ X �→ π((xn)∞ +n=1) ∈ +∞ +� +n=1 +T −n+1(J(xn)) ⊂ [0, 1) +is a well-defined homeomorphism onto its image satisfying T ◦ π = π ◦ σ. We +consider the potential φ = − log |T ′ ◦ π|, where T ′ denotes the derivative of T +which is one-sided at boundary points of the Markov partition. From the mean + +LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES +7 +value theorem applied to the inverse branches of T, for any p ≥ 1 and all ξ, η ∈ Jp +we have +log |T ′(ξ)| +|T ′(η)| ≤ 2|T(ξ) − T(η)| < 2. +In particular, φ is acceptable. Since sup[p] eφ is comparable to p−2, P(βφ) < ∞ +holds if and only if � +p∈N p−2β is finite, which is equivalent to β > 1/2. It is easy +to see that for n ≥ 1, T n maps [0, 1/(n + 1)) diffeomorphically onto [0, 1). The +mean value theorem implies limn→∞ sup[0,1/(n+1)) |(T n)′| = ∞, while |(T n)′(0)| = 1 +for n ≥ 1. It follows that for any β > 1/2 there is no Bowen’s Gibbs state for +the potential βφ. Meanwhile, it is known [13] that for β > 1/2, the equilibrium +state for βφ is unique, which we denote by µβφ. For 1/2 < β < 1, µβφ has positive +entropy and fully supported. For β ≥ 1, µβφ is the unit point mass at π−1(0). +Let I denote the set of irrational numbers in (0, 1). The set En corresponds to +the set of numbers in I ∪ {0} for which the continued fraction (1.12) is periodic +of period n. As in the proof of [14, Theorem 28], one can show that any number +in �∞ +n=1{ξ ∈ I: T n(ξ) = ξ} is a quadratic irrational, i.e., an irrational root of a +quadratic polynomial with integer coefficients. Conversely, any quadratic irrational +in I has an eventually periodic continued fraction of the form (1.12), see [20, +Theorem 3]. +An induced system as in Theorem A is obtained from the first return map to +the interval (1/2, 1) not containing the neutral fixed point. From Theorems A and +B we obtain the following. For ξ ∈ [0, 1) and n ≥ 1, let δn +ξ denote the empirical +measure (1/n) �n−1 +k=0 δT k(ξ) on [0, 1). +Theorem C. For any 1/2 < β ≤ 1, the sequence of Borel probability measures on +M(π(X)) given by +1 +Zn(βφ) +� +ξ∈I∪{0} +T n(ξ)=ξ +|(T n)′(ξ)|−βδδn +ξ +for n = 1, 2, . . . +satisfies the LDP. The minimizer is unique and it is the unit point mass at µβφ◦π−1. +Moreover, for any bounded continuous function ˜ϕ: M(π(X)) → R we have +lim +n→∞ +1 +Zn(βφ) +� +ξ∈I∪{0} +T n(ξ)=ξ +|(T n)′(ξ)|−β ˜ϕ(δn +ξ ) = ˜ϕ(µβφ ◦ π−1). +1.3. Structure of the paper. The rest of this paper consists of two sections. In +Section 2 we verify the exponential tightness of the sequence in (1.4) under the +assumption of Theorem A. In Section 3 we complete proofs of all the theorems. +We close with a remark on possible generalizations of the main results. +2. Exponential tightness +The aim of this section is to verify the exponential tightness of the sequence in +(1.4). In Section 2.1 we start with a symbolic representation of the induced system. +In Section 2.2 we introduce the notion of local Gibbs states. In Section 2.3 we prove +a main technical estimate assuming the existence of a local Gibbs state. Using this + +8 +HIROKI TAKAHASI +estimate, we verify the exponential tightness in Section 2.4. In Section 2.5 we show +that the assumption of Theorem A implies the existence of a local Gibbs state. +2.1. Symbolic representation of the induced system. For a set S and an +integer j ≥ 1, let Sj denote the set of words of elements of S of word length j. We +introduce an empty word ∅ and set S0 = {∅}, a∅ = a = ∅a, a∅b = ab for a, b ∈ S. +We set W(S) = � +j≥1 Sj, N0 = N ∪ {0} and W0(S) = W(S) ∪ S0. +Let (X∗, R) be an inducing scheme. Let +N∗ = N ∩ [p∗, ∞) and N∗ = N ∩ [1, p∗ − 1). +For each p ∈ N∗ and ω ∈ W0(N∗), the set � +q∈N∗[pωq] is mapped by the induced +map τ in (1.7) bijectively onto X∗. Since the domain X∗∩�∞ +k=1 σ−kX∗ of definition +of τ is partitioned into countably infinite sets of this form, the induced system τ|Σ +is represented as the countable full shift over the infinite alphabet +(2.1) +A = +� � +q∈N∗ +[pωq]: p ∈ N∗ and ω ∈ W0(N∗) +� +. +To make this statement into a rigorous one, we endow A with the discrete +topology, and consider the countable full shift +AN = {z = (zn)∞ +n=1: zn ∈ A for every n ≥ 1}. +We use bold letters to denote elements of W(A), and a double square bracket [[·]] +to denote cylinders in AN: the n-cylinder (n ≥ 1) spanned by a = a1 · · · an ∈ An is +[[a]] = {z = (zk)∞ +k=1 ∈ AN : zk = ak for every k ∈ {1, . . . , n}}. +By definition, for each k ∈ {1, . . . , n} we have ak = � +q∈N∗[pkωjkq] where pk ∈ N∗, +jk ∈ N0, ωjk ∈ Njk +∗ . Let ∥a∥ denote the word length of p1ωj1p2ωj2 · · · pnωjn in +W(N), namely +∥a∥ = n + j1 + · · · + jn. +Let [a] denote the corresponding ∥a∥-cylinder in X, namely +[a] = [p1ωj1p2ωj2 · · · pnωjn] ⊂ X. +It is easy to check that a coding map Π: AN → Σ given by +(2.2) +Π: (zn)∞ +n=1 ∈ AN �→ Π((zn)∞ +n=1) ∈ +∞ +� +n=1 +[z1 · · · zn] ⊂ Σ +is well-defined, and is a homeomorphism. Let θ denote the left shift acting on AN. +Clearly we have Π ◦ θ = τ|Σ ◦ Π. +The following notation will be frequently used later. For a = a1 · · · an ∈ An as +above with [a] = [p1ωj1p2ωj2 · · · pnωjn] and q ∈ N∗, let +aq = p1ωj1p2ωj2 · · · pnωjnq ∈ W(N). +Lemma 2.1. Let (Σ, τ|Σ) be an induced system and let Π be the coding map in +(2.2). + +LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES +9 +(a) For every a ∈ W(A), +Π[[a]] = Σ ∩ +� +q∈N∗ +[aq]. +(b) If a, b ∈ W(A) satisfy ∥a∥ = ∥b∥, then a = b or [[a]] ∩ [[b]] = ∅. +Proof. If n ≥ 1, a ∈ An then �n−1 +k=0 R ◦ τ k equals ∥a∥ on Π[[a]], which implies (a). +If ∥a∥ = ∥b∥ and a ̸= b then (a) implies Π[[a]] ∩ Π[[b]] = ∅, and so [[a]] ∩ [[b]] = ∅, +which verifies (b). +□ +2.2. Local Gibbs states. Let φ: X → R satisfy P(φ) < ∞ and let (Σ, τ|Σ) be +an induced system. A Borel probability measure λφ on AN is called a local Gibbs +state for the potential φ associated with (Σ, τ|Σ), if there exist constants C ≥ 1, +γ0 ∈ R such that for any a ∈ W(A) and any x ∈ Π[[a]] we have +(2.3) +C−1 ≤ +λφ[[a]] +exp +� +S∥a∥φ(x) − γ0∥a∥ +� ≤ C. +We do not require the θ-invariance. If the context is clear, we simply call λφ a +local Gibbs state (associated with (Σ, τ|Σ)). +If λφ is a local Gibbs state, then for any a ∈ W(A), the λφ-measure of the +cylinder [[a]] in AN is given (up to multiplicative constants) by the Birkhoff sum of +φ along the orbit of x of length ∥a∥ and the word length ∥a∥. Since the word length +of a as a word in W(A) does not appear in the formula (2.3), λφ well captures part +of the original dynamics (X, σ). +If λφ is a local Gibbs state, the Borel probability measure λφ ◦ Π−1 on Σ is +τ|Σ-invariant. These two measures are related as follows. +Lemma 2.2. Let φ: X → R satisfy P(φ) < ∞, let (Σ, τ|Σ) be an induced system +and let λφ be a local Gibbs state associated with (Σ, τ|Σ). For any a ∈ W(A) we +have +λφ[[a]] = λφ ◦ Π−1(Σ ∩ [a]). +Proof. We write νφ for λφ ◦ Π−1, and {R = n} for {z ∈ Σ: R(z) = n} for each +n ≥ 1. We have νφ{R = n} > 0 for every n ≥ 1. Let νφ|{R=n} denote the restriction +of νφ to {R = n}. The measure +µφ = +∞ +� +n=1 +n−1 +� +k=0 +νφ|{R=n} ◦ σ−k +is a finite measure if and only if +� +Rdνφ < ∞. Since {R = n} is disjoint from +�n−1 +k=1 σ−k(Σ) for n ≥ 2, we have +(2.4) +µφ|Σ = +∞ +� +n=1 +νφ|{R=n} = νφ = λφ ◦ Π−1. +For any a ∈ W(A) we have Π[[a]] ⊂ Σ, and so +(2.5) +λφ[[a]] = µφΠ[[a]]. + +10 +HIROKI TAKAHASI +By µφ|Σ = νφ in (2.4) and Lemma 2.1(a), for any a ∈ W(A) we have +(2.6) +µφΠ[[a]] = νφΠ[[a]] = +� +q∈N∗ +νφ(Σ ∩ [aq]). +Let j ≥ 1 be such that a ∈ Aj. Then σ∥a∥ and τ j coincide on Σ ∩ [a]. Since +� +q∈N∗(Σ∩[aq]) ⊂ (τ|Σ)−j(� +q∈N∗[q]) = ∅ by τ(Σ) ⊂ Σ, we have νφ(� +q∈N∗ Σ∩[aq]) = +0. Combining this with (2.5), (2.6) we obtain +λφ[[a]] = +� +q∈N∗ +νφ(Σ ∩ [aq]) + +� +q∈N∗ +νφ(Σ ∩ [aq]) = νφ(Σ ∩ [a]), +as required. +□ +2.3. Exponential decay on partition functions. The next proposition pro- +vides a main technical estimate under the existence of a local Gibbs state. +Proposition 2.3. Let φ: X → R be acceptable and satisfy P(φ) < ∞. Assume +there exist an induced system (Σ, τ|Σ) and an associated local Gibbs state λφ. There +exist δ′ ∈ (0, 1/5] and n0 ≥ 1 such that if δ ∈ (0, δ′] and {Ni}∞ +i=1 is a non-decreasing +integer sequence such that +(2.7) +max N∗ ≤ N1 and +(2.8) +∞ +� +k=Ni+1 +� +a∈A +Π[[a]]⊂[k] +λφ[[a]] ≤ δ2i for every i ≥ 1, +then for every n ≥ n0 and every m ∈ {1, . . . , n} we have +(2.9) +� +x∈En +δn +x (X\Γ)=m/n +exp Snφ(x) ≤ eγ0n2nn(4δ)m +1 − 4δ , +where +(2.10) +Γ = {x = (xi)∞ +i=1 ∈ X : xi ≤ Ni for every i ≥ 1}. +Proposition 2.3 asserts that contributions of elements of En to Zn(φ) whose orbit +escape from the compact set Γ exactly m times within period n is exponentially +small in m. Similar estimates were obtained in [34] under the existence of Bowen’s +Gibbs states. +Proof of Proposition 2.3. Since λφ is a local Gibbs state, there exist constants C ≥ +1 and γ0 ∈ R such that for any a ∈ W(A) and any x ∈ Π[[a]] we have +(2.11) +C−1 ≤ +λφ[[a]] +exp +� +S∥a∥φ(x) − γ0∥a∥ +� ≤ C. +For the rest of the proof of Proposition 2.3, we shall use the notation a ≪ b for +two positive reals a, b to indicate that a/b is bounded from infinity by a constant +which depends only on C. If a ≪ b and b ≪ a, we shall write a ≍ b. + +LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES +11 +The first inequality in (2.11) will be used to bound a partial sum of Zn(φ) from +above by a sum of λφ-measures of cylinders in AN. Further, we will bound this +sum using the product property which is a consequence of (2.11): +(2.12) +λφ[[ab]] ≍ λφ[[a]]λφ[[b]] for a, b ∈ W(A). +Let δ ∈ (0, 1/5] and let {Ni}∞ +i=1 be a non-decreasing integer sequence satisfying +(2.7) and (2.8). Let Γ = Γ({Ni}∞ +i=1) be the compact subset of X given by (2.10). +If x ∈ En \ �n−1 +i=0 σ−i(Σ) then xi ≤ max N∗ = N1 ≤ Ni for 1 ≤ i ≤ n, and so +δn +x(Γ) = 1. By this and the periodicity of elements of En, for 1 ≤ m ≤ n we have +� +x∈En +δn +x (X\Γ)=m/n +exp Snφ(x) = +� +x∈En∩�n−1 +i=0 σ−i(Σ) +δn +x (X\Γ)=m/n +exp Snφ(x) +≤ n +� +x∈En∩Σ +δn +x (X\Γ)=m/n +exp Snφ(x). +(2.13) +To bound the last sum in (2.13), we decompose the set {x ∈ En ∩ Σ: δn +x(X \ +Γ) = m/n} into subsets sharing the same itinerary up to time n, and estimate +a contribution from each subset separately, and finally unify all these estimates +counting the total number of possible itineraries. +Define a function r: X \ Γ → N by +r(x) = min{i ≥ 1: xi > Ni}. +From (2.7) we have Σ ⊂ X \ Γ. Hence, for each x ∈ Σ there are infinitely many +i ≥ 0 with σix /∈ Γ. By an itinerary of x ∈ Σ we mean two sequences {nj(x)}∞ +j=1, +{rj(x)}∞ +j=1 in N0 given by the recursion formulas +n1(x) = min{i ≥ 0: σix /∈ Γ}, and +rj(x) = r(σnj(x)x), nj+1(x) = min{i ≥ nj(x) + rj(x): σix /∈ Γ} for j ≥ 1. +Lemma 2.4. Let x ∈ Σ. +(a) {i ≥ 0: σix /∈ Γ} = �∞ +j=1[nj(x), nj(x) + rj(x) − 1] ∩ N0. +(b) xnj(x)+rj(x) ∈ N∗ for every j ≥ 1. +Proof. Since {Ni}∞ +i=1 is non-decreasing, if x /∈ Γ then σix /∈ Γ for 0 ≤ i ≤ r(x) − +1. +This implies (a). +Since σnj(x)x = xnj(x)+1xnj(x)+2 · · · and σnj(x)x /∈ Γ with +r(σnj(x)x) = rj(x), we obtain xnj(x)+rj(x) > Nrj(x) ≥ N1, which together with (2.7) +yields xnj(x)+rj(x) ∈ N∗ as in (b). +□ +For each j ∈ {1, . . . , m} and n1 · · · nj ∈ Nj +0, r1 · · · rj ∈ Nj with n1 < · · · < nj ≤ +n, we put +(2.14) +∆ +r1···rj +n1···nj = {x ∈ En ∩ Σ: (ni(x), ri(x)) = (ni, ri) for every i ∈ {1, . . . , j}}. +Lemma 2.5. If δ > 0 is sufficiently small, then for j ∈ {1, . . . , m}, n1 · · · nj ∈ Nj +0 +and r1 · · · rj ∈ Nj such that ∆ +r1···rj +n1···nj ̸= ∅ we have +� +x∈∆ +r1···rj +n1···nj +exp Snφ(x) ≤ eγ0nδr1+···+rj. + +12 +HIROKI TAKAHASI +Proof. We start with the case j = 1. We introduce two sets of induced words +B0 = {b ∈ W(A): ∥b∥ = n − n1 − r1 + 1, Π[[b]] ⊂ ∪∞ +k=Nr1+1[k]} and +D0 = {d ∈ W(A): ∥d∥ = n1 + r1 − 1}. +For each x ∈ ∆r1 +n1 we have xn+1 = x1 ∈ N∗ by the definition (2.14), and xn1+r1 ∈ +N∗ by Lemma 2.4(b). Hence there exist d ∈ D0 and b ∈ B0 such that [d] = +[x1 · · ·xn1+r1−1] and [b] = [xn1+r1 · · · xn], and so x ∈ Π[[db]]. By (2.11) and (2.12), +exp Snφ(x) ≪ eγ0nλφ[[db]] ≪ eγ0nλφ[[d]]λφ[[b]]. +Summing this inequality over all x ∈ ∆r1 +n1 and then using � +b∈B0 λφ[[b]] ≤ δ2r1 from +(2.8) and � +b∈D0 λφ[[d]] ≤ 1 which follows from Lemma 2.1(b), we obtain +� +x∈∆r1 +n1 +exp Snφ(x) ≪eγ0n � +d∈D0 +λφ[[d]] +� +b∈B0 +λφ[[b]] ≤ eγ0nδ2r1 ≤ eγ0nδr1, +(2.15) +provided δ is small enough. In case m = 1 we are done. +To proceed, suppose m ≥ 2. let j, j+1 ∈ {1, . . . , m} and let n1 · · · njnj+1 ∈ Nj+1 +0 +, +r1 · · · rjrj+1 ∈ Nj+1 be such that ∆ +r1···rjrj+1 +n1···njnj+1 ̸= ∅. Define +Aj = {a ∈ W(A): ∥a∥ = nj + rj, Π[[a]] ∩ ∆ +r1···rj+1 +n1···nj+1 ̸= ∅}, +Bj = {b ∈ W(A): ∥b∥ = n − nj+1 − rj+1 + 1, Π[[b]] ⊂ ∪∞ +k=Nrj+1+1[k]}, +Cj = {c ∈ W(A): ∥c∥ = n − nj − rj} , +Dj = {d ∈ W(A): ∥d∥ = nj+1 + rj+1 − 1} . +Let a ∈ Aj. +For each c ∈ Cj we have ∥ac∥ = n. +Since X is the full shift, +Π[[ac]] contains a unique element of En ∩ Σ which we denote by ac. Then we have +ac ∈ Π[[a]] ∩ ∆ +r1···rj +n1···nj. By (2.11) and (2.12), +exp Snφ(ac) ≫ eγ0nλφ[[a]]λφ[[c]]. +This implies +� +x∈Π[[a]]∩∆ +r1···rj +n1···nj +exp Snφ(x) ≫ eγ0nλφ[[a]] +� +c∈Cj +λφ[[c]]. +(2.16) +By Lemma 2.2, for each c ∈ Cj we have +λφ[[c]] = λφ ◦ Π−1(Σ ∩ [c]). +Since the sets Σ ∩ [c], c ∈ Cj are pairwise disjoint and their union equals Σ, +(2.17) +� +c∈Cj +λφ[[c]] = 1. +Combining (2.16) and (2.17) yields +� +x∈Π[[a]]∩∆ +r1···rj +n1···nj +exp Snφ(x) ≫ eγ0nλφ[[a]]. +(2.18) +Similarly to the case j = 1, for each x ∈ Π[[a]]∩∆ +r1···rj+1 +n1···nj+1 we have xn+1 = x1 ∈ N∗ +by the definition (2.14) and xnj+1+rj+1 ∈ N∗ by Lemma 2.4(b). Hence there exist + +LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES +13 +d ∈ Dj, b ∈ Bj such that [d] = [x1 · · · xnj+1+rj+1−1] and [b] = [xnj+1+rj+1 · · ·xn]. +We have [[d]] ⊂ [[a]] and x ∈ Π[[db]], and by (2.11) and (2.12), +exp Snφ(x) ≪ eγ0nλφ[[d]]λφ[[b]]. +Summing this inequality over all x ∈ Π[[a]]∩∆ +r1···rj+1 +n1···nj+1, and then using � +b∈Bj λφ[[b]] ≤ +δ2rj+1 from (2.8) and � +d∈Dj +[[d]]⊂[[a]] +λφ[[d]] ≤ λφ[[a]] from Lemma 2.1(b), we obtain +� +x∈Π[[a]]∩∆ +r1···rj+1 +n1···nj+1 +exp Snφ(x) ≪ eγ0n � +d∈Dj +[[d]]⊂[[a]] +λφ[[d]] +� +b∈Bj +λφ[[b]] +≤ eγ0nλφ[[a]]δ2rj+1. +(2.19) +The two estimates in (2.18) and (2.19) yield +� +x∈Π[[a]]∩∆ +r1···rj+1 +n1···nj+1 exp Snφ(x) +� +x∈Π[[a]]∩∆ +r1···rj +n1···nj exp Snφ(x) ≤ δrj+1, +provided δ is small enough. Rearranging this inequality and summing the result +over all a ∈ Aj yields +� +x∈∆ +r1···rj+1 +n1···nj+1 +exp Snφ(x) ≤ δrj+1 +� +x∈∆ +r1···rj +n1···nj +exp Snφ(x). +Applying this inequality recursively and combining the final result with (2.15) +yields the desired inequality in Lemma 2.5. +□ +For integers L ≥ m and s ∈ {1, . . . , m}, we denote by KL,s the set of elements +(n1 · · · ns, r1 · · · rs) of Ns +0×Ns such that 0 ≤ n1 < · · · < ns ≤ n and r1+· · ·+rs = L. +The number of ways of locating n1, . . . , ns in [0, n] does not exceed ( n +s ), and for +each location (n1, . . . , ns) the number of all feasible combinations of (r1, . . . , rs) +with r1 +· · ·+rs = L is bounded by the number of ways of dividing L objects into +s groups, not exceeding +� L+s−1 +s−1 +� +≤ 2L+s−1. This yields #KL,s ≤ ( n +s ) +� L+s−1 +s−1 +� +≤ +2n2L+s−1. Clearly, for each x ∈ En ∩ Σ satisfying δn +x(X \ Γ) = m/n there exist +L ≥ m, s ∈ {1, . . . , m} and (n1 · · · ns, r1 · · · rs) ∈ KL,s such that x ∈ ∆r1···rs +n1···ns. If +δ ∈ (0, 1/5] is sufficiently small, then together with Lemma 2.5 we obtain +� +x∈En∩Σ +δn +x (X\Γ)=m/n +exp Snφ(x) ≤ +m +� +s=1 +∞ +� +L=m +� +(n1···ns,r1···rs)∈KL,s +� +x∈∆r1···rs +n1···ns +exp Snφ(x) +≤ 2n +m +� +s=1 +∞ +� +L=m +2L+s−1δL ≤ 2n +∞ +� +L=m +(4δ)L = 2n(4δ)m +1 − 4δ . +From this and (2.13), (2.9) follows and the proof of Proposition 2.3 is complete. +□ + +14 +HIROKI TAKAHASI +2.4. Verifying exponential tightness. We now use Proposition 2.3 to show the +desired exponential tightness. +Proposition 2.6. Let φ: X → R be acceptable such that P(φ) < ∞ and let +(Σ, τ|Σ) be an induced system. If there exists a local Gibbs state for the potential +φ associated with (Σ, τ|Σ), then {˜µn}∞ +n=1 is exponentially tight. +Proof. The argument below is an adaptation of the proof of Sanov’s theorem (see +e.g., [7]) to our setup. For each integer ℓ ≥ 1, we fix δℓ ∈ (0, 1/5] such that +(2.20) +1 +1 − 4δℓ +∞ +� +m=0 +e2ℓ2m(4δℓ)m ≤ 2. +We apply Proposition 2.3 and fix a non-decreasing integer sequence {Ni}∞ +i=1 such +that (2.7) and (2.8) with δ = δℓ hold. We define a compact subset Γℓ = Γ({Ni}∞ +i=1) +of X by (2.10), and set +Kℓ = +� +ν ∈ M(X): ν(Γℓ) ≥ 1 − 1 +ℓ +� +. +Since M(X) is a Polish space and Γℓ is a closed set, the weak* convergence µk → µ +for a sequence {µk}∞ +k=1 in Kℓ implies lim supk→∞ µk(Γℓ) ≤ µ(Γℓ). Hence, Kℓ is a +closed set. For an integer L ≥ 1 we define +KL = +∞ +� +ℓ=L +Kℓ. +This set is tight, and by Prohorov’s theorem any sequence in KL has a limit point. +Hence it is sequentially compact. Since the weak* topology is metrizable with the +bounded Lipschitz metric, KL is compact. By Chebyshev’s inequality, for n ≥ 1 +we have +� +x∈En +exp(ℓ2nδn +x (X\Γℓ))≥eℓn +exp Snφ(x) ≤ e−2ℓn +� +x∈En +δn +x (X\Γℓ)≥1/n +exp +� +2ℓ2nδn +x(X \ Γℓ) +� +exp Snφ(x) += e−2ℓn +n +� +m=1 +e2ℓ2m +� +x∈En +δn +x (X\Γ)=m/n +exp Snφ(x). +Combining this inequality with Proposition 2.3 and (2.20), we have +� +x∈En +δn +x /∈Kℓ +exp Snφ(x) = +� +x∈En +δn +x (X\Γℓ)≥ 1 +ℓ +exp Snφ(x) = +� +x∈En +exp(ℓ2nδn +x (X\Γℓ))≥eℓn +exp Snφ(x) +≤ +2nn +1 − 4δℓ +eγ0ne−2ℓn +n +� +m=0 +e2ℓ2m(4δℓ)m +≤ 10 · 2nneγ0ne−2ℓn. + +LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES +15 +If L ≥ 1 is large enough, then +˜µn(M(X) \ KL) ≤ +∞ +� +ℓ=L +˜µn(M(X) \ Kℓ) = +∞ +� +ℓ=L +1 +Zn(φ) +� +x∈En +δn +x /∈Kℓ +exp Snφ(x) +≤ 10 · 2nneγ0n +Zn(φ) +∞ +� +ℓ=L +e−2ℓn ≤ 2neγ0n +Zn(φ) e−Ln. +Combining this with the equality limn→∞(1/n) log Zn(φ) = P(φ) < ∞ which fol- +lows from the uniform continuity of φ, we obtain +lim sup +n→∞ +1 +n log ˜µn(M(X) \ KL) ≤ −L + log 2. +Since L ≥ 1 is an arbitrary large integer, {˜µn}∞ +n=1 is exponentially tight. +□ +2.5. Existence of a local Gibbs state. The next proposition ensures the exis- +tence of a local Gibbs state under the assumption of Theorem A. +Proposition 2.7. Let φ: X → R satisfy P(φ) < ∞. Assume there exists an +induced system (Σ, τ|Σ) for which the induced potentials Φγ, γ ∈ R associated with +φ are locally H¨older continuous, and there exists γ0 ∈ R such that P(Φγ0) = 0. +Then there exists a local Gibbs state for the potential φ associated with (Σ, τ|Σ). +Proof. Note that P(Φγ0 ◦ Π) = P(Φγ0) = 0. Since AN is the countable full shift, +the finiteness of P(Φγ0 ◦ Π) implies the summability of the potential Φγ0 ◦ Π. By +[18, Corollary 2.7.5] together with the summability and the local H¨older continuity +of Φγ0 ◦ Π, there exists a unique θ-invariant Bowen’s Gibbs state for the potential +Φγ0 ◦ Π, which we denote by λφ. There exists C ≥ 1 such that for every m ≥ 1, +any a ∈ Am and any z ∈ [[a]] we have +C−1 ≤ +λφ[[a]] +exp +� +−P(Φγ0 ◦ Π)m + �m−1 +k=0 Φγ0 ◦ Π(θkz) +� ≤ C. +For the series in the denominator of the fraction, for x ∈ Π[[a]] we have +m−1 +� +k=0 +Φγ0 ◦ Π(θkΠ−1(x)) = S�m−1 +k=0 R(τ kx)φ(x) − γ0 +m−1 +� +k=0 +R(τ kx) += S∥a∥φ(x) − γ0∥a∥. +Substituting this and P(Φγ0 ◦ Π) = 0 into the denominator of the fraction implies +that λφ is a local Gibbs state for the potential φ. +□ +Remark 2.8. Under the assumption and notation of Proposition 2.7 and its proof, +if γ0 = P(φ), λφ is θ-invariant and +� +Rd(λφ ◦ Π−1) < ∞, then the measure +1 +� +Rd(λφ ◦ Π−1) +∞ +� +n=0 +(λφ ◦ Π−1)|{R>n} ◦ σ−n +is in Mφ(X, σ), and it is an equilibrium state for the potential φ. The normalized +restriction of this measure to Σ is λφ ◦ Π−1. + +16 +HIROKI TAKAHASI +3. Proofs of the main results +In this section we complete the proofs of all the theorems. In Sections 3.1 and +3.2, we prove lower and upper bounds for certain fundamental open and closed +subsets of M(X) respectively. In Section 3.3, we combine these bounds and the +exponential tightness verified in Section 2 to complete the proof of Theorem A. +In Sections 3.4 we complete the proof of Theorem B. In view of applications, in +Section 3.5 we give a sufficient condition for the vanishing of the pressure of the +induced potential that is assumed in Theorem A. Using this, we complete the proof +of Theorem C in Section 3.6. +3.1. Lower bound for fundamental open sets. We introduce notations in this +and the next two subsections. Let Cu(X) denote the set of real-valued bounded +uniformly continuous functions on X. For an integer ℓ ≥ 1 we define +Cu(X)ℓ = {⃗ϕ = (ϕ1, . . . , ϕℓ): ϕj ∈ Cu(X) for every j ∈ {1, . . . , ℓ}}. +For ⃗ϕ = (ϕ1, . . . , ϕℓ) ∈ Cu(X)ℓ, ⃗α = (α1, . . . , αℓ) ∈ Rℓ and µ ∈ M(X), the +expression +� +⃗ϕdµ > ⃗α indicates that +� +ϕjdµ > αj holds for all j ∈ {1, . . . , ℓ}. The +meaning of +� +⃗ϕdµ ≥ ⃗α is analogous. Put ∥⃗α∥ = max1≤j≤ℓ |αj|. For ε ∈ R we write +⃗ε = (ε, . . . , ε) ∈ Rℓ. For n ≥ 1 and p1 · · · pn ∈ Nn, let p1 · · · pn denote the element +of En that is contained in [p1 · · · pn]. +Proposition 3.1. Let φ: X → R be acceptable and satisfy P(φ) < ∞. Let ℓ ≥ 1, +⃗ϕ ∈ Cu(X)ℓ and ⃗α ∈ Rℓ. Let G ⊂ M(X) be an open set of the form +G = +� +µ ∈ M(X): +� +⃗ϕdµ > ⃗α +� +. +For any measure µ ∈ Mφ(X, σ) ∩ G, we have +lim inf +n→∞ +1 +n log ˜µn(G) ≥ Fφ(µ). +Proof. By virtue of the definition of the pressure P(φ), it suffices to show that +(3.1) +lim inf +n→∞ +1 +n log +� +x∈En +δn +x ∈G +exp Snφ(x) ≥ h(µ) + +� +φdµ. +The proof of [36, Main Theorem] works verbatim to show the next lemma that +approximates non-ergodic measures with ergodic ones in a particular sense. +Lemma 3.2. For any µ ∈ Mφ(X, σ) and any ε > 0 there exists an ergodic measure +µ′ ∈ Mφ(X, σ) which is supported on a compact set and satisfies +|h(µ) − h(µ′)| < ε, +���� +� +⃗ϕdµ − +� +⃗ϕdµ′ +���� < ε and +���� +� +φdµ − +� +φdµ′ +���� < ε. +By Lemma 3.2, it suffices to show (3.1) for all µ ∈ Mφ(X, σ) which is ergodic. Let +ε > 0 be such that +(3.2) +� +⃗ϕdµ > ⃗α + ⃗ε. + +LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES +17 +From the uniform continuity of each component of ⃗ϕ and that of φ, from Birkhoff’s +ergodic theorem and Shannon-McMillan-Breiman’s theorem, for any sufficiently +large n ≥ 1 there is a finite subset Gn of Nn such that +(3.3) +���� +1 +n log #Gn − h(µ) +���� < ε +2, +and for every p1 · · · pn ∈ Gn, +(3.4) +sup +x∈[p1···pn] +���� +� +⃗ϕdδn +x − +� +⃗ϕdµ +���� < ε +2 and +sup +x∈[p1···pn] +���� +1 +nSnφ(x) − +� +φdµ +���� < ε +2. +Then (3.2) and the first inequality in (3.4) yield +� +⃗ϕdδn +p1···pn > ⃗α, and the second +inequality in (3.4) yields (1/n)Snφ(p1 · · · pn) > +� +φdµ − ε/2. Therefore +� +x∈En +δn +x ∈G +exp Snφ(x) ≥ +� +p1···pn∈Gn +exp Snφ(p1 · · · pn) ≥ #Gn exp +� +n +� +φdµ − εn +2 +� +. +Taking logarithms and dividing by a sufficiently large n we have +1 +n log +� +x∈En +δn +x ∈G +exp Snφ(x) ≥ 1 +n log #Gn + +� +φdµ − ε +2 > h(µ) + +� +φdµ − ε. +Letting n → ∞ and then ε → 0 yields (3.1). +□ +3.2. Upper bound for fundamental closed sets. We proceed to upper bounds +on fundamental closed sets, which are not necessarily compact. +Proposition 3.3. Let φ: X → R be acceptable and satisfy P(φ) < ∞. Let ℓ ≥ 1, +⃗ϕ ∈ Cu(X)ℓ, ⃗α ∈ Rℓ and let C ⊂ M(X) be a non-empty closed set of the form +C = +� +µ ∈ M(X): +� +⃗ϕdµ ≥ ⃗α +� +. +For any ε > 0 there exists µ ∈ Mφ(X, σ) such that +� +⃗ϕdµ > ⃗α − ⃗ε and +lim sup +n→∞ +1 +n log ˜µn(C) ≤ Fφ(µ). +Proof. A main ingredient is the next lemma, the proof of which is analogous to the +standard proof of the variational principle [40]. For n ≥ 1 we put +Dn(φ) = +sup +p1···pn∈Nn +sup +x,y∈[p1···pn] +Snφ(x) − Snφ(y). +Lemma 3.4. For any ε > 0 there exists n0 ≥ 1 such that if n ≥ n0 then for any +non-empty finite subset Cn of Nn satisfying δn +p1···pn ∈ C for every p1 · · · pn ∈ Cn, +there exists a measure µ0 ∈ Mφ(X, σ) such that +log +� +p1···pn∈Cn +sup +[p1···pn] +exp Snφ ≤ +� +h(µ0) + +� +φdµ0 +� +n+Dn(φ) +and +� +⃗ϕdµ0 > ⃗α−⃗ε. + +18 +HIROKI TAKAHASI +Proof. Since all components of ⃗ϕ are bounded uniformly continuous, for any ε > 0 +there exists n0 ≥ 1 such that if n ≥ n0 then for any p1 · · · pn ∈ Nn satisfying +δn +p1···pn ∈ C, +� +⃗ϕdδn +x ≥ ⃗α − (1/2)⃗ε holds for any x ∈ [p1 · · · pn]. In what follows we +assume n ≥ n0. +Set Λ = �∞ +k=0 σ−nk(� +p1···pn∈Cn[p1 · · · pn]). Then σn|Λ : Λ → Λ is topologically +conjugate to the left shift acting on the finite full shift space +CN +n = {(ˆpm)∞ +m=1 : ˆpm ∈ Cn for every m ≥ 1}. +Since the function ˆφ = Snφ induces a continuous potential on CN +n , the variational +principle [4] yields +sup +ˆµ∈M(Λ,σn|Λ) +� +h(ˆµ) + +� +ˆφdˆµ +� += lim +m→∞ +1 +m log +� +ˆp1···ˆpm∈Cm +n +sup +[ˆp1···ˆpm] +� +exp +m−1 +� +k=0 +ˆφ ◦ σnk +� +, +where M(Λ, σn|Λ) denotes the space of σn|Λ-invariant Borel probability measures +endowed with the weak* topology, and h(ˆµ) denotes the measure-theoretic entropy +of ˆµ ∈ M(Λ, σn|Λ) with respect to σn|Λ. For the series in the right-hand side, we +have +� +ˆp1···ˆpm∈Cm +n +sup +[ˆp1···ˆpm] +exp +�m−1 +� +k=0 +ˆφ ◦ σnk +� +≥ +� +� +p1···pn∈Cn +inf +[p1···pn] exp Snφ +�m +≥ +� +exp(−Dn(φ)) +� +p1···pn∈Cn +sup +[p1···pn] +exp Snφ +�m +. +Taking logarithms of both sides, dividing by m and then letting m → ∞ gives +lim +m→∞ +1 +m log +� +ˆp1···ˆpm∈Cm +n +sup +[ˆp1···ˆpm] +exp +�m−1 +� +k=0 +ˆφ ◦ σnk +� +≥ log +� +p1···pn∈Cn +sup +[p1···pn] +exp Snφ−Dn(φ). +Plugging this into the previous inequality yields +sup +ˆµ∈M(Λ,σn|Λ) +� +h(ˆµ) + +� +ˆφdˆµ +� +≥ log +� +p1···pn∈Cn +sup +[p1···pn] +exp Snφ − Dn(φ). +By the compactness of the space M(Λ, σn|Λ) and the upper semicontinuity of the +map ˆµ �→ h(ˆµ) + +� ˆφdˆµ on this space, the supremum is attained, say by ˆµ0. The +measure µ0 = (1/n) �n−1 +j=0 ˆµ0 ◦ σ−j is in Mφ(X, σ) and satisfies +� +h(µ0) + +� +φdµ0 +� +n = +sup +ˆµ∈M(Λ,σn|Λ) +� +h(ˆµ) + +� +ˆφdˆµ +� +. +Since the support of µ0 is contained in set {x ∈ X : +� +⃗ϕdδn +x > ⃗α − ⃗ε/2} by the +choice of n0 and the assumption n ≥ n0, we obtain +� +⃗ϕdµ0 > ⃗α−⃗ε as required. +□ +Continuing the proof of Proposition 3.3, we note that P(φ) < ∞ implies Zn(φ) < +∞ for every n ≥ 1. Hence it is possible to choose a finite subset Cn of the countable + +LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES +19 +set +� +p1 · · · pn ∈ Nn : δn +p1···pn ∈ C +� +such that +� +p1···pn∈Nn +δn +p1···pn∈C +exp Snφ(p1 · · · pn) ≤ 2 +� +p1···pn∈Cn +exp Snφ(p1 · · · pn). +By this inequality and Lemma 3.4, there exists µ0 ∈ Mφ(X, σ) such that +� +⃗ϕdµ0 > +⃗α − ⃗ε and +log +� +x∈En +δn +x ∈C +exp Snφ(x) = log +� +p1···pn∈Nn +δn +p1···pn∈C +exp Snφ(p1 · · · pn) +≤ log +� +p1···pn∈Cn +exp Snφ(p1 · · · pn) + log 2 +≤ log +� +p1···pn∈Cn +sup +[p1···pn] +exp Snφ + log 2 +≤ +� +h(µ0) + +� +φdµ0 +� +n + Dn(φ) + log 2. +Since φ is acceptable, it is uniformly continuous and so Dn(φ) = o(n) (n → ∞). +Dividing both sides of the above last displayed inequality by n, letting n → ∞ +and combining the result with P(φ) = limn→∞(1/n) log Zn(φ) yields the desired +inequality. +□ +3.3. Proof of Theorem A. Let φ: X → R be acceptable and satisfy P(φ) < ∞. +Assume there exists an induced system for which the induced potentials Φγ, γ ∈ R +associated with φ are locally H¨older continuous, and there exists γ0 ∈ R such that +P(Φγ0) = 0. +Let G be a non-empty open subset of M(X). Since subsets of M(X) of the +form +� +µ ∈ M(X): +� +⃗ϕdµ > ⃗α +� +with ℓ ≥ 1, ⃗ϕ ∈ Cu(X)ℓ, ⃗α ∈ Rℓ constitute a base +of the weak* topology of M(X), G is written as the union G = � +λ Gλ of sets Gλ of +this form. For each Gλ, Proposition 3.1 gives +lim inf +n→∞ +1 +n log ˜µn(Gλ) ≥ sup +Gλ +Fφ, +and hence +lim inf +n→∞ +1 +n log ˜µn(G) ≥ sup +λ +sup +Gλ +Fφ = sup +G +Fφ = − inf +G Iφ, +as required in (1.1). +Let C be a compact closed subset of M(X). Let G be an arbitrary open set +containing C. Since M(X) is metrizable by the bounded Lipschitz metric and C is +compact, we can choose ε > 0 and finitely many closed sets C1, . . . , Cs of the form +Ck = +� +µ ∈ M(X): +� +⃗ϕkdµ ≥ ⃗αk +� +with ℓk ≥ 1, ⃗ϕk ∈ Cu(X)ℓk, ⃗αk ∈ Rℓk, so that +C ⊂ �s +k=1 Ck ⊂ �s +k=1 Ck(ε) ⊂ G where Ck(ε) = {µ ∈ M(X): +� +⃗ϕkdµ > ⃗αk − ⃗ε}. +By Lemma 3.4 and Fφ ≤ −Iφ, for 1 ≤ k ≤ s we have +lim sup +n→∞ +1 +n log ˜µn(Ck) ≤ − inf +Ck(ε) Iφ + ε. + +20 +HIROKI TAKAHASI +Then we have +lim sup +n→∞ +1 +n log ˜µn(C) ≤ max +1≤k≤s +� +− inf +Ck(ε) Iφ +� ++ ε ≤ − inf +G Iφ + ε. +Since ε > 0 is arbitrary and G is an arbitrary open set containing C, it follows that +lim sup +n→∞ +1 +n log ˜µn(C) ≤ inf +G⊃C +� +− inf +G Iφ +� += − inf +C Iφ, +as required in (1.2). The last equality is due to the lower semicontinuity of Iφ. +Since {˜µn}∞ +n=1 is exponentially tight by Proposition 2.6, the standard arguments +as in [7] show the upper bound (1.2) for any non-compact closed subset of M(X), +and that Iφ is a good rate function. This completes the proof of Theorem A. +□ +3.4. Proof of Theorem B. Let φ: X → R be acceptable and satisfy P(φ) < ∞. +Assume there exists an induced system for which the associated induced potentials +Φγ, γ ∈ R are locally H¨older continuous, and there exists γ0 ∈ R such that +P(Φγ0) = 0. Assume that the minimizer of the rate function Iφ in (1.10) is unique, +denoted by µmin. Let {˜µnj}∞ +j=1 be an arbitrary convergent subsequence of {˜µn}∞ +n=1 +with the limit measure ˜µ. It suffices to show that ˜µ is the unit point mass at µmin. +We fix a metric which generates the weak* topology on M(X). Since the rate +function Iφ in (1.10) is the good rate function by Theorem A, for any α > 0 the +level set +L(α) = {µ ∈ M(X): Iφ(µ) ≤ α} +is a compact set. Let µ ∈ M(X) \ {µmin}. By the lower semicontinuity of the +rate function and Iφ(µ) > 0, it is possible to take r > 0 such that the closure of +the open ball Br(µ) of radius r about µ does not intersect L(Iφ(µ)/2). The weak* +convergence ˜µnj → ˜µ gives +˜µ(Br(µ)) ≤ lim inf +j→∞ ˜µnj(Br(µ)). +By this and the large deviations upper bound for closed sets (1.2), we have +˜µ(Br(µ)) ≤ lim sup +j→∞ +˜µnj(Br(µ)) ≤ lim sup +j→∞ +exp +� +−Iφ(µ)nj +2 +� += 0. +Hence, the support of ˜µ does not contain µ. Since µ is an arbitrary element of +M(X) \ {µmin}, it follows that ˜µ is the unit point mass at µmin. This completes +the proof of Theorem B. +□ +3.5. Sufficient condition for vanishing of pressure. A direct check of the +condition P(Φγ0) = 0 in Theorem A may be cumbersome, while checking the +finiteness of induced pressures is considered to be easier. In view of applications, +we give a sufficient condition for the second assumption in Theorem A on the +induced potential. +Lemma 3.5. Let φ: X → R be acceptable and satisfy P(φ) < ∞. Let (Σ, τ|Σ) be +an induced system and let Φγ : Σ → R (γ ∈ R) be the associated family of induced +potentials. If there exists δ ∈ R such that 0 < P(Φδ) < ∞, then there exists γ0 ∈ R +such that P(Φγ0) = 0. + +LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES +21 +Proof. Let γ ∈ R and suppose P(Φγ) < ∞. Let n ≥ 1. Since ∥a∥ ≥ n for any +a ∈ An and P(Φγ) is finite, for any γ′ > γ we have +� +a∈An +sup +[a] +exp(S∥a∥φ − γ′∥a∥) ≤ exp(−(γ′ − γ)n) +� +a∈An +sup +[a] +exp(S∥a∥φ − γ∥a∥). +Taking logarithms, dividing by n and letting n → ∞ yields P(Φγ′) ≤ −γ′ + +γ + P(Φγ). +This and the assumption in the lemma together imply that both +γ∞ = inf{γ ∈ R: P(Φγ) < ∞} and γ0 = inf{γ > γ∞: P(Φγ) ≥ 0} are finite. By +the variational principle [18, Theorem 2.1.8], γ ∈ (γ∞, ∞) �→ P(Φγ) is convex and +so continuous. Hence P(Φγ0) = 0 holds. +□ +3.6. Proof of Theorem C. Recall that T : [0, 1) → [0, 1) denotes the R´enyi map +(1.11). +For a bounded interval J ⊂ R let |J| denote its Euclidean length. +In +consideration of the neutral fixed point 0 of T, we set N∗ = N \ {1}, N∗ = {1} and +define an inducing scheme (X∗, R) by (1.5), (1.6), the induced system (Σ, τ|Σ) by +(1.7), (1.8), and define an infinite alphabet A and a coding map Π by (2.1), (2.2) +respectively, keeping the notation in Section 2.1. We have Σ = (1/2, 1)∩I and A = +��∞ +q=2[p1n−1q]: p ≥ 2 and n ≥ 1 +� +. +For simplicity we will denote by C various +positive constants which depend only on T. +For each a = �∞ +q=2[p1n−1q] ∈ A, we put +J(a) = T −1([1/(∥a∥ + 1), 1/∥a∥)) ∩ Jp. +Then R equals ∥a∥ on the set Π[[a]] = π−1(J(a)). There exists C ≥ 1 such that for +any a ∈ W(A) we have +(3.5) +C−1 ≤ |J(a)| · ∥a∥2 ≤ C. +Define an induced map U : � +a∈A J(a) → [0, 1) by U|J(a) = T ∥a∥|J(a). Recall that +φ = − log |T ′ ◦ π|. For β, γ ∈ R define Φβ,γ : Σ → R by +Φβ,γ(x) = βSR(x)φ(x) − γR(x), +which is the induced potential associated with βφ. +Lemma 3.6. For all β, γ ∈ R, Φβ,γ is locally H¨older continuous. +Proof. From the bounded distortion near the neutral fixed point [19, Lemma 2.2], +there exists C > 0 such that for any a ∈ A and all x, y ∈ [a] we have +(3.6) +Φβ,γ(x) − Φβ,γ(y) ≤ Cβ|U(ξ) − U(η)| ≤ Cβ, +where ξ = π(x) and η = π(y). If x ̸= y then d(x, y) = e−n holds for some n ≥ 2, +and there exists a1 · · ·an ∈ An such that x, y ∈ [[a1 · · · an]]. Since there is ρ > 1 +such that inf[0,1)\J1 |T ′| ≥ ρ, if n ≥ 3 then we have +(3.7) +|U(ξ) − U(η)| ≤ +|Un−1(ξ) − Un−1(η)| +inf�n−1 +k=2 U−k(J(ak)) |(Un−2)′| ≤ ρ2−n. +The local H¨older continuity of Φβ,γ follows from (3.6) and (3.7). +□ +Lemma 3.7. For any β ∈ (1/2, 1] there exists γ ∈ R such that 0 ≤ P(Φβ,γ) < ∞. + +22 +HIROKI TAKAHASI +Proof. From Lemma 3.6, |T(Jp)| = 1 for p ≥ 2 and (3.5), there exists C ≥ 1 such +that for a = �∞ +q=2[p1n−1q] ∈ A and all β, γ ∈ R we have +1 +|Jp|β sup +[a] +exp Φβ,γ = e−γn sup +Π[a] +exp(βSnφ) +|Jp|β +≤ Ce−γnn−2β. +Summing the result over all a ∈ A, we have +∞ +� +n=1 +� +a∈A +∥a∥=n +sup +[a] +exp Φβ,γ ≤ C +∞ +� +n=1 +e−γnn−2β +∞ +� +p=2 +|Jp|β ≤ C +∞ +� +n=1 +e−γnn−2β +∞ +� +p=2 +p−2β. +Let β ∈ (1/2, 1). Then we have P(βφ) > 0 [13], and the above series is finite +for all γ ∈ (0, P(βφ)]. In particular, P(Φβ,P (βφ)) is finite. Since any measure in +M(X, σ) other than the unit point mass at 1∞ = 111 · · · charges Σ, the equilibrium +state µβφ for the potential βφ satisfies µβφ(Σ) > 0. Let ˆµβφ denote the normalized +restriction of µβφ to Σ. Since τ is the first return map to Σ, ˆµβφ is τ|Σ-invariant +and satisfies +� +Rdˆµβφ < ∞. By the variational principle for Φβ,P (βφ) and Abramov- +Kac’s formula [44, Theorem 5.1], we obtain +∞ > P(Φβ,P (βφ)) ≥ h(ˆµβφ) + +� +(Φβ − P(βφ)R)dˆµβφ += (Fβφ(µβφ) + P(βφ) − P(βφ)) +� +Rdˆµβφ = 0. +We have verified1 that 0 ≤ P(Φβ,P (βφ)) < ∞ as reqiured in the lemma. +For the remaining case β = 1, we have P(φ) = 0 [13]. From Lemma 3.6 there is +C ≥ 1 such that for n ≥ 1 and a = a1 · · · an ∈ An, +C−1 ≤ +���n +k=1 U−k(J(ak)) +�� +sup[a] exp +��n−1 +k=0 Φ1,0 ◦ τ k� ≤ C. +Summing this double inequalities over all a ∈ An, taking logarithms, dividing by +n and then letting n → ∞ yields P(Φ1,0) = 0 as required in the lemma. +□ +Lemma 3.6 and Lemmas 3.5, 3.7 together verify the assumption in Theorem A +for the potential βφ, β ∈ (1/2, 1]. It follows from [35] that for any β ∈ (1/2, 1], +any minimizer of the rate function Iβφ is an equilibrium state for βφ. Since the +equilibrium state for βφ is unique [13], the minimizer of the rate function Iβφ is +unique. Since the map π in (1.13) is continuous, the assertions in Theorem C +follow from Theorems A and B. +□ +3.7. Some generalizations. We have worked on two full shift spaces X and Σ +(or AN), the latter obtained from the former via inducing (recall Section 2.1). The +assumption that X is the full shift has been used to construct sets of periodic +points of the same period, in the proofs of exponential tightness (Lemma 2.5) and +the lower large deviation bound (Proposition 3.1). For the induced system (Σ, τ|Σ), +we have effected the thermodynamic formalism for countable Markov shifts [18]. +1In fact, one can show P(Φβ,P (βφ)) = 0. See [23] for example. + +LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES +23 +The setup in this paper can be slightly generalized. The above-mentioned con- +structions of sets of periodic points can be done even if X is replaced by a finitely +primitive shift (see [18] for the definition). Then the induced shift space becomes +finitely irreducible, for which the thermodynamic formalism works too [27]. +Acknowledgments. 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Soc. 133 (2005) 2283–2295. +Keio Institute of Pure and Applied Sciences (KiPAS), Department of Mathe- +matics, Keio University, Yokohama, 223-8522, JAPAN +Email address: hiroki@math.keio.ac.jp + diff --git a/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf b/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..9b788f35cdd00d1e31ea689c6cefa923e5c3faaa --- /dev/null +++ b/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9b419642c988ce06ee7d59e7088e6df402afaac6ca0bab327ad65a7c81311b81 +size 256100 diff --git a/4tFAT4oBgHgl3EQfmB2k/vector_store/index.faiss b/4tFAT4oBgHgl3EQfmB2k/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..5672dae801bfcd4785b5429d4e8585cf41ab9ee6 --- /dev/null +++ b/4tFAT4oBgHgl3EQfmB2k/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d96994319a72fd2f6d02a77a7ff6d6b4d6b8c8a3ceaf3cbb743ccda0fa55b977 +size 2228269 diff --git a/4tFAT4oBgHgl3EQfmB2k/vector_store/index.pkl b/4tFAT4oBgHgl3EQfmB2k/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..98d792f17dd13058f42f290a405a13d271568961 --- /dev/null +++ b/4tFAT4oBgHgl3EQfmB2k/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f85fb201d6d181a9a98a153c85aacb63b7a32cf98e9669f01327fe460a4e4f2c +size 91167 diff --git a/69E1T4oBgHgl3EQfnASE/vector_store/index.pkl b/69E1T4oBgHgl3EQfnASE/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..6564696b6ed8bee19b21d9fc3df9e8dc95df121d --- /dev/null +++ b/69E1T4oBgHgl3EQfnASE/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4c6857cc44ef704dd40f9428a5bec896acaf9bf1b1bca4819e6f517536c1b870 +size 128015 diff --git a/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf b/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..2e7c3dc42fa192c9b3d7772d7d7dbfe677177589 --- /dev/null +++ b/6NFKT4oBgHgl3EQf_C4s/content/2301.11960v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a5ed0d42a34b344b6123d917be2a957b00d11cbfb2defaf47927284802285603 +size 454363 diff --git a/6NFKT4oBgHgl3EQf_C4s/vector_store/index.pkl b/6NFKT4oBgHgl3EQf_C4s/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..887b146ff30d0f9f4a9c03d43114b4ef8e953440 --- /dev/null +++ b/6NFKT4oBgHgl3EQf_C4s/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:eaa8152e3c2485eb88962a3a0b4d3d71c845a47314feed0e407e8082d9dba58d +size 78815 diff --git a/6dFKT4oBgHgl3EQf_S4j/content/tmp_files/2301.11961v1.pdf.txt b/6dFKT4oBgHgl3EQf_S4j/content/tmp_files/2301.11961v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2ba09cff1fb5fc6a64961d8ef6c2afeb60b8b291 --- /dev/null +++ b/6dFKT4oBgHgl3EQf_S4j/content/tmp_files/2301.11961v1.pdf.txt @@ -0,0 +1,3632 @@ +Reduced-Order Autodifferentiable Ensemble Kalman Filters +Yuming Chen∗ +Daniel Sanz-Alonso∗ +Rebecca Willett∗ +University of Chicago +Abstract +This paper introduces a computational framework to reconstruct and forecast a partially observed +state that evolves according to an unknown or expensive-to-simulate dynamical system. Our reduced- +order autodifferentiable ensemble Kalman filters (ROAD-EnKFs) learn a latent low-dimensional surrogate +model for the dynamics and a decoder that maps from the latent space to the state space. The learned +dynamics and decoder are then used within an ensemble Kalman filter to reconstruct and forecast the +state. Numerical experiments show that if the state dynamics exhibit a hidden low-dimensional structure, +ROAD-EnKFs achieve higher accuracy at lower computational cost compared to existing methods. If +such structure is not expressed in the latent state dynamics, ROAD-EnKFs achieve similar accuracy at +lower cost, making them a promising approach for surrogate state reconstruction and forecasting. +1 +Introduction +Reconstructing and forecasting a time-evolving state given partial and noisy time-series data is a fundamental +problem in science and engineering, with far-ranging applications in numerical weather forecasting, climate, +econometrics, signal processing, stochastic control, and beyond. Two common challenges are the presence of +model error in the dynamics governing the evolution of the state, and the high computational cost to simulate +operational model dynamics. Model error hinders the accuracy of forecasts, while the computational cost +to simulate the dynamics hinders the quantification of uncertainties in these forecasts. +Both challenges +can be alleviated by leveraging data to learn a surrogate model for the dynamics. Data-driven methods +enable learning closure terms and unresolved scales in the dynamics, thus enhancing the forecast skill of +existing models. In addition, surrogate models are inexpensive to simulate and enable using a large number +of particles within ensemble Kalman or Monte Carlo methods for state reconstruction and forecasting, thus +enhancing the uncertainty quantification. +This paper investigates a framework for state reconstruction and forecasting that relies on data-driven +surrogate modeling of the dynamics in a low-dimensional latent space. Our reduced-order autodifferentiable +ensemble Kalman filters (ROAD-EnKFs) leverage the EnKF algorithm to estimate by maximum likelihood +the latent dynamics as well as a decoder from latent space to state space. The learned latent dynamics +and decoder are subsequently used to reconstruct and forecast the state. Numerical experiments show that, +compared to existing methods, ROAD-EnKFs achieve higher accuracy at lower computational cost provided +that the state dynamics exhibit a hidden low-dimensional structure. When such structure is not expressed +in the latent dynamics, ROAD-EnKFs achieve similar accuracy at lower cost, making them a promising +approach for surrogate state reconstruction and forecasting. +Our work blends in an original way several techniques and insights from inverse problems, data assimila- +tion, machine learning, and reduced-order modeling. First, if the state dynamics were known and inexpensive +to simulate, a variety of filtering and smoothing algorithms from data assimilation (e.g. extended, ensemble, +and unscented Kalman filters and smoothers, as well as particle filters) can be used to reconstruct and fore- +cast the state. These algorithms often build on a Bayesian formulation, where posterior inference on the state +∗University of Chicago, Chicago, IL (ymchen@uchicago.edu, sanzalonso@uchicago.edu, willett@uchicago.edu) +1 +arXiv:2301.11961v1 [stat.ML] 27 Jan 2023 + +combines the observed data with a prior distribution defined using the model dynamics. Hence, learning a +surrogate model for the dynamics can be interpreted as learning a prior regularization for state reconstruction +and forecasting. Second, the task of learning the regularization can be viewed as an inverse problem: we seek +to recover the state dynamics from partially and noisily observed trajectories. Data assimilation facilitates +the numerical solution of this inverse problem by providing estimates of the hidden state. Third, our work +leverages machine learning and reduced-order modeling to parameterize the dynamics in a low-dimensional +latent space and learn a decoder from latent space to state space. In particular, we parameterize the decoder +using recent ideas from discretization-invariant operator learning. Our numerical experiments demonstrate +the computational advantage of co-learning an inexpensive surrogate model in latent space together with a +decoder, rather than a more expensive-to-simulate dynamics in state space. +1.1 +Related Work +Ensemble Kalman Filters in Data Assimilation +The EnKF algorithm, reviewed in [44,47,72,75], is a +popular method for state reconstruction and forecasting in data assimilation, with applications in numerical +weather forecasting, the geophysical sciences, and signal processing [29,30,83,91]. The EnKF propagates N +equally-weighted particles through the dynamics, and assimilates new observations via Kalman-type updates +computed with empirical moments. +If the state dynamics are known, the EnKF can achieve accurate +reconstruction with a small ensemble size N even in applications where the state and the observations are +high-dimensional, provided that the effective dimension is moderate [32]; EnKFs with a small ensemble size +have a low computational and memory cost compared to traditional Kalman filters [72]. Ensemble Kalman +methods are also successful solvers for inverse problems, as reviewed in [14]. In this paper, we employ the +EnKF to approximate the data log-likelihood of surrogate models for unknown or expensive-to-simulate +dynamics. The use of the EnKF for maximum likelihood estimation (MLE) was first proposed in [81], which +adopted a derivative-free optimization approach; see also [70]. Empirical studies on the likelihood computed +with EnKFs and other data assimilation techniques can be found in [12,64]. The application of the EnKF to +approximate the data log-likelihood within pseudo-marginal Markov chain Monte Carlo methods for Bayesian +parameter estimation was investigated in [28]; see also [79,80]. The paper [20] introduced derivative-based +optimization of an EnKF approximation of the log-likelihood to perform state and parameter estimation +in high-dimensional nonlinear systems. However, to the best of our knowledge, no prior work combines +estimation of the log-likelihood via EnKFs with learning low-dimensional surrogate models, including both +surrogate latent dynamics and a decoder from latent space to state space. +Blending Data Assimilation with Reduced-Order Models +Model reduction techniques have been +employed in data assimilation to improve the state reconstruction accuracy in high-dimensional dynamical +systems. The assimilation in the unstable subspace (AUS) method [13,51,67,74,87] projects the dynamics +onto a time-dependent subspace of the tangent space where the dynamics are unstable, and assimilates the +observations therein. The unstable directions are determined by the Lyapunov vectors with nonnegative +Lyapunov exponents, and can be approximated using discrete QR algorithms [24, 25]. The observations +can also be projected onto the unstable directions to reduce the data dimension [59]. We refer to [5] for +a review of projection-based model reduction techniques. However, these methods rely on prior knowledge +about the dynamics to identify the unstable subspaces and to construct the latent dynamics, and data +assimilation is performed after the subspaces are found. In contrast, our paper introduces a framework +that uses data assimilation as a tool to build surrogate latent dynamics from data. Another approach to +reduce the dimension of data assimilation problems exploits the conditionally Gaussian distribution of slow +variables arising in the stochastic parameterization of a wide range of dynamical systems [17, 18, 61]. This +conditional Gaussian structure can be exploited to obtain adequate uncertainty quantification of forecasts +with a moderate sample size. A caveat, however, is that identifying the slow variables can be challenging +in practice. As in our approach, these techniques often rely on machine learning to learn closure terms for +the dynamics [18]. Finally, we refer to [78] for a discussion on how the effective dimension of transport map +methods for data assimilation can be reduced by exploiting the conditional independence structure of the +2 + +reference-target pair. +Merging Data Assimilation with Machine Learning +Recent developments in machine learning to +model dynamical systems from data are reviewed in [55]. One line of work [27,70,84,88] embeds the EnKF +and the ensemble Kalman smoother (EnKS) into the expectation-maximization (EM) algorithm for MLE [23], +with a special focus on estimation of error covariance matrices. The expectation step (E-step) is approximated +by EnKF/EnKS with the Monte Carlo EM objective [90]. A subsequent line of work [7,9,31,66,92] introduces +training of a neural network (NN) surrogate model in the maximization step (M-step) based on the states +filtered by the E-step. Unfortunately, it can be hard to achieve an accurate approximation of the E-step using +EnKF/EnKS [20]. Another line of work [21, 52, 60, 65] approximates the data log-likelihood with particle +filters (PFs) [26,35] and performs MLE using derivative-based optimization. However, the resampling step +in PFs is not readily differentiable, and, in addition, PFs often collapse when the dimensions of the state +and the observations are large [2,4]. Finally, techniques that leverage machine learning to obtain inexpensive +analog ensembles for data assimilation are starting to emerge [95]. +Data-Driven Modeling of Dynamical Systems with Machine Learning +Machine learning is also +useful for dimensionality reduction in time-series modeling. +As an important example, recurrent neural +networks (RNNs) [57,96] assimilate data into the time-evolving latent states using NN updates. The paper +[73] models the latent state evolution in recurrent networks with NN-embedded differential equations [19]. +Other types of NN updates to incorporate the data into latent states include gated recurrent units (GRU) +[22,45], long short-term memory (LSTM) [39,54], and controlled differential equations (CDEs) [48]. Another +approach is to directly model the differential equation governing the state dynamics from observation data +using regression. +Such methods include sparse regression over a dictionary of candidate functions using +L1-regularization [10, 76, 86]. These techniques rely on full observation of the state, and, importantly, on +time-derivative data that are rarely available in practice and are challenging to approximate from noisy +discrete-time observation data [40, 41]. When the data are not guaranteed to lie in the same space as the +underlying dynamics, an autoencoder structure can be jointly learned with the latent state dynamics [16]. +Different modeling techniques can be applied to learn the latent state dynamics, including sparse dictionary +regression [16], recurrent networks [34,63], and the Koopman operator learning [58]. It is important to notice +that, in contrast to, e.g., [10, 42], the focus of this paper is on state reconstruction and forecasting, rather +than on obtaining an interpretable model for the dynamics. +1.2 +Outline and Main Contributions +• Section 2 formalizes the problem setting and goals. We introduce a reduced-order state-space model +(SSM) framework, where the dynamics are modeled in a low-dimensional latent space and learned +jointly with a decoder from latent space to state space. +• Section 3 introduces our main algorithm, the reduced-order autodifferentiable ensemble Kalman filter +(ROAD-EnKF). As part of the derivation of the algorithm, we discuss the use of EnKFs to estimate +the data log-likelihood within reduced-order SSMs. +• Section 4 contains important implementation considerations, including the design of the decoder, the +use of truncated backpropagation to enhance the scalability for large windows of data, and the choice +of regularization in latent space. +• Section 5 demonstrates the performance of our method in three examples: (i) a Lorenz 63 model +embedded in a high-dimensional space, where we compare our approach to the SINDy-AE algorithm +[16]; (ii) Burgers equation, where we showcase that ROAD-EnKFs are able to forecast the emergence of +shocks, a phenomenon not included in our training data-set; and (iii) Kuramoto-Sivashinky equation, +a common test problem for filtering methods due to its chaotic behavior, where the ROAD-EnKF +framework provides a computational benefit over state-of-the-art methods with similar accuracies. +3 + +• Section 6 closes with a summary of the paper and open questions for further research. +Notation +We denote by t ∈ {0, 1, . . . } a discrete-time index and by n ∈ {1, . . . , N} a particle index. Time indices +will be denoted with subscripts and particles with superscripts, so that un +t represents a generic particle n at +time t. We denote the particle dimension by du. We denote ut0:t1 := {ut}t1 +t=t0 and un1:n2 := {un}n1 +n=n0. The +collection un0:n1 +t0:t1 is defined similarly. The Gaussian density with mean m and covariance C evaluated at u is +denoted by N(u; m, C). The corresponding Gaussian distribution is denoted by N(m, C). +2 +Problem Formulation +In this section, we formalize and motivate our goals: reconstructing and forecasting a time-evolving, partially- +observed state with unknown or expensive-to-simulate dynamics. An important step towards these goals is to +learn a surrogate model for the dynamics. In Subsection 2.1 we consider an SSM framework where the state +dynamics are parameterized and learned in order to reconstruct and forecast the state. Next, in Subsection +2.2, we introduce a reduced-order SSM framework where the dynamics are modeled in a latent space and +a decoder from latent space to state space is learned along with the latent dynamics. Our ROAD-EnKF +algorithm, introduced in Section 3, operates in this reduced-order SSM. +2.1 +Setting and Motivation +Consider a parameterized SSM of the form: +(dynamics) +ut = Fα(ut−1) + ξt, +ξt ∼ N(0, Qβ), +1 ≤ t ≤ T, +(2.1) +(observation) +yt = Htut + ηt, +ηt ∼ N(0, Rt), +1 ≤ t ≤ T, +(2.2) +(initialization) +u0 ∼ pu(u0). +(2.3) +The state dynamics map Fα : Rdu → Rdu and error covariance matrix Qβ ∈ Rdu×du depend on unknown +parameter θ := (α⊤, β⊤)⊤ ∈ Rdθ. The observation matrices Ht ∈ Rdy×du and error covariance matrices +Rt ∈ Rdy×dy are assumed to be known and possibly time-varying. We further assume independence of all +random variables u0, ξ1:T , and η1:T . +Given observation data y1:T drawn from the SSM (2.1)-(2.3), we aim to accomplish two goals: +Goal 1: Reconstruct the states u1:T . +Goal 2: Forecast the states uT +1:T +Tf for some forecast lead time Tf ≥ 1. +state space +u0 +u1 +u2 +· · · +uT +uT +1 +· · · +uT +Tf +observation space +y1 +y2 +· · · +yT +Figure 1: Structure of data under SSM (2.1)-(2.3), where we assume only observations y1:T := {y1, . . . , yT } are +available. Our goals are to reconstruct the states u1:T (Goal 1) and to forecast future states uT +1:T +Tf for some +Tf ≥ 1 (Goal 2). +If the true parameter θ ∈ Rdθ was known and the dynamics were inexpensive to simulate, the first goal can +be accomplished by applying a filtering (or smoothing) algorithm on the SSM (2.1)-(2.3), while the second +4 + +goal can be accomplished by iteratively applying the dynamics model (2.1) to the reconstructed state uT . +We are interested in the case where θ needs to be estimated in order to reconstruct and forecast the state. +The covariance Qβ in the dynamics model (2.1) may represent model error or stochastic forcing in the +dynamics; in either case, estimating Qβ from data can improve the reconstruction and forecast of the state. +In this paper, we are motivated by applications where Fα represents a surrogate model for the flow between +observations of an autonomous ordinary differential equation (ODE). Letting ∆s be the equally-spaced time +between observations and fα : Rdu �→ Rdu be the parameterized vector field of the differential equation, we +then have +(ODE) +du +ds = fα(u), +Fα : u(s) �→ u(s + ∆s), +(2.4) +where u(s) ∈ Rdu is the state as a function of continuous-time variable s ≥ 0. The ODE (2.4) may arise +from spatial discretization of a system of partial differential equations (PDEs). For instance, we will consider +1-dimensional partial differential equations for u(x, s) of order κ ≥ 1, where u is a function of the spatial +variable x ∈ [0, L] and continuous-time variable s ≥ 0: +(PDE) +∂u +∂s = fα +� +u, ∂u +∂x, . . . , ∂κu +∂xκ +� +, +Fα : u(·, s) �→ u(·, s + ∆s), +(2.5) +with suitable boundary conditions. After discretizing this equation on a spatial domain with grid points +0 = x1 < x2 < · · · < xM = L, (2.5) can be expressed in the form of (2.4) by replacing the spatial derivatives +with their finite difference approximations, and u, fα, Fα with their finite-dimensional approximations on the +grid. As a result, du equals the number of grid points M. Several examples and additional details will be +given in Section 5. +2.2 +Reduced-Order Modeling +When the state is high-dimensional (i.e., du is large), direct reconstruction and forecast of the state is +computationally expensive, and surrogate modeling of the state dynamics map Fα becomes challenging. We +then advocate reconstructing and forecasting the state ut through a low-dimensional latent representation +zt, modeling the state dynamics within the low-dimensional latent space. This idea is formalized via the +following reduced-order parameterized SSM: +(latent dynamics) +zt = Gα(zt−1) + ζt, +ζt ∼ N(0, Sβ), +1 ≤ t ≤ T, +(2.6) +(decoding) +ut = Dγ(zt), +1 ≤ t ≤ T, +(2.7) +(observation) +yt = Htut + ηt, +ηt ∼ N(0, Rt), +1 ≤ t ≤ T, +(2.8) +(latent initialization) +z0 ∼ pz(z0). +(2.9) +The latent dynamics map Gα : Rdz �→ Rdz and error covariance matrix Sβ ∈ Rdz×dz are defined on a dz +dimensional latent space with dz < du, and the decoder function Dγ : Rdz �→ Rdu maps from latent space +to state space. The reduced-order SSM depends on an unknown parameter θ := (α⊤, β⊤, γ⊤)⊤ ∈ Rdθ. The +remaining assumptions are the same as in Subsection 2.1. +Writing Hγ,t(·) := HtDγ(·), the reduced-order SSM (2.6)-(2.9) can be combined into +(latent dynamics) +zt = Gα(zt−1) + ζt, +ζt ∼ N(0, Sβ), +1 ≤ t ≤ T, +(2.10) +(observation) +yt = Hγ,t(zt) + ηt, +ηt ∼ N(0, Rt), +1 ≤ t ≤ T, +(2.11) +(latent initialization) +z0 ∼ p(z0), +(2.12) +where the observation function Hγ,t(·) is nonlinear if the decoder Dγ(·) is nonlinear. As in Subsection 2.1, the +map Gα may be interpreted as the flow between observations of an ODE with vector field gα : Rdz �→ Rdz. +If the true parameter θ ∈ Rdθ was known, given observation data y1:T drawn from the reduced-order SSM +(2.6)-(2.9), we can reconstruct the states u1:T (Goal 1) by first applying a filtering (or smoothing) algorithm +5 + +latent space +state space +z0 +z1 +z2 +· · · +zT +zT +1 +· · · +zT +Tf +u1 +u2 +· · · +uT +uT +1 +· · · +uT +Tf +observation space +y1 +y2 +· · · +yT +Figure 2: +Structure of data under reduced-order SSM (2.6)-(2.9), where we assume only observations y1:T := +{y1, . . . , yT } are available. Our goals are to reconstruct the states u1:T (Goal 1) and to forecast future states uT +1:T +Tf +for some Tf ≥ 1 (Goal 2). +on (2.10)-(2.12) to estimate z1:T , and then applying the decoder Dγ. We can forecast the states uT +1:T +Tf +(Goal 2) by first applying iteratively the latent dynamics model (2.6) to the reconstructed latent state zT , +and then applying the decoder Dγ. As in Subsection 2.1, we are interested in the case where θ needs to be +estimated from the given data y1:T . +3 +Reduced-Order Autodifferentiable Ensemble Kalman Filters +As discussed in the previous section, to achieve both goals of state reconstruction and forecast, it is essential +to obtain a suitable surrogate model for the dynamics by learning the parameter θ. The general approach we +take is the following: (1) estimate θ with maximum likelihood; (2) apply a filtering algorithm with estimated +parameter θ to reconstruct and forecast the states u1:T . As we shall see, the maximum likelihood estimation +of θ will rely itself on a filtering algorithm. For the SSM in Subsection 2.1, this approach was introduced +in [20] via AD-EnKF (Algorithm 4.1 in [20]). +Here we focus on the reduced-order SSM in Subsection +2.2, namely (2.10)-(2.12), which is a more general case than the SSM in Subsection 2.1; this explains the +terminology reduced-order AD-EnKF (ROAD-EnKF). +In Subsection 3.1, we describe how the log-likelihood L(θ) = log pθ(y1:T ) can be expressed in terms of +the normalizing constants that arise from sequential filtering. In Subsection 3.2, we give background on +EnKFs and on how to use these filtering algorithms to estimate L(θ). In Subsection 3.3, we introduce our +ROAD-EnKF method that takes as input multiple independent instances of observation data yI +1:T across the +same time range, and performs both state reconstruction and forecasting. +3.1 +Sequential Filtering and Data Log-Likelihood +Suppose that θ = (α⊤, β⊤, γ⊤)⊤ is known. We recall that, for 1 ≤ t ≤ T, the filtering distributions pθ(zt|y1:t) +of the SSM (2.10)-(2.12) can be obtained sequentially, alternating between prediction and analysis steps: +(prediction) +pθ(zt|y1:t−1) = +� +N +� +zt; Gα(zt−1), Sβ +� +pθ(zt−1|y1:t−1) dzt−1, +(3.1) +(analysis) +pθ(zt|y1:t) = +1 +Et(θ)N +� +yt; Hγ,t(zt), Rt +� +pθ(zt|y1:t−1), +(3.2) +with the convention pθ(·|y1:0) := pθ(·). Here Et(θ) is a normalizing constant which does not depend on zt. +It can be shown that +Et(θ) = pθ(yt|y1:t−1) = +� +N +� +yt; Hγ,t(zt), Rt +� +pθ(zt|y1:t−1) dzt, +(3.3) +6 + +and therefore the data log-likelihood admits the characterization +L(θ) := log pθ(y1:T ) = +T +� +t=1 +log pθ(yt|y1:t−1) = +T +� +t=1 +log Et(θ). +(3.4) +Analytical expressions of the filtering distributions pθ(zt|y1:t) and the data log-likelihood L(θ) are only +available for a small class of SSMs, which includes linear-Gaussian and discrete SSMs [46,68]. Outside these +special cases, filtering algorithms need to be employed to approximate the filtering distributions, and these +algorithms can be leveraged to estimate the log-likelihood. +3.2 +Estimation of the Log-Likelihood with Ensemble Kalman Filters +Given θ = (α⊤, β⊤, γ⊤)⊤, the EnKF algorithm [29,30] sequentially approximates the filtering distributions +pθ(zt|y1:t) using N equally-weighted particles z1:N +t +. At prediction steps, each particle zn +t is propagated using +the latent dynamics model (2.10), while at analysis steps a Kalman-type update is performed for each +particle: +(prediction step) +�z n +t = Gα(zn +t−1) + ζn +t , +ζn +t +i.i.d. +∼ N(0, Sβ), +(3.5) +(analysis step) +zn +t = �z n +t + �Kt +� +yt + ηn +t − Hγ,t(�z n +t ) +� +, +ηn +t +i.i.d. +∼ N(0, Rt). +(3.6) +The Kalman gain �Kt := �Czy,t( �Cyy,t + Rt)−1 is defined using empirical covariances given by +�Czy,t = +1 +N − 1 +N +� +n=1 +(�z n +t − �mt) +� +Hγ,t(�z n +t )− � +Ht +�⊤, +�Cyy,t = +1 +N − 1 +N +� +n=1 +� +Hγ,t(�z n +t )− � +Ht +�� +Hγ,t(�z n +t )− � +Ht +�⊤, (3.7) +where +�mt = 1 +N +N +� +n=1 +�z n +t , +� +Ht = 1 +N +N +� +n=1 +Hγ,t(�z n +t ). +(3.8) +The empirical moments �Cyy,t, � +Ht defined in equations (3.7) and (3.8) provide a Gaussian approximation +to the predictive distribution for Hγ,t(zt): +pθ(Hγ,t(zt)|y1:t−1) ≈ N(Hγ,t(zt); � +Ht, �Cyy,t). +(3.9) +By applying the change of variables formula to (3.3), we have +Et(θ) = +� +N +� +yt; Hγ,t(zt), Rt +� +pθ(zt|y1:t−1) dzt += +� +N +� +yt; Hγ,t(zt), Rt +� +pθ(Hγ,t(zt)|y1:t−1) dHγ,t(zt) +≈ N(yt; � +Ht, �Cyy,t + Rt), +(3.10) +where the approximation step follows from (3.9) and the formula for convolution of two Gaussians. From +(3.4), we obtain the following estimate of the data log-likelihood: +LEnKF(θ) := +T +� +t=1 +log N +� +yt; � +Ht, �Cyy,t + Rt +� +≈ L(θ). +(3.11) +The estimate LEnKF(θ) can be computed online with EnKF, and is stochastic as it depends on the randomness +used to propagate the particles, e.g., the choice of random seed. The whole procedure is summarized in +7 + +Algorithm 3.1, which implicitly defines a stochastic map θ �→ LEnKF(θ). +Algorithm 3.1 Ensemble Kalman Filter and Log-likelihood Estimation +Input: θ = (α⊤, β⊤, γ⊤)⊤, y1:T . (If multiple input instances yI +1:T are provided, run the following proce- +dure for each instance yi +1:T .) +1: Initialize LEnKF(θ) = 0. Draw zn +0 +i.i.d. +∼ pz(z0). +2: for t = 1, . . . , T do +3: +Set �z n +t = Gα(zn +t−1) + ζn +t , where ζn +t +i.i.d. +∼ N(0, Sβ). +▷ Prediction step +4: +Compute �mt, � +Ht, �Czy,t, �Cyy,t by equations (3.7) and (3.8) and set �Kt = �Czy,t( �Cyy,t + Rt)−1. +5: +Set zn +t = �z n +t + �Kt +� +yt + ηn +t − Hγ,t(�z n +t ) +� +, where ηn +t +i.i.d. +∼ N(0, Rt). +▷ Analysis step +6: +Set LEnKF(θ) ← LEnKF(θ) + log N +� +yt; � +Ht, �Cyy,t + Rt +� +. +7: end for +Output: EnKF particles z1:N +0:T . Log-likelihood estimate LEnKF(θ).(If multiple input instances yI +1:T are +provided, return instead the average of log-likelihood estimates.) +3.3 +Main Algorithm +The main idea of our algorithm is to perform maximum likelihood estimation on the parameter θ by gradient +ascent, via differentiation through the map θ �→ LEnKF(θ). Our core method is summarized in Algorithm 3.2, +which includes estimation of θ as well as reconstruction and forecast of states. Our PyTorch implementa- +tion is at https://github.com/ymchen0/ROAD-EnKF. The gradient of the map θk �→ LEnKF(θk) can be +evaluated using autodiff libraries [1, 8, 69] that support auto-differentiation of common matrix operations, +e.g. matrix multiplication, inverse, and determinant [33]. We use the “reparameterization trick” [49,71] to +auto-differentiate through the stochasticity in the EnKF algorithm, as in Subsection 4.1 of [20]. +In Section 5, we consider numerical examples where the data are generated from an unknown SSM in the +form of (2.6)-(2.9) with no explicit knowledge of the reduced-order structure; we also consider examples where +the data are generated directly from (2.1)-(2.3). In practice, multiple independent instances of observation +data yI +1:T may be available across the same time range, where each superscript i ∈ I corresponds to one +instance of observation data y1:T . We assume that each instance yi +1:T is drawn i.i.d. from the same SSM, +with different realizations of initial state, model error, and observation error for each instance. We assume +that data are split into training and test sets yItrain +1:T +and yItest +1:T . During training, we randomly select a small +batch of data from yItrain +1:T +at each iteration, and evaluate the averaged log-likelihood and its gradient over +the batch to perform a parameter update. The idea is reminiscent of stochastic gradient descent in the +optimization literature: matrix operations of EnKF can be parallelized within a batch to utilize the data +more efficiently, reducing the computational and memory cost compared to using the full training set at each +iteration. The state reconstruction and forecast performance are evaluated on the unseen test set yItest +1:T . +State reconstruction and forecast via Algorithm 3.2 can be interpreted from a probabilistic point of view. +For convenience, we drop the superscripts I and k in this discussion. For 0 ≤ t ≤ T, since the particles z1:N +t +form an approximation of the filtering distribution pθ(zt|y1:t) for latent state zt, it follows from (2.7) that +the output particles u1:N +t +of the algorithm form an approximation of the filtering distribution pθ(ut|y1:t) for +state ut. For T + 1 ≤ t ≤ T + Tf, it follows from (2.10) that the particles z1:N +t +form an approximation of +the predictive distribution pθ(zt|y1:T ). Therefore, by (2.7) the output particles u1:N +t +of the algorithm form +an approximation of the predictive distribution pθ(ut|y1:T ) for future state ut. +4 +Implementation Details +This section considers the practical implementation of ROAD-EnKF Algorithm 3.2, including parameteri- +zation of the surrogate latent dynamics map gα and decoder Dγ (Subsection 4.1), computational efficiency +8 + +Algorithm 3.2 Reduced-Order Autodifferentiable Ensemble Kalman Filter (ROAD-EnKF) +Input: Observations yI +1:T , split into yItrain +1:T +and yItest +1:T . Learning rate η. Batch size B. +1: Initialize SSM parameter θ0 and set k = 0. Write Hγ,t(·) = HtDγ(·). +// Training phase +2: while not converging do +3: +Randomly select B indices from Itrain, denoted as IB. +4: +Compute zIB,1:N +0:T +, LEnKF(θk) = EnsembleKalmanFilter(θk, yIB +1:T ) using Algorithm 3.1. +5: +Compute ∇θLEnKF(θk) by auto-differentiating the map θk �→ LEnKF(θk). +6: +Set θk+1 = θk + η∇θLEnKF(θk) and k ← k + 1. +7: end while +// Test phase +8: zItest,1:N +0:T +, LEnKF(θk) = EnsembleKalmanFilter(θk, yItest +1:T ). +▷ State reconstruction +9: Simulate zItest,1:N +t +using (2.10) with α = αk, β = βk for t = T + 1, . . . , T + Tf. +▷ State forecast +10: Compute uItest,1:N +0:T +Tf += Dγk(zItest,1:N +0:T +Tf ) . +Output: Learned reduced-order SSM parameter θk and particles uItest,1:N +0:T +Tf . +for high-dimensional observations (Subsection 4.2), and regularization on latent states (Subsection 4.3). +4.1 +Surrogate Latent Dynamics and Decoder Design +In our numerical experiments, we adopt a simple parameterization for the surrogate latent dynamics map gα +using a two-layer fully connected NN. For our design of the decoder Dγ, the idea stems from the literature +on convolutional autoencoders for computer vision tasks (e.g., [62]), where both the encoder and decoder +networks consist of multiple convolutional layers with residual connections that map between the image +space and latent space. Here, to suit our setting, we replace the kernel-based local convolutional layers with +Fourier-based spectral convolutional layers (‘Fourier layers’) introduced in [36,56]. The latter treat a finite- +dimensional vector as a spatial discretization of a function on a grid, and learn a finite-dimensional mapping +that approximates an operator between function spaces. The learning accuracy is known empirically to not +depend on the level of the discretization [56], determined by du in our case. Using Fourier layers to learn +dynamical systems and differential equations was originally proposed in [56]. For the sake of completeness, +we describe below the definition of spectral convolutional layers and how they are incorporated into our +decoder design. +Spectral Convolutional Layer +Given an input vin ∈ Rnin×du where nin is the number of input channels +and du is the input dimension, which is also the size of the grid where the function is discretized, we +first apply a discrete Fourier transform (DFT) in spatial domain to get λin := DFT(vin) ∈ Cnin×du. We +then multiply it by a learned complex weight tensor W ∈ Cnout×nin×du that is even symmetric1 to get +λout := W × λin ∈ Cnout×du. The multiplication is defined by +(W × λin)i,k = +nin +� +j=1 +Wi,j,k(λin)j,k. +(4.1) +This can be regarded as ‘channel mixing’, since for the k-th Fourier mode (1 ≤ k ≤ du), all nin input channels +of λ are linearly mixed to produce nout output channels through the matrix W·,·,k. Other types of (possibly +nonlinear) mixing introduced in [36] can also be applied, and we leave them to future work. We then apply an +inverse discrete Fourier transform (IDFT) in spatial domain to get the output vout = IDFT(λout) ∈ Rnout×du. +We call the mapping vin �→ vout a spectral convolutional layer (SpecConv). +1That is, W satisfies Wi,j,k = W i,j,du+2−k ∀i, j and ∀k ≥ 2. This ensures that the inverse discrete Fourier transform of +λout is real. In practice, the parameterization of W requires up to nin × nout × (⌊du/2⌋ + 1) complex entries. +9 + +Fourier Neural Decoder +Given latent variable z ∈ Rdz (where we omit the subscript t for convenience), +we first apply a complex linear layer f0(·) to get z0 = f0(z) := W0z + b0 ∈ Ch for W0 ∈ Ch×dz and b0 ∈ Ch, +where h is the dimension of z0 to be specified. We then apply an IDFT that treats z0 as a one-sided Hermitian +signal in Fourier domain2 to get v0 := IDFT(z0) ∈ Rdu. We then apply L spectral convolutional layers to +get vL, with proper choices of channel numbers as well as residual connections, normalization layers, and +activation functions. More specifically, vL is defined by iteratively applying the following +vℓ = fℓ(vℓ−1) := Act +� +Norm +� +SpecConv(vℓ−1) + 1x1Conv(vℓ−1) +�� +, +1 ≤ ℓ ≤ L, +(4.2) +where vℓ ∈ Rnℓ×du, Act and Norm refer to the activation function and the normalization layer, 1x1Conv +refers to the one-by-one convolutional layer which can be viewed as a generalization of residual connection, +and n0 = 1. We refer to fℓ as a ‘Fourier layer’. The final part of the decoder is a two-layer fully connected +NN that is applied to vL ∈ RnL×du over channel dimension to get u ∈ Rdu. See Figure 3 for the architecture. +Notice that the learned variables γ of the decoder include W0, b0 of the initial linear layer, complex weight +tensors W’s of SpecConv layers, weights and biases of 1x1Conv layers, as well as the final fully-connected +NN. +z +Linear +IDFT +Fourier Layer 1 +Fourier Layer L +⋯ +vℓ−1 +vL +u +v0 +DFT +Linear +Channel +Mixing +IDFT +1x1Conv ++ +MLP +Channel +Mixing +Norm +Act +vℓ +SpecConv +(a) +(b) +Figure 3: (a) Network architecture of the decoder Dγ. Starting from z ∈ Rdz in a low-dimensional latent +space, we first apply a complex linear layer followed by an IDFT to lift it to v0 ∈ Rdu in a high-dimensional state +space. We then apply L Fourier layers iteratively to get vL ∈ RnL×du where nL is the channel dimension. We project +it back to the state space by applying a two-layer fully-connected NN to mix the channels and output u ∈ Rdu. (b) +Fourier layer: The design was first proposed in [56], and we describe it here for the sake of completeness. The +upper half represents a spectral convolutional layer, where we transform the input vℓ−1 ∈ Rnℓ−1×du into the frequency +space with DFT, mix the channels with a complex linear map, and transform back with IDFT. The lower half is a +one-by-one convolutional layer, which is a generalization of residual connection. The outputs from both layers are +summed up and passed through a normalization and an activation layer to produce the output vℓ ∈ Rnℓ×du. +2z0 is either truncated or zero-padded to a signal of dimension C⌊du/2⌋+1. +10 + +4.2 +Algorithmic Design for Computational Efficiency +If the time-window length T is large, we follow [20] and use truncated backpropagation to auto-differentiate +the map θ �→ LEnKF(θ): we divide the sequence into multiple short subsequences and backpropagate within +each subsequence. The idea stems from Truncated Backpropagation Through Time (TBPTT) for RNNs +[82,93] and the recursive maximum likelihood method for hidden Markov models [53]. By doing so, multiple +gradient ascent steps can be performed for each single filtering pass, and thus the data can be utilized more +efficiently. Moreover, gradient explosion/vanishing [3] are less likely to happen. We refer to [20] for more +details. We choose this variant of ROAD-EnKF in our experiments. +In this work we are mostly interested in the case where du and dy are large, and dz is small. Moreover, +the ensemble size N that we consider is moderate, i.e., du ≥ dy > N > dz. Therefore, we do not pursue +the covariance localization approach as in [20] (see also [38, 43]), which is most effective when N < dz. +Instead, we notice that the computational bottlenecks of the analysis step in the EnKF Algorithm 3.1 are +the O(d3 +y) operations of computing the Kalman gain (Line 4) as well as updating the data log-likelihood +(Line 6), where we need to compute the matrix inverse and log-determinant of a dy ×dy matrix ( �Cyy,t +Rt). +If dy > N, the number of operations can be improved to O(N 3) as follows. +Let Yt ∈ Rdy×N be the +matrix representation of the centered ensemble after applying the observation function, i.e., its n-th column +is Y n +t +:= +1 +√N−1 +� +Hγ,t(�zn +t ) − 1 +N +�N +m=1 Hγ,t(�zm +t ) +� +(we drop the parameter γ for convenience). This leads to +�Cyy,t = YtY ⊤ +t . By the matrix inversion lemma [94], +( �Cyy,t + Rt)−1 = R−1 +t +− R−1 +t Yt(I + Y ⊤ +t R−1 +t Yt)−1Y ⊤ +t R−1 +t , +(4.3) +logdet( �Cyy,t + Rt) = logdet(I + Y ⊤ +t R−1 +t Yt) + logdet(Rt), +(4.4) +where I + Y ⊤ +t R−1 +t Yt ∈ RN×N. The computational cost can be further reduced if the quantities R−1 +t +and +logdet(Rt) can be pre-computed, for instance when Rt = rI for some scalar r ∈ R. +Moreover, in practice, to update the ensemble in Line 5 of Algorithm 3.1, instead of inverting I+Y ⊤ +t R−1 +t Yt +directly in (4.3) followed by a matrix multiplication, we find it more numerically stable to first solve the +following linear system: +(I + Y ⊤ +t R−1 +t Yt)un +t = Y ⊤ +t R−1 +t +� +yt + γn +t − Hγ,t(�zn +t ) +� +(4.5) +for un +t ∈ RN, and then perform the analysis step (Line 5 of Algorithm 3.1) by +zn +t = �zn +t + �Czy,t( �Cyy,t + Rt)−1� +yt + γn +t − Hγ,t(�zn +t ) +� += �zn +t + �Czy,t +� +R−1 +t +− R−1 +t Yt(I + Y ⊤ +t R−1 +t Yt)−1Y ⊤ +t R−1 +t +�� +yt + γn +t − Hγ,t(�zn +t ) +� += �zn +t + �Czy,tR−1 +t +� +yt + γn +t − Hγ,t(�zn +t ) − Ytun +t +� +. +(4.6) +Similar ideas and computational cost analysis can be found in [85]. +For the benchmark experiments in +Section 5, we modify the AD-EnKF algorithm as presented in [20] to incorporate the above ideas. +4.3 +Latent Space Regularization +Since the estimation of u is given by Dγ(z), where both Dγ(·) and z need to be identified from data, we +overcome potential identifiability issues by regularizing z in the latent space. To further motivate the need for +latent space regularization, consider the following example: if the pair (z, Dγ(·)) provides a good estimation +of ut, then so does (cz, 1 +cDγ(·)) for any constant c ̸= 0. Therefore, the norm of z can be arbitrarily large, +and thus we regularize z’s in the latent space so that their norms do not explode. +We perform regularization by extending the observation model (2.11) to impose additional constraints on +the latent state variable zt’s. The idea stems from regularization in ensemble Kalman methods for inverse +11 + +problems [15,37]. We first extend (2.11) to the equations: +�yt = Hγ,t(zt) + ηt, +ηt ∼ N(0, Rt), +0 = zt + ϵt, +ϵt ∼ N(0, σ2Idz), +(4.7) +where σ is a parameter to be chosen that incorporates the prior information that each coordinate of zt is an +independent centered Gaussian random variable with standard deviation σ. Define +yaug +t += +�yt +0 +� +, +Haug +γ,t (zt) = +�Hγ,t(zt) +zt +� +, +ηaug +t +∼ N(0, Raug +t +), +Raug +t += +�Rt +0 +0 +σ2Idz +� +. +(4.8) +We then write (4.7) into an augmented observation model +yaug +t += Haug +γ,t (zt) + ηaug +t +, +ηaug +t +∼ N(0, Raug +t +). +(4.9) +To perform latent space regularization in ROAD-EnKF, during the training stage we run EnKF (Line 4 +of Algorithm 3.2) with augmented data yaug +1:T and SSM with the augmented observation model, i.e., (2.10)- +(4.9)-(2.12). During test stage, we run EnKF (Line 8 of Algorithm 3.2) with the original data and SSM, i.e., +(2.10)-(2.11)-(2.12). +5 +Numerical Experiments +In this section, we compare our ROAD-EnKF method to the SINDy autoencoder [16], which we abbreviate +as SINDy-AE. It learns an encoder-decoder pair that maps between observation space (yt’s) and latent space +(zt’s), and simultaneously performs a sparse dictionary learning in the latent space to discover the latent +dynamics. Similar to SINDy-AE, our ROAD-EnKF method jointly discovers a latent space and the dynamics +therein that is a low-dimensional representation of the data. However, our method differs from SINDy-AE +in four main aspects: (1) No time-derivative data for y1:T are required; (2) No encoder is required; (3) State +reconstruction and forecast can be performed even when the data y1:T are noisy and partial observation of +u1:T , while SINDy-AE is targeted at noiseless and fully observed data that are dense in time; (4) Stochastic +representation of latent dynamics model can be learned, and uncertainty quantification can be performed in +state reconstruction and forecast tasks through the use of particles, while SINDy-AE only provides a point +estimate in both tasks. +We also compare our ROAD-EnKF method to AD-EnKF [20]. +Although AD-EnKF enjoys some of +the benefits of ROAD-EnKF, including the capability to learn from noisy, partially observed data and +perform uncertainty quantification, it directly learns the dynamics model in high-dimensional state space +(i.e., on ut’s instead of zt’s), which leads to higher model complexity, as well as additional computational +and memory costs when performing the EnKF step. Moreover, AD-EnKF does not take advantage of the +possible low-dimensional representation of the state. We compare in Table 5.1 below the capabilities of the +three algorithms under different scenarios. +Learn from noisy +and partially observed data +Uncertainty +quantification +No need of +time-derivative data +Low-dimensional +state representation +SINDy-AE [16] + + + + +AD-EnKF [20] + + + + +ROAD-EnKF (this paper) + + + + +Table 5.1: Comparison of SINDy-AE, AD-EnKF, and ROAD-EnKF under different scenarios. +Other alternative methods include EnKF-embedded EM algorithms (e.g. [9]) and autodifferentiable PF +algorithms (e.g., [65]). Since [20] already establishes AD-EnKF’s superiority to those approaches, we do not +include them in these experiments, and we refer to [20] for more details. +The training procedure is the following: We first specify a forecast lead time Tf. We then generate +12 + +training data yItrain +0:T +and test data with extended time range (uItest,∗ +0:T +Tf , yItest +0:T ) with Ntrain := |Itrain| and +Ntest := |Itest|. The data are either generated from a reduced-order SSM (2.6)-(2.9) with explicit knowledge +of true parameter θ (Subsection 5.1), or from an SSM (2.1)-(2.3) with no explicit knowledge of the exact +reduced-order structure (Subsections 5.2 and 5.3). The data yItrain +0:T +and yItest +0:T +are then passed into ROAD- +EnKF (Algorithm 3.2), and we evaluate the following: +Reconstruction-RMSE (RMSE-r): +Measures the state reconstruction error of the algorithm. We take +the particle mean of uItest,1:N +0:T +as a point estimate of the true states uItest,∗ +0:T +, and evaluate the RMSE: +RMSE-r = +� +� +� +� +1 +duNtest(T − Tb) +T +� +t=Tb +� +i∈Itest +���ui +t − ui,∗ +t +��� +2 +, +where ui +t = 1 +N +N +� +n=1 +ui,n +t . +(5.1) +Here Tb is a number of burn-in steps to remove transient errors in the reconstruction that stem from the +choice of initialization. For simplicity, we set Tb = ⌊T/5⌋ as in [20]. +Forecast-RMSE (RMSE-f): +Measures the t-step state forecast error of the algorithm, for lead time +t ∈ {1, . . . , Tf}. We take the particle mean of uItest,1:N +T +t +as a point estimate of the true future states uItest,∗ +T +t +: +RMSE-f(t) = +� +1 +duNtest +� +i∈Itest +���ui +T +t − ui,∗ +T +t +��� +2 +, +where ui +T +t = 1 +N +N +� +n=1 +ui,n +T +t . +(5.2) +Test Log-Likelihood: +Measures the averaged log-likelihood of the learned reduced-order SSM over test +observation data yItest +0:T , which is LEnKF(θk) defined in Line 8 of Algorithm 3.2. +For AD-EnKF, the above metrics can be similarly computed, following [20]. For SINDy-AE, as uncer- +tainty quantification is not performed, we use its decoder output as the point estimate of the state in both +reconstruction and forecast. Moreover, log-likelihood computation is not available for SINDy-AE. +5.1 +Embedding of Chaotic Dynamics (Lorenz 63) +In this subsection, we reconstruct and forecast a state defined by embedding a Lorenz 63 (L63) model in a +high-dimensional state space. A similar experiment was used in [16] to motivate the SINDy-AE algorithm, +and hence this example provides a good point of comparison. The data are generated using the L63 system +as the true latent state dynamics model: +dz +ds = g(z), +� +� +� +� +� +g(1)(z) = 10(z(2) − z(1)), +g(2)(z) = z(1)(28 − z(3)) − z(2), +g(3)(z) = z(1)z(2) − 8 +3z(3), +G : z(s) �→ z(s + ∆s), +(5.3) +where z(i) and g(i) denote the i-th coordinate of z and component of g, and ∆s is the time between ob- +servations. We further assume there is no noise in the true latent state dynamics model, i.e., S = 0. To +construct the true reduced-order SSM, we define D ∈ Rdu×6 such that its i-th column Di ∈ Rdu is given by +the discretized i-th Legendre polynomial over du grid points. The true states ut ∈ Rdu are defined by +ut := D +� +z(1) +t +/40 +z(2) +t +/40 +z(3) +t +/40 +(z(1) +t +/40)3 +(z(2) +t +/40)3 +(z(3) +t +/40)3 +�⊤ +. +(5.4) +We consider two cases of the observation model (2.8): (1) full observation, where all coordinates of ut are +observed, i.e., Ht = Idu and dy = du; (2) partial observation, where for each t, only a fixed portion c < 1 of +all coordinates of ut are observed, and the coordinate indices are chosen randomly without replacement. In +this case, Ht ∈ Rdy×du is a submatrix of Idu and varies across time, and dy = cdu. This partial observation +13 + +set-up has been studied in the literature (e.g., [7, 9]) for data assimilation problems. For both cases, we +assume Rt = 0.01Idy and z0 ∼ N(0, 4Idz). +We consider full observation with du = dy = 128 and partial observation with du = 128, dy = 64 +(i.e., c = 1/2). We generate Ntrain = 1024 training data and Ntest = 20 test data with the true reduced- +order SSM defined by (5.3) and (5.4). +We set the number of observations T = 250 with time between +observations ∆s = 0.1. +We set the forecast lead time Tf = 10. +The latent flow map G is integrated +using the Runge–Kutta–Fehlberg method. The surrogate latent dynamics map gα is parameterized as a +two-layer fully connected NN, and is integrated using a fourth-order Runge-Kutta method with step size +∆int +s += 0.05. The error covariance matrix Sβ in the latent dynamics is parametrized using a diagonal matrix +with positive diagonal elements β ∈ Rdz. The decoder Dγ is parameterized as a Fourier Neural Decoder +(FND) discussed in Subsection 4.1. Details of the network hyperparameters for this and subsequent examples +are summarized in Table 5.2, obtained through cross-validation experiments on the training dataset. The +latent space dimension for both SINDy-AE and ROAD-EnKF is set to dz = 3. The ensemble size for both +AD-EnKF and ROAD-EnKF is set to N = 100. +L63 +Burgers +KS +FND +L +4 +2 +4 +h +6 +40 +40 +(n0, . . . , nL) +(1, 20, 20, 20, 20) +(1, 20, 20) +(1, 20, 20, 20, 20) +Norm +LayerNorm +Activation +ReLU +Latent space reg. +σ +2 +4 +4 +Optimization +Optimizer +Adam +Learning rate (η) +1e-3 +Batch size (B) +16 +4 +4 +TBPTT length +10 +Table 5.2: Choices of hyperparameters for ROAD-EnKF on different numerical examples. +In Table 5.3 we list the performance metrics of each method with full and partial observation. The state +reconstruction and forecast performance on a single instance of test data are plotted in Figure 4 and 5 for +the full observation case, and in Figure 6 for the partial observation case. For the full observation case, we +compare ROAD-EnKF with AD-EnKF and SINDy-AE, adopting for the latter the implementation in [16]. +Since SINDy-AE requires time-derivative data as input, we use a finite difference approximation computed +from data y1:T . We also include the results for SINDy-AE where the exact time-derivative data are used. +We find that ROAD-EnKF is able to reconstruct and forecast the states consistently with the lowest RMSE, +and the performance is not affected by whether the state is fully or partially observed. AD-EnKF is able +to reconstruct and forecast the state with a higher RMSE than that of ROAD-EnKF, and the performance +deteriorates in the partially observed setting. SINDy-AE with finite difference approximation of derivative +data also achieves higher reconstruction RMSE than that of ROAD-EnKF, and does not give accurate state +forecasts. This is likely due to the fact that data are sparse in time (i.e., ∆s is large) which leads to a larger +error when approximating the true time-derivative, and hence it is more difficult to extract meaningful +dynamics from the data. Even when the true time-derivative data are used (which is not available unless +we have explicit knowledge of the true reduced-order SSM), SINDy-AE has a higher reconstruction RMSE +compared to ROAD-EnKF, and its forecast performance is still worse than the other two methods. Moreover, +it cannot handle partial observation. +In terms of computational cost, ROAD-EnKF is more efficient than AD-EnKF since the surrogate dy- +namics are cheaper to simulate and the EnKF algorithm is more efficient to perform in both training and +testing. However, ROAD-EnKF takes more time than SINDy-AE, since the latter does not rely on a filtering +algorithm, but rather an encoder, to reconstruct the states and perform learning. +14 + +SINDy-AE +(full) +SINDy-AE +(w/ derivative, full) +AD-EnKF +(full) +ROAD-EnKF +(full) +AD-EnKF +(partial) +ROAD-EnKF +(partial) +RMSE-r +0.0142 +0.0148 +0.0168 +0.0078 +0.0368 +0.0079 +RMSE-f(1) +0.1310 +0.0191 +0.0156 +0.0069 +0.0315 +0.0069 +RMSE-f(5) +1.6580 +0.0333 +0.0335 +0.0141 +0.0729 +0.0125 +Log-likelihood +− +2.25 × 104 +2.58 × 104 +1.28 × 104 +1.40 × 104 +Training time (per epoch) +5.15s +9.74s +6.15s +8.86s +5.62s +Test time +2.35s +4.57s +2.95s +4.52s +2.73s +Table 5.3: Performance metrics for different algorithms at convergence. (Embedded L63 example, Subsection 5.1.) +0 +32 +64 +96 +128 +−0.5 +0.0 +0.5 +1.0 +1.5 +SINDy-AE +rmse: 0.0095 +t=40, +Reconstruction +0 +32 +64 +96 +128 +rmse: 0.0120 +t=80, +Reconstruction +0 +32 +64 +96 +128 +rmse: 0.0102 +t=120, +Reconstruction +0 +32 +64 +96 +128 +rmse: 0.0204 +t=160, +Reconstruction +0 +32 +64 +96 +128 +rmse: 0.0193 +t=200, +Reconstruction +0 +32 +64 +96 +128 +−0.5 +0.0 +0.5 +1.0 +1.5 +SINDy-AE +w/ derivative data +rmse: 0.0137 +0 +32 +64 +96 +128 +rmse: 0.0097 +0 +32 +64 +96 +128 +rmse: 0.0125 +0 +32 +64 +96 +128 +rmse: 0.0251 +0 +32 +64 +96 +128 +rmse: 0.0182 +0 +32 +64 +96 +128 +−0.5 +0.0 +0.5 +1.0 +1.5 +AD-EnKF +rmse: 0.0217 +0 +32 +64 +96 +128 +rmse: 0.0107 +0 +32 +64 +96 +128 +rmse: 0.0360 +0 +32 +64 +96 +128 +rmse: 0.0103 +0 +32 +64 +96 +128 +rmse: 0.0120 +0 +32 +64 +96 +128 +−0.5 +0.0 +0.5 +1.0 +1.5 +ROAD-EnKF +rmse: 0.0068 +0 +32 +64 +96 +128 +rmse: 0.0029 +0 +32 +64 +96 +128 +rmse: 0.0181 +0 +32 +64 +96 +128 +rmse: 0.0025 +0 +32 +64 +96 +128 +rmse: 0.0032 +Observation +Reconstruction +Truth +Figure 4: State reconstruction performance with full observation (du = dy = 128) on the embedded L63 example +in Subsection 5.1. For each method (row), the reconstructed states ut (blue) for a single test sequence are plotted +for t = 40, 80, 120, 160, 200 (column). +The true values of the 128-dimensional states are plotted in red dashed +lines, along with the noisy observations in black dots. +Both AD-EnKF and ROAD-EnKF perform probabilistic +state reconstructions through particles (all plotted in blue), while SINDy-AE only provides point estimates. The +reconstruction RMSE’s are computed for each plot. For SINDy-AE, even with derivative data (not required for AD- +EnKF and ROAD-EnKF), the reconstruction performance is similar to that of AD-EnKF, while being worse than +that of ROAD-EnKF. +5.2 +Burgers Equation +In this subsection and the following one, we learn high-dimensional SSMs without explicit reference to a +true model for low-dimensional latent dynamics. We first consider the 1-dimensional Burgers equation for +15 + +0 +32 +64 +96 +128 +−0.5 +0.0 +0.5 +1.0 +SINDy-AE +rmse: 0.0215 +t=250, +Forecast (Start) +0 +32 +64 +96 +128 +rmse: 0.1282 +t=252, +Forecast +0 +32 +64 +96 +128 +rmse: 0.3571 +t=254, +Forecast +0 +32 +64 +96 +128 +rmse: 0.4359 +t=256, +Forecast +0 +32 +64 +96 +128 +rmse: 0.2722 +t=258, +Forecast +0 +32 +64 +96 +128 +−0.5 +0.0 +0.5 +1.0 +SINDy-AE +w/ derivative data +rmse: 0.0167 +0 +32 +64 +96 +128 +rmse: 0.0209 +0 +32 +64 +96 +128 +rmse: 0.0509 +0 +32 +64 +96 +128 +rmse: 0.0620 +0 +32 +64 +96 +128 +rmse: 0.0316 +0 +32 +64 +96 +128 +−0.5 +0.0 +0.5 +1.0 +AD-EnKF +rmse: 0.0149 +0 +32 +64 +96 +128 +rmse: 0.0148 +0 +32 +64 +96 +128 +rmse: 0.0119 +0 +32 +64 +96 +128 +rmse: 0.0170 +0 +32 +64 +96 +128 +rmse: 0.0235 +0 +32 +64 +96 +128 +−0.5 +0.0 +0.5 +1.0 +ROAD-EnKF +rmse: 0.0035 +0 +32 +64 +96 +128 +rmse: 0.0026 +0 +32 +64 +96 +128 +rmse: 0.0034 +0 +32 +64 +96 +128 +rmse: 0.0037 +0 +32 +64 +96 +128 +rmse: 0.0032 +Forecast +Truth +Figure 5: Forecast performance with full observation (du = dy = 128) on the embedded L63 example in Subsection +5.1. For each method (row), the forecasted states ut (blue) for a single test sequence are plotted for t = 250 (start of +forecast), 252, 254, 256, 258 (column). The true values of the du = 128 dimensional states are plotted in red dashed +lines. Both AD-EnKF and ROAD-EnKF perform probabilistic forecast through particles (all plotted in blue), while +SINDy-AE only provides point estimates. The forecast RMSE’s are computed for each plot. For SINDy-AE, even +with derivative data (not required for AD-EnKF and ROAD-EnKF), the forecast performance is similar to that of +AD-EnKF, while being worse than that of ROAD-EnKF. +u(x, s), where u is a function of the spatial variable x ∈ [0, L] and continuous-time variable s > 0: +∂u +∂s = −u∂u +∂x + ν ∂2u +∂x2 , +u(0, s) = u(L, s) = 0, +u(x, 0) = u0(x). +(5.5) +Here ν is the viscosity parameter, and we set ν = 1/150, L = 2. Burgers equation [11] has various applications +in fluid dynamics, including modeling of viscous flows. We are interested in reconstructing solution states, +as well as in the challenging problem of forecasting shocks that emerge outside the time range covered by +the training data. Equation (5.5) is discretized on [0, L] with equally-spaced grid points 0 = x1 < x2 < · · · < +xM = L, using a second-order finite difference method. Setting ∆x := xi − xi−1 = +L +M−1, we obtain the +16 + +0 +32 +64 +96 +128 +−1.0 +−0.5 +0.0 +0.5 +1.0 +AD-EnKF +rmse: 0.0394 +t=40, +Reconstruction +0 +32 +64 +96 +128 +rmse: 0.0298 +t=80, +Reconstruction +0 +32 +64 +96 +128 +rmse: 0.0375 +t=120, +Reconstruction +0 +32 +64 +96 +128 +rmse: 0.0362 +t=160, +Reconstruction +0 +32 +64 +96 +128 +rmse: 0.0378 +t=200, +Reconstruction +0 +32 +64 +96 +128 +−1.0 +−0.5 +0.0 +0.5 +1.0 +ROAD-EnKF +rmse: 0.0042 +0 +32 +64 +96 +128 +rmse: 0.0033 +0 +32 +64 +96 +128 +rmse: 0.0097 +0 +32 +64 +96 +128 +rmse: 0.0100 +0 +32 +64 +96 +128 +rmse: 0.0039 +Observation +Reconstruction +Truth +0 +32 +64 +96 +128 +0 +1 +2 +AD-EnKF +rmse: 0.0357 +t=250, +Forecast (Start) +0 +32 +64 +96 +128 +rmse: 0.0315 +t=252, +Forecast +0 +32 +64 +96 +128 +rmse: 0.0164 +t=254, +Forecast +0 +32 +64 +96 +128 +rmse: 0.0335 +t=256, +Forecast +0 +32 +64 +96 +128 +rmse: 0.1820 +t=258, +Forecast +0 +32 +64 +96 +128 +0 +1 +2 +ROAD-EnKF +rmse: 0.0064 +0 +32 +64 +96 +128 +rmse: 0.0048 +0 +32 +64 +96 +128 +rmse: 0.0024 +0 +32 +64 +96 +128 +rmse: 0.0037 +0 +32 +64 +96 +128 +rmse: 0.0108 +Forecast +Truth +Figure 6: State reconstruction (upper half) and forecast (lower half) performance with partial observation (du = 128, +dy = 64) on the embedded L63 example in Subsection 5.1. For each method, the reconstructed states ut (blue) for +a single test sequence are plotted for t = 40, 80, 120, 160, 200 (column), and the forecasted states ut (blue) for a +single test sequence are plotted for t = 250 (start of forecast), 252, 254, 256, 258 (column). The true values of the +128-dimensional states are plotted in red dashed lines, along with the noisy observations in black dots. SINDy-AE +is inapplicable here because it cannot handle partial observations, while both AD-EnKF and ROAD-EnKF perform +probabilistic state reconstructions and forecast through particles (all plotted in blue). The reconstruction/forecast +RMSEs are computed for each plot. +following ODE system: +du(i) +ds += − +� +u(i+1)�2 − +� +u(i−1)�2 +4∆x ++ ν u(i+1) − 2u(i) + u(i−1) +∆x2 +, +i = 2, . . . , M − 1, +u(1)(s) = u(M)(s) = 0, +u(i)(0) = u0(i∆x). +(5.6) +Here u(i)(s) is an approximation of u(i∆x, s), the value of u at the i-th spatial node at time s. Equation +(5.6) defines a flow map F : u(s) �→ u(s + ∆s) for state variable u with du = M, which we refer to as the +true state dynamics model. We assume there is no noise in the dynamics, i.e., Q = 0. +17 + +Similar to Subsection 5.1, we consider two cases: full observation with du = dy = 256 and partial +observation with du = 256, dy = 128 (i.e., c = 1/2). The initial conditions u0 are generated in the following +way: +u(i) +0 += U sin 2πi∆x +L +, +U ∼ Uniform(0.5, 1.5). +(5.7) +We generate Ntrain = 1024 training data and Ntest = 20 test data with the true state dynamics model defined +through equation (5.6) and (5.7) with Rt = 0.01Idy. We set the number of observations T = 300 with time +between observations ∆s = 0.001. We set the forecast lead time Tf = 300. The flow map F is integrated +using the fourth-order Runge–Kutta method with a fine step size ∆s/20. The surrogate latent dynamics map +gα is parameterized as a two-layer fully connected NN, and is integrated using a fourth-order Runge-Kutta +method with step size ∆int +s += 0.001. The error covariance matrix Sβ in the latent dynamics is parametrized +using a diagonal matrix with positive diagonal elements β ∈ Rdz. The decoder Dγ is parameterized as an +FND, discussed in Subsection 4.1. Details of the network hyperparameters are listed in Table 5.2. The latent +space dimension for ROAD-EnKF is set to dz = 40. The ensemble size for both AD-EnKF and ROAD-EnKF +is set to N = 100. In this example and the following one, we set z0 ∼ N(0, σ2Idz) with the same σ defined +in Subsection 4.3. +In Table 5.4, we list the performance metrics of each method with full and partial observation. The state +reconstruction and forecast performance on a single instance of test data are plotted in Figures 7 (snapshots) +and 8 (contour plot) for the partial observation case. Corresponding plots with full observation are shown in +Figures 11 and 12 in the appendix. We find that ROAD-EnKF is able to reconstruct and forecast the states +with the lowest RMSE, in both full and partial observation scenarios. More importantly, the emergence +of shocks is accurately forecasted even though this phenomenon is not included in the time range covered +by the training data. AD-EnKF achieves a higher RMSE than ROAD-EnKF for both state reconstruction +and forecast tasks. AD-EnKF forecasts the emergence of shocks with lower accuracy than ROAD-EnKF, +which indicates that AD-EnKF fails to fully learn the state dynamics. SINDy-AE with finite difference +approximation of derivative data has the highest reconstruction RMSE among the three methods, and is +not able to produce meaningful long-time state forecasts. This is remarkable, given that in this example the +data are relatively dense (∆s is small) which facilitates, in principle, the approximation of time derivatives. +In terms of computational cost, ROAD-EnKF is more efficient than AD-EnKF during both training and +testing, but takes more time than SINDy-AE for the same reason as in Subsection 5.1. +In Table 5.5, we list the performance metrics of ROAD-EnKF with full observation and different choices of +latent space dimension dz ranging from 1 to 240. The results for partial observation show a similar trend and +are not shown. We find that, as dz increases, the state reconstruction performance stabilizes when dz ≥ 4. +In order to achieve better long-time state forecast performance, dz needs to be further increased, and the +forecast performance stabilizes when dz ≥ 10. Both training and testing time slightly increase as dz grows, +which can be explained by the following: The computational time for both training and testing can be divided +into the prediction step and the analysis step. We have shown in Subsection 4.2 that the computational +bottleneck of the analysis step depends on the choices of ensemble size N and dy, and is less affected by the +increase of dz. Moreover, the computational time of the prediction step depends on the complexity of the +surrogate latent dynamics (two-layer NNs), which are relatively cheap to simulate for ROAD-EnKF. On the +other hand, AD-EnKF enjoys similar computational complexity as ROAD-EnKF during the analysis step, +but requires a more complicated surrogate model (NNs with Fourier layers) to capture the dynamics, which +is more expensive to simulate. More experimental results on different parameterization methods of surrogate +dynamics can be found in Table B.1 in the appendix. +18 + +SINDy-AE +(full) +AD-EnKF +(full) +ROAD-EnKF +(full) +AD-EnKF +(partial) +ROAD-EnKF +(partial) +RMSE-r +0.1433 +0.0102 +0.0044 +0.0122 +0.0081 +RMSE-f(30) +4.4579 +0.0212 +0.0096 +0.0228 +0.0160 +RMSE-f(150) +4.4906 +0.0763 +0.0514 +0.0724 +0.0581 +Log-likelihood +− +6.40 × 104 +6.60 × 104 +3.24 × 104 +3.27 × 104 +Training time (per epoch) +11.78s +26.75s +12.10s +27.08s +12.20s +Test time +2.78s +11.54s +4.21s +7.76s +3.24s +Table 5.4: Performance metrics for different algorithms at convergence. (Burgers example, Subsection 5.2.) +0 +64 +128 +192 +256 +−1.0 +−0.5 +0.0 +0.5 +1.0 +AD-EnKF +rmse: 0.0168 +t=50, +Reconstruction +0 +64 +128 +192 +256 +rmse: 0.0104 +t=100, +Reconstruction +0 +64 +128 +192 +256 +rmse: 0.0136 +t=150, +Reconstruction +0 +64 +128 +192 +256 +rmse: 0.0148 +t=200, +Reconstruction +0 +64 +128 +192 +256 +rmse: 0.0092 +t=250, +Reconstruction +0 +64 +128 +192 +256 +−1.0 +−0.5 +0.0 +0.5 +1.0 +ROAD-EnKF +rmse: 0.0089 +0 +64 +128 +192 +256 +rmse: 0.0072 +0 +64 +128 +192 +256 +rmse: 0.0055 +0 +64 +128 +192 +256 +rmse: 0.0049 +0 +64 +128 +192 +256 +rmse: 0.0056 +Observation +Reconstruction +Truth +0 +64 +128 +192 +256 +−0.5 +0.0 +0.5 +1.0 +AD-EnKF +rmse: 0.0095 +t=300, +Forecast (Start) +0 +64 +128 +192 +256 +rmse: 0.0345 +t=375, +Forecast +0 +64 +128 +192 +256 +rmse: 0.0537 +t=450, +Forecast +0 +64 +128 +192 +256 +rmse: 0.0849 +t=525, +Forecast +0 +64 +128 +192 +256 +rmse: 0.1171 +t=600, +Forecast +0 +64 +128 +192 +256 +−0.5 +0.0 +0.5 +1.0 +ROAD-EnKF +rmse: 0.0070 +0 +64 +128 +192 +256 +rmse: 0.0130 +0 +64 +128 +192 +256 +rmse: 0.0251 +0 +64 +128 +192 +256 +rmse: 0.0415 +0 +64 +128 +192 +256 +rmse: 0.0616 +Forecast +Truth +Figure 7: State reconstruction (upper half) and forecast (lower half) performance with partial observation (du = 256, +dy = 128) on the Burgers example in Subsection 5.2. For each method, the reconstructed states ut (blue) for a +single test sequence are plotted for t = 50, 100, 150, 200, 250 (column), and the forecasted states (blue) for a single +test sequence are plotted for t = 300 (start of forecast), 375, 450, 525, 600 (column). The true values of the 256- +dimensional states are plotted in red dashed lines, along with the noisy observations in black dots. Both AD-EnKF +and ROAD-EnKF perform probabilistic state reconstructions and forecast through particles (all plotted in blue). The +reconstruction/forecast RMSEs are computed for each plot. +19 + +(a) Ground truth. +(b) Reconstruction and forecast. +Figure 8: Contour plot of state reconstruction and forecast output with partial observation (du = 256, dy = 128) on +the Burgers example in Subsection 5.2, as well as the ground truth (top). For each method (row), the reconstructed +and forecasted states (left column) for a single test sequence are plotted, for each state dimension (y-axis) and +time (x-axis). The error compared to the ground truth are plotted in the right column. For both AD-EnKF and +ROAD-EnKF we use particle means as point estimates. +ROAD-EnKF +AD-EnKF +dz = 1 +dz = 2 +dz = 4 +dz = 10 +dz = 20 +dz = 40 +dz = 120 +dz = 240 +RMSE-r +0.2293 +0.0316 +0.0035 +0.0058 +0.0039 +0.0044 +0.0048 +0.0059 +0.0102 +RMSE-f(30) +0.2593 +0.0761 +0.0165 +0.0108 +0.0112 +0.0096 +0.0100 +0.0103 +0.0212 +RMSE-f(150) +0.2690 +0.1827 +0.1313 +0.0501 +0.0607 +0.0514 +0.0373 +0.0382 +0.0763 +Log-likelihood (×104) +-14.4 +6.03 +6.63 +6.59 +6.61 +6.60 +6.61 +6.63 +6.40 +Training time +(per epoch) +11.70s +11.78s +11.79s +11.98s +11.98s +12.10s +12.44s +13.40s +26.75s +Test time +3.36s +3.80s +4.13s +4.14s +4.08s +4.21s +4.53s +4.90s +11.54s +Table 5.5: Performance metrics for ROAD-EnKF at convergence with full observation (du = dy = 256) and different +latent space dimension dz. (Burgers example, Subsection 5.2.) +20 + +250 +0.6 +200 +0.4 +0.2 +Truth +150 +0.0 +100 +-0.2 +-0.4 +50 +-0.6 +0 +0 +100 +200 +300 +400 +500t<300 +t>300 +t<300 +t>300 +Reconstruction +Forecast +Reconstruction Error +Forecast Error +250 +250 + 0.6 + 0.6 +200 +200 +0.4 + 0.4 +0.2 +0.2 +150 +150 +0.0 +0.0 +2100 +100 +-0.2 +-0.2 +-0.4 +-0.4 +50 +50 +0.6 +0.6 +100 +200 +300 +400 +500 +100 +200 +300 +400 +500 +250 +250 +0.6 +0.6 +200 +200 + 0.4 +0.4 + 0.2 +0.2 +150 +150 +0.0 +0.0 +100 +100 +0.2 +0.2 +0.4 +-0.4 +50 +50 +-0.6 +-0.6 +0 + +0 +100 +200 +300 +400 +500 +100 +200 +300 +400 +5005.3 +Kuramoto-Sivashinsky Equation +In this subsection, we consider the Kuramoto-Sivashinsky (KS) equation for u(x, s), where u is a function of +the spatial variable x ∈ [0, L] and continuous-time variable s > 0: +∂u +∂s = −ν ∂4u +∂x4 − ∂2u +∂x2 − u∂u +∂x, +u(0, s) = u(L, s) = 0, +∂u +∂x(0, s) = ∂u +∂x(L, s) = 0, +u(x, 0) = u0(x), +(5.8) +Here ν is the viscosity parameter, and we set ν = 0.05, L = 2. We impose Dirichlet and Neumann boundary +conditions to ensure ergodicity of the system [6]. The KS equation was originally introduced by Kuramoto and +Sivashinsky to model turbulence of reaction-diffusion systems [50] and propagation of flame [77]. Equation +(5.8) is discretized on [0, L] with equally-spaced grid points 0 = x1 < x2 < · · · < xM = L, using a second-order +finite difference method. Setting ∆x := xi − xi−1 = +L +M−1, we obtain the following ODE system: +∂u(i) +∂s += −ν u(i−2) − 4u(i−1) + 6u(i) − 4u(i+2) + u(i+2) +∆x4 +− u(i+1) − 2u(i) + u(i−1) +∆x2 +− +� +u(i+1)�2 − +� +u(i−1)�2 +4∆x +, +i = 2, . . . , du − 1, +u(1)(s) = u(du)(s) = 0, +u(0)(s) = u(2)(s), u(du+1)(s) = u(du−1)(s), +u(i)(0) = u0(i∆x). +(5.9) +The discretization method follows [89]. Here u(i)(s) is an approximation of u(i∆x, s), the value of u at the +i-th spatial node and time s. Two ghost nodes u(0) and u(du+1) are added to account for Neumann boundary +conditions, and are not regarded as part of the state. Equation (5.9) defines a flow map F : u(s) �→ u(s+∆s) +for state variable u with du = M, which we refer to as the true state dynamics model. We assume there is +no noise in the dynamics, i.e., Q = 0. +Similar to Subsection 5.1, we consider two cases: full observation with du = dy = 256 and partial +observation with du = 256, dy = 128 (i.e., c = 1/2). The initial conditions u0 are generated at random +from the attractor of the dynamical system, by simulating a long run beforehand. We generate Ntrain = 512 +training data and Ntest = 20 test data with the true state dynamics model defined through (5.9) with +Rt = Idy. We set the number of observations T = 450 with time between observations ∆s = 0.1. We set +the forecast lead time Tf = 50. The flow map F is integrated using the fourth-order Runge–Kutta method +with a fine step size ∆s/10000. The surrogate latent dynamics map gα is parameterized as a two-layer +fully connected NN, and is integrated using a fourth-order Runge-Kutta method with step size ∆int +s += 0.05. +The error covariance matrix Sβ in the latent dynamics is parametrized using a diagonal matrix with positive +diagonal elements β ∈ Rdz. The decoder Dγ is parameterized as an FND, discussed in Subsection 4.1. Details +of the network hyperparameters are listed in Table 5.2. The latent space dimension for ROAD-EnKF is set +to dz = 40. The ensemble size for both AD-EnKF and ROAD-EnKF is set to N = 100. +In Table 5.6 we list the performance metrics of AD-EnKF and ROAD-EnKF with full and partial obser- +vation. SINDy-AE is not listed here as we find it unable to capture the dynamics for any choice of latent +space dimension. The state reconstruction and forecast performance on a single instance of test data are +plotted in Figure 9 (snapshots), and Figure 10 (contour plot, ROAD-EnKF) for the partial observation case. +Corresponding plots with full observation are shown in Figures 13 and 14 in the appendix. We find that +ROAD-EnKF is able to reconstruct the states with lower RMSE than AD-EnKF in both full observation and +partial observation cases. Both methods can produce meaningful forecast multiple steps forward into the +21 + +future. ROAD-EnKF achieves a higher forecast RMSE than AD-EnKF in full observation case, while having +a lower forecast RMSE in partial observation case. Although ROAD-EnKF does not consistently have a +better forecast performance than AD-EnKF due to the difficulty of finding a reduced-order representation +for the highly chaotic system, we find that its performance is not much impacted by partial observation. +Moreover, it is two times more efficient than AD-EnKF in both training and testing, due to the times saved +for simulating a cheaper surrogate model and running the EnKF algorithm in a lower dimensional space. +Notice in Figure 10(b) and Figure 14(b) that, although the predictive means of all particles are ‘smoothed’ +when passing a certain time threshold, each particle individually produces nontrivial forecasts for a larger +number of time steps into the future, thus illustrating the variability of particle forecasts and the stochastic +nature of state reconstruction and forecast in our ROAD-EnKF framework. +AD-EnKF +(full) +ROAD-EnKF +(full) +AD-EnKF +(partial) +ROAD-EnKF +(partial) +RMSE-r +0.4658 +0.3552 +0.4686 +0.3589 +RMSE-f(1) +0.5137 +0.5626 +0.6231 +0.5644 +RMSE-f(5) +1.0910 +1.2734 +1.4669 +1.3780 +Log-likelihood +−1.89 × 106 +−1.88 × 106 +−9.33 × 105 +−9.07 × 105 +Training time (per epoch) +28.92s +12.53s +28.72s +12.61s +Test time +12.35s +5.11s +6.22s +4.39s +Table 5.6: Performance metrics for different algorithms at convergence. (KS example, Subsection 5.3.) +6 +Conclusions and Future Directions +This paper introduced a computational framework to reconstruct and forecast a partially observed state +that evolves according to an unknown or expensive-to-simulate dynamical system. Our ROAD-EnKFs use +an EnKF algorithm to estimate by maximum likelihood a surrogate model for the dynamics in a latent +space, as well as a decoder from latent space to state space. Our numerical experiments demonstrate the +computational advantage of co-learning an inexpensive surrogate model in latent space together with a +decoder, rather than a more expensive-to-simulate dynamics in state space. +The proposed computational framework accommodates partial observation of the state, does not require +time derivative data, and enables uncertainty quantification. In addition, it provides significant algorithmic +flexibility through the choice of latent space, surrogate model for the latent dynamics, and decoder design. +In this work, we showed that accurate and cheap reconstructions and forecasts can be obtained by choosing +an inexpensive NN surrogate model, and a decoder inspired by recent ideas from operator learning. While +adequate choice of NN architecture and decoder may be problem-specific, an important question for further +research is to derive guidelines and physics-informed NNs that are well-suited for certain classes of problems. +Acknowledgments +The authors are grateful to Melissa Adrian for her generous feedback on an earlier version of this manuscript. +YC was partially supported by NSF DMS-2027056 and NSF OAC-1934637. DSA is grateful for the support of +NSF DMS-2237628, NSF DMS-2027056, DOE DE-SC0022232, and the BBVA Foundation. RW is grateful for +the support of DOD FA9550-18-1-0166, DOE DE-AC02-06CH11357, NSF OAC-1934637, NSF DMS-1930049, +and NSF DMS-2023109. +22 + +0 +64 +128 +192 +256 +−15 +−10 +−5 +0 +5 +10 +AD-EnKF +rmse: 0.4478 +t=50, +Reconstruction +0 +64 +128 +192 +256 +rmse: 0.5089 +t=150, +Reconstruction +0 +64 +128 +192 +256 +rmse: 0.4132 +t=250, +Reconstruction +0 +64 +128 +192 +256 +rmse: 0.3472 +t=350, +Reconstruction +0 +64 +128 +192 +256 +rmse: 0.5187 +t=450, +Reconstruction +0 +64 +128 +192 +256 +−15 +−10 +−5 +0 +5 +10 +ROAD-EnKF +rmse: 0.3306 +0 +64 +128 +192 +256 +rmse: 0.4324 +0 +64 +128 +192 +256 +rmse: 0.3293 +0 +64 +128 +192 +256 +rmse: 0.3488 +0 +64 +128 +192 +256 +rmse: 0.2349 +Observation +Reconstruction +Truth +0 +64 +128 +192 +256 +−10 +−5 +0 +5 +10 +AD-EnKF +rmse: 0.5187 +t=450, +Forecast (Start) +0 +64 +128 +192 +256 +rmse: 0.9386 +t=452, +Forecast +0 +64 +128 +192 +256 +rmse: 0.9549 +t=454, +Forecast +0 +64 +128 +192 +256 +rmse: 1.7769 +t=456, +Forecast +0 +64 +128 +192 +256 +rmse: 2.5097 +t=458, +Forecast +0 +64 +128 +192 +256 +−10 +−5 +0 +5 +10 +ROAD-EnKF +rmse: 0.2349 +0 +64 +128 +192 +256 +rmse: 0.4737 +0 +64 +128 +192 +256 +rmse: 0.6419 +0 +64 +128 +192 +256 +rmse: 1.0449 +0 +64 +128 +192 +256 +rmse: 1.6044 +Forecast +Truth +Figure 9: State reconstruction (upper half) and forecast (lower half) performance with partial observation (du = 256, +dy = 128) on the KS example in Subsection 5.3. 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Zhang, A Review of Recurrent Neural Networks: LSTM Cells and +Network Architectures, Neural Computation, 31 (2019), pp. 1235–1270. +A +Improving AD-EnKF with Spectral Convolutional Layers +This appendix discusses an enhancement of the AD-EnKF algorithm [20], used for numerical comparisons in +Section 5. AD-EnKF runs EnKF on the full-order SSM (2.1)-(2.3) and learns the parameter θ = (α⊤, β⊤)⊤ +by auto-differentiating through a similarly defined log-likelihood objective, as in Subsection 3.2. A high +dimension of u makes challenging the NN parameterization of Fα (resp. fα in the ODE case) in the state +dynamics model (2.1). +In particular, the local convolutional NN used in [20] does not perform well in +the high-dimensional numerical experiments considered in Section 5. We thus propose a more flexible NN +parameterization of Fα (resp. fα) using the idea of spectral convolutional layers. +We design Fα (resp. +fα) in a way similar to the Fourier Neural Decoder, but without the complex +linear layer and IDFT step at the beginning. That is, we start with a state variable u′ ∈ Rdu as the input, +iteratively apply (4.2) with v0 = u′ to get vL ∈ RnL×du, followed by a fully-connected network applied over +the channel dimension to get the output u ∈ Rdu. The architecture is the same as Figure 3(a) but we start +at v0 instead of z. +B +Additional Materials: Burgers Example +For SINDy-AE, we use a finite difference approximation computed from data y1:T to approximate the exact +time-derivative. The latent space dimension for SINDy-AE is set to 6. Increasing it does not further enhance +the performance, but increases the computational cost. +AD-EnKF +(FC, Euler) +AD-EnKF +(FC, RK4) +AD-EnKF +(Fourier, Euler) +AD-EnKF +(Fourier, RK4) +ROAD-EnKF +(FC, Euler) +ROAD-EnKF +(FC, RK4) +RMSE-r +0.1023 +0.0934 +0.0537 +0.0102 +0.0045 +0.0044 +RMSE-f(30) +0.0999 +0.0831 +0.1302 +0.0212 +0.0100 +0.0096 +RMSE-f(150) +0.1971 +0.1608 +0.2908 +0.0763 +0.0664 +0.0514 +Log-likelihood +2.31 × 104 +2.87 × 104 +5.41 × 104 +6.40 × 104 +6.60 × 104 +6.57 × 104 +Training time (per epoch) +4.97s +5.80s +12.31s +26.75s +11.21s +12.10s +Test time +2.29s +2.97s +4.79s +11.54s +3.28s +4.21s +Table B.1: Ablation study: AD-EnKF versus ROAD-EnKF with different NN parameterization and numerical +integration methods for surrogate dynamics (FC: NN with fully-connected layers; Fourier: NN with Fourier layers; +Euler: Euler method for ODE integration; RK4: fourth-order Runge Kutta method for ODE integration). Switching +from RK4 to Euler method while keeping the same NN configuration gives a computational speed-up, and the speed- +up is more noticeable when the NN involves Fourier layers. However, after the switch, the accuracy drops more +significantly for AD-EnKF than for ROAD-EnKF. The best configuration for AD-EnKF (Fourier with RK4) still +yields a lower accuracy compared to both ROAD-EnKF configurations, while taking more time to compute. (Burgers +example, full observation case, Subsection 5.2.) +30 + +0 +64 +128 +192 +256 +−1.0 +−0.5 +0.0 +0.5 +1.0 +SINDy-AE +rmse: 0.1433 +t=50, +Reconstruction +0 +64 +128 +192 +256 +rmse: 0.1188 +t=100, +Reconstruction +0 +64 +128 +192 +256 +rmse: 0.0946 +t=150, +Reconstruction +0 +64 +128 +192 +256 +rmse: 0.0598 +t=200, +Reconstruction +0 +64 +128 +192 +256 +rmse: 0.0290 +t=250, +Reconstruction +0 +64 +128 +192 +256 +−1.0 +−0.5 +0.0 +0.5 +1.0 +AD-EnKF +rmse: 0.0107 +0 +64 +128 +192 +256 +rmse: 0.0089 +0 +64 +128 +192 +256 +rmse: 0.0093 +0 +64 +128 +192 +256 +rmse: 0.0080 +0 +64 +128 +192 +256 +rmse: 0.0086 +0 +64 +128 +192 +256 +−1.0 +−0.5 +0.0 +0.5 +1.0 +ROAD-EnKF +rmse: 0.0083 +0 +64 +128 +192 +256 +rmse: 0.0082 +0 +64 +128 +192 +256 +rmse: 0.0066 +0 +64 +128 +192 +256 +rmse: 0.0052 +0 +64 +128 +192 +256 +rmse: 0.0042 +Observation +Reconstruction +Truth +0 +64 +128 +192 +256 +−0.5 +0.0 +0.5 +SINDy-AE +rmse: 0.0192 +t=300, +Forecast (Start) +0 +64 +128 +192 +256 +rmse: 4.5660 +t=375, +Forecast +0 +64 +128 +192 +256 +rmse: 4.5667 +t=450, +Forecast +0 +64 +128 +192 +256 +rmse: 4.5739 +t=525, +Forecast +0 +64 +128 +192 +256 +rmse: 4.5883 +t=600, +Forecast +0 +64 +128 +192 +256 +−0.5 +0.0 +0.5 +AD-EnKF +rmse: 0.0085 +0 +64 +128 +192 +256 +rmse: 0.0330 +0 +64 +128 +192 +256 +rmse: 0.0643 +0 +64 +128 +192 +256 +rmse: 0.0908 +0 +64 +128 +192 +256 +rmse: 0.1101 +0 +64 +128 +192 +256 +−0.5 +0.0 +0.5 +ROAD-EnKF +rmse: 0.0045 +0 +64 +128 +192 +256 +rmse: 0.0100 +0 +64 +128 +192 +256 +rmse: 0.0213 +0 +64 +128 +192 +256 +rmse: 0.0335 +0 +64 +128 +192 +256 +rmse: 0.0453 +Forecast +Truth +Figure 11: State reconstruction (upper half) and forecast (lower half) performance with full observation (du = dy = +256) on the Burgers example in Subsection 5.2. For each method, the reconstructed states ut (blue) for a single test +sequence are plotted for t = 50, 100, 150, 200, 250 (column), and the forecasted states (blue) for a single test sequence +are plotted for t = 300 (start of forecast), 375, 450, 525, 600 (column). The true values of the 256-dimensional states +are plotted in red dashed lines, along with the noisy observations in black dots. Both AD-EnKF and ROAD-EnKF +perform probabilistic state reconstructions and forecast through particles (all plotted in blue), while SINDy-AE only +provides point estimates. The reconstruction/forecast RMSEs are computed for each plot. +31 + +(a) Ground truth. +(b) Reconstruction and forecast. +Figure 12: Contour plot of state reconstruction and forecast output with full observation (du = dy = 256) on the +Burgers example in Subsection 5.2, as well as the ground truth (top). For each method (row), the reconstructed and +forecasted states (left column) for a single test sequence are plotted, for each state dimension (y-axis) and time (x- +axis). The error compared to the ground truth are plotted in the right column. For both AD-EnKF and ROAD-EnKF +we use particle means as point estimates. +C +Additional Materials: Kuramoto-Sivashinky Example +32 + +250 +0.6 +200 +0.4 +0.2 +Truth +150 +0.0 +100 +-0.2 +-0.4 +50 +-0.6 +0 +0 +100 +200 +300 +400 +500t<300 +t>300 +t<300 +t>300 +Reconstruction +Forecast +Reconstruction Error +Forecast Error +250 +250 +0.6 +0.6 +200 +200 +0.4 +0.4 +0.2 +0.2 +150 +0.0 +0.0 +100 +100 - +0.2 +-0.2 +S +0.4 +0.4 +50 +50 +-0.6 +0.6 +100 +200 +300 +400 +500 +100 +200 +300 +400 +500 +250 +250 - +0.6 +0.6 +200 +200 +0.4 +0.4 +0.2 +0.2 +150 +150 +0.0 +0.0 +100 - +-0.2 +-0.2 +-0.4 +-0.4 +50 +50 +0.6 +0.6 +以o +100 +200 +300 +400 +500 +100 +200 +300 +400 +500 +250 +250 +0.6 +0.6 +200 +200 +0.4 +0.4 +0.2 +0.2 +150 +150 +0.0 +0.0 +100 +100 +-0.2 +0.2 +-0.4 +-0.4 +50 +50 +0.6 +-0.6 +01 +0- +0 +100 +200 +300 +400 +500 +0 +100 +200 +300 +400 +5000 +64 +128 +192 +256 +−10 +0 +10 +AD-EnKF +rmse: 0.4430 +t=50, +Reconstruction +0 +64 +128 +192 +256 +rmse: 0.3721 +t=150, +Reconstruction +0 +64 +128 +192 +256 +rmse: 0.4419 +t=250, +Reconstruction +0 +64 +128 +192 +256 +rmse: 0.3058 +t=350, +Reconstruction +0 +64 +128 +192 +256 +rmse: 0.4504 +t=450, +Reconstruction +0 +64 +128 +192 +256 +−10 +0 +10 +ROAD-EnKF +rmse: 0.2746 +0 +64 +128 +192 +256 +rmse: 0.4711 +0 +64 +128 +192 +256 +rmse: 0.3168 +0 +64 +128 +192 +256 +rmse: 0.2995 +0 +64 +128 +192 +256 +rmse: 0.3667 +Observation +Reconstruction +Truth +0 +64 +128 +192 +256 +−10 +−5 +0 +5 +10 +AD-EnKF +rmse: 0.4504 +t=450, +Forecast (Start) +0 +64 +128 +192 +256 +rmse: 1.0364 +t=452, +Forecast +0 +64 +128 +192 +256 +rmse: 0.9103 +t=454, +Forecast +0 +64 +128 +192 +256 +rmse: 1.0975 +t=456, +Forecast +0 +64 +128 +192 +256 +rmse: 2.8994 +t=458, +Forecast +0 +64 +128 +192 +256 +−10 +−5 +0 +5 +10 +ROAD-EnKF +rmse: 0.3667 +0 +64 +128 +192 +256 +rmse: 1.0875 +0 +64 +128 +192 +256 +rmse: 0.6654 +0 +64 +128 +192 +256 +rmse: 0.8397 +0 +64 +128 +192 +256 +rmse: 1.7079 +Forecast +Truth +Figure 13: State reconstruction (upper half) and forecast (lower half) performance with full observation (du = +dy = 256) on the KS example in Subsection 5.3. For each method, the reconstructed states ut (blue) are plotted +for t = 50, 150, 250, 350, 450 (column), and the forecasted states (blue) are plotted for t = 450 (start of forecast), +452, 454, 456, 458 (column). The true values of the 256-dimensional states are plotted in red dashed lines, along with +the noisy observations in black dots. Both AD-EnKF and ROAD-EnKF perform probabilistic state reconstructions +and forecast through particles (all plotted in blue). The reconstruction/forecast RMSEs are computed for each plot. +33 + +(a) Ground truth. +(b) Reconstruction and forecast. +Figure 14: Contour plot of state reconstruction and forecast output of ROAD-EnKF with full observation (du = +dy = 256) on the KS example in Subsection 5.3, as well as the ground truth (top). The particle means of reconstructed +and forecasted states are plotted, for each state dimension (y-axis) and time (x-axis). The individual reconstructed +and forecasted states of three randomly chosen particles are also plotted. +34 + +250 +10 +200 +5 +150 +Truth +0 +100 +50 +-10 +400 +420 +440 +460 +480t<450 +t>450 +t<450 +t>450 +Reconstruction +Forecast +Reconstruction +Forecast +250 +250 +10 +-10 +(Mean) +3200 +200 +5 +5 +150 +150 +0 +ROAD-EnKF ( +0 +100 +ROAD-E +-5 +50 - +50 +-10 +-10 +420 +440 +460 +480 +420 +440 +460 +480 +250 +250 +Particle #3), +10 +-10 +(Particle +2200 +200 +5 +5 +150 +150 +0 +ROAD-EnKF ( +0 +ROAD-EnKF +100 +100 +-5 +50 +50 +-10 +-10 +R +420 +440 +460 +480 +400 +420 +440 +460 +480 \ No newline at end of file diff --git a/6dFKT4oBgHgl3EQf_S4j/content/tmp_files/load_file.txt b/6dFKT4oBgHgl3EQf_S4j/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7855649fc0f5c6bf550d86224459e9a023bbc7b4 --- /dev/null +++ b/6dFKT4oBgHgl3EQf_S4j/content/tmp_files/load_file.txt @@ -0,0 +1,1930 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf,len=1929 +page_content='Reduced-Order Autodifferentiable Ensemble Kalman Filters Yuming Chen∗ Daniel Sanz-Alonso∗ Rebecca Willett∗ University of Chicago Abstract This paper introduces a computational framework to reconstruct and forecast a partially observed state that evolves according to an unknown or expensive-to-simulate dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Our reduced- order autodifferentiable ensemble Kalman filters (ROAD-EnKFs) learn a latent low-dimensional surrogate model for the dynamics and a decoder that maps from the latent space to the state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The learned dynamics and decoder are then used within an ensemble Kalman filter to reconstruct and forecast the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Numerical experiments show that if the state dynamics exhibit a hidden low-dimensional structure, ROAD-EnKFs achieve higher accuracy at lower computational cost compared to existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' If such structure is not expressed in the latent state dynamics, ROAD-EnKFs achieve similar accuracy at lower cost, making them a promising approach for surrogate state reconstruction and forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 1 Introduction Reconstructing and forecasting a time-evolving state given partial and noisy time-series data is a fundamental problem in science and engineering, with far-ranging applications in numerical weather forecasting, climate, econometrics, signal processing, stochastic control, and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Two common challenges are the presence of model error in the dynamics governing the evolution of the state, and the high computational cost to simulate operational model dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Model error hinders the accuracy of forecasts, while the computational cost to simulate the dynamics hinders the quantification of uncertainties in these forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Both challenges can be alleviated by leveraging data to learn a surrogate model for the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Data-driven methods enable learning closure terms and unresolved scales in the dynamics, thus enhancing the forecast skill of existing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' In addition, surrogate models are inexpensive to simulate and enable using a large number of particles within ensemble Kalman or Monte Carlo methods for state reconstruction and forecasting, thus enhancing the uncertainty quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' This paper investigates a framework for state reconstruction and forecasting that relies on data-driven surrogate modeling of the dynamics in a low-dimensional latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Our reduced-order autodifferentiable ensemble Kalman filters (ROAD-EnKFs) leverage the EnKF algorithm to estimate by maximum likelihood the latent dynamics as well as a decoder from latent space to state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The learned latent dynamics and decoder are subsequently used to reconstruct and forecast the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Numerical experiments show that, compared to existing methods, ROAD-EnKFs achieve higher accuracy at lower computational cost provided that the state dynamics exhibit a hidden low-dimensional structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' When such structure is not expressed in the latent dynamics, ROAD-EnKFs achieve similar accuracy at lower cost, making them a promising approach for surrogate state reconstruction and forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Our work blends in an original way several techniques and insights from inverse problems, data assimila- tion, machine learning, and reduced-order modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' First, if the state dynamics were known and inexpensive to simulate, a variety of filtering and smoothing algorithms from data assimilation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' extended, ensemble, and unscented Kalman filters and smoothers, as well as particle filters) can be used to reconstruct and fore- cast the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' These algorithms often build on a Bayesian formulation, where posterior inference on the state ∗University of Chicago, Chicago, IL (ymchen@uchicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='edu, sanzalonso@uchicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='edu, willett@uchicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='edu) 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='11961v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='ML] 27 Jan 2023 combines the observed data with a prior distribution defined using the model dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Hence, learning a surrogate model for the dynamics can be interpreted as learning a prior regularization for state reconstruction and forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Second, the task of learning the regularization can be viewed as an inverse problem: we seek to recover the state dynamics from partially and noisily observed trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Data assimilation facilitates the numerical solution of this inverse problem by providing estimates of the hidden state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Third, our work leverages machine learning and reduced-order modeling to parameterize the dynamics in a low-dimensional latent space and learn a decoder from latent space to state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' In particular, we parameterize the decoder using recent ideas from discretization-invariant operator learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Our numerical experiments demonstrate the computational advantage of co-learning an inexpensive surrogate model in latent space together with a decoder, rather than a more expensive-to-simulate dynamics in state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1 Related Work Ensemble Kalman Filters in Data Assimilation The EnKF algorithm, reviewed in [44,47,72,75], is a popular method for state reconstruction and forecasting in data assimilation, with applications in numerical weather forecasting, the geophysical sciences, and signal processing [29,30,83,91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The EnKF propagates N equally-weighted particles through the dynamics, and assimilates new observations via Kalman-type updates computed with empirical moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' If the state dynamics are known, the EnKF can achieve accurate reconstruction with a small ensemble size N even in applications where the state and the observations are high-dimensional, provided that the effective dimension is moderate [32];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' EnKFs with a small ensemble size have a low computational and memory cost compared to traditional Kalman filters [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Ensemble Kalman methods are also successful solvers for inverse problems, as reviewed in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' In this paper, we employ the EnKF to approximate the data log-likelihood of surrogate models for unknown or expensive-to-simulate dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The use of the EnKF for maximum likelihood estimation (MLE) was first proposed in [81], which adopted a derivative-free optimization approach;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' see also [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Empirical studies on the likelihood computed with EnKFs and other data assimilation techniques can be found in [12,64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The application of the EnKF to approximate the data log-likelihood within pseudo-marginal Markov chain Monte Carlo methods for Bayesian parameter estimation was investigated in [28];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' see also [79,80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The paper [20] introduced derivative-based optimization of an EnKF approximation of the log-likelihood to perform state and parameter estimation in high-dimensional nonlinear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' However, to the best of our knowledge, no prior work combines estimation of the log-likelihood via EnKFs with learning low-dimensional surrogate models, including both surrogate latent dynamics and a decoder from latent space to state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Blending Data Assimilation with Reduced-Order Models Model reduction techniques have been employed in data assimilation to improve the state reconstruction accuracy in high-dimensional dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The assimilation in the unstable subspace (AUS) method [13,51,67,74,87] projects the dynamics onto a time-dependent subspace of the tangent space where the dynamics are unstable, and assimilates the observations therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The unstable directions are determined by the Lyapunov vectors with nonnegative Lyapunov exponents, and can be approximated using discrete QR algorithms [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The observations can also be projected onto the unstable directions to reduce the data dimension [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We refer to [5] for a review of projection-based model reduction techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' However, these methods rely on prior knowledge about the dynamics to identify the unstable subspaces and to construct the latent dynamics, and data assimilation is performed after the subspaces are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' In contrast, our paper introduces a framework that uses data assimilation as a tool to build surrogate latent dynamics from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Another approach to reduce the dimension of data assimilation problems exploits the conditionally Gaussian distribution of slow variables arising in the stochastic parameterization of a wide range of dynamical systems [17, 18, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' This conditional Gaussian structure can be exploited to obtain adequate uncertainty quantification of forecasts with a moderate sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' A caveat, however, is that identifying the slow variables can be challenging in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' As in our approach, these techniques often rely on machine learning to learn closure terms for the dynamics [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Finally, we refer to [78] for a discussion on how the effective dimension of transport map methods for data assimilation can be reduced by exploiting the conditional independence structure of the 2 reference-target pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Merging Data Assimilation with Machine Learning Recent developments in machine learning to model dynamical systems from data are reviewed in [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' One line of work [27,70,84,88] embeds the EnKF and the ensemble Kalman smoother (EnKS) into the expectation-maximization (EM) algorithm for MLE [23], with a special focus on estimation of error covariance matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The expectation step (E-step) is approximated by EnKF/EnKS with the Monte Carlo EM objective [90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' A subsequent line of work [7,9,31,66,92] introduces training of a neural network (NN) surrogate model in the maximization step (M-step) based on the states filtered by the E-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Unfortunately, it can be hard to achieve an accurate approximation of the E-step using EnKF/EnKS [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Another line of work [21, 52, 60, 65] approximates the data log-likelihood with particle filters (PFs) [26,35] and performs MLE using derivative-based optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' However, the resampling step in PFs is not readily differentiable, and, in addition, PFs often collapse when the dimensions of the state and the observations are large [2,4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Finally, techniques that leverage machine learning to obtain inexpensive analog ensembles for data assimilation are starting to emerge [95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Data-Driven Modeling of Dynamical Systems with Machine Learning Machine learning is also useful for dimensionality reduction in time-series modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' As an important example, recurrent neural networks (RNNs) [57,96] assimilate data into the time-evolving latent states using NN updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The paper [73] models the latent state evolution in recurrent networks with NN-embedded differential equations [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Other types of NN updates to incorporate the data into latent states include gated recurrent units (GRU) [22,45], long short-term memory (LSTM) [39,54], and controlled differential equations (CDEs) [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Another approach is to directly model the differential equation governing the state dynamics from observation data using regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Such methods include sparse regression over a dictionary of candidate functions using L1-regularization [10, 76, 86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' These techniques rely on full observation of the state, and, importantly, on time-derivative data that are rarely available in practice and are challenging to approximate from noisy discrete-time observation data [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' When the data are not guaranteed to lie in the same space as the underlying dynamics, an autoencoder structure can be jointly learned with the latent state dynamics [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Different modeling techniques can be applied to learn the latent state dynamics, including sparse dictionary regression [16], recurrent networks [34,63], and the Koopman operator learning [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' It is important to notice that, in contrast to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=', [10, 42], the focus of this paper is on state reconstruction and forecasting, rather than on obtaining an interpretable model for the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2 Outline and Main Contributions Section 2 formalizes the problem setting and goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We introduce a reduced-order state-space model (SSM) framework, where the dynamics are modeled in a low-dimensional latent space and learned jointly with a decoder from latent space to state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Section 3 introduces our main algorithm, the reduced-order autodifferentiable ensemble Kalman filter (ROAD-EnKF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' As part of the derivation of the algorithm, we discuss the use of EnKFs to estimate the data log-likelihood within reduced-order SSMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Section 4 contains important implementation considerations, including the design of the decoder, the use of truncated backpropagation to enhance the scalability for large windows of data, and the choice of regularization in latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Section 5 demonstrates the performance of our method in three examples: (i) a Lorenz 63 model embedded in a high-dimensional space, where we compare our approach to the SINDy-AE algorithm [16];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (ii) Burgers equation, where we showcase that ROAD-EnKFs are able to forecast the emergence of shocks, a phenomenon not included in our training data-set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' and (iii) Kuramoto-Sivashinky equation, a common test problem for filtering methods due to its chaotic behavior, where the ROAD-EnKF framework provides a computational benefit over state-of-the-art methods with similar accuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 3 Section 6 closes with a summary of the paper and open questions for further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Notation We denote by t ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' } a discrete-time index and by n ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' , N} a particle index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Time indices will be denoted with subscripts and particles with superscripts, so that un t represents a generic particle n at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We denote the particle dimension by du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We denote ut0:t1 := {ut}t1 t=t0 and un1:n2 := {un}n1 n=n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The collection un0:n1 t0:t1 is defined similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The Gaussian density with mean m and covariance C evaluated at u is denoted by N(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' m, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The corresponding Gaussian distribution is denoted by N(m, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 2 Problem Formulation In this section, we formalize and motivate our goals: reconstructing and forecasting a time-evolving, partially- observed state with unknown or expensive-to-simulate dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' An important step towards these goals is to learn a surrogate model for the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' In Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1 we consider an SSM framework where the state dynamics are parameterized and learned in order to reconstruct and forecast the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Next, in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2, we introduce a reduced-order SSM framework where the dynamics are modeled in a latent space and a decoder from latent space to state space is learned along with the latent dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Our ROAD-EnKF algorithm, introduced in Section 3, operates in this reduced-order SSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1 Setting and Motivation Consider a parameterized SSM of the form: (dynamics) ut = Fα(ut−1) + ξt, ξt ∼ N(0, Qβ), 1 ≤ t ≤ T, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1) (observation) yt = Htut + ηt, ηt ∼ N(0, Rt), 1 ≤ t ≤ T, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2) (initialization) u0 ∼ pu(u0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3) The state dynamics map Fα : Rdu → Rdu and error covariance matrix Qβ ∈ Rdu×du depend on unknown parameter θ := (α⊤, β⊤)⊤ ∈ Rdθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The observation matrices Ht ∈ Rdy×du and error covariance matrices Rt ∈ Rdy×dy are assumed to be known and possibly time-varying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We further assume independence of all random variables u0, ξ1:T , and η1:T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Given observation data y1:T drawn from the SSM (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3), we aim to accomplish two goals: Goal 1: Reconstruct the states u1:T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Goal 2: Forecast the states uT +1:T +Tf for some forecast lead time Tf ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' state space u0 u1 u2 · · uT uT +1 · · uT +Tf observation space y1 y2 · · yT Figure 1: Structure of data under SSM (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3), where we assume only observations y1:T := {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' , yT } are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Our goals are to reconstruct the states u1:T (Goal 1) and to forecast future states uT +1:T +Tf for some Tf ≥ 1 (Goal 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' If the true parameter θ ∈ Rdθ was known and the dynamics were inexpensive to simulate, the first goal can be accomplished by applying a filtering (or smoothing) algorithm on the SSM (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3), while the second 4 goal can be accomplished by iteratively applying the dynamics model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1) to the reconstructed state uT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We are interested in the case where θ needs to be estimated in order to reconstruct and forecast the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The covariance Qβ in the dynamics model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1) may represent model error or stochastic forcing in the dynamics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' in either case, estimating Qβ from data can improve the reconstruction and forecast of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' In this paper, we are motivated by applications where Fα represents a surrogate model for the flow between observations of an autonomous ordinary differential equation (ODE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Letting ∆s be the equally-spaced time between observations and fα : Rdu �→ Rdu be the parameterized vector field of the differential equation, we then have (ODE) du ds = fα(u), Fα : u(s) �→ u(s + ∆s), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4) where u(s) ∈ Rdu is the state as a function of continuous-time variable s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The ODE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4) may arise from spatial discretization of a system of partial differential equations (PDEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' For instance, we will consider 1-dimensional partial differential equations for u(x, s) of order κ ≥ 1, where u is a function of the spatial variable x ∈ [0, L] and continuous-time variable s ≥ 0: (PDE) ∂u ∂s = fα � u, ∂u ∂x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' , ∂κu ∂xκ � , Fα : u(·, s) �→ u(·, s + ∆s), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5) with suitable boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' After discretizing this equation on a spatial domain with grid points 0 = x1 < x2 < · · · < xM = L, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5) can be expressed in the form of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4) by replacing the spatial derivatives with their finite difference approximations, and u, fα, Fα with their finite-dimensional approximations on the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' As a result, du equals the number of grid points M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Several examples and additional details will be given in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2 Reduced-Order Modeling When the state is high-dimensional (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=', du is large), direct reconstruction and forecast of the state is computationally expensive, and surrogate modeling of the state dynamics map Fα becomes challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We then advocate reconstructing and forecasting the state ut through a low-dimensional latent representation zt, modeling the state dynamics within the low-dimensional latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' This idea is formalized via the following reduced-order parameterized SSM: (latent dynamics) zt = Gα(zt−1) + ζt, ζt ∼ N(0, Sβ), 1 ≤ t ≤ T, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6) (decoding) ut = Dγ(zt), 1 ≤ t ≤ T, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='7) (observation) yt = Htut + ηt, ηt ∼ N(0, Rt), 1 ≤ t ≤ T, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='8) (latent initialization) z0 ∼ pz(z0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='9) The latent dynamics map Gα : Rdz �→ Rdz and error covariance matrix Sβ ∈ Rdz×dz are defined on a dz dimensional latent space with dz < du, and the decoder function Dγ : Rdz �→ Rdu maps from latent space to state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The reduced-order SSM depends on an unknown parameter θ := (α⊤, β⊤, γ⊤)⊤ ∈ Rdθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The remaining assumptions are the same as in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Writing Hγ,t(·) := HtDγ(·), the reduced-order SSM (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='9) can be combined into (latent dynamics) zt = Gα(zt−1) + ζt, ζt ∼ N(0, Sβ), 1 ≤ t ≤ T, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='10) (observation) yt = Hγ,t(zt) + ηt, ηt ∼ N(0, Rt), 1 ≤ t ≤ T, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='11) (latent initialization) z0 ∼ p(z0), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='12) where the observation function Hγ,t(·) is nonlinear if the decoder Dγ(·) is nonlinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' As in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1, the map Gα may be interpreted as the flow between observations of an ODE with vector field gα : Rdz �→ Rdz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' If the true parameter θ ∈ Rdθ was known, given observation data y1:T drawn from the reduced-order SSM (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='9), we can reconstruct the states u1:T (Goal 1) by first applying a filtering (or smoothing) algorithm 5 latent space state space z0 z1 z2 · · zT zT +1 · · zT +Tf u1 u2 · · uT uT +1 · · uT +Tf observation space y1 y2 · · yT Figure 2: Structure of data under reduced-order SSM (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='9), where we assume only observations y1:T := {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' , yT } are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Our goals are to reconstruct the states u1:T (Goal 1) and to forecast future states uT +1:T +Tf for some Tf ≥ 1 (Goal 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' on (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='10)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='12) to estimate z1:T , and then applying the decoder Dγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We can forecast the states uT +1:T +Tf (Goal 2) by first applying iteratively the latent dynamics model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6) to the reconstructed latent state zT , and then applying the decoder Dγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' As in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1, we are interested in the case where θ needs to be estimated from the given data y1:T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 3 Reduced-Order Autodifferentiable Ensemble Kalman Filters As discussed in the previous section, to achieve both goals of state reconstruction and forecast, it is essential to obtain a suitable surrogate model for the dynamics by learning the parameter θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The general approach we take is the following: (1) estimate θ with maximum likelihood;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (2) apply a filtering algorithm with estimated parameter θ to reconstruct and forecast the states u1:T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' As we shall see, the maximum likelihood estimation of θ will rely itself on a filtering algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' For the SSM in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1, this approach was introduced in [20] via AD-EnKF (Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1 in [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Here we focus on the reduced-order SSM in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2, namely (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='10)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='12), which is a more general case than the SSM in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' this explains the terminology reduced-order AD-EnKF (ROAD-EnKF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' In Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1, we describe how the log-likelihood L(θ) = log pθ(y1:T ) can be expressed in terms of the normalizing constants that arise from sequential filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' In Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2, we give background on EnKFs and on how to use these filtering algorithms to estimate L(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' In Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3, we introduce our ROAD-EnKF method that takes as input multiple independent instances of observation data yI 1:T across the same time range, and performs both state reconstruction and forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1 Sequential Filtering and Data Log-Likelihood Suppose that θ = (α⊤, β⊤, γ⊤)⊤ is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We recall that, for 1 ≤ t ≤ T, the filtering distributions pθ(zt|y1:t) of the SSM (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='10)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='12) can be obtained sequentially, alternating between prediction and analysis steps: (prediction) pθ(zt|y1:t−1) = � N � zt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Gα(zt−1), Sβ � pθ(zt−1|y1:t−1) dzt−1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1) (analysis) pθ(zt|y1:t) = 1 Et(θ)N � yt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Hγ,t(zt), Rt � pθ(zt|y1:t−1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2) with the convention pθ(·|y1:0) := pθ(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Here Et(θ) is a normalizing constant which does not depend on zt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' It can be shown that Et(θ) = pθ(yt|y1:t−1) = � N � yt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Hγ,t(zt), Rt � pθ(zt|y1:t−1) dzt, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3) 6 and therefore the data log-likelihood admits the characterization L(θ) := log pθ(y1:T ) = T � t=1 log pθ(yt|y1:t−1) = T � t=1 log Et(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4) Analytical expressions of the filtering distributions pθ(zt|y1:t) and the data log-likelihood L(θ) are only available for a small class of SSMs, which includes linear-Gaussian and discrete SSMs [46,68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Outside these special cases, filtering algorithms need to be employed to approximate the filtering distributions, and these algorithms can be leveraged to estimate the log-likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2 Estimation of the Log-Likelihood with Ensemble Kalman Filters Given θ = (α⊤, β⊤, γ⊤)⊤, the EnKF algorithm [29,30] sequentially approximates the filtering distributions pθ(zt|y1:t) using N equally-weighted particles z1:N t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' At prediction steps, each particle zn t is propagated using the latent dynamics model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='10), while at analysis steps a Kalman-type update is performed for each particle: (prediction step) �z n t = Gα(zn t−1) + ζn t , ζn t i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' ∼ N(0, Sβ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5) (analysis step) zn t = �z n t + �Kt � yt + ηn t − Hγ,t(�z n t ) � , ηn t i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' ∼ N(0, Rt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6) The Kalman gain �Kt := �Czy,t( �Cyy,t + Rt)−1 is defined using empirical covariances given by �Czy,t = 1 N − 1 N � n=1 (�z n t − �mt) � Hγ,t(�z n t )− � Ht �⊤, �Cyy,t = 1 N − 1 N � n=1 � Hγ,t(�z n t )− � Ht �� Hγ,t(�z n t )− � Ht �⊤, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='7) where �mt = 1 N N � n=1 �z n t , � Ht = 1 N N � n=1 Hγ,t(�z n t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='8) The empirical moments �Cyy,t, � Ht defined in equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='8) provide a Gaussian approximation to the predictive distribution for Hγ,t(zt): pθ(Hγ,t(zt)|y1:t−1) ≈ N(Hγ,t(zt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' � Ht, �Cyy,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='9) By applying the change of variables formula to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3), we have Et(θ) = � N � yt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Hγ,t(zt), Rt � pθ(zt|y1:t−1) dzt = � N � yt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Hγ,t(zt), Rt � pθ(Hγ,t(zt)|y1:t−1) dHγ,t(zt) ≈ N(yt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' � Ht, �Cyy,t + Rt), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='10) where the approximation step follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='9) and the formula for convolution of two Gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4), we obtain the following estimate of the data log-likelihood: LEnKF(θ) := T � t=1 log N � yt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' � Ht, �Cyy,t + Rt � ≈ L(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='11) The estimate LEnKF(θ) can be computed online with EnKF, and is stochastic as it depends on the randomness used to propagate the particles, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=', the choice of random seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The whole procedure is summarized in 7 Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1, which implicitly defines a stochastic map θ �→ LEnKF(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1 Ensemble Kalman Filter and Log-likelihood Estimation Input: θ = (α⊤, β⊤, γ⊤)⊤, y1:T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (If multiple input instances yI 1:T are provided, run the following proce- dure for each instance yi 1:T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=') 1: Initialize LEnKF(θ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Draw zn 0 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' ∼ pz(z0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 2: for t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' , T do 3: Set �z n t = Gα(zn t−1) + ζn t , where ζn t i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' ∼ N(0, Sβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' ▷ Prediction step 4: Compute �mt, � Ht, �Czy,t, �Cyy,t by equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='8) and set �Kt = �Czy,t( �Cyy,t + Rt)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 5: Set zn t = �z n t + �Kt � yt + ηn t − Hγ,t(�z n t ) � , where ηn t i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' ∼ N(0, Rt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' ▷ Analysis step 6: Set LEnKF(θ) ← LEnKF(θ) + log N � yt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' � Ht, �Cyy,t + Rt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 7: end for Output: EnKF particles z1:N 0:T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Log-likelihood estimate LEnKF(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (If multiple input instances yI 1:T are provided, return instead the average of log-likelihood estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=') 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3 Main Algorithm The main idea of our algorithm is to perform maximum likelihood estimation on the parameter θ by gradient ascent, via differentiation through the map θ �→ LEnKF(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Our core method is summarized in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2, which includes estimation of θ as well as reconstruction and forecast of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Our PyTorch implementa- tion is at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='com/ymchen0/ROAD-EnKF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The gradient of the map θk �→ LEnKF(θk) can be evaluated using autodiff libraries [1, 8, 69] that support auto-differentiation of common matrix operations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' matrix multiplication, inverse, and determinant [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We use the “reparameterization trick” [49,71] to auto-differentiate through the stochasticity in the EnKF algorithm, as in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1 of [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' In Section 5, we consider numerical examples where the data are generated from an unknown SSM in the form of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='9) with no explicit knowledge of the reduced-order structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' we also consider examples where the data are generated directly from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' In practice, multiple independent instances of observation data yI 1:T may be available across the same time range, where each superscript i ∈ I corresponds to one instance of observation data y1:T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We assume that each instance yi 1:T is drawn i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' from the same SSM, with different realizations of initial state, model error, and observation error for each instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We assume that data are split into training and test sets yItrain 1:T and yItest 1:T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' During training, we randomly select a small batch of data from yItrain 1:T at each iteration, and evaluate the averaged log-likelihood and its gradient over the batch to perform a parameter update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The idea is reminiscent of stochastic gradient descent in the optimization literature: matrix operations of EnKF can be parallelized within a batch to utilize the data more efficiently, reducing the computational and memory cost compared to using the full training set at each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The state reconstruction and forecast performance are evaluated on the unseen test set yItest 1:T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' State reconstruction and forecast via Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2 can be interpreted from a probabilistic point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' For convenience, we drop the superscripts I and k in this discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' For 0 ≤ t ≤ T, since the particles z1:N t form an approximation of the filtering distribution pθ(zt|y1:t) for latent state zt, it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='7) that the output particles u1:N t of the algorithm form an approximation of the filtering distribution pθ(ut|y1:t) for state ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' For T + 1 ≤ t ≤ T + Tf, it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='10) that the particles z1:N t form an approximation of the predictive distribution pθ(zt|y1:T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Therefore, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='7) the output particles u1:N t of the algorithm form an approximation of the predictive distribution pθ(ut|y1:T ) for future state ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 4 Implementation Details This section considers the practical implementation of ROAD-EnKF Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2, including parameteri- zation of the surrogate latent dynamics map gα and decoder Dγ (Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1), computational efficiency 8 Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2 Reduced-Order Autodifferentiable Ensemble Kalman Filter (ROAD-EnKF) Input: Observations yI 1:T , split into yItrain 1:T and yItest 1:T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Learning rate η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Batch size B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 1: Initialize SSM parameter θ0 and set k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Write Hγ,t(·) = HtDγ(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' // Training phase 2: while not converging do 3: Randomly select B indices from Itrain, denoted as IB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 4: Compute zIB,1:N 0:T , LEnKF(θk) = EnsembleKalmanFilter(θk, yIB 1:T ) using Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 5: Compute ∇θLEnKF(θk) by auto-differentiating the map θk �→ LEnKF(θk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 6: Set θk+1 = θk + η∇θLEnKF(θk) and k ← k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 7: end while // Test phase 8: zItest,1:N 0:T , LEnKF(θk) = EnsembleKalmanFilter(θk, yItest 1:T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' ▷ State reconstruction 9: Simulate zItest,1:N t using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='10) with α = αk, β = βk for t = T + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' , T + Tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' ▷ State forecast 10: Compute uItest,1:N 0:T +Tf = Dγk(zItest,1:N 0:T +Tf ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Output: Learned reduced-order SSM parameter θk and particles uItest,1:N 0:T +Tf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' for high-dimensional observations (Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2), and regularization on latent states (Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1 Surrogate Latent Dynamics and Decoder Design In our numerical experiments, we adopt a simple parameterization for the surrogate latent dynamics map gα using a two-layer fully connected NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' For our design of the decoder Dγ, the idea stems from the literature on convolutional autoencoders for computer vision tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=', [62]), where both the encoder and decoder networks consist of multiple convolutional layers with residual connections that map between the image space and latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Here, to suit our setting, we replace the kernel-based local convolutional layers with Fourier-based spectral convolutional layers (‘Fourier layers’) introduced in [36,56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The latter treat a finite- dimensional vector as a spatial discretization of a function on a grid, and learn a finite-dimensional mapping that approximates an operator between function spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The learning accuracy is known empirically to not depend on the level of the discretization [56], determined by du in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Using Fourier layers to learn dynamical systems and differential equations was originally proposed in [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' For the sake of completeness, we describe below the definition of spectral convolutional layers and how they are incorporated into our decoder design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Spectral Convolutional Layer Given an input vin ∈ Rnin×du where nin is the number of input channels and du is the input dimension, which is also the size of the grid where the function is discretized, we first apply a discrete Fourier transform (DFT) in spatial domain to get λin := DFT(vin) ∈ Cnin×du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We then multiply it by a learned complex weight tensor W ∈ Cnout×nin×du that is even symmetric1 to get λout := W × λin ∈ Cnout×du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The multiplication is defined by (W × λin)i,k = nin � j=1 Wi,j,k(λin)j,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1) This can be regarded as ‘channel mixing’, since for the k-th Fourier mode (1 ≤ k ≤ du), all nin input channels of λ are linearly mixed to produce nout output channels through the matrix W·,·,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Other types of (possibly nonlinear) mixing introduced in [36] can also be applied, and we leave them to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We then apply an inverse discrete Fourier transform (IDFT) in spatial domain to get the output vout = IDFT(λout) ∈ Rnout×du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We call the mapping vin �→ vout a spectral convolutional layer (SpecConv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 1That is, W satisfies Wi,j,k = W i,j,du+2−k ∀i, j and ∀k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' This ensures that the inverse discrete Fourier transform of λout is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' In practice, the parameterization of W requires up to nin × nout × (⌊du/2⌋ + 1) complex entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 9 Fourier Neural Decoder Given latent variable z ∈ Rdz (where we omit the subscript t for convenience), we first apply a complex linear layer f0(·) to get z0 = f0(z) := W0z + b0 ∈ Ch for W0 ∈ Ch×dz and b0 ∈ Ch, where h is the dimension of z0 to be specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We then apply an IDFT that treats z0 as a one-sided Hermitian signal in Fourier domain2 to get v0 := IDFT(z0) ∈ Rdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We then apply L spectral convolutional layers to get vL, with proper choices of channel numbers as well as residual connections, normalization layers, and activation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' More specifically, vL is defined by iteratively applying the following vℓ = fℓ(vℓ−1) := Act � Norm � SpecConv(vℓ−1) + 1x1Conv(vℓ−1) �� , 1 ≤ ℓ ≤ L, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2) where vℓ ∈ Rnℓ×du, Act and Norm refer to the activation function and the normalization layer, 1x1Conv refers to the one-by-one convolutional layer which can be viewed as a generalization of residual connection, and n0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We refer to fℓ as a ‘Fourier layer’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The final part of the decoder is a two-layer fully connected NN that is applied to vL ∈ RnL×du over channel dimension to get u ∈ Rdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' See Figure 3 for the architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Notice that the learned variables γ of the decoder include W0, b0 of the initial linear layer, complex weight tensors W’s of SpecConv layers, weights and biases of 1x1Conv layers, as well as the final fully-connected NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' z Linear IDFT Fourier Layer 1 Fourier Layer L ⋯ vℓ−1 vL u v0 DFT Linear Channel Mixing IDFT 1x1Conv + MLP Channel Mixing Norm Act vℓ SpecConv (a) (b) Figure 3: (a) Network architecture of the decoder Dγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Starting from z ∈ Rdz in a low-dimensional latent space, we first apply a complex linear layer followed by an IDFT to lift it to v0 ∈ Rdu in a high-dimensional state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We then apply L Fourier layers iteratively to get vL ∈ RnL×du where nL is the channel dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We project it back to the state space by applying a two-layer fully-connected NN to mix the channels and output u ∈ Rdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (b) Fourier layer: The design was first proposed in [56], and we describe it here for the sake of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The upper half represents a spectral convolutional layer, where we transform the input vℓ−1 ∈ Rnℓ−1×du into the frequency space with DFT, mix the channels with a complex linear map, and transform back with IDFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The lower half is a one-by-one convolutional layer, which is a generalization of residual connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The outputs from both layers are summed up and passed through a normalization and an activation layer to produce the output vℓ ∈ Rnℓ×du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 2z0 is either truncated or zero-padded to a signal of dimension C⌊du/2⌋+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2 Algorithmic Design for Computational Efficiency If the time-window length T is large, we follow [20] and use truncated backpropagation to auto-differentiate the map θ �→ LEnKF(θ): we divide the sequence into multiple short subsequences and backpropagate within each subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The idea stems from Truncated Backpropagation Through Time (TBPTT) for RNNs [82,93] and the recursive maximum likelihood method for hidden Markov models [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' By doing so, multiple gradient ascent steps can be performed for each single filtering pass, and thus the data can be utilized more efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Moreover, gradient explosion/vanishing [3] are less likely to happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We refer to [20] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We choose this variant of ROAD-EnKF in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' In this work we are mostly interested in the case where du and dy are large, and dz is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Moreover, the ensemble size N that we consider is moderate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=', du ≥ dy > N > dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Therefore, we do not pursue the covariance localization approach as in [20] (see also [38, 43]), which is most effective when N < dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Instead, we notice that the computational bottlenecks of the analysis step in the EnKF Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1 are the O(d3 y) operations of computing the Kalman gain (Line 4) as well as updating the data log-likelihood (Line 6), where we need to compute the matrix inverse and log-determinant of a dy ×dy matrix ( �Cyy,t +Rt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' If dy > N, the number of operations can be improved to O(N 3) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Let Yt ∈ Rdy×N be the matrix representation of the centered ensemble after applying the observation function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=', its n-th column is Y n t := 1 √N−1 � Hγ,t(�zn t ) − 1 N �N m=1 Hγ,t(�zm t ) � (we drop the parameter γ for convenience).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' This leads to �Cyy,t = YtY ⊤ t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' By the matrix inversion lemma [94], ( �Cyy,t + Rt)−1 = R−1 t − R−1 t Yt(I + Y ⊤ t R−1 t Yt)−1Y ⊤ t R−1 t , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3) logdet( �Cyy,t + Rt) = logdet(I + Y ⊤ t R−1 t Yt) + logdet(Rt), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4) where I + Y ⊤ t R−1 t Yt ∈ RN×N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The computational cost can be further reduced if the quantities R−1 t and logdet(Rt) can be pre-computed, for instance when Rt = rI for some scalar r ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Moreover, in practice, to update the ensemble in Line 5 of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1, instead of inverting I+Y ⊤ t R−1 t Yt directly in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3) followed by a matrix multiplication, we find it more numerically stable to first solve the following linear system: (I + Y ⊤ t R−1 t Yt)un t = Y ⊤ t R−1 t � yt + γn t − Hγ,t(�zn t ) � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5) for un t ∈ RN, and then perform the analysis step (Line 5 of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1) by zn t = �zn t + �Czy,t( �Cyy,t + Rt)−1� yt + γn t − Hγ,t(�zn t ) � = �zn t + �Czy,t � R−1 t − R−1 t Yt(I + Y ⊤ t R−1 t Yt)−1Y ⊤ t R−1 t �� yt + γn t − Hγ,t(�zn t ) � = �zn t + �Czy,tR−1 t � yt + γn t − Hγ,t(�zn t ) − Ytun t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6) Similar ideas and computational cost analysis can be found in [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' For the benchmark experiments in Section 5, we modify the AD-EnKF algorithm as presented in [20] to incorporate the above ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3 Latent Space Regularization Since the estimation of u is given by Dγ(z), where both Dγ(·) and z need to be identified from data, we overcome potential identifiability issues by regularizing z in the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' To further motivate the need for latent space regularization, consider the following example: if the pair (z, Dγ(·)) provides a good estimation of ut, then so does (cz, 1 cDγ(·)) for any constant c ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Therefore, the norm of z can be arbitrarily large, and thus we regularize z’s in the latent space so that their norms do not explode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We perform regularization by extending the observation model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='11) to impose additional constraints on the latent state variable zt’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The idea stems from regularization in ensemble Kalman methods for inverse 11 problems [15,37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We first extend (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='11) to the equations: �yt = Hγ,t(zt) + ηt, ηt ∼ N(0, Rt), 0 = zt + ϵt, ϵt ∼ N(0, σ2Idz), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='7) where σ is a parameter to be chosen that incorporates the prior information that each coordinate of zt is an independent centered Gaussian random variable with standard deviation σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Define yaug t = �yt 0 � , Haug γ,t (zt) = �Hγ,t(zt) zt � , ηaug t ∼ N(0, Raug t ), Raug t = �Rt 0 0 σ2Idz � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='8) We then write (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='7) into an augmented observation model yaug t = Haug γ,t (zt) + ηaug t , ηaug t ∼ N(0, Raug t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='9) To perform latent space regularization in ROAD-EnKF, during the training stage we run EnKF (Line 4 of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2) with augmented data yaug 1:T and SSM with the augmented observation model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=', (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='10)- (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='9)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' During test stage, we run EnKF (Line 8 of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2) with the original data and SSM, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=', (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='10)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='11)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 5 Numerical Experiments In this section, we compare our ROAD-EnKF method to the SINDy autoencoder [16], which we abbreviate as SINDy-AE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' It learns an encoder-decoder pair that maps between observation space (yt’s) and latent space (zt’s), and simultaneously performs a sparse dictionary learning in the latent space to discover the latent dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Similar to SINDy-AE, our ROAD-EnKF method jointly discovers a latent space and the dynamics therein that is a low-dimensional representation of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' However, our method differs from SINDy-AE in four main aspects: (1) No time-derivative data for y1:T are required;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (2) No encoder is required;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (3) State reconstruction and forecast can be performed even when the data y1:T are noisy and partial observation of u1:T , while SINDy-AE is targeted at noiseless and fully observed data that are dense in time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (4) Stochastic representation of latent dynamics model can be learned, and uncertainty quantification can be performed in state reconstruction and forecast tasks through the use of particles, while SINDy-AE only provides a point estimate in both tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We also compare our ROAD-EnKF method to AD-EnKF [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Although AD-EnKF enjoys some of the benefits of ROAD-EnKF, including the capability to learn from noisy, partially observed data and perform uncertainty quantification, it directly learns the dynamics model in high-dimensional state space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=', on ut’s instead of zt’s), which leads to higher model complexity, as well as additional computational and memory costs when performing the EnKF step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Moreover, AD-EnKF does not take advantage of the possible low-dimensional representation of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We compare in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1 below the capabilities of the three algorithms under different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Learn from noisy and partially observed data Uncertainty quantification No need of time-derivative data Low-dimensional state representation SINDy-AE [16] \x17 \x17 \x17 \x13 AD-EnKF [20] \x13 \x13 \x13 \x17 ROAD-EnKF (this paper) \x13 \x13 \x13 \x13 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1: Comparison of SINDy-AE, AD-EnKF, and ROAD-EnKF under different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Other alternative methods include EnKF-embedded EM algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' [9]) and autodifferentiable PF algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=', [65]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Since [20] already establishes AD-EnKF’s superiority to those approaches, we do not include them in these experiments, and we refer to [20] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The training procedure is the following: We first specify a forecast lead time Tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We then generate 12 training data yItrain 0:T and test data with extended time range (uItest,∗ 0:T +Tf , yItest 0:T ) with Ntrain := |Itrain| and Ntest := |Itest|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The data are either generated from a reduced-order SSM (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='9) with explicit knowledge of true parameter θ (Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1), or from an SSM (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3) with no explicit knowledge of the exact reduced-order structure (Subsections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The data yItrain 0:T and yItest 0:T are then passed into ROAD- EnKF (Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2), and we evaluate the following: Reconstruction-RMSE (RMSE-r): Measures the state reconstruction error of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We take the particle mean of uItest,1:N 0:T as a point estimate of the true states uItest,∗ 0:T , and evaluate the RMSE: RMSE-r = � � � � 1 duNtest(T − Tb) T � t=Tb � i∈Itest ���ui t − ui,∗ t ��� 2 , where ui t = 1 N N � n=1 ui,n t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1) Here Tb is a number of burn-in steps to remove transient errors in the reconstruction that stem from the choice of initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' For simplicity, we set Tb = ⌊T/5⌋ as in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Forecast-RMSE (RMSE-f): Measures the t-step state forecast error of the algorithm, for lead time t ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' , Tf}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We take the particle mean of uItest,1:N T +t as a point estimate of the true future states uItest,∗ T +t : RMSE-f(t) = � 1 duNtest � i∈Itest ���ui T +t − ui,∗ T +t ��� 2 , where ui T +t = 1 N N � n=1 ui,n T +t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2) Test Log-Likelihood: Measures the averaged log-likelihood of the learned reduced-order SSM over test observation data yItest 0:T , which is LEnKF(θk) defined in Line 8 of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' For AD-EnKF, the above metrics can be similarly computed, following [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' For SINDy-AE, as uncer- tainty quantification is not performed, we use its decoder output as the point estimate of the state in both reconstruction and forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Moreover, log-likelihood computation is not available for SINDy-AE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1 Embedding of Chaotic Dynamics (Lorenz 63) In this subsection, we reconstruct and forecast a state defined by embedding a Lorenz 63 (L63) model in a high-dimensional state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' A similar experiment was used in [16] to motivate the SINDy-AE algorithm, and hence this example provides a good point of comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The data are generated using the L63 system as the true latent state dynamics model: dz ds = g(z), � � � � � g(1)(z) = 10(z(2) − z(1)), g(2)(z) = z(1)(28 − z(3)) − z(2), g(3)(z) = z(1)z(2) − 8 3z(3), G : z(s) �→ z(s + ∆s), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3) where z(i) and g(i) denote the i-th coordinate of z and component of g, and ∆s is the time between ob- servations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We further assume there is no noise in the true latent state dynamics model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=', S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' To construct the true reduced-order SSM, we define D ∈ Rdu×6 such that its i-th column Di ∈ Rdu is given by the discretized i-th Legendre polynomial over du grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The true states ut ∈ Rdu are defined by ut := D � z(1) t /40 z(2) t /40 z(3) t /40 (z(1) t /40)3 (z(2) t /40)3 (z(3) t /40)3 �⊤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4) We consider two cases of the observation model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='8): (1) full observation, where all coordinates of ut are observed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=', Ht = Idu and dy = du;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (2) partial observation, where for each t, only a fixed portion c < 1 of all coordinates of ut are observed, and the coordinate indices are chosen randomly without replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' In this case, Ht ∈ Rdy×du is a submatrix of Idu and varies across time, and dy = cdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' This partial observation 13 set-up has been studied in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=', [7, 9]) for data assimilation problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' For both cases, we assume Rt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='01Idy and z0 ∼ N(0, 4Idz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We consider full observation with du = dy = 128 and partial observation with du = 128, dy = 64 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=', c = 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We generate Ntrain = 1024 training data and Ntest = 20 test data with the true reduced- order SSM defined by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We set the number of observations T = 250 with time between observations ∆s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We set the forecast lead time Tf = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The latent flow map G is integrated using the Runge–Kutta–Fehlberg method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The surrogate latent dynamics map gα is parameterized as a two-layer fully connected NN, and is integrated using a fourth-order Runge-Kutta method with step size ∆int s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The error covariance matrix Sβ in the latent dynamics is parametrized using a diagonal matrix with positive diagonal elements β ∈ Rdz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The decoder Dγ is parameterized as a Fourier Neural Decoder (FND) discussed in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Details of the network hyperparameters for this and subsequent examples are summarized in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2, obtained through cross-validation experiments on the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The latent space dimension for both SINDy-AE and ROAD-EnKF is set to dz = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The ensemble size for both AD-EnKF and ROAD-EnKF is set to N = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' L63 Burgers KS FND L 4 2 4 h 6 40 40 (n0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' , nL) (1, 20, 20, 20, 20) (1, 20, 20) (1, 20, 20, 20, 20) Norm LayerNorm Activation ReLU Latent space reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' σ 2 4 4 Optimization Optimizer Adam Learning rate (η) 1e-3 Batch size (B) 16 4 4 TBPTT length 10 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2: Choices of hyperparameters for ROAD-EnKF on different numerical examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' In Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3 we list the performance metrics of each method with full and partial observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The state reconstruction and forecast performance on a single instance of test data are plotted in Figure 4 and 5 for the full observation case, and in Figure 6 for the partial observation case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' For the full observation case, we compare ROAD-EnKF with AD-EnKF and SINDy-AE, adopting for the latter the implementation in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Since SINDy-AE requires time-derivative data as input, we use a finite difference approximation computed from data y1:T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We also include the results for SINDy-AE where the exact time-derivative data are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We find that ROAD-EnKF is able to reconstruct and forecast the states consistently with the lowest RMSE, and the performance is not affected by whether the state is fully or partially observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' AD-EnKF is able to reconstruct and forecast the state with a higher RMSE than that of ROAD-EnKF, and the performance deteriorates in the partially observed setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' SINDy-AE with finite difference approximation of derivative data also achieves higher reconstruction RMSE than that of ROAD-EnKF, and does not give accurate state forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' This is likely due to the fact that data are sparse in time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=', ∆s is large) which leads to a larger error when approximating the true time-derivative, and hence it is more difficult to extract meaningful dynamics from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Even when the true time-derivative data are used (which is not available unless we have explicit knowledge of the true reduced-order SSM), SINDy-AE has a higher reconstruction RMSE compared to ROAD-EnKF, and its forecast performance is still worse than the other two methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Moreover, it cannot handle partial observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' In terms of computational cost, ROAD-EnKF is more efficient than AD-EnKF since the surrogate dy- namics are cheaper to simulate and the EnKF algorithm is more efficient to perform in both training and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' However, ROAD-EnKF takes more time than SINDy-AE, since the latter does not rely on a filtering algorithm, but rather an encoder, to reconstruct the states and perform learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 14 SINDy-AE (full) SINDy-AE (w/ derivative, full) AD-EnKF (full) ROAD-EnKF (full) AD-EnKF (partial) ROAD-EnKF (partial) RMSE-r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0142 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0148 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0168 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0078 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0368 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0079 RMSE-f(1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1310 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0191 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0156 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0069 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0315 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0069 RMSE-f(5) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6580 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0333 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0335 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0141 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0729 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0125 Log-likelihood − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='25 × 104 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='58 × 104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='28 × 104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='40 × 104 Training time (per epoch) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='15s 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='74s 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='15s 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='86s 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='62s Test time 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='35s 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='57s 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='95s 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='52s 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='73s Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3: Performance metrics for different algorithms at convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (Embedded L63 example, Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=') 0 32 64 96 128 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 SINDy-AE rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0095 t=40, Reconstruction 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0120 t=80, Reconstruction 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0102 t=120, Reconstruction 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0204 t=160, Reconstruction 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0193 t=200, Reconstruction 0 32 64 96 128 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 SINDy-AE w/ derivative data rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0137 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0097 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0125 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0251 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0182 0 32 64 96 128 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 AD-EnKF rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0217 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0107 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0360 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0103 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0120 0 32 64 96 128 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 ROAD-EnKF rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0068 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0029 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0181 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0025 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0032 Observation Reconstruction Truth Figure 4: State reconstruction performance with full observation (du = dy = 128) on the embedded L63 example in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' For each method (row), the reconstructed states ut (blue) for a single test sequence are plotted for t = 40, 80, 120, 160, 200 (column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The true values of the 128-dimensional states are plotted in red dashed lines, along with the noisy observations in black dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Both AD-EnKF and ROAD-EnKF perform probabilistic state reconstructions through particles (all plotted in blue), while SINDy-AE only provides point estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The reconstruction RMSE’s are computed for each plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' For SINDy-AE, even with derivative data (not required for AD- EnKF and ROAD-EnKF), the reconstruction performance is similar to that of AD-EnKF, while being worse than that of ROAD-EnKF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2 Burgers Equation In this subsection and the following one, we learn high-dimensional SSMs without explicit reference to a true model for low-dimensional latent dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We first consider the 1-dimensional Burgers equation for 15 0 32 64 96 128 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 SINDy-AE rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0215 t=250, Forecast (Start) 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1282 t=252, Forecast 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3571 t=254, Forecast 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4359 t=256, Forecast 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2722 t=258, Forecast 0 32 64 96 128 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 SINDy-AE w/ derivative data rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0167 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0209 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0509 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0620 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0316 0 32 64 96 128 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 AD-EnKF rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0149 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0148 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0119 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0170 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0235 0 32 64 96 128 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 ROAD-EnKF rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0035 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0026 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0034 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0037 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0032 Forecast Truth Figure 5: Forecast performance with full observation (du = dy = 128) on the embedded L63 example in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' For each method (row), the forecasted states ut (blue) for a single test sequence are plotted for t = 250 (start of forecast), 252, 254, 256, 258 (column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The true values of the du = 128 dimensional states are plotted in red dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Both AD-EnKF and ROAD-EnKF perform probabilistic forecast through particles (all plotted in blue), while SINDy-AE only provides point estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The forecast RMSE’s are computed for each plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' For SINDy-AE, even with derivative data (not required for AD-EnKF and ROAD-EnKF), the forecast performance is similar to that of AD-EnKF, while being worse than that of ROAD-EnKF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' u(x, s), where u is a function of the spatial variable x ∈ [0, L] and continuous-time variable s > 0: ∂u ∂s = −u∂u ∂x + ν ∂2u ∂x2 , u(0, s) = u(L, s) = 0, u(x, 0) = u0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5) Here ν is the viscosity parameter, and we set ν = 1/150, L = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Burgers equation [11] has various applications in fluid dynamics, including modeling of viscous flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We are interested in reconstructing solution states, as well as in the challenging problem of forecasting shocks that emerge outside the time range covered by the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5) is discretized on [0, L] with equally-spaced grid points 0 = x1 < x2 < · · · < xM = L, using a second-order finite difference method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Setting ∆x := xi − xi−1 = L M−1, we obtain the 16 0 32 64 96 128 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 AD-EnKF rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0394 t=40, Reconstruction 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0298 t=80, Reconstruction 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0375 t=120, Reconstruction 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0362 t=160, Reconstruction 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0378 t=200, Reconstruction 0 32 64 96 128 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 ROAD-EnKF rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0042 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0033 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0097 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0100 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0039 Observation Reconstruction Truth 0 32 64 96 128 0 1 2 AD-EnKF rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0357 t=250, Forecast (Start) 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0315 t=252, Forecast 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0164 t=254, Forecast 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0335 t=256, Forecast 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1820 t=258, Forecast 0 32 64 96 128 0 1 2 ROAD-EnKF rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0064 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0048 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0024 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0037 0 32 64 96 128 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0108 Forecast Truth Figure 6: State reconstruction (upper half) and forecast (lower half) performance with partial observation (du = 128, dy = 64) on the embedded L63 example in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' For each method, the reconstructed states ut (blue) for a single test sequence are plotted for t = 40, 80, 120, 160, 200 (column), and the forecasted states ut (blue) for a single test sequence are plotted for t = 250 (start of forecast), 252, 254, 256, 258 (column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The true values of the 128-dimensional states are plotted in red dashed lines, along with the noisy observations in black dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' SINDy-AE is inapplicable here because it cannot handle partial observations, while both AD-EnKF and ROAD-EnKF perform probabilistic state reconstructions and forecast through particles (all plotted in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The reconstruction/forecast RMSEs are computed for each plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' following ODE system: du(i) ds = − � u(i+1)�2 − � u(i−1)�2 4∆x + ν u(i+1) − 2u(i) + u(i−1) ∆x2 , i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' , M − 1, u(1)(s) = u(M)(s) = 0, u(i)(0) = u0(i∆x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6) Here u(i)(s) is an approximation of u(i∆x, s), the value of u at the i-th spatial node at time s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6) defines a flow map F : u(s) �→ u(s + ∆s) for state variable u with du = M, which we refer to as the true state dynamics model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We assume there is no noise in the dynamics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=', Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 17 Similar to Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1, we consider two cases: full observation with du = dy = 256 and partial observation with du = 256, dy = 128 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=', c = 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The initial conditions u0 are generated in the following way: u(i) 0 = U sin 2πi∆x L , U ∼ Uniform(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='7) We generate Ntrain = 1024 training data and Ntest = 20 test data with the true state dynamics model defined through equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='7) with Rt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='01Idy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We set the number of observations T = 300 with time between observations ∆s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We set the forecast lead time Tf = 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The flow map F is integrated using the fourth-order Runge–Kutta method with a fine step size ∆s/20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The surrogate latent dynamics map gα is parameterized as a two-layer fully connected NN, and is integrated using a fourth-order Runge-Kutta method with step size ∆int s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The error covariance matrix Sβ in the latent dynamics is parametrized using a diagonal matrix with positive diagonal elements β ∈ Rdz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The decoder Dγ is parameterized as an FND, discussed in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Details of the network hyperparameters are listed in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The latent space dimension for ROAD-EnKF is set to dz = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The ensemble size for both AD-EnKF and ROAD-EnKF is set to N = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' In this example and the following one, we set z0 ∼ N(0, σ2Idz) with the same σ defined in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' In Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4, we list the performance metrics of each method with full and partial observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The state reconstruction and forecast performance on a single instance of test data are plotted in Figures 7 (snapshots) and 8 (contour plot) for the partial observation case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Corresponding plots with full observation are shown in Figures 11 and 12 in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We find that ROAD-EnKF is able to reconstruct and forecast the states with the lowest RMSE, in both full and partial observation scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' More importantly, the emergence of shocks is accurately forecasted even though this phenomenon is not included in the time range covered by the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' AD-EnKF achieves a higher RMSE than ROAD-EnKF for both state reconstruction and forecast tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' AD-EnKF forecasts the emergence of shocks with lower accuracy than ROAD-EnKF, which indicates that AD-EnKF fails to fully learn the state dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' SINDy-AE with finite difference approximation of derivative data has the highest reconstruction RMSE among the three methods, and is not able to produce meaningful long-time state forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' This is remarkable, given that in this example the data are relatively dense (∆s is small) which facilitates, in principle, the approximation of time derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' In terms of computational cost, ROAD-EnKF is more efficient than AD-EnKF during both training and testing, but takes more time than SINDy-AE for the same reason as in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' In Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5, we list the performance metrics of ROAD-EnKF with full observation and different choices of latent space dimension dz ranging from 1 to 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The results for partial observation show a similar trend and are not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We find that, as dz increases, the state reconstruction performance stabilizes when dz ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' In order to achieve better long-time state forecast performance, dz needs to be further increased, and the forecast performance stabilizes when dz ≥ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Both training and testing time slightly increase as dz grows, which can be explained by the following: The computational time for both training and testing can be divided into the prediction step and the analysis step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We have shown in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2 that the computational bottleneck of the analysis step depends on the choices of ensemble size N and dy, and is less affected by the increase of dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Moreover, the computational time of the prediction step depends on the complexity of the surrogate latent dynamics (two-layer NNs), which are relatively cheap to simulate for ROAD-EnKF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' On the other hand, AD-EnKF enjoys similar computational complexity as ROAD-EnKF during the analysis step, but requires a more complicated surrogate model (NNs with Fourier layers) to capture the dynamics, which is more expensive to simulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' More experimental results on different parameterization methods of surrogate dynamics can be found in Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1 in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 18 SINDy-AE (full) AD-EnKF (full) ROAD-EnKF (full) AD-EnKF (partial) ROAD-EnKF (partial) RMSE-r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1433 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0044 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0122 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0081 RMSE-f(30) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4579 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0212 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0228 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0160 RMSE-f(150) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4906 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0763 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0514 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0724 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0581 Log-likelihood − 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='40 × 104 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='60 × 104 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='24 × 104 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='27 × 104 Training time (per epoch) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='78s 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='75s 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='10s 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='08s 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='20s Test time 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='78s 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='54s 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='21s 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='76s 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='24s Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4: Performance metrics for different algorithms at convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (Burgers example, Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=') 0 64 128 192 256 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 AD-EnKF rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0168 t=50, Reconstruction 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0104 t=100, Reconstruction 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0136 t=150, Reconstruction 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0148 t=200, Reconstruction 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0092 t=250, Reconstruction 0 64 128 192 256 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 ROAD-EnKF rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0089 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0072 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0055 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0049 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0056 Observation Reconstruction Truth 0 64 128 192 256 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 AD-EnKF rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0095 t=300, Forecast (Start) 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0345 t=375, Forecast 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0537 t=450, Forecast 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0849 t=525, Forecast 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1171 t=600, Forecast 0 64 128 192 256 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 ROAD-EnKF rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0070 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0130 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0251 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0415 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0616 Forecast Truth Figure 7: State reconstruction (upper half) and forecast (lower half) performance with partial observation (du = 256, dy = 128) on the Burgers example in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' For each method, the reconstructed states ut (blue) for a single test sequence are plotted for t = 50, 100, 150, 200, 250 (column), and the forecasted states (blue) for a single test sequence are plotted for t = 300 (start of forecast), 375, 450, 525, 600 (column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The true values of the 256- dimensional states are plotted in red dashed lines, along with the noisy observations in black dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Both AD-EnKF and ROAD-EnKF perform probabilistic state reconstructions and forecast through particles (all plotted in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The reconstruction/forecast RMSEs are computed for each plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 19 (a) Ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (b) Reconstruction and forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Figure 8: Contour plot of state reconstruction and forecast output with partial observation (du = 256, dy = 128) on the Burgers example in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2, as well as the ground truth (top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' For each method (row), the reconstructed and forecasted states (left column) for a single test sequence are plotted, for each state dimension (y-axis) and time (x-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The error compared to the ground truth are plotted in the right column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' For both AD-EnKF and ROAD-EnKF we use particle means as point estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' ROAD-EnKF AD-EnKF dz = 1 dz = 2 dz = 4 dz = 10 dz = 20 dz = 40 dz = 120 dz = 240 RMSE-r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2293 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0316 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0058 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0044 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0048 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0059 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0102 RMSE-f(30) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2593 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0761 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0165 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0112 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0212 RMSE-f(150) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2690 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1827 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1313 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0501 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0607 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0514 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0373 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0382 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0763 Log-likelihood (×104) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='03 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='63 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='59 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='61 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='60 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='61 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='63 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='40 Training time (per epoch) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='70s 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='78s 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='79s 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='98s 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='98s 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='10s 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='44s 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='40s 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='75s Test time 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='36s 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='80s 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='13s 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='14s 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='08s 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='21s 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='53s 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='90s 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='54s Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5: Performance metrics for ROAD-EnKF at convergence with full observation (du = dy = 256) and different latent space dimension dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (Burgers example, Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=') 20 250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2 Truth 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6 0 0 100 200 300 400 500t<300 t>300 t<300 t>300 Reconstruction Forecast Reconstruction Error Forecast Error 250 250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6 200 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4 50 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6 100 200 300 400 500 100 200 300 400 500 250 250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6 200 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2 150 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 100 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4 50 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6 0 + 0 100 200 300 400 500 100 200 300 400 5005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3 Kuramoto-Sivashinsky Equation In this subsection, we consider the Kuramoto-Sivashinsky (KS) equation for u(x, s), where u is a function of the spatial variable x ∈ [0, L] and continuous-time variable s > 0: ∂u ∂s = −ν ∂4u ∂x4 − ∂2u ∂x2 − u∂u ∂x, u(0, s) = u(L, s) = 0, ∂u ∂x(0, s) = ∂u ∂x(L, s) = 0, u(x, 0) = u0(x), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='8) Here ν is the viscosity parameter, and we set ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='05, L = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We impose Dirichlet and Neumann boundary conditions to ensure ergodicity of the system [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The KS equation was originally introduced by Kuramoto and Sivashinsky to model turbulence of reaction-diffusion systems [50] and propagation of flame [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='8) is discretized on [0, L] with equally-spaced grid points 0 = x1 < x2 < · · · < xM = L, using a second-order finite difference method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Setting ∆x := xi − xi−1 = L M−1, we obtain the following ODE system: ∂u(i) ∂s = −ν u(i−2) − 4u(i−1) + 6u(i) − 4u(i+2) + u(i+2) ∆x4 − u(i+1) − 2u(i) + u(i−1) ∆x2 − � u(i+1)�2 − � u(i−1)�2 4∆x , i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' , du − 1, u(1)(s) = u(du)(s) = 0, u(0)(s) = u(2)(s), u(du+1)(s) = u(du−1)(s), u(i)(0) = u0(i∆x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='9) The discretization method follows [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Here u(i)(s) is an approximation of u(i∆x, s), the value of u at the i-th spatial node and time s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Two ghost nodes u(0) and u(du+1) are added to account for Neumann boundary conditions, and are not regarded as part of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='9) defines a flow map F : u(s) �→ u(s+∆s) for state variable u with du = M, which we refer to as the true state dynamics model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We assume there is no noise in the dynamics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=', Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Similar to Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1, we consider two cases: full observation with du = dy = 256 and partial observation with du = 256, dy = 128 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=', c = 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The initial conditions u0 are generated at random from the attractor of the dynamical system, by simulating a long run beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We generate Ntrain = 512 training data and Ntest = 20 test data with the true state dynamics model defined through (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='9) with Rt = Idy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We set the number of observations T = 450 with time between observations ∆s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We set the forecast lead time Tf = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The flow map F is integrated using the fourth-order Runge–Kutta method with a fine step size ∆s/10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The surrogate latent dynamics map gα is parameterized as a two-layer fully connected NN, and is integrated using a fourth-order Runge-Kutta method with step size ∆int s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The error covariance matrix Sβ in the latent dynamics is parametrized using a diagonal matrix with positive diagonal elements β ∈ Rdz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The decoder Dγ is parameterized as an FND, discussed in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Details of the network hyperparameters are listed in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The latent space dimension for ROAD-EnKF is set to dz = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The ensemble size for both AD-EnKF and ROAD-EnKF is set to N = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' In Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6 we list the performance metrics of AD-EnKF and ROAD-EnKF with full and partial obser- vation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' SINDy-AE is not listed here as we find it unable to capture the dynamics for any choice of latent space dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The state reconstruction and forecast performance on a single instance of test data are plotted in Figure 9 (snapshots), and Figure 10 (contour plot, ROAD-EnKF) for the partial observation case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Corresponding plots with full observation are shown in Figures 13 and 14 in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We find that ROAD-EnKF is able to reconstruct the states with lower RMSE than AD-EnKF in both full observation and partial observation cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Both methods can produce meaningful forecast multiple steps forward into the 21 future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' ROAD-EnKF achieves a higher forecast RMSE than AD-EnKF in full observation case, while having a lower forecast RMSE in partial observation case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Although ROAD-EnKF does not consistently have a better forecast performance than AD-EnKF due to the difficulty of finding a reduced-order representation for the highly chaotic system, we find that its performance is not much impacted by partial observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Moreover, it is two times more efficient than AD-EnKF in both training and testing, due to the times saved for simulating a cheaper surrogate model and running the EnKF algorithm in a lower dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Notice in Figure 10(b) and Figure 14(b) that, although the predictive means of all particles are ‘smoothed’ when passing a certain time threshold, each particle individually produces nontrivial forecasts for a larger number of time steps into the future, thus illustrating the variability of particle forecasts and the stochastic nature of state reconstruction and forecast in our ROAD-EnKF framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' AD-EnKF (full) ROAD-EnKF (full) AD-EnKF (partial) ROAD-EnKF (partial) RMSE-r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4658 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3552 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4686 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3589 RMSE-f(1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5137 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5626 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6231 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5644 RMSE-f(5) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0910 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2734 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4669 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3780 Log-likelihood −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='89 × 106 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='88 × 106 −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='33 × 105 −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='07 × 105 Training time (per epoch) 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='92s 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='53s 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='72s 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='61s Test time 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='35s 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='11s 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='22s 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='39s Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6: Performance metrics for different algorithms at convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (KS example, Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=') 6 Conclusions and Future Directions This paper introduced a computational framework to reconstruct and forecast a partially observed state that evolves according to an unknown or expensive-to-simulate dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Our ROAD-EnKFs use an EnKF algorithm to estimate by maximum likelihood a surrogate model for the dynamics in a latent space, as well as a decoder from latent space to state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Our numerical experiments demonstrate the computational advantage of co-learning an inexpensive surrogate model in latent space together with a decoder, rather than a more expensive-to-simulate dynamics in state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The proposed computational framework accommodates partial observation of the state, does not require time derivative data, and enables uncertainty quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' In addition, it provides significant algorithmic flexibility through the choice of latent space, surrogate model for the latent dynamics, and decoder design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' In this work, we showed that accurate and cheap reconstructions and forecasts can be obtained by choosing an inexpensive NN surrogate model, and a decoder inspired by recent ideas from operator learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' While adequate choice of NN architecture and decoder may be problem-specific, an important question for further research is to derive guidelines and physics-informed NNs that are well-suited for certain classes of problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Acknowledgments The authors are grateful to Melissa Adrian for her generous feedback on an earlier version of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' YC was partially supported by NSF DMS-2027056 and NSF OAC-1934637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' DSA is grateful for the support of NSF DMS-2237628, NSF DMS-2027056, DOE DE-SC0022232, and the BBVA Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' RW is grateful for the support of DOD FA9550-18-1-0166, DOE DE-AC02-06CH11357, NSF OAC-1934637, NSF DMS-1930049, and NSF DMS-2023109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 22 0 64 128 192 256 −15 −10 −5 0 5 10 AD-EnKF rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4478 t=50, Reconstruction 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5089 t=150, Reconstruction 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4132 t=250, Reconstruction 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3472 t=350, Reconstruction 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5187 t=450, Reconstruction 0 64 128 192 256 −15 −10 −5 0 5 10 ROAD-EnKF rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3306 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4324 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3293 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3488 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2349 Observation Reconstruction Truth 0 64 128 192 256 −10 −5 0 5 10 AD-EnKF rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5187 t=450, Forecast (Start) 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='9386 t=452, Forecast 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='9549 t=454, Forecast 0 64 128 192 256 rmse: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='7769 t=456, Forecast 0 64 128 192 256 rmse: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5097 t=458, Forecast 0 64 128 192 256 −10 −5 0 5 10 ROAD-EnKF rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2349 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4737 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6419 0 64 128 192 256 rmse: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0449 0 64 128 192 256 rmse: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6044 Forecast Truth Figure 9: State reconstruction (upper half) and forecast (lower half) performance with partial observation (du = 256, dy = 128) on the KS example in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' For each method, the reconstructed states ut (blue) for a single test sequence are plotted for t = 50, 150, 250, 350, 450 (column), and the forecasted states (blue) for a single test sequence are plotted for t = 450 (start of forecast), 452, 454, 456, 458 (column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The true values of the 256-dimensional states are plotted in red dashed lines, along with the noisy observations in black dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Both AD-EnKF and ROAD-EnKF perform probabilistic state reconstructions and forecast through particles (all plotted in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The reconstruction/forecast RMSEs are computed for each plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Abadi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Barham, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Chen, A.' metadata={'source': 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system for large-scale machine learning, in 12th {USENIX} Symposium on Operating Systems Design and Implementation ({OSDI} 16), 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 265–283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Agapiou, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Papaspiliopoulos, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Sanz-Alonso, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Stuart, Importance sampling: Intrinsic dimension and computational cost, Statistical Science, 32 (2017), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 405–431.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' [3] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Bengio, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Simard, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Frasconi, Learning long-term dependencies with gradient descent is difficult, IEEE Transactions on Neural Networks, 5 (1994), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 157–166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 23 (a) Ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (b) Reconstruction and forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Figure 10: Contour plot of state reconstruction and forecast output of ROAD-EnKF with partial observation (du = 256, dy = 128) on the KS example in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3, as well as the ground truth (top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The particle means of reconstructed and forecasted states for a single test sequence are plotted, for each state dimension (y-axis) and time (x-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The reconstructed and forecasted states of three randomly chosen particles are also plotted individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' [4] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Bengtsson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Hu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Zhang, A Review of Recurrent Neural Networks: LSTM Cells and Network Architectures, Neural Computation, 31 (2019), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 1235–1270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' A Improving AD-EnKF with Spectral Convolutional Layers This appendix discusses an enhancement of the AD-EnKF algorithm [20], used for numerical comparisons in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' AD-EnKF runs EnKF on the full-order SSM (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3) and learns the parameter θ = (α⊤, β⊤)⊤ by auto-differentiating through a similarly defined log-likelihood objective, as in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' A high dimension of u makes challenging the NN parameterization of Fα (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' fα in the ODE case) in the state dynamics model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' In particular, the local convolutional NN used in [20] does not perform well in the high-dimensional numerical experiments considered in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We thus propose a more flexible NN parameterization of Fα (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' fα) using the idea of spectral convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' We design Fα (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' fα) in a way similar to the Fourier Neural Decoder, but without the complex linear layer and IDFT step at the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' That is, we start with a state variable u′ ∈ Rdu as the input, iteratively apply (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2) with v0 = u′ to get vL ∈ RnL×du, followed by a fully-connected network applied over the channel dimension to get the output u ∈ Rdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The architecture is the same as Figure 3(a) but we start at v0 instead of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' B Additional Materials: Burgers Example For SINDy-AE, we use a finite difference approximation computed from data y1:T to approximate the exact time-derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The latent space dimension for SINDy-AE is set to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Increasing it does not further enhance the performance, but increases the computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' AD-EnKF (FC, Euler) AD-EnKF (FC, RK4) AD-EnKF (Fourier, Euler) AD-EnKF (Fourier, RK4) ROAD-EnKF (FC, Euler) ROAD-EnKF (FC, RK4) RMSE-r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0934 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0537 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0044 RMSE-f(30) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0831 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1302 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0212 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0096 RMSE-f(150) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1971 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1608 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2908 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0763 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0664 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0514 Log-likelihood 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='31 × 104 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='87 × 104 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='41 × 104 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='40 × 104 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='60 × 104 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='57 × 104 Training time (per epoch) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='97s 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='80s 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='31s 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='75s 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='21s 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='10s Test time 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='29s 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='97s 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='79s 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='54s 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='28s 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='21s Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1: Ablation study: AD-EnKF versus ROAD-EnKF with different NN parameterization and numerical integration methods for surrogate dynamics (FC: NN with fully-connected layers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Fourier: NN with Fourier layers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Euler: Euler method for ODE integration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' RK4: fourth-order Runge Kutta method for ODE integration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Switching from RK4 to Euler method while keeping the same NN configuration gives a computational speed-up, and the speed- up is more noticeable when the NN involves Fourier layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' However, after the switch, the accuracy drops more significantly for AD-EnKF than for ROAD-EnKF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The best configuration for AD-EnKF (Fourier with RK4) still yields a lower accuracy compared to both ROAD-EnKF configurations, while taking more time to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (Burgers example, full observation case, Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=') 30 0 64 128 192 256 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 SINDy-AE rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1433 t=50, Reconstruction 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1188 t=100, Reconstruction 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0946 t=150, Reconstruction 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0598 t=200, Reconstruction 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0290 t=250, Reconstruction 0 64 128 192 256 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 AD-EnKF rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0107 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0089 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0093 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0080 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0086 0 64 128 192 256 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 ROAD-EnKF rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0083 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0082 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0066 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0052 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0042 Observation Reconstruction Truth 0 64 128 192 256 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 SINDy-AE rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0192 t=300, Forecast (Start) 0 64 128 192 256 rmse: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5660 t=375, Forecast 0 64 128 192 256 rmse: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5667 t=450, Forecast 0 64 128 192 256 rmse: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5739 t=525, Forecast 0 64 128 192 256 rmse: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5883 t=600, Forecast 0 64 128 192 256 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 AD-EnKF rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0085 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0330 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0643 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0908 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='1101 0 64 128 192 256 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='5 ROAD-EnKF rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0045 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0100 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0213 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0335 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0453 Forecast Truth Figure 11: State reconstruction (upper half) and forecast (lower half) performance with full observation (du = dy = 256) on the Burgers example in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' For each method, the reconstructed states ut (blue) for a single test sequence are plotted for t = 50, 100, 150, 200, 250 (column), and the forecasted states (blue) for a single test sequence are plotted for t = 300 (start of forecast), 375, 450, 525, 600 (column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The true values of the 256-dimensional states are plotted in red dashed lines, along with the noisy observations in black dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Both AD-EnKF and ROAD-EnKF perform probabilistic state reconstructions and forecast through particles (all plotted in blue), while SINDy-AE only provides point estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The reconstruction/forecast RMSEs are computed for each plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 31 (a) Ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (b) Reconstruction and forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Figure 12: Contour plot of state reconstruction and forecast output with full observation (du = dy = 256) on the Burgers example in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2, as well as the ground truth (top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' For each method (row), the reconstructed and forecasted states (left column) for a single test sequence are plotted, for each state dimension (y-axis) and time (x- axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The error compared to the ground truth are plotted in the right column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' For both AD-EnKF and ROAD-EnKF we use particle means as point estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' C Additional Materials: Kuramoto-Sivashinky Example 32 250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2 Truth 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6 0 0 100 200 300 400 500t<300 t>300 t<300 t>300 Reconstruction Forecast Reconstruction Error Forecast Error 250 250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6 200 200 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4 50 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6 01 0- 0 100 200 300 400 500 0 100 200 300 400 5000 64 128 192 256 −10 0 10 AD-EnKF rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4430 t=50, Reconstruction 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3721 t=150, Reconstruction 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4419 t=250, Reconstruction 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3058 t=350, Reconstruction 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4504 t=450, Reconstruction 0 64 128 192 256 −10 0 10 ROAD-EnKF rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2746 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4711 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3168 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='2995 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3667 Observation Reconstruction Truth 0 64 128 192 256 −10 −5 0 5 10 AD-EnKF rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='4504 t=450, Forecast (Start) 0 64 128 192 256 rmse: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0364 t=452, Forecast 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='9103 t=454, Forecast 0 64 128 192 256 rmse: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0975 t=456, Forecast 0 64 128 192 256 rmse: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='8994 t=458, Forecast 0 64 128 192 256 −10 −5 0 5 10 ROAD-EnKF rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3667 0 64 128 192 256 rmse: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='0875 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='6654 0 64 128 192 256 rmse: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='8397 0 64 128 192 256 rmse: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='7079 Forecast Truth Figure 13: State reconstruction (upper half) and forecast (lower half) performance with full observation (du = dy = 256) on the KS example in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' For each method, the reconstructed states ut (blue) are plotted for t = 50, 150, 250, 350, 450 (column), and the forecasted states (blue) are plotted for t = 450 (start of forecast), 452, 454, 456, 458 (column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The true values of the 256-dimensional states are plotted in red dashed lines, along with the noisy observations in black dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Both AD-EnKF and ROAD-EnKF perform probabilistic state reconstructions and forecast through particles (all plotted in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The reconstruction/forecast RMSEs are computed for each plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 33 (a) Ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' (b) Reconstruction and forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' Figure 14: Contour plot of state reconstruction and forecast output of ROAD-EnKF with full observation (du = dy = 256) on the KS example in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content='3, as well as the ground truth (top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The particle means of reconstructed and forecasted states are plotted, for each state dimension (y-axis) and time (x-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' The individual reconstructed and forecasted states of three randomly chosen particles are also plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} +page_content=' 34 250 10 200 5 150 Truth 0 100 50 10 400 420 440 460 480t<450 t>450 t<450 t>450 Reconstruction Forecast Reconstruction Forecast 250 250 10 10 (Mean) 3200 200 5 5 150 150 0 ROAD-EnKF ( 0 100 ROAD-E 5 50 - 50 10 10 420 440 460 480 420 440 460 480 250 250 Particle #3), 10 10 (Particle 2200 200 5 5 150 150 0 ROAD-EnKF ( 0 ROAD-EnKF 100 100 5 50 50 10 10 R 420 440 460 480 400 420 440 460 480' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFKT4oBgHgl3EQf_S4j/content/2301.11961v1.pdf'} diff --git 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large number of comets. These pose a long-standing +conundrum for solar system formation models as they can only be created in the in- +ner hot disk at temperatures higher than 800 K, and there is no obvious mechanism +to transport them out into the comets formation region. Here we propose that these +particles could have formed inside the hydrostatic envelopes surrounding young pro- +toplanets still embedded in the protoplanetary disk. Using a simplified 1D model we +investigate the thermal structure of these envelopes, and find that for core masses +ranging from 0.08 to 1.5 M⊕, located anywhere between 1 and 30 AU, the temper- +ature and pressure at the base of the envelopes are high enough to quickly vaporize +silicate particles of various sizes. Moreover, if the grain abundance is atleast solar, these +envelopes become fully convective, allowing for dust ejection across the Bondi radius +back into the disk. Amorphous silicates are hence thermally processed into crystalline +particles in these envelopes, and then transported back to disk through convective +diffusion to be finally incorporated into the cometary building blocks. +Key words: planets and satellites: formation – comets: general – planets and satellites: +composition +1 +INTRODUCTION +High temperature minerals are ubiquitous in the cold outer +solar system small bodies. One of the earliest remote sensing +detections was for crystalline silicates (CSs) such as olivines +and pyroxenes in the grains of comets 1P/Halley, D/1993 +F2 (Shoemaker-Levy), C/1987 P1 (Bradfield) (Hanner et al. +1994), and C/1993 A1 (Mueller) (Hanner, Lynch, & Russell +1994). Subsequent detections were made in comets C/1995 +O1 (Hale-Bopp) (Hayward, Hanner, & Sekanina 2000; Cro- +visier et al. 1997; Wooden et al. 1999), 103P/Hartley (Cro- +visier et al. 2000), and more recently 17P/Holmes (Shinnaka +et al. 2018). On the other hand Calcium-Aluminum inclu- +sions (CAIs) that form at even higher temperatures were +found in the dust collected by Stardust in comet 81P/Wild +(Brownlee et al. 2006). We refer the reader to the observa- +tional review of Mumma & Charnley (2011) for more infor- +mations. +The presence of CSs have been a primary challenge to +solar system formation models for decades, as the thermal +conditions in the outer protoplanetary disk are not con- +ducive to their formation locally. CSs can form starting +from amorphous silicates through either direct vaporization +followed by re-condensation at temperatures higher than +⋆ E-mail: malidib@nyu.edu +∼1800 K, or thermal annealing for T> 800 K. Annealing is +a physical process where sufficiently energetic molecules of a +solid slowly regroup into a crystal lattice. This mechanism is +not instantaneous and necessitates high temperature expo- +sure for a period of few weeks followed by slow cooling (Gail +1998, 2001). In typical protoplanetary disks, direct conden- +sation can be active only inside ∼ 0.1 AU, and annealing is +inefficient outside ∼ 1.5 AU. +Moreover, there is no acceptable mechanism to trans- +port these particles from the inner disk to the comets for- +mation region. Earlier transport models relied on turbulent +diffusion (Bockel´ee-Morvan et al. 2002), but recent ALMA +observations suggest that protoplanetary disks are laminar +(Flaherty et al. 2015, 2018), and thus this mechanism is un- +likely to be efficient. Large scale outward advection in the +disk’s midplane has also been proposed (Hughes & Armitage +2010), but 3D MHD simulations ruled out the presence of +such advection (Fromang, Lyra, & Masset 2011). Another +possible transport mechanism is photophoresis (Mousis et +al. 2007), but this necessitates a relatively large (1-2 AU) +central hole in the disk. Finally, Ali-Dib et al. (2015) pro- +posed that FU-Ori outbursts might form these particles in +situ, but this depends on the outburst trigger radius being +large enough, which is uncertain. +Here we show how high temperature minerals form nat- +urally, and in-situ, in the envelopes surrounding low mass +© 2023 The Authors +arXiv:2301.01472v1 [astro-ph.EP] 4 Jan 2023 + +2 +M. Ali-Dib +proto-planets embedded in the disk. We present models +showing that the temperatures and pressures at the base +of these envelopes easily reach conditions that allow for +the formation of crystalline silicates through direct vapor- +ization and re-condensation. Primordial amorphous silicates +are thus accreted and then thermally processed in these en- +velopes, before finally getting ejected back to the disk as +crystalline particles via convective diffusion. We emphasize +that this work concerns the formation of generic CSs such as +olivines and pyroxenes, and not necessarily chondrules and +CAIs due to their additional formation-time constrains that +are outside the scope of this work. We present our model in +section 2, results in section 3, and conclude in section 4. +2 +MODEL +We model the atmosphere using the standard atmospheric +structure equations. The equation of hydrostatic equilibrium +is given by: +dP +dr = −GM +r2 ρ(r) +(1) +and we define the temperature gradient equation starting +from the standard assumption that heat can be transported +using either radiation (if the local envelope is convectively +stable) or adiabatic convection (if unstable). It is hence writ- +ten as: +dT +dr = ∇ T +P +dP +dr +(2) +where ∇ is defined, starting from the Schwarzschild con- +vective stability criterion (∇rad +< +∇ad), to be ∇ += +min(∇ad, ∇rad). Here ∇ad is the adiabatic gradient: +∇ad ≡ +� d ln T +d ln P +� +ad += γ − 1 +γ +(3) +where the adiabatic constant γ = 1.5. ∇rad is the radiative +gradient: +∇rad ≡ +3κP +64πGMσT 4 L +(4) +where L is the envelope’s luminosity generated by accretion +at a rate +˙Macc : +L = GM ˙Macc +Rc +(5) +Rc is the core radius, and κ is the opacity that we define +following Ormel (2014) as: +κ = κgas + κgr +(6) +with: +κgr = κgeomQe = 3Zgr +4ρsa × min(0.6πa +λmax , 2) +(7) +where Zgr is the grains abundance, as their size, and ρs their +internal density. we use ρs = 3 g/cm3 for both the core and +the dust particles. +The equilibrium dust size in the envelope as is set by +two competing processes: grain growth through coagulation +(Ormel 2014) and grain collisional destruction (Ali-Dib & +Thompson 2020). The relative relevance of these two pro- +cesses is decided mainly by whether the collisional speeds +reach the silicate fragmentation threshold (Vf ∼ 100 cm/s, +Blum & Wurm (2008)). The collisional speed is approxi- +mated here as the largest among the dust’s convective ve- +locity Vcon,d (eq. 19) and the dust’s radial drift velocity: +Vdrift,d = τstop GM +r2 +(8) +where τstop is the stopping time. +As discussed in (Ali-Dib & Thompson 2020), collisions +in these envelopes are likely to be destructive. This leads +to a small characteristic dust size, increasing the opac- +ity (thus growing the convective zone), and decreasing the +vaporization timescale. Here we only select models where +max(Vcon,d,Vdrift,d) is higher than 100 cm/s everywhere in +the disk. +The convective fragmentation dust size is hence calcu- +lated following (Ali-Dib & Thompson 2020) as: +as,conv = 4πV 2 +f r3ρ2 +gcg +Lρs +(9) +where ρg and cg are the gas’ density and sound speed. The +drift fragmentation dust size is given by: +as,drift = Vfr2ρgcg +GMρs +(10) +with finally as=min(as,conv,as,drift). +Note that as,conv is defined everywhere in the envelope, +since, as discussed below, we also only select fully convec- +tive envelopes. For this approach to be applicable, the par- +ticles need to reach the local fragmentation threshold at ev- +ery point in the envelope. Therefore, for self-consistency, we +only keep models where the mean free time for collisions +is shorter than the convective timescale. In the convective +fragmentation regime this can be written as: +as +as + 4ℓg/9 < 9Z2 +gr +ρg +ρs Mcon r +ℓg +(11) +where Mcon is the convective Mach number, and ℓg the mean +free path of the gas. In the drift regime this is replaced by : +3 +4Zgr +c2 +gr +GM +Mcon +1 + 9as/4ℓg < 1 +(12) +The gas opacity is given by: +κgas = 10−8ρ2/3 +g +T 3 +(13) +Finally we close the system with the ideal gas equation of +state P = ρgkBT/µ. We solve these equations by integrating +inwards from the outer boundary at Rout, the minimum of +the Bondi and Hill radii, to the core. We assumed the disk is +radiative and calculate its temperature and density following +Ali-Dib, Cumming, & Lin (2020): +Td = 373 r−9/10 +au +K and ρd = 1.7×10−10 r−33/20 +au +g/cm3 (14) +Once we have the envelope’s thermal structure, we can +calculate additional quantities needed for the subsequent +analysis. We calculate the silicate particles vaporization rate +as : +1 +as +das +dt = − +� µSil +2πkT +�1/2 P sat +Sil +ρsas +(15) +MNRAS 000, 1–5 (2023) + +Origins of hot minerals +3 +with (Krieger 1967): +P sat +sil (T) = 3.2 × 1014e−(6×104 K)/T +(16) +and hence the silicate grains vaporization timescale is given +by : +τvap,sil = +� 1 +as +das +dt +�−1 +(17) +We define the gas and dust convective velocities respec- +tively as: +Vcon,g = +� +L +4πr2ρg +�1/3 +(18) +where we assumed that in the convective zone the energy is +entirely transported through adiabatic convection, and +Vcon,d ∼ Vcon (Vconτstop/r)1/2 +(19) +We +finally +calculate +the +dust’s +convective +mixing +timescale as: +τmix,d = H/Vcon,d +(20) +3 +RESULTS +We start by exploring parameter space in order to find the +values that allow for the creation of CSs in proto-envelopes. +We explore core masses ranging from Pluto’s mass (0.002 +M⊕) to a hypothetical giant planet’s core (10 M⊕), placed +between 1 and 30 AU where ambient temperatures are too +low to create CSs in the disk. The grains abundance Zgr +ranges from subsolar (10−3) to supersolar (1.0). +Our results are summarized in Fig. 1. In this plot we +show only the areas of parameter space leading to envelopes +conducive to the creation of CSs and that are self-consistent +to our model assumptions. This is defined by these condi- +tions: +(i) τvap,sil is less than τmix,d at the base of the envelope. +This simply constrain the envelopes to those where solid +silicates at their base can get vaporized faster than they are +transported back into the upper cooler zones. +(ii) The envelope is fully convective. This ensures that the +newly created CSs can be convectively diffused all the way +back into the disk. This condition is inspired by the results of +Ali-Dib & Thompson (2020) who considered a similar setup +with a 0.3 M⊕ core embedded in the disk, and showed that, +for typical accretion rates, pebble fragmentation and dust +loading increases the opacity and push the convective zone +out till it reaches the Bondi radius. Dust particles in these +steady-state envelopes are then diffusively ejected back to +the disk. Our results rely on this mechanism to transport +the newly created CSs from the hot inner envelope back to +the disk to be incorporated in proto-comets. +(iii) The collisional velocity is higher than 100 cm/s +throughout the envelope, and conditions 11 and 12 are sat- +isfied. This ensures that our dust size prescription is self- +consistent. +3.1 +Core mass +Figure 1 shows that, while CSs can be created under a va- +riety of parameter ranges, trends do exist. We start with +our nominal model, for +˙Macc = 10−6M⊕/yr. First, there +is a relatively narrow range of masses that extends from +around 0.08 M⊕ (40 times Pluto’s mass) to 1.5 M⊕, beyond +which the chances of creating CSs drops drastically. This +implies that CSs might have formed in the proto-envelopes +of Mars to Earth mass protoplanets that have since disap- +peared via giant collisions or dynamical ejection, or possibly +grown into giant planetary cores. The lower limit on core +masses is mainly due to their envelopes’ relatively cooler +temperatures, increasing τvap,sil considerably. On the other +hand, cores with masses higher than 1.5 M⊕ have dust par- +ticles large enough in their middle and inner envelopes to +switch from the Rosseland mean opacity regime to the ge- +ometric opacity regime, as can be seen in Fig. 2 (left hand +panel). This decreases the radiative gradient, creating an in- +ner radiative zone that prevents these envelopes from being +fully convective. It is worth noting that, while our model +considers the smallest of the Hill and Bondi radii to be the +envelope’s outer boundary, all of our acceptable cases that +form CSs are in the Bondi regime. This is expected as the +Hill regime dominates for higher mass cores (5-10 M⊕) that +were excluded above. The Bondi radius RB = 2GM/c2 +s is +obtained by equating the local sound speed to the gravi- +tational escape velocity, and thus describes a usually light +but bound envelope where gas particles do not have enough +thermal energy to escape. For higher mass cores, the Bondi +radius is large enough for the Hill stability criteria to become +the more stringent constrain. +3.2 +Semimajor axis +A complementary piece of information is the semimajor axis, +where we find that CSs can form almost anywhere in the disk +if the envelope’s grain abundance is high enough as discussed +further below. Semimajor axis controls the temperature and +density at the outer boundary, which seems to be important +only in the marginal cases, for example for low core masses +where the envelopes would be too cold if placed further out +in the disk. The wide range of possible semimajor axis allows +for the possibility of creating CSs in the comets formation +region. Classically, Oort cloud comets were thought to form +among the giant planets all the way down to 5 AU, while +Jupiter family comets were thought to form in the scattered +disk (Duncan, Quinn, & Tremaine 1987; Duncan & Levison +1997; Dones et al. 2015). Alternatively, Brasser & Morbidelli +(2013) proposed that both could have formed in the same +region beyond Neptune. +3.3 +Grains abundance +We moreover find that creating CSs necessitate solar to su- +persolar grain abundance in the envelope (Zgr >= 0.01). +This result is not consistent with the subsolar grain abun- +dances found in models that incorporate dust growth & set- +tling to the core but omit dust fragmentation with convec- +tive mixing. Ormel (2014) for example added a simple grain +growth equation to the atmospheric structure equations, and +found that Zgr can be as low as 10−4 in parts of the envelope. +Mordasini (2014) also created an atmospheric model incor- +porating dust settling and coagulation, and found that this +mostly results in subsolar opacities. The main role of Zgr is +MNRAS 000, 1–5 (2023) + +4 +M. Ali-Dib +to increase the opacity and extend the convective zone all the +way to the outer boundary (Bondi or Hill radius). Ali-Dib +& Thompson (2020) discussed the gradual buildup of Zgr in +the envelope through accretion and fragmentation, and de- +rived a lower limit on Zgr in order to get a fully convective +envelope: +Zgr > 0.12 T 2 +d,2 +ρd,−11 +�tacc,c +Myr +� � +Mc +0.3M⊕ +�−2/3 +(21) +which is generally consistent with our Zgr values. In order to +get supersolar Zgr, multiple conditions need to be satisfied: +• The dust size need to be fragmentation-limited, which +is a pre-requisite for our dust-size prescription. This depends +on many factors, including the accretion rate (setting the lu- +minosity and thus convective speeds) and particles’ porosity +and chemical composition (Blum & Wurm 2008; Okuzumi +et al. 2012; Wada et al. 2008, 2009). +• A significant fraction of the dust should not get accreted +by the core, but remain mixed in the envelope. This is an +open question with many complications. In our cases, sili- +cates are in vapor form at the base of the envelope which +should stop accretion from taking place unless the tempera- +ture is low enough for the inner envelope to reach saturation +pressure. This also depends on the nature of convection in +these envelopes, whether it is diffusive as we are assuming, +or whether it is dominated by large scale eddies that can +enhance accretion by the core (Johansen & Nordlund 2020). +3.4 +Accretion rate +Finally we investigate the effects of using a lower accretion +rate. Our results for +˙Macc = 10−7M⊕/yr are shown in Fig. +1. In this case we find that while the semimajor axis range +remains the same and the lower mass limit does not change +(∼ 0.08M⊕), the upper limit decreases by over a factor 2 to +∼ 0.6M⊕. This is expected since, lower accretion rate leads +to lower luminosities. As seen in Fig. 2 (right hand side), +this decreases the radiative gradient and allows for a radia- +tive zone in the inner envelope even though the dust size in +the 2 cases converge to the same inner value. In some cases +Zgr can compensate for the lower luminosity and increases +the opacity enough to create fully convective envelopes, ex- +plaining the overall larger Zgr we find for the lower accretion +rate cases. +4 +SUMMARY & CONCLUSIONS +Crystalline silicates are ubiquitous in comets, but can only +form at very high temperatures. Here we investigated the +possibility of transforming amorphous silicates into crys- +talline particles inside the envelopes of protoplanets through +vaporization followed by re-condensation, and then ejecting +them back to the disk through diffusion in the fully con- +vective envelopes. Using a simplified 1D envelope structure +model that incorporates a dust size prescription accounting +for fragmentation and growth, we showed that crystalline +silicates can be created from a diverse set of parameters. +Cores need to be between 0.08 to 1.5 M⊕ in mass, as lighter +cores do not allow for temperatures high enough to vaporize +silicates, and the envelopes of more massive cores are often +not fully convective. We finally found that the location in +the disk (1 to 30 AU) has little influence on the results, ex- +cept in marginal cases, and that a solar to supersolar grain +abundance is needed, but this can be achieved through dust +fragmentation and accumulation. Our mechanism is simple +and does not rely on assumptions about the disk, although +it depends on the assumed diffusive nature of 1D convection. +Whether this is realistic needs to be investigated further us- +ing 3D hydrodynamic simulations. +ACKNOWLEDGEMENTS +We thank the anonymous referee for their constructive com- +ments that greatly improved this manuscript. This work is +supported by Tamkeen under the NYU Abu Dhabi Research +Institute grant CAP3. +DATA AVAILABILITY +The data underlying this article (numerical simulations out- +put files) will be shared on reasonable request to the corre- +sponding author. +REFERENCES +Ali-Dib M., Martin R. G., Petit J.-M., Mousis O., Vernazza +P., Lunine J. I., 2015, A&A, 583, A58. doi:10.1051/0004- +6361/201526453 +Ali-Dib M., Cumming A., Lin D. N. 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Note the different color scales for +the two panels. +109 +1010 +1011 +r [cm] +10-4 +10-3 +10-2 +10-1 +100 +101 +102 +103 +104 +105 +2π ad / λmax +2π ad / λmax, 5M⊕ +2π ad / λmax, 1M⊕ +Geometric opacity limit +10-1 +100 +101 +102 +Temperature gradient +∇rad, 1M⊕ +∇rad, 5M⊕ +∇ad +109 +1010 +1011 +r [cm] +10-6 +10-5 +10-4 +10-3 +10-2 +10-1 +Dust size [cm] +ad, ˙M=10−6 +ad, ˙M=10−7 +10-1 +100 +101 +102 +Temperature gradient +∇rad, ˙M=10−6 +∇rad, ˙M=10−7 +∇ad +Figure 2. Left: solid lines are the opacity efficiency factors Qe (eq. 7) for 2 different core masses with all other parameters being equal +(15 AU, +˙Macc = 10−6M⊕/yr). These reach the regime switch value of 2 (solid blue line) at different radii, creating a radiative zone in +the inner envelope for the 5 M⊕ case but not for 1 M⊕ due to its smaller dust size. The dashed lines are the radiative and adiabatic +gradients, indicating the radiative and convective zones. 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W., 1999, ApJ, 517, +1034 +This paper has been typeset from a TEX/LATEX file prepared by +the author. +MNRAS 000, 1–5 (2023) + +Macc = 10-6 Ma /yr +0.0 +-0.2 +100 +Core Mass [M +-0.6 +-0.8 +-1.0 + grai +-1.2 +6 +10-1 +Lo +-1.4 +-1.6 +0 +5 +10 +15 +20 +25 +30 +Semimajor axis [AU]Macc = 10-7 Mg / yr +100 +0.00 +-0.08 +-0.16 +b- +N +Core Mass [M] +-0.24 +abundance +-0.32 +-0.40 +og grain +10-1 +-0.48 +-0.56 +-0.64 +-0.72 +0 +5 +10 +15 +20 +25 +30 +Semimajor axis [AU] \ No newline at end of file diff --git a/8tAzT4oBgHgl3EQfgvwn/content/tmp_files/load_file.txt b/8tAzT4oBgHgl3EQfgvwn/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ca95f636ec45f19aa164b6944a26bfb773e3dc85 --- /dev/null +++ b/8tAzT4oBgHgl3EQfgvwn/content/tmp_files/load_file.txt @@ -0,0 +1,410 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf,len=409 +page_content='MNRAS 000, 1–5 (2023) Preprint 5 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='0 Dust processing in protoplanetary envelopes as the origin of hot minerals in comets Mohamad Ali-Dib1⋆ 1Center for Astro, Particle and Planetary Physics (CAP3), New York University Abu Dhabi, UAE Accepted 2022-12-30 ABSTRACT Crystalline silicates are found in a large number of comets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' These pose a long-standing conundrum for solar system formation models as they can only be created in the in- ner hot disk at temperatures higher than 800 K, and there is no obvious mechanism to transport them out into the comets formation region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Here we propose that these particles could have formed inside the hydrostatic envelopes surrounding young pro- toplanets still embedded in the protoplanetary disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Using a simplified 1D model we investigate the thermal structure of these envelopes, and find that for core masses ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='08 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='5 M⊕, located anywhere between 1 and 30 AU, the temper- ature and pressure at the base of the envelopes are high enough to quickly vaporize silicate particles of various sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Moreover, if the grain abundance is atleast solar, these envelopes become fully convective, allowing for dust ejection across the Bondi radius back into the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Amorphous silicates are hence thermally processed into crystalline particles in these envelopes, and then transported back to disk through convective diffusion to be finally incorporated into the cometary building blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Key words: planets and satellites: formation – comets: general – planets and satellites: composition 1 INTRODUCTION High temperature minerals are ubiquitous in the cold outer solar system small bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' One of the earliest remote sensing detections was for crystalline silicates (CSs) such as olivines and pyroxenes in the grains of comets 1P/Halley, D/1993 F2 (Shoemaker-Levy), C/1987 P1 (Bradfield) (Hanner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' 1994), and C/1993 A1 (Mueller) (Hanner, Lynch, & Russell 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Subsequent detections were made in comets C/1995 O1 (Hale-Bopp) (Hayward, Hanner, & Sekanina 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Cro- visier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Wooden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' 1999), 103P/Hartley (Cro- visier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' 2000), and more recently 17P/Holmes (Shinnaka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' On the other hand Calcium-Aluminum inclu- sions (CAIs) that form at even higher temperatures were found in the dust collected by Stardust in comet 81P/Wild (Brownlee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' We refer the reader to the observa- tional review of Mumma & Charnley (2011) for more infor- mations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' The presence of CSs have been a primary challenge to solar system formation models for decades, as the thermal conditions in the outer protoplanetary disk are not con- ducive to their formation locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' CSs can form starting from amorphous silicates through either direct vaporization followed by re-condensation at temperatures higher than ⋆ E-mail: malidib@nyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='edu ∼1800 K, or thermal annealing for T> 800 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Annealing is a physical process where sufficiently energetic molecules of a solid slowly regroup into a crystal lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' This mechanism is not instantaneous and necessitates high temperature expo- sure for a period of few weeks followed by slow cooling (Gail 1998, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' In typical protoplanetary disks, direct conden- sation can be active only inside ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='1 AU, and annealing is inefficient outside ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='5 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Moreover, there is no acceptable mechanism to trans- port these particles from the inner disk to the comets for- mation region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Earlier transport models relied on turbulent diffusion (Bockel´ee-Morvan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' 2002), but recent ALMA observations suggest that protoplanetary disks are laminar (Flaherty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' 2015, 2018), and thus this mechanism is un- likely to be efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Large scale outward advection in the disk’s midplane has also been proposed (Hughes & Armitage 2010), but 3D MHD simulations ruled out the presence of such advection (Fromang, Lyra, & Masset 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Another possible transport mechanism is photophoresis (Mousis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' 2007), but this necessitates a relatively large (1-2 AU) central hole in the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Finally, Ali-Dib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' (2015) pro- posed that FU-Ori outbursts might form these particles in situ, but this depends on the outburst trigger radius being large enough, which is uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Here we show how high temperature minerals form nat- urally, and in-situ, in the envelopes surrounding low mass © 2023 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='01472v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='EP] 4 Jan 2023 2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Ali-Dib proto-planets embedded in the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' We present models showing that the temperatures and pressures at the base of these envelopes easily reach conditions that allow for the formation of crystalline silicates through direct vapor- ization and re-condensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Primordial amorphous silicates are thus accreted and then thermally processed in these en- velopes, before finally getting ejected back to the disk as crystalline particles via convective diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' We emphasize that this work concerns the formation of generic CSs such as olivines and pyroxenes, and not necessarily chondrules and CAIs due to their additional formation-time constrains that are outside the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' We present our model in section 2, results in section 3, and conclude in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' 2 MODEL We model the atmosphere using the standard atmospheric structure equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' The equation of hydrostatic equilibrium is given by: dP dr = −GM r2 ρ(r) (1) and we define the temperature gradient equation starting from the standard assumption that heat can be transported using either radiation (if the local envelope is convectively stable) or adiabatic convection (if unstable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' It is hence writ- ten as: dT dr = ∇ T P dP dr (2) where ∇ is defined, starting from the Schwarzschild con- vective stability criterion (∇rad < ∇ad), to be ∇ = min(∇ad, ∇rad).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Here ∇ad is the adiabatic gradient: ∇ad ≡ � d ln T d ln P � ad = γ − 1 γ (3) where the adiabatic constant γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' ∇rad is the radiative gradient: ∇rad ≡ 3κP 64πGMσT 4 L (4) where L is the envelope’s luminosity generated by accretion at a rate ˙Macc : L = GM ˙Macc Rc (5) Rc is the core radius, and κ is the opacity that we define following Ormel (2014) as: κ = κgas + κgr (6) with: κgr = κgeomQe = 3Zgr 4ρsa × min(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='6πa λmax , 2) (7) where Zgr is the grains abundance, as their size, and ρs their internal density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' we use ρs = 3 g/cm3 for both the core and the dust particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' The equilibrium dust size in the envelope as is set by two competing processes: grain growth through coagulation (Ormel 2014) and grain collisional destruction (Ali-Dib & Thompson 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' The relative relevance of these two pro- cesses is decided mainly by whether the collisional speeds reach the silicate fragmentation threshold (Vf ∼ 100 cm/s, Blum & Wurm (2008)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' The collisional speed is approxi- mated here as the largest among the dust’s convective ve- locity Vcon,d (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' 19) and the dust’s radial drift velocity: Vdrift,d = τstop GM r2 (8) where τstop is the stopping time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' As discussed in (Ali-Dib & Thompson 2020), collisions in these envelopes are likely to be destructive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' This leads to a small characteristic dust size, increasing the opac- ity (thus growing the convective zone), and decreasing the vaporization timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Here we only select models where max(Vcon,d,Vdrift,d) is higher than 100 cm/s everywhere in the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' The convective fragmentation dust size is hence calcu- lated following (Ali-Dib & Thompson 2020) as: as,conv = 4πV 2 f r3ρ2 gcg Lρs (9) where ρg and cg are the gas’ density and sound speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' The drift fragmentation dust size is given by: as,drift = Vfr2ρgcg GMρs (10) with finally as=min(as,conv,as,drift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Note that as,conv is defined everywhere in the envelope, since, as discussed below, we also only select fully convec- tive envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' For this approach to be applicable, the par- ticles need to reach the local fragmentation threshold at ev- ery point in the envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Therefore, for self-consistency, we only keep models where the mean free time for collisions is shorter than the convective timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' In the convective fragmentation regime this can be written as: as as + 4ℓg/9 < 9Z2 gr ρg ρs Mcon r ℓg (11) where Mcon is the convective Mach number, and ℓg the mean free path of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' In the drift regime this is replaced by : 3 4Zgr c2 gr GM Mcon 1 + 9as/4ℓg < 1 (12) The gas opacity is given by: κgas = 10−8ρ2/3 g T 3 (13) Finally we close the system with the ideal gas equation of state P = ρgkBT/µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' We solve these equations by integrating inwards from the outer boundary at Rout, the minimum of the Bondi and Hill radii, to the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' We assumed the disk is radiative and calculate its temperature and density following Ali-Dib, Cumming, & Lin (2020): Td = 373 r−9/10 au K and ρd = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='7×10−10 r−33/20 au g/cm3 (14) Once we have the envelope’s thermal structure, we can calculate additional quantities needed for the subsequent analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' We calculate the silicate particles vaporization rate as : 1 as das dt = − � µSil 2πkT �1/2 P sat Sil ρsas (15) MNRAS 000, 1–5 (2023) Origins of hot minerals 3 with (Krieger 1967): P sat sil (T) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='2 × 1014e−(6×104 K)/T (16) and hence the silicate grains vaporization timescale is given by : τvap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='sil = � 1 as das dt �−1 (17) We define the gas and dust convective velocities respec- tively as: Vcon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='g = � L 4πr2ρg �1/3 (18) where we assumed that in the convective zone the energy is entirely transported through adiabatic convection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' and Vcon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='d ∼ Vcon (Vconτstop/r)1/2 (19) We finally calculate the dust’s convective mixing timescale as: τmix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='d = H/Vcon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='d (20) 3 RESULTS We start by exploring parameter space in order to find the values that allow for the creation of CSs in proto-envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' We explore core masses ranging from Pluto’s mass (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='002 M⊕) to a hypothetical giant planet’s core (10 M⊕), placed between 1 and 30 AU where ambient temperatures are too low to create CSs in the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' The grains abundance Zgr ranges from subsolar (10−3) to supersolar (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Our results are summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' In this plot we show only the areas of parameter space leading to envelopes conducive to the creation of CSs and that are self-consistent to our model assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' This is defined by these condi- tions: (i) τvap,sil is less than τmix,d at the base of the envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' This simply constrain the envelopes to those where solid silicates at their base can get vaporized faster than they are transported back into the upper cooler zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' (ii) The envelope is fully convective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' This ensures that the newly created CSs can be convectively diffused all the way back into the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' This condition is inspired by the results of Ali-Dib & Thompson (2020) who considered a similar setup with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='3 M⊕ core embedded in the disk, and showed that, for typical accretion rates, pebble fragmentation and dust loading increases the opacity and push the convective zone out till it reaches the Bondi radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Dust particles in these steady-state envelopes are then diffusively ejected back to the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Our results rely on this mechanism to transport the newly created CSs from the hot inner envelope back to the disk to be incorporated in proto-comets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' (iii) The collisional velocity is higher than 100 cm/s throughout the envelope, and conditions 11 and 12 are sat- isfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' This ensures that our dust size prescription is self- consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='1 Core mass Figure 1 shows that, while CSs can be created under a va- riety of parameter ranges, trends do exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' We start with our nominal model, for ˙Macc = 10−6M⊕/yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' First, there is a relatively narrow range of masses that extends from around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='08 M⊕ (40 times Pluto’s mass) to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='5 M⊕, beyond which the chances of creating CSs drops drastically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' This implies that CSs might have formed in the proto-envelopes of Mars to Earth mass protoplanets that have since disap- peared via giant collisions or dynamical ejection, or possibly grown into giant planetary cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' The lower limit on core masses is mainly due to their envelopes’ relatively cooler temperatures, increasing τvap,sil considerably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' On the other hand, cores with masses higher than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='5 M⊕ have dust par- ticles large enough in their middle and inner envelopes to switch from the Rosseland mean opacity regime to the ge- ometric opacity regime, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' 2 (left hand panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' This decreases the radiative gradient, creating an in- ner radiative zone that prevents these envelopes from being fully convective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' It is worth noting that, while our model considers the smallest of the Hill and Bondi radii to be the envelope’s outer boundary, all of our acceptable cases that form CSs are in the Bondi regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' This is expected as the Hill regime dominates for higher mass cores (5-10 M⊕) that were excluded above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' The Bondi radius RB = 2GM/c2 s is obtained by equating the local sound speed to the gravi- tational escape velocity, and thus describes a usually light but bound envelope where gas particles do not have enough thermal energy to escape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' For higher mass cores, the Bondi radius is large enough for the Hill stability criteria to become the more stringent constrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='2 Semimajor axis A complementary piece of information is the semimajor axis, where we find that CSs can form almost anywhere in the disk if the envelope’s grain abundance is high enough as discussed further below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Semimajor axis controls the temperature and density at the outer boundary, which seems to be important only in the marginal cases, for example for low core masses where the envelopes would be too cold if placed further out in the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' The wide range of possible semimajor axis allows for the possibility of creating CSs in the comets formation region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Classically, Oort cloud comets were thought to form among the giant planets all the way down to 5 AU, while Jupiter family comets were thought to form in the scattered disk (Duncan, Quinn, & Tremaine 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Duncan & Levison 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Dones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Alternatively, Brasser & Morbidelli (2013) proposed that both could have formed in the same region beyond Neptune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='3 Grains abundance We moreover find that creating CSs necessitate solar to su- persolar grain abundance in the envelope (Zgr >= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' This result is not consistent with the subsolar grain abun- dances found in models that incorporate dust growth & set- tling to the core but omit dust fragmentation with convec- tive mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Ormel (2014) for example added a simple grain growth equation to the atmospheric structure equations, and found that Zgr can be as low as 10−4 in parts of the envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Mordasini (2014) also created an atmospheric model incor- porating dust settling and coagulation, and found that this mostly results in subsolar opacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' The main role of Zgr is MNRAS 000, 1–5 (2023) 4 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Ali-Dib to increase the opacity and extend the convective zone all the way to the outer boundary (Bondi or Hill radius).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Ali-Dib & Thompson (2020) discussed the gradual buildup of Zgr in the envelope through accretion and fragmentation, and de- rived a lower limit on Zgr in order to get a fully convective envelope: Zgr > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='12 T 2 d,2 ρd,−11 �tacc,c Myr � � Mc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='3M⊕ �−2/3 (21) which is generally consistent with our Zgr values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' In order to get supersolar Zgr, multiple conditions need to be satisfied: The dust size need to be fragmentation-limited, which is a pre-requisite for our dust-size prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' This depends on many factors, including the accretion rate (setting the lu- minosity and thus convective speeds) and particles’ porosity and chemical composition (Blum & Wurm 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Okuzumi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Wada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' 2008, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' A significant fraction of the dust should not get accreted by the core, but remain mixed in the envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' This is an open question with many complications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' In our cases, sili- cates are in vapor form at the base of the envelope which should stop accretion from taking place unless the tempera- ture is low enough for the inner envelope to reach saturation pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' This also depends on the nature of convection in these envelopes, whether it is diffusive as we are assuming, or whether it is dominated by large scale eddies that can enhance accretion by the core (Johansen & Nordlund 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='4 Accretion rate Finally we investigate the effects of using a lower accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Our results for ˙Macc = 10−7M⊕/yr are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' In this case we find that while the semimajor axis range remains the same and the lower mass limit does not change (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='08M⊕), the upper limit decreases by over a factor 2 to ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='6M⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' This is expected since, lower accretion rate leads to lower luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' 2 (right hand side), this decreases the radiative gradient and allows for a radia- tive zone in the inner envelope even though the dust size in the 2 cases converge to the same inner value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' In some cases Zgr can compensate for the lower luminosity and increases the opacity enough to create fully convective envelopes, ex- plaining the overall larger Zgr we find for the lower accretion rate cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' 4 SUMMARY & CONCLUSIONS Crystalline silicates are ubiquitous in comets, but can only form at very high temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Here we investigated the possibility of transforming amorphous silicates into crys- talline particles inside the envelopes of protoplanets through vaporization followed by re-condensation, and then ejecting them back to the disk through diffusion in the fully con- vective envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Using a simplified 1D envelope structure model that incorporates a dust size prescription accounting for fragmentation and growth, we showed that crystalline silicates can be created from a diverse set of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Cores need to be between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='08 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content='5 M⊕ in mass, as lighter cores do not allow for temperatures high enough to vaporize silicates, and the envelopes of more massive cores are often not fully convective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' We finally found that the location in the disk (1 to 30 AU) has little influence on the results, ex- cept in marginal cases, and that a solar to supersolar grain abundance is needed, but this can be achieved through dust fragmentation and accumulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Our mechanism is simple and does not rely on assumptions about the disk, although it depends on the assumed diffusive nature of 1D convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Whether this is realistic needs to be investigated further us- ing 3D hydrodynamic simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We thank the anonymous referee for their constructive com- ments that greatly improved this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' This work is supported by Tamkeen under the NYU Abu Dhabi Research Institute grant CAP3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' DATA AVAILABILITY The data underlying this article (numerical simulations out- put files) will be shared on reasonable request to the corre- sponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' REFERENCES Ali-Dib M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=', Martin R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=', Petit J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' The semimajor axis, core mass, and envelope grain abundance for all the cases that satisfy our conditions to form and eject crystal silicates as enumerated in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Left: ˙Macc = 10−6M⊕/yr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Right: ˙Macc = 10−7M⊕/yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Note the different color scales for the two panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' 109 1010 1011 r [cm] 10-4 10-3 10-2 10-1 100 101 102 103 104 105 2π ad / λmax 2π ad / λmax, 5M⊕ 2π ad / λmax, 1M⊕ Geometric opacity limit 10-1 100 101 102 Temperature gradient ∇rad, 1M⊕ ∇rad, 5M⊕ ∇ad 109 1010 1011 r [cm] 10-6 10-5 10-4 10-3 10-2 10-1 Dust size [cm] ad, ˙M=10−6 ad, ˙M=10−7 10-1 100 101 102 Temperature gradient ∇rad, ˙M=10−6 ∇rad, ˙M=10−7 ∇ad Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Left: solid lines are the opacity efficiency factors Qe (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' 7) for 2 different core masses with all other parameters being equal (15 AU, ˙Macc = 10−6M⊕/yr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' These reach the regime switch value of 2 (solid blue line) at different radii, creating a radiative zone in the inner envelope for the 5 M⊕ case but not for 1 M⊕ due to its smaller dust size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' The dashed lines are the radiative and adiabatic gradients, indicating the radiative and convective zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Right: solid lines are the dust size ad for cases with 2 different accretion rates but all other parameters being equal (15 AU, 1 M⊕).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Dashed lines are the radiative and adiabatic gradients for the same cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' In all plots, the x-axis is the radius from the core, extending from the core to the envelope’s outer boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' Hanner M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=', Lynch D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=', Russell R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'} +page_content=', 1994, ApJ, 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index 0000000000000000000000000000000000000000..a57a933c26d54bb8df2cdc86f07b48db7a5ac64e --- /dev/null +++ b/B9FQT4oBgHgl3EQf-TfR/content/tmp_files/2301.13454v1.pdf.txt @@ -0,0 +1,318 @@ + +Compliance Costs of AI Technology +Commercialization: A Field Deployment +Perspective + +Weiyue Wu and Shaoshan Liu + +1. Introduction + +While Artificial Intelligence (AI) technologies are progressing fast, compliance costs have +become a huge financial burden for AI startups, which are already constrained on research & +development budgets. This situation creates a compliance trap, as many AI startups are not +financially prepared to cope with a broad spectrum of regulatory requirements. Particularly, +the complex and varying regulatory processes across the globe subtly give advantages to well- +established and resourceful technology firms over resource-constrained AI startups [1]. The +continuation of this trend may phase out the majority of AI startups and lead to giant +technology firms' monopolies of AI technologies. To demonstrate the reality of the +compliance trap, from a field deployment perspective, we delve into the details of compliance +costs of AI commercial operations. +2. Financial Vulnerability: Tech Giants vs. AI Startups + +Compared to established tech giants, AI startups are much more financially vulnerable. +Based on the OECD Regulatory Compliance Cost Assessment Guidance [2], we quantitatively +compare the financial vulnerability of tech giants versus AI startups. Conducting a financial +statement simulation provides a glimpse at the impact of changes in compliance costs. The +simulation in Table 1 assumes that gross profit and expense are a fixed percentage of revenue. +The compliance cost is split into a fixed cost regardless of revenue and a variable cost that +takes 5% of revenue. When the fixed compliance cost increases by 200%, the operating +margin of the startup changes from 13% to -7%, turning the company from a profitable +business into a money-losing one. By contrast, such a change only brings a slight drop in +operating margin in tech giants. + +Table 1: Sensitivity Analysis: Impact of Change in Compliance Cost on Operating Income + +Start-ups +Tech Giants +200% increase in +200% increase in +Base case +fixed compliance cost +Base case +fixed compliance cost +Revenue +$2,000,000 +$2,000,000 +$20,000,000 +$20,000,000 +Gross Margin +40% +40% +40% +40% +Gross Profit +800,000 +800,000 +8,000,000 +8,000,000 +Total Compliance Cost +300,000 +700,000 +1,200,000 +1,600,000 +Fixed compliance cost +200,000 +600,000 +200,000 +600,000 +Variable compliance cost +100,000 +100,000 +1,000,000 +1,000,000 +Expenses +240,000 +240,000 +2,400,000 +2,400,000 +Operating Income +260,000 +(140,000) +4,400,000 +4,000,000 +Operating Margin +13% +-7% +22% +20% +The actual situation of AI startups is more complex than this estimation, as it is nearly +impossible for external analysts to estimate the compliance costs. According to Accounting +Standards, companies are not required to disclose compliance cost explicitly in financial +reports when the cost is not material [3]. Thus, compliance costs may become hidden by +nature and classified into other categories, such as Research and Development expenses and +other administrative expenses. Lacking first-hand information, analysts on the macro- +economic level tend to underestimate the costs of AI regulations. For instance, in an impact +assessment report of the Europe AI Act, the estimated annual compliance cost of one AI +product that averagely costs EUR 170,000 to develop is EUR 29,277, we believe this study has +underestimated the actual costs of AI compliance [4]. + +3. The Compliance Trap + +AI is a highly regulated industry, but unfortunately, there is no standardized AI regulation +framework, and hence compliance costs often become a financial trap for AI startups [1]. +Most AI entrepreneurs may not even be aware of the existence of compliance costs, let alone +the severe impact compliance costs may have on the company's overall financial health. As +illustrated in Figure 1, we summarize the following challenges. + + +Figure 1: the compliance trap for AI startups + +First, unlike R&D budgeting, due to varying AI regulatory frameworks across the globe or +even across multiple regions within a country, there is no standard method to budget for AI +compliance costs. Even estimating the range of AI compliance costs is infeasible. + +Second, even with an AI compliance budget, the actual costs may significantly deviate from +the budget. AI startups often encounter new compliance issues as they progress through +commercialization. In addition, opportunity costs arise as regulators inspect AI products on +safety and privacy issues, causing delays in commercial deployments. + +Third, varying AI regulations often introduce indirect costs. For instance, a strict compliance +environment demands engineers deal with regulatory issues such as responding to various +compliance technical inquiries instead of spending time developing products. Such a shift of +focus does not reflect in financial reports, as engineers' costs are categorized as R&D costs. + + +No Standard Method to Budget +Deviation from Compliance Budget +Indirect Costs +Al start-upsfacevarying Al regulatory +Al start-ups encounter new compliance issues and +Al start-ups might work around compliance +frameworks across theglobe +delay of commercial deployments +by other activities such as R&D and marketing +Compliancebudgetplanningoftechcompanies +Rough budget-overrun disaggregation,% +Classification ofCompliance Costs +- Standard waterfall method +Pro-forma +Certification cost +Situation +Revise +Finalize +budget +budget +budget +Regulatory complexity +orecast +Actual +CompliancebudgetplanningofAl companies +Shifting requirements +cost +Marketing cost +Total Cost +R&Dcost +- On an Ad-Hoc basis +of +50 +Lack of standards +Compliance +Logistics cost +Training cost +Testing and +Qualityassurance +100 +Insurancecost +Budget +30 +validation cost +cost +Documentation +Marketing cost +cost +15 +Labor cost +Direct cost +Indirect cost4. A Field Deployment Perspective + +In this section, with more than six years of first-hand experience in deploying commercial +autonomous driving services, we delve into the details of compliance costs from a field +deployment perspective, in the hope that the insights we provide can raise awareness of the +adverse impact of the lack of standardized AI regulations. + +4.1. Background + + PerceptIn is an autonomous driving startup founded in 2016. It offers autonomous micro- +mobility solutions to customers from the United States, European Union, and Asia. The +company only budgeted for ordinary compliance expenses, such as the direct labor cost of a +safety driver on board and the equipment cost of a waterproof surveillance camera. While +facing a broad spectrum of regulatory obstacles across different countries, PerceptIn had +fallen into the compliance trap. Many financial and human resources have been spent out of +budget to comply with various regulatory frameworks in different regions. + +4.2. Scenario 1 – No Standard Method to Budget + +The AI regulation framework in China was blurry, and when the company first launched the +autonomous micro-mobility project in China, it was impossible to budget for compliance costs. +For instance, since relevant regulations were absent back then, the company needed to +develop its own testing plan to obtain deployment approval. Without detailed testing +standards, the company had to spend $25,000 per month to simulate real-world scenarios at +the initial stage for a testing site for testing and demonstration purposes. The testing process +is to obtain detailed validation results, whereas the demonstration process is to convince the +regulatory body regarding the safety and reliability of the service. The $300,000 annual cost +was not included in the company’s original budget and imposed a heavy burden on the +company's financial health. + +4.3. Scenario 2 – Deviation from Compliance Budget + +The company was invited to launch an autonomous driving pilot program in a European +city. Before rolling out the project, the company was asked to prepare a risk mitigation plan +for 40 different scenarios. To cope with the regulatory process, the R&D team shifted its focus +to responding to scenarios-based functional specifications and supplemented the mitigation +plan with real-time data. During the project budgeting phase, the company had prepared 20 +man-days to cope with the AI regulatory process. Nevertheless, the process turned out to +consume 400 man-days to complete the process. While the original budget was $10,000, in +the end, the process consumed $200,000. Such a severe mismatch was caused by the lack of +a standardized regulatory process, as any response from our technical team would bring on +the next round of regulatory questions. + +4.4. Scenario 3 – Indirect Costs + + +Japan is famous for its rigid structure in organizations. Thus without an established +compliance process in place, gaining the confidence of the Japanese government is essential +to commercial deployment. To gain the confidence of the Japanese government, the company +first debuted a marketing campaign to promote safe autonomous micro-mobility services in +a smart city project [5]. With a successful local case and globally established brand, the +company then discussed operation permits with the Ministry of Land, Infrastructure, +Transport, and Tourism (MLIT) [6]. The preparation and initiation of the project took over 24 +months, costing $500,000 in promotion, material preparation, and marketing campaigns. +Traditionally marketing activities were not meant to cope with compliance requirements. +However, in this case, marketing was a tool to convince the regulatory body to further +autonomous driving operation permits. + +In the case of PerceptIn, the compliance cost of one deployment project is $ 344,000 on +average, whereas the average R&D cost is around $150,000, making the compliance costs 2.3 +times the amount of R&D costs, far exceeding the 17.6% estimation of the Europe AI Act. + +5. The Silver Linings + +The root cause of the compliance trap is the lack of a standardized AI regulatory framework. +An ultimate solution to this problem lies in creating a global golden standard for AI regulation. +A study of the Food and Drug Administration's (FDA's) history sheds light on properly +regulating a new field. First, pharmaceutical products a century ago and AI today are both +viewed as black boxes. Even the most sophisticated scientists could not predict their +development trajectory, let alone legal experts. Second, pharmaceutical and AI technologies +can potentially cause severe public risks if they are not adequately regulated. Third, both +industries have enormous potential for improving people's well-being. Like the FDA, a +consumer protection agency shall be established to ensure that AI is developed for people's +well-being. Such an agency should develop the expertise and capability to scientifically judge +whether an AI product is ethical and legal. Such an agency should provide guidance to various +governments within the U.S. and worldwide on AI regulations. +However, before a consensus can be reached regarding the golden standard, a new +business model, Compliance-as-a-Service (CaaS), can specialize in dealing with varying AI +regulatory frameworks and thus amortize compliance costs across different AI startups. In +addition, CaaS reduces the friction between regulatory bodies and AI startups by providing an +interface to compile legal terms into technical and operational plans. With the new business +model, AI entrepreneurs can adequately budget for compliance when evaluating the potential +of an innovative idea. + +6. Summary + +AI is a promising industry mainly filled with startups exploring the applications of AI +technologies in different aspects of our daily lives. Compared to well-established tech giants, +AI startups are financially vulnerable. Unfortunately, the lack of standardized AI regulatory +frameworks creates a compliance trap that may destroy an AI startup financially, which could +lead to a more profound impact of creating a competitive advantage for tech giants over AI + +startups. We have examined the details of compliance costs from a field deployment +perspective to demonstrate the reality of the compliance trap. Ideally, if a global golden +standard on AI regulation could be developed, then AI startups could accurately budget for +compliance costs. However, before a consensus can be reached regarding the golden +standard, we believe that a new business model, compliance as a service, can specialize in +dealing with varying AI regulatory frameworks and thus amortize compliance costs across +different AI startups. + +References: +1. Wu, W. and Liu, S., 2021. Dilemma of the Artificial Intelligence Regulatory +Landscape. Communications of the ACM, 2023. +2. OECD. Publishing, 2014. OECD Regulatory Compliance Cost Assessment Guidance. OECD Publishing. +3. PricewaterhouseCoppers, 2022. Illustrative IFRS consolidated financial statements. +4. Renda, A., Arroyo, J., Fanni, R., Laurer, M., Sipiczki, A., Yeung, T., Maridis, G., Fernandes, M., Endrodi, +G. and Milio, S., 2021. Study to support an impact assessment of regulatory requirements for artificial +intelligence in Europe. European Commission: Brussels, Belgium. +5. Fukuoka City conducts demonstration test of compact self-driving car by US company PerceptIn, Inc.. +Nikkei, accessed 2023-01-05, https://www.nikkei.com/article/DGXLRSP518592_W9A900C1000000/ +6. List of Proposal Sectors and Private Companies, etc. . In Seeds proposal for realization of smart island, +Japanese Ministry of Land, Infrastructure, Transport and Tourism, accessed 2023-01-05, +https://www.mlit.go.jp/kokudoseisaku/chirit/kokudoseisaku_chirit_tk_000309.html + + +Biography: + +Weiyue Wu is Chief Operating Officer of PerceptIn, an autonomous driving startup founded +in 2016. At PerceptIn, she has been in charge of commercial autonomous driving service +deployments in the US, Europe, Japan, and China. Before PerceptIn, she served as Investment +Director of Oxford Seed Fund and Investment Advisor of ARM Accelerator. She began her +career as a Multi-National Corporation Compliance Auditor at KPMG and a Senior Automobile +Consultant at Deloitte. She received her MBA from the University of Oxford. She is a founding +member of IEEE Special Technical Community on Autonomous Driving Technologies, a +Certified Public Accountant and a practicing lawyer in China. + +Dr. Shaoshan Liu’s background is a unique combination of technology, entrepreneurship, and +public policy, which enables him to take on great global challenges. On technology, Dr. Liu +has published 4 textbooks, more than 100 research papers, and holds more than 150 patents +in autonomous systems. On entrepreneurship, Dr. Shaoshan Liu is CEO of PerceptIn and has +commercially deployed autonomous micro-mobility services in the U.S., Europe, Japan, and +China etc. He is the Asia Chair of IEEE Entrepreneurship. On public policy, Dr. Liu has served +on the World Economic Forum’s panel on Industry Response to Government Procurement +Policy, is leading the Autonomous Machine Computing roadmap under IEEE International +Roadmap of Devices and Systems (IRDS) and is a member of the ACM U.S. Technology Policy +Committee. Dr. Liu’s educational background includes a M.S. in Biomedical Engineering, a +Ph.D. in Computer Engineering from the U.C. Irvine, and a Master of Public Administration +(MPA) from Harvard University. He is an IEEE Senior Member, an IEEE Computer Society +Distinguished Speaker, an ACM Distinguished Speaker, an Advisory Council member of + +Harvard Business Review, a member of MIT Technology Review’s Global Insights Panel, and a +member of the Forbes Technology Council. + diff --git a/B9FQT4oBgHgl3EQf-TfR/content/tmp_files/load_file.txt b/B9FQT4oBgHgl3EQf-TfR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0c1d35dc85c221c7a1b913bb81de01cf93f573be --- /dev/null +++ b/B9FQT4oBgHgl3EQf-TfR/content/tmp_files/load_file.txt @@ -0,0 +1,270 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf,len=269 +page_content='Compliance Costs of AI Technology Commercialization: A Field Deployment Perspective Weiyue Wu and Shaoshan Liu 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Introduction While Artificial Intelligence (AI) technologies are progressing fast, compliance costs have become a huge financial burden for AI startups, which are already constrained on research & development budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' This situation creates a compliance trap, as many AI startups are not financially prepared to cope with a broad spectrum of regulatory requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Particularly, the complex and varying regulatory processes across the globe subtly give advantages to well- established and resourceful technology firms over resource-constrained AI startups [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=" The continuation of this trend may phase out the majority of AI startups and lead to giant technology firms' monopolies of AI technologies." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' To demonstrate the reality of the compliance trap, from a field deployment perspective, we delve into the details of compliance costs of AI commercial operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Financial Vulnerability: Tech Giants vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' AI Startups Compared to established tech giants, AI startups are much more financially vulnerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Based on the OECD Regulatory Compliance Cost Assessment Guidance [2], we quantitatively compare the financial vulnerability of tech giants versus AI startups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Conducting a financial statement simulation provides a glimpse at the impact of changes in compliance costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' The simulation in Table 1 assumes that gross profit and expense are a fixed percentage of revenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' The compliance cost is split into a fixed cost regardless of revenue and a variable cost that takes 5% of revenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' When the fixed compliance cost increases by 200%, the operating margin of the startup changes from 13% to -7%, turning the company from a profitable business into a money-losing one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' By contrast, such a change only brings a slight drop in operating margin in tech giants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Table 1: Sensitivity Analysis: Impact of Change in Compliance Cost on Operating Income Start-ups Tech Giants 200% increase in 200% increase in Base case fixed compliance cost Base case fixed compliance cost Revenue $2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='000 $2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='000 $20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='000 $20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='000 Gross Margin 40% 40% 40% 40% Gross Profit 800,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='000 800,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='000 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='000 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='000 Total Compliance Cost 300,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='000 700,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='000 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='200,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='000 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='600,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='000 Fixed compliance cost 200,' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='000 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='000 Expenses 240,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='000 240,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='000 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='400,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='000 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='400,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='000 Operating Income 260,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='000 (140,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='000) 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='400,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='000 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='000 Operating Margin 13% -7% 22% 20% The actual situation of AI startups is more complex than this estimation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' as it is nearly impossible for external analysts to estimate the compliance costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' According to Accounting Standards, companies are not required to disclose compliance cost explicitly in financial reports when the cost is not material [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Thus, compliance costs may become hidden by nature and classified into other categories, such as Research and Development expenses and other administrative expenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Lacking first-hand information, analysts on the macro- economic level tend to underestimate the costs of AI regulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' For instance, in an impact assessment report of the Europe AI Act, the estimated annual compliance cost of one AI product that averagely costs EUR 170,000 to develop is EUR 29,277, we believe this study has underestimated the actual costs of AI compliance [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' The Compliance Trap AI is a highly regulated industry, but unfortunately, there is no standardized AI regulation framework, and hence compliance costs often become a financial trap for AI startups [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=" Most AI entrepreneurs may not even be aware of the existence of compliance costs, let alone the severe impact compliance costs may have on the company's overall financial health." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' As illustrated in Figure 1, we summarize the following challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Figure 1: the compliance trap for AI startups First, unlike R&D budgeting, due to varying AI regulatory frameworks across the globe or even across multiple regions within a country, there is no standard method to budget for AI compliance costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Even estimating the range of AI compliance costs is infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Second, even with an AI compliance budget, the actual costs may significantly deviate from the budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' AI startups often encounter new compliance issues as they progress through commercialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' In addition, opportunity costs arise as regulators inspect AI products on safety and privacy issues, causing delays in commercial deployments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Third, varying AI regulations often introduce indirect costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' For instance, a strict compliance environment demands engineers deal with regulatory issues such as responding to various compliance technical inquiries instead of spending time developing products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=" Such a shift of focus does not reflect in financial reports, as engineers' costs are categorized as R&D costs." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' No Standard Method to Budget Deviation from Compliance Budget Indirect Costs Al start-upsfacevarying Al regulatory Al start-ups encounter new compliance issues and Al start-ups might work around compliance frameworks across theglobe delay of commercial deployments by other activities such as R&D and marketing Compliancebudgetplanningoftechcompanies Rough budget-overrun disaggregation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='Classification ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='ofCompliance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='Costs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='Standard ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='waterfall ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='method ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='Pro-forma ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='Certification ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='cost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='Situation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='Revise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='Finalize ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='budget ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='budget ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='budget ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='Regulatory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='complexity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='orecast ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='Actual ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='CompliancebudgetplanningofAl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='companies ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='Shifting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='requirements ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='cost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='Marketing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='cost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='Total ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='Cost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='R&Dcost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='On ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='an ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='Ad-Hoc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='basis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='Lack ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='Testing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='Qualityassurance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='Insurancecost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='Budget ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='validation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='cost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='cost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='Documentation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='Marketing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='cost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='cost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='Labor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='cost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='Direct ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='cost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='Indirect ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='cost4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' A Field Deployment Perspective In this section, with more than six years of first-hand experience in deploying commercial autonomous driving services, we delve into the details of compliance costs from a field deployment perspective, in the hope that the insights we provide can raise awareness of the adverse impact of the lack of standardized AI regulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Background PerceptIn is an autonomous driving startup founded in 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' It offers autonomous micro- mobility solutions to customers from the United States, European Union, and Asia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' The company only budgeted for ordinary compliance expenses, such as the direct labor cost of a safety driver on board and the equipment cost of a waterproof surveillance camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' While facing a broad spectrum of regulatory obstacles across different countries, PerceptIn had fallen into the compliance trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Many financial and human resources have been spent out of budget to comply with various regulatory frameworks in different regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Scenario 1 – No Standard Method to Budget The AI regulation framework in China was blurry, and when the company first launched the autonomous micro-mobility project in China, it was impossible to budget for compliance costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' For instance, since relevant regulations were absent back then, the company needed to develop its own testing plan to obtain deployment approval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Without detailed testing standards, the company had to spend $25,000 per month to simulate real-world scenarios at the initial stage for a testing site for testing and demonstration purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' The testing process is to obtain detailed validation results, whereas the demonstration process is to convince the regulatory body regarding the safety and reliability of the service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=" The $300,000 annual cost was not included in the company’s original budget and imposed a heavy burden on the company's financial health." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Scenario 2 – Deviation from Compliance Budget The company was invited to launch an autonomous driving pilot program in a European city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Before rolling out the project, the company was asked to prepare a risk mitigation plan for 40 different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' To cope with the regulatory process, the R&D team shifted its focus to responding to scenarios-based functional specifications and supplemented the mitigation plan with real-time data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' During the project budgeting phase, the company had prepared 20 man-days to cope with the AI regulatory process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Nevertheless, the process turned out to consume 400 man-days to complete the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' While the original budget was $10,000, in the end, the process consumed $200,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Such a severe mismatch was caused by the lack of a standardized regulatory process, as any response from our technical team would bring on the next round of regulatory questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Scenario 3 – Indirect Costs Japan is famous for its rigid structure in organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Thus without an established compliance process in place, gaining the confidence of the Japanese government is essential to commercial deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' To gain the confidence of the Japanese government, the company first debuted a marketing campaign to promote safe autonomous micro-mobility services in a smart city project [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' With a successful local case and globally established brand, the company then discussed operation permits with the Ministry of Land, Infrastructure, Transport, and Tourism (MLIT) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' The preparation and initiation of the project took over 24 months, costing $500,000 in promotion, material preparation, and marketing campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Traditionally marketing activities were not meant to cope with compliance requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' However, in this case, marketing was a tool to convince the regulatory body to further autonomous driving operation permits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' In the case of PerceptIn, the compliance cost of one deployment project is $ 344,000 on average, whereas the average R&D cost is around $150,000, making the compliance costs 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='3 times the amount of R&D costs, far exceeding the 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='6% estimation of the Europe AI Act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' The Silver Linings The root cause of the compliance trap is the lack of a standardized AI regulatory framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' An ultimate solution to this problem lies in creating a global golden standard for AI regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=" A study of the Food and Drug Administration's (FDA's) history sheds light on properly regulating a new field." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' First, pharmaceutical products a century ago and AI today are both viewed as black boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Even the most sophisticated scientists could not predict their development trajectory, let alone legal experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Second, pharmaceutical and AI technologies can potentially cause severe public risks if they are not adequately regulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=" Third, both industries have enormous potential for improving people's well-being." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=" Like the FDA, a consumer protection agency shall be established to ensure that AI is developed for people's well-being." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Such an agency should develop the expertise and capability to scientifically judge whether an AI product is ethical and legal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Such an agency should provide guidance to various governments within the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' and worldwide on AI regulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' However, before a consensus can be reached regarding the golden standard, a new business model, Compliance-as-a-Service (CaaS), can specialize in dealing with varying AI regulatory frameworks and thus amortize compliance costs across different AI startups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' In addition, CaaS reduces the friction between regulatory bodies and AI startups by providing an interface to compile legal terms into technical and operational plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' With the new business model, AI entrepreneurs can adequately budget for compliance when evaluating the potential of an innovative idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Summary AI is a promising industry mainly filled with startups exploring the applications of AI technologies in different aspects of our daily lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Compared to well-established tech giants, AI startups are financially vulnerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Unfortunately, the lack of standardized AI regulatory frameworks creates a compliance trap that may destroy an AI startup financially, which could lead to a more profound impact of creating a competitive advantage for tech giants over AI startups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' We have examined the details of compliance costs from a field deployment perspective to demonstrate the reality of the compliance trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Ideally, if a global golden standard on AI regulation could be developed, then AI startups could accurately budget for compliance costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' However, before a consensus can be reached regarding the golden standard, we believe that a new business model, compliance as a service, can specialize in dealing with varying AI regulatory frameworks and thus amortize compliance costs across different AI startups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' References: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Wu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' and Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Dilemma of the Artificial Intelligence Regulatory Landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Communications of the ACM, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' OECD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Publishing, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' OECD Regulatory Compliance Cost Assessment Guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' OECD Publishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' PricewaterhouseCoppers, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Illustrative IFRS consolidated financial statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Renda, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=', Arroyo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=', Fanni, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=', Laurer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=', Sipiczki, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=', Yeung, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=', Maridis, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=', Fernandes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=', Endrodi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' and Milio, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Study to support an impact assessment of regulatory requirements for artificial intelligence in Europe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' European Commission: Brussels, Belgium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Fukuoka City conducts demonstration test of compact self-driving car by US company PerceptIn, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='. Nikkei, accessed 2023-01-05, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='nikkei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='com/article/DGXLRSP518592_W9A900C1000000/ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' List of Proposal Sectors and Private Companies, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' In Seeds proposal for realization of smart island, Japanese Ministry of Land, Infrastructure, Transport and Tourism, accessed 2023-01-05, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='mlit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='jp/kokudoseisaku/chirit/kokudoseisaku_chirit_tk_000309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='html Biography: Weiyue Wu is Chief Operating Officer of PerceptIn, an autonomous driving startup founded in 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' At PerceptIn, she has been in charge of commercial autonomous driving service deployments in the US, Europe, Japan, and China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Before PerceptIn, she served as Investment Director of Oxford Seed Fund and Investment Advisor of ARM Accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' She began her career as a Multi-National Corporation Compliance Auditor at KPMG and a Senior Automobile Consultant at Deloitte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' She received her MBA from the University of Oxford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' She is a founding member of IEEE Special Technical Community on Autonomous Driving Technologies, a Certified Public Accountant and a practicing lawyer in China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Shaoshan Liu’s background is a unique combination of technology, entrepreneurship, and public policy, which enables him to take on great global challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' On technology, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Liu has published 4 textbooks, more than 100 research papers, and holds more than 150 patents in autonomous systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' On entrepreneurship, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Shaoshan Liu is CEO of PerceptIn and has commercially deployed autonomous micro-mobility services in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=', Europe, Japan, and China etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' He is the Asia Chair of IEEE Entrepreneurship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' On public policy, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Liu has served on the World Economic Forum’s panel on Industry Response to Government Procurement Policy, is leading the Autonomous Machine Computing roadmap under IEEE International Roadmap of Devices and Systems (IRDS) and is a member of the ACM U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Technology Policy Committee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Liu’s educational background includes a M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' in Biomedical Engineering, a Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' in Computer Engineering from the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' Irvine, and a Master of Public Administration (MPA) from Harvard University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} +page_content=' He is an IEEE Senior Member, an IEEE Computer Society Distinguished Speaker, an ACM Distinguished Speaker, an Advisory Council member of Harvard Business Review, a member of MIT Technology Review’s Global Insights Panel, and a member of the Forbes Technology Council.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'} diff --git a/BtE1T4oBgHgl3EQfVwQu/content/tmp_files/2301.03105v1.pdf.txt b/BtE1T4oBgHgl3EQfVwQu/content/tmp_files/2301.03105v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..99c03df0965aa87083a1307f510ec5556a99f273 --- /dev/null +++ b/BtE1T4oBgHgl3EQfVwQu/content/tmp_files/2301.03105v1.pdf.txt @@ -0,0 +1,1425 @@ +arXiv:2301.03105v1 [math.GT] 8 Jan 2023 +FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT +BUNDLES +NIMA ANVARI AND IAN HAMBLETON +Abstract. Given a 4-manifold with a homologically trivial and locally-linear cyclic +group action, we obtain necessary and sufficient conditions for the existence of equi- +variant bundles. +The conditions are derived from the twisted signature formula and +are in the form of congruence relations between the fixed point data and the isotropy +representations. +1. Introduction +Finite group actions on 4-manifolds can be studied in various settings. We are mainly +interested in comparing smooth actions with those which are topological and locally linear, +but important examples arise for symplectic 4-manifolds and complex surfaces. Here is a +sampling of survey articles and recent work on aspects of this general theme: [1, 2, 3, 5, +6, 7, 10, 15, 22, 20, 21, 23, 26, 31, 36]. We will focus on the existence and classification +of equivariant bundles, and their applications in Yang-Mills gauge theory to the study of +finite group actions. +We begin by recalling some standard definitions. Let (X, π) denote a simply-connected, +closed 4-manifold X together with a locally linear and homologically trivial action of a +cyclic group π = Z/p of prime order. The fixed point set Xπ has Euler characteristic +χ(Xπ) = b2(X) + 2, by the Lefschetz fixed point formula. In general Xπ will consist of +isolated fixed points and a disjoint union of fixed 2-spheres (see [9, §2]). +Definition 1.1. At each isolated fixed point x ∈ Xπ, the tangent space admits an equi- +variant decomposition (TxiX, π) = C(ai) ⊕ C(bi) of complex representation spaces. Let +t · (z1, z2) = (ζaiz1, ζbiz2) +denote the action for t a fixed generator in the cyclic group π, ζ = e2πi/p, and with integers +(ai, bi), both non-zero modulo p. +(i) The integers (ai, bi) are the local tangential rotation data, and are well-defined up +to order and simultaneous change in sign. +(ii) Similarly, for each point x on a π-fixed 2-sphere Fj, there is a representation +C ⊕ C(cj) corresponding to the equivariant splitting +TX | Fj = TFj ⊕ N(Fj) +where N(Fj) is the normal bundle with rotation ζcj, and cj ̸≡ 0 (mod p). +Date: January 4, 2023. +This research was partially supported by NSERC Discovery Grant A4000. +1 + +2 +NIMA ANVARI AND IAN HAMBLETON +(iii) The total fixed point rotation data is the collection +F = {(ai, bi), (cj, αj) | i ∈ I, j ∈ J}. +where I and J index the isolated fixed points and 2-spheres respectively and +αj = [Fj]·[Fj] is the self-intersection number of the fixed spheres [Fj] ∈ H2(X; Z). +(iv) By an equivariant line bundle (L, π) → (X, π) we mean a principal U(1)-bundle +L over X together with a lift of the π-action to the total space. Given such a +lift, there exists a set of isotropy representations of the π-action on each fiber +over the fixed point set which we denote by L | xi = tλi over isolated fixed points +and L | Fj = tλj over a fixed 2-sphere. Denote the collection of these isotropy +representations by I = {λi, λj | i ∈ I, j ∈ J}. +With this notation we have our main result. +Theorem A. Let (X, π) denote a simply-connected, closed 4-manifold with a locally linear, +homologically trivial action of a cyclic group π = Z/p of odd prime order p, with fixed-point +rotation data F = {(ai, bi), (cj, αj) | i ∈ I, j ∈ J}. +A collection I of integers {λi, λj | i ∈ I, j ∈ J} can be realized (modulo p) as the isotropy +representations of an equivariant line bundle (L, π) → (X, π) if and only if there is a +collection of integers {mj | j ∈ J} such that +(1.1) +� +i∈I +λi +aibi ++ +� +j∈J +cjmj − λjαj +c2 +j +≡ 0 +(mod p). +When a solution exists, the integers mj = c1(i∗L)[Fj] satisfy equation (1.1). +Remark 1.2. A special case of this result can be found in [22, Proposition 1.4], for a +cyclic group π of odd order acting locally linearly and semi-freely on the complex projective +plane CP 2 with has three isolated fixed points. The necessary condition (1.1) in Theorem +A is established by extending the methods of [22, §2] to more general actions. +Remark 1.3. The definitions above generalize directly to actions (X, π) where π is any +finite group, and the structural group G of the principal bundle is any compact Lie group. +For applications in gauge theory, G = SU(2) is an important example. The details will +be left to the reader (see also [18, 19]). +2. Some motivating questions +Here are some questions related to the general theme (all actions will be assumed to +preserve orientation). More information about some of these directions can be found in +the references. +1. Does there exist a smooth Z/p-action on a (homotopy) K3 surface, which induces the +identity on integral homology ? +This is a well-known question of Allan Edmonds. Note that results of Edmonds and +Ewing [12] imply that topological locally linear examples exist for odd primes. +2. Does there exist a smooth Z/p-action on a homotopy K3 surface, which contains an +invariant embedded Brieskorn homology 3-sphere Σ(2, 3, 7) ? + +FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES +3 +Fintushel and Stern [14] showed that many homotopy K3 surfaces admit embeddings +of Σ(2, 3, 7). The question concerns the possible existence of an equivariant splitting of +the K3 surface along the Brieskorn sphere. +3. A Brieskorn homology 3-sphere Σ(a, b, c) admits a free Z/p-action if p ∤ abc. Does there +exist a smooth, homologically trivial extension of this action with isolated fixed points to +any smooth simply connected negative definite 4-manifold X with boundary Σ(a, b, c) ? +Anvari [1] proved that the free Z/7 on Σ(2, 3, 5) does not extend in this way over the min- +imal negative definite 4-manifold obtained by resolving the link singularity. The method of +proof involves studying equivariant Yang-Mills gauge theory on the non-compact manifold +with cylindrical end obtained from X by attaching the end Σ(2, 3, 5)×[0, ∞). Information +about the Floer homology of Brieskorn spheres is an essential ingredient in tackling this +problem (see Saveliev [32, 33, 34]). +4. What sets of rotation numbers can be realized by a smooth, pseudo-free Z/n-action +on X = S2 × S2 ? +A pseudo-free action is one with isolated singular points. If the action is semi-free, +there are “standard models” with rotation data {(a, b), (c, d), (a, −b), (c, −d)} at the four +fixed points. This question for X = CP 2 was answered in [11, 20], where it turned out +that the rotation data was the same for locally linear actions. +5. Let (X, π) denote a smooth action of a finite group π on a closed, simply connected +smooth 4-manifold. Under what conditins does there exist a π-equivariant principal G- +bundle over X with prescribed Chern classes, for G = U(1) or G = SU(2) ? +Some information about this question was provided in [22, 20] for X = CP 2 and π +finite cyclic, or more generally for X negative definite (see also [19] for the connection +between Chern classes and the isotropy representations). +For X = S4, the existence +and classification of such bundles was applied by Austin [4] and Furuta [15, 17, 16] to +study group actions via instanton gauge theory. The compactification of an equivariant +version of Donaldson’s Yang-Mills moduli space [8], [20] involves “bubbling” convergence +to equivariant instantons over the 4-sphere. Further applications of equivariant bundles +arise in studying the equivariant compactification of moduli spaces over cylindrical end +4-manifolds. +3. The G-Signature Formula +In this section we review the terms of the G-signature formula that we will need in +deriving congruence relations. +The G-signature of a closed 4-manifold X with an orientation preserving action of a +finite group G acting as isometries on X is defined as the virtual representation +(3.1) +Sign(X, G) = [H2 ++(X; C)] − [H2 +−(X; C)] +where H2 +±(X; C) are the maximal positive/negative definite G-invariant subspaces of +H2(X; C). Taking characters gives the g-signatures +(3.2) +Sign(g, X) = trg H2 ++(X) − trg H2 +−(X) + +4 +NIMA ANVARI AND IAN HAMBLETON +which by the G-signature formula can be computed from the fixed point set Xg. +Let D : C∞(Λ+) → C∞(Λ−) denote the signature operator. The Lefschetz numbers +are computed as follows. +(3.3) +L(g, D) = (−1)n(n+1)/2 chg(Λ+ − Λ−)(TX | Xg ⊗ C)Td(TXg ⊗ C) +e(TXg) chg(Λ−1Ng ⊗ C) +[Xg] +Where n = dim Xg and note that TX | Xg = TXg ⊕ Ng and +(3.4) +chg(Λ+ − Λ−)(TX | Xg ⊗ C) = chg(Λ+ − Λ−)(TXg ⊗ C) chg(Λ+ − Λ−)(Ng ⊗ C). +Let F denote a fixed surface, then the contribution to the g-signature is given by +L(g, D) | F = (−1)(e−x − ex)(e−y−iθ − ey+iθ) +x(1 − ey+iθ)(1 − e−y−iθ) +x(−x) +(1 − e−x)(1 − ex)[F] += coth +�y + iθ +2 +� +x cot +�x +2 +� +[F] +where the following trigonometric identity is used +(3.5) +coth +�x +2 +� += +e−x − ex +(1 − e−x)(1 − ex). +To evaluate on [F], we use the Taylor expansions +x coth (x/2) = 2 + 1/6x2 + · · · +coth +�y + iθ +2 +� += coth (iθ/2) − 1 +2 csch2 +�iθ +2 +� +y +Thus the contribution to the Lefschetz number is given by +L(g, D) | F = {2 coth(iθ/2) − csch2(iθ/2)y}[F] += − csch2(iθ/2)[F]2 = csc2(θ/2)[F]2 += +−4tcF +(tcF − 1)2[F]2 +where θ = 2πcF +p +and cF is the rotation number on the normal fiber of F and t = e2πi/p is +a primitive pth root of unity. Similarly we can compute the contribution from isolated +fixed points: +L(g, D) | pt = +(e−iθ1 − eiθ1)(e−iθ2 − eiθ2) +(1 − eiθ1)(1 − e−iθ1)(1 − eiθ2)(1 − e−iθ2) += coth(iθ1/2) coth(iθ2/2) += − cot(θ1/2) cot(θ2/2) += (ta + 1)(tb + 1) +(ta − 1)(tb − 1) + +FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES +5 +where θ1 = 2πa +p +and θ2 = 2πb +p +are the rotation numbers at the fixed point. By summing +over the fixed point set the G-signature formula is given by +(3.6) +Sign(X) = +� +i +(tai + 1) +(tai − 1) +(tbi + 1) +(tbi − 1) + +� +j +−4αjtcj +(tcj − 1)2 +where αj denotes the self-intersection [Fj] · [Fj] of the fixed 2-spheres {Fj}. This formula +can be viewed as an equation in the cyclotomic field Q[ζ] = Q[t]/Φp(t) where Φp(t) is the +cyclotomic polynomial 1 + t + t2 + · · · + tp−1. +4. Congruence Relations for the G-Signature Formula +In this section we derive congruence relations satisfied by the rotation data F using the +G-signature formula. We first note that since (ta−1)/(t−1) is a unit in ring of cyclotomic +integers Z[ζ] when (a, p) = 1, multiplying both sides of the G-signature formula by (t−1)2 +induces an equation in the ring R = Z[ζ]. The I-adic expansion of the resulting right- +hand side leads to congruence relations relating the rotation data, where I denote the +ideal generated by (t − 1) in R. +Following the method of [22] we lift the equation to Z[t], compute the Taylor expan- +sions about t = 1 and reduce the coefficients modulo p. Since the indeterminacy of the +coefficients are determined from the expansion of the cyclotomic polynomial Φp(t) (for +which p divides the coefficients of its Taylor expansion about t = 1 up to order p − 1) we +obtain valid congruence relations by equating coefficients modulo p up to order p − 1. +The expansion arising from contributions from isolated fixed points are given by +(ta + 1) +(ta − 1) +(tb + 1) +(tb − 1)(t − 1)2 = 4 +ab + 4 +ab(t − 1) + 1 +3 +�a2 + b2 + 1 +ab +� +(t − 1)2 +− +1 +180 +�a4 + b4 − 5a2b2 + 3 +ab +� +(t − 1)4 + · · · +Similarly the expansion of the second term is given by expressions of the form +−4αtc +(tc − 1)2(t − 1)2 = −4α +c2 ++ −4α +c2 (t − 1) + 1 +3 +α(c2 − 1) +c2 +(t − 1)2 +− 1 +60 +α(c − 1)(1 + c + c2 + c3) +c2 +(t − 1)4 + · · · . +Equating both sides of the expansion (mod p) from the resulting equation in R we thus +obtain the following congruence relations: +Theorem 4.1. [22, p. 625] Let (X, π) denote a simply connected, closed 4-manifold with +a homologically trivial, locally-linear group action of a finite cyclic group π = Z/p of odd + +6 +NIMA ANVARI AND IAN HAMBLETON +order. Then the following congruence relations hold +(1) +� +i +1 +aibi +− +� +j +αj +c2 +j +≡ 0 +(mod p) +(2) +� +i +a2 +i + b2 +i +aibi ++ +� +j +αj ≡ 3 Sign(X) +(mod p) +(3) +� +i +a4 +i + b4 +i − 5a2 +i b2 +i +aibi ++ 3 +� +j +αjc2 +j ≡ 0 +(mod p) +(4) +� +i +2a6 +i − 7a4 +i b2 +i − 7a2 +i b4 +i + 2b6 +i +aibi ++ 10 +� +j +αjc4 +j ≡ 0 +(mod p) +Higher-order relations are valid up to and including terms of order p − 1. +Example 4.2 (Linear models on CP 2). Let G = Z/p with odd prime p act linearly +on X = CP 2 by t · [z1 : z2 : z3] = [ζaz1 : z2 : z3] for 0 < a < p. +The fixed point +set consists of one isolated fixed point [1 : 0 : 0] with tangential rotation number (a, a) +and a fixed projective line F = {[z1 : z2 : z3] | z1 = 0} with self-intersection +1 and +a rotation of cF ≡ a (mod p) on the normal bundle. +Then it is easy to check that +the congruence relations are satisfied. Similarly, in the case when the action is given by +t·[z1 : z2 : z3] = [ζaz1 : ζbz2 : z3] for 0 < a < b < p the action consists of three isolated fixed +points [0 : 0 : 1], [1 : 0 : 0], [0 : 1 : 0] with rotation numbers (a, b), (b − a, −a), (a − b, −b) +and with some algebra it can be checked that the congruence relations hold. Additional +examples can be obtained for #nCP 2 by equivariant connected sums along fixed point +sets using the linear models. +5. Equivariant Line Bundles +Let (X, π) denote a simply connected, closed 4-manifold with a homologically trivial +action of a finite cyclic group π = Z/p of odd prime p and L → X an equivariant line +bundle. We compute the contribution of a fixed surface F to the twisted G-signature +formula: +L(g, D) | F = {2 coth(iθ/2) − csch2(iθ/2)y}{chg(i∗L)}[F] += {2 coth(iθ/2) − csch2(iθ/2)y}{ez+iφ}[F] += {2 coth(iθ/2) − csch2(iθ/2)y}{eiφ + eiφz}[F] += {2 coth(iθ/2)eiφz − csch2(iθ/2)yeiφ += 2c1(i∗L)[F](tcF + 1) +(tcF − 1)tλ + −4[F]2tcF +(tcF − 1)2 tλ +where φ = 2πλ +p +and z = c1(i∗L). Similarly for the contribution to the isolated fixed points. +To summarize, given an equivariant line bundle (L, π) → (X, π), the index of the twisted +G-signature operator gives a virtual character given by + +FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES +7 +χ(t) = +� +i +(tai + 1) +(tai − 1) +(tbi + 1) +(tbi − 1)tλi + +� +j +−4αjtcj +(tcj − 1)2tλj + +� +j +2c1(i∗L)[Fj](tcj + 1) +(tcj − 1)tλj +where {Fj} are fixed 2-spheres of the action on X with αj denoting the self-intersection +[Fj] · [Fj]. Note that +χ(1) = ch(L)L(X)[X] = +� +1 + c1(L) + 1 +2c1(L)2 +� � +4 + p1 +3 +� +[X] +(5.1) += +�p1 +3 + 2c1(L)2� +[X] = Sign(X) + 2c1(L)2[X]. +(5.2) +Since χ(t) is a virtual character for G = Z/p we may write χ(t) = �p−1 +i=0 aiti for some +ai ∈ Z and +χ(t)(t − 1)2 = χ(1)(t − 1)2 + higher order terms +(mod p) +We can then take the Taylor expansion of the right-hand side of the twisted G-signature +formula after multiplying by (t−1)2 and equate the first and second order terms to obtain +two additional congruence relations. The first order term vanishes while the second order +term is congruent to χ(1) (mod p). +Taking Taylor expansions for these terms in the +G-signature formula give: +(ta + 1) +(ta − 1) +(tb + 1) +(tb − 1)(t − 1)2tλ = 4 +ab + 4(λ + 1) +ab +(t − 1)+ +1 +3 +(a2 + b2 + 1 + 6λ2 + 6λ) +ab +(t − 1)2 + · · · . +Similarly for the second term: +−4αtc +(tc − 1)2(t − 1)2tλ = −4α +c2 ++ −4α(1 + λ) +c2 +(t − 1)+ +1 +3 +α(c2 − 1 − 6λ − 6λ2) +c2 +(t − 1)2 + · · · . +and for the third term: +2m(tc + 1) +(tc − 1)(t − 1)2tλ = 4m +c (t − 1) + 2m(2λ + 1) +c +(t − 1)2 + · · · +where m = c1(i∗L)[F]. Combining these expressions we obtain the following theorem: +Theorem 5.1. Let (L, π) → (X, π) denote an equivariant line bundle over a simply +connected, closed 4-manifold with a homologically trivial group action of a finite cyclic + +8 +NIMA ANVARI AND IAN HAMBLETON +group π = Z/p of odd order. Then the following congruence relation holds +(i) +� +i +λi +aibi +− +� +j +λjαj +c2 +j ++ +� +j +c1(i∗L)[Fj] +cj +≡ 0 +(mod p) +(ii) +� +i +λ2 +i +aibi +− +� +j +λ2 +jαj +c2 +j ++ 2 +� +j +λjc1(i∗L)[Fj] +cj +≡ c1(L)2[X] +(mod p). +Example 5.2. Let π = Z/p act on X = CP 2 preserving an almost complex structure. +Then the complexified second exterior power of the tangent bundle is an equivariant line +bundle see [22, Proposition 1.8]). +Example 5.3 (Linear models on CP 2). Let p denote an odd prime and consider X = CP 2 +and a linear action t·[z1 : z2 : z3] = [ζaz1 : ζbz2 : z3] for 0 < a < b < p. We give an explicit +construction of equivariant line bundles over X. Consider a finite dimensional complex +representation space V = C(λ1) ⊕ C(λ2) ⊕ C(λ3) with action given by ρ ∈ GL3(C): +(5.4) +ρ = + + +tλ1 +tλ2 +tλ3 + + : C3 \ {0} −→ C3 \ {0} +with the λi positive integer weights. Let S(V ) denote the unit sphere in V then ρ com- +mutes with the free S1-action on S(V ) and +CP 2 = S(V )/S1 = S5/S1. +The ρ-action on S(V ) is a lift of the linear action on X if the following system of linear +congruences +λ1 − λ3 ≡ a, +λ2 − λ3 ≡ b +λ2 − λ1 ≡ b − a, +λ3 − λ1 ≡ −a +λ1 − λ2 ≡ a − b, +λ3 − λ2 ≡ −b. +has a solution. This system has one degree of freedom; let λ3 = λ be a fixed parameter, +then the isotropy representations over the three isolated fixed points are given by: +fixed point p1 = [0 : 0 : 1], rotation number (a, b), with isotropyλ3 ≡ λ +fixed point p2 = [1 : 0 : 0], rotation number (b − a, −a), with isotropyλ1 ≡ λ + a +fixed point p3 = [0 : 1 : 0], rotation number (−b, a − b)with isotropyλ2 ≡ λ + b. +The equivariant line bundle +L = S(V ) ×S1 C +is the canonical bundle over CP 2 and the congruence relations of the theorem are satisfied: +� +i +λi +aibi +≡ 0 +� +i +λ2 +i +aibi +≡ 1. + +FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES +9 +In the case when b = 0 the action is given by t · [z1 : z2 : z3] = [ζaz1 : z2 : z3] and has +a fixed projective line F = {z1 = 0} with normal rotation number cF ≡ a (mod p) and +self-intersection +1, while the isolated fixed point [1 : 0 : 0] has rotation number (a, a). +The compatibility for a lift of the linear action is the congruence relation: +λ − λF ≡ a +(mod p). +It is easily seen that the congruence relation of the theorem are satisfied: +λ +a2 − λF +a2 + c1(i∗L)[F] +a +≡ 0. +where we used c1(i∗L)[F] = −1 since the first Chern class of the canonical line bundle +over CP 2 is negative of the preferred generator [F] ∈ H2(CP 2; Z) [30, Theorem 14.10, +p. 169]. +Similarly, the second relation is +λ2 +a2 − λ2 +F +a2 + 2λFc1(i∗L)[F] +a +≡ (a + λF)2 − λ2 +F +a2 ++ 2λFc1(i∗L)[F] +a +≡ 1 ≡ c1(L)2[X]. +Definition 5.5. Let (X, π) be a homologically trivial of π = Z/p in the setting of Theorem +A, with rotation data F = {(ai, bi), (cj, αj) | i ∈ I, j ∈ J}. We say that (X, π) satisfies the +condition of Theorem A if there exists a set of isotropy data I = {λi, λj | i ∈ I, j ∈ J}, +and a set of integers {mj | j ∈ J} so that the equation given in Theorem A holds. +In the next statement, we apply the equivariant connected sum operation to line bun- +dles. +Lemma 5.6. Suppose that (X, π) satisfies the condition of Theorem A. There there is an +equivariant connected sum (X♯ CP 2, π) with a linear π-action on CP 2 which also satisfies +the condition of Theorem A. +Proof. We first suppose that (X, π) contains a fixed 2-sphere Fj with data {αj, cj}. Let +(CP 2, π) be the linear action given by t · [z1 : z2 : z3] = [ζ−cjz1 : z2 : z3] as in Example 4.2. +We do the equivariant connected sum (preserving the orientations) along a point in Fj +and a point in the fixed 2-sphere of CP 2. The new data is obtained by (i) adding the data +{(−cj, −cj); λj} for the newly created isolated fixed point (on CP 2), and (ii) the data +{(cj, αj + 1); λj} for the new fixed 2-sphere. With these choices, it follows that the action +(X♯ CP 2, π) satisfies the condition of Theorem A. The proof in case the action (X, π) has +only isolated fixed points is easier, and will be left to the reader. +□ +Remark 5.7. Suppose that (X, π) has data satisfying the condition of Theorem A, and +contains a fixed 2-sphere F. Let X0 ⊂ X denote the complement of a linear π-invariant +4-ball neighbourhood of a point x ∈ F. If L is an equivariant line bundle over (X♯ CP 2, π), +then the restriction of L to X0 extends to an equivariant line bundle over (X, π) realizing +the given data. + +10 +NIMA ANVARI AND IAN HAMBLETON +6. The proof of Theorem A +The first relation in Theorem 5.1 proves the necessary conditions of Theorem A. To +prove sufficiency we will need the following lemmas. +Note that in a standard lens space Y = L(n; a, b), a generator µ ∈ H1(Y ; Z) is repre- +sented by a circle fibre in the fibration S1 → L(n; a, b) → S2 given by the quotient of a +free Z/n action on S3. +Lemma 6.1. The linking paring lk: H1(Y ) × H1(Y ) → Q/Z in the lens space Y = +L(n; a, b) is given by lk(µ, µ) = ab +p where µ is a generator of H1(Y ; Z) = Z/n. +Proof. In the usual representation of lens spaces L(n; q) as the quotient of the free Z/n +action t · (z1, z2) = (ζz1, ζqz2) on S3, the linking pairing is given by lk(µ, µ) = q +n. The +diffeomorphism L(n; a∗b) → L(n; a, b) arising from changing the generator in Z/n induces +a map on first homology given by multiplication by a. It follows that the linking pairing +on Y is given by +lk(aµ, aµ) = a2 · +�a∗b +n +� += ab +n ∈ Q/Z. +□ +Lemma 6.2 ((See [22, Proposition 1.4, p. 621], [25, Lemma 2.11, p. 95])). Let φ: π1(Y ) −→ +U(1) be the holonomy representation of a flat U(1)-bundle over the lens space Y += +L(n; a, b) that sends a generator µ to exp(2πiλ/n). Then the Poincar´e dual of the first +Chern class PD(c1(L)) is given by λ +ab[µ] in H1(Y ; Z). +Proof. The adjoint to the linking form Φ: H1(Y ; Z) → Hom(H1(Y ; Z), Q/Z) sends m[µ] +to lk(mµ, −) which can be identified with the holonomy representation of the flat bundle +via : +e2πi·lk(mµ,−) : H1(Y ; Z) → U(1) +and this maps the generator to exp(2πi · mab/n). It follows that m ≡ λ +ab (mod n). +□ +If Y is a lens space, we let ˆµ ∈ H2(Y ; Z) denote a standard cohomology generator: the +Poincar´e dual to the circle fibre class µ ∈ H1(Y ; Z). +Lemma 6.3. Let u: Y ′ → Y be a d-fold regular covering of lens space, where H1(Y ; Z) ∼= +Z/dn and H1(Y ′; Z) ∼= Z/n, with gcd(d, n) = 1. Then u∗(ˆµ) = ˆµ′ ∈ H2(Y ′; Z). +Proof. For a d-fold regular covering u: Y ′ → Y of lens spaces, we have u∗[µ′] = d[µ] ∈ +H1(Y, Z) and u∗[Y ′] = d[Y ] ∈ H3(Y ; Z). +The cohomology generator ˆµ = [Y ] ∩ µ ∈ +H2(Y ; Z) is the Poincar´e dual of µ, and similarly for ˆµ′ ∈ H2(Y ′; Z). We have the formula +u∗([Y ′] ∩ u∗(ˆµ)) = u∗(µ′) = dµ. +If H1(Y ; Z) = Z/dn and H1(Y ′; Z) = Z/n, where gcd(d, n) = 1, then u∗(ˆµ) = kˆµ′ implies +that k ≡ 1 (mod n). Hence u∗(ˆµ) = ˆµ′ ∈ H2(Y ′; Z). +□ + +FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES +11 +Suppose that (X, π) satisfies the assumptions of Theorem 5.1. Let Σ ⊂ X denote the +singular set of the action, and let X0 := X − Σ. Write αj = pajαj, where αj is prime to +p (for each fixed 2-sphere Fj). +Lemma 6.4. If the singular set Σ ⊂ X contains an isolated point, then H1(X0; Z) is a +quotient of � Z/αj and has order prime to p. +Proof. Let F ⊂ X denote the disjoint union of the fixed 2-spheres, so F = � +j Fj. First +note that H1(X0) ∼= H3(X, F) and we have an exact sequence +· · · → H2(X) → H2(F) → H3(X, F) → H3(X) → . . . +Since H3(X) = 0, and the homology classes of the fixed 2-spheres are linearly independent +mod p (by [9, Corollary 2.6]), it follows that H3(X, F) ∼= H1(X0) is a torsion group of +order prime to p. +Moreover, the exact Mayer-Vietoris sequence +0 → H2(X0) ⊕ H2(F) → H2(X) → H1(∂X0) → H1(X0) → 0 +and the equality H1(∂X0) = H1(∂ν(F)) ∼= � Z/αj completes the proof. +□ +The proof of Theorem A. By Theorem 5.1(i), the indicated formulas hold if (X, π) admits +an equivariant line bundle. It remains to prove that a solution {λi, λj, mj | i ∈ I, j ∈ J} +to the congruence relation +� +i +λi +aibi +− +� +j +λjαj +c2 +j ++ +� +j +mj +cj +≡ 0 +(mod p) +is sufficient for the existence of an equivariant line bundle with {λi, λj} isotropy repre- +sentations over the isolated fixed points and 2-spheres respectively, and mj ≡ c1(i∗L | Fj) +mod αj. For simplicity, we will assume that the action (X, π) contains at least one iso- +lated fixed point. This may always be arranged by taking the equivariant connected sum +of (X, π) along a fixed 2-sphere with a suitable linear π-action on CP 2 (see Lemma 5.6). +Let X0 = X−N, where N = ν(Σ) is a π-invariant tubular neighbourhood of the singular +set Σ ⊂ X. More explicitly, X0 is the compact 4-manifold with boundary obtained by +removing π-invariant 4-balls around each isolated fixed point and π-invariant tubular +neighbourhoods D2 → ν(Fj) → Fj around each π-fixed 2-sphere Fj with rotation tcj on +D2-fibers. Then ν(Fj) is a 2-disk bundle over S2 with Euler class αj[F] ∈ H2(F; Z) ∼= Z, +and the lens space ∂ν(Fj) = L(αj, 1) inherits a free Z/p action with rotation number cj +on the circle fibre. +If W0 := X0/π denotes the quotient manifold with (regular) covering map q: X0 → W0 +classified by u: W0 → Bπ, then the boundary ∂W0 consists of lens spaces Yi = L(p; ai, bi) +and Yj = L(pαj; cj, cj). Note that +H1(W0; Z) ∼= Z/p ⊕ H1(X0, Z) +by the spectral sequence of the covering. +By Lemma 6.4, H1(X0; Z) is a quotient of +� Z/αj and has order prime to p. +Recall that π-equivariant line bundles L over (X, π) are classified by an element +θ(L) ∈ H2 +π(X; Z) = H2(X ×π Eπ; Z) + +12 +NIMA ANVARI AND IAN HAMBLETON +in the Borel requivariant cohomology of X (see [27]). Since the π-action on X0 is free, for +the restriction L0 ց X0 we have θ(L0) ∈ H2 +π(X0; Z) ∼= H2(W0; Z), and θ(L0) = c1(¯L0), +where ¯L0 is the line bundle over W0 obtained by dividing out the free π-action on the +total space of L0. Moreover, in the short exact sequence +0 → H2(Z/p; Z) +c∗ +−→ H2(W0; Z) +q∗ +−→ H2(X0; Z) → 0 +the pullback q∗(θ(L0)) = c1(q∗(¯L0)) = c1(L0) ∈ H2(X0; Z). +The strategy will be to find a suitable element θ(L) ∈ H2 +π(X; Z) by studying the Mayer- +Vietoris sequence +· · · → H2 +π(X) → H2 +π(X0) ⊕ H2 +π(N) → H2 +π(∂X0) +δ−→ H3 +π(X) → H3 +π(X0) ⊕ H3 +π(N) → . . . +in Borel cohomology associated to the π-equivariant decomposition X = X0 ∪ N. +We observe the following: +(i) The Mayer-Vietoris coboundary map H2 +π(∂X0) +δ−→ H3 +π(X) factors +H2 +π(∂X0) +δ−→ H3 +π(X0, ∂X0) ∼= H3 +π(X, N) → H3 +π(X). +(ii) The cokernel of the map H2 +π(N) → H2 +π(∂X0) has exponent p. This follows from +the commutative diagram of restriction maps +H2 +π(N) +� +� +H2 +π(∂X0) +� +� � Z/pαj +� +H2(N) +� H2(∂X0) +∼ += � � Z/αj +since the map H2 +π(N) → H2(N) is surjective (by the Borel spectral sequence) and +the map H2(N, ∂N) → H2(N) is adjoint to the (diagonal) intersection form on +N, with cokernel H2(∂N) = H2(∂X0), hence determined by the self-intersection +numbers {αj}. +(iii) We have H3 +π(X0, ∂X0) ∼= H3(W0, ∂W0) ∼= H1(W0) ∼= Z/p⊕H1(X0), where H1(X0) +is a quotient of � Z/αj and has order prime to p (by Lemma 6.4). +To complete the proof of Theorem A, it is now enough to produce a class +θ0 ∈ H2 +π(X0) ∼= H2(W0), +which added together with the classes already found in H2 +π(N) will have image zero in +H2 +π(∂X0). By the observations above, this amounts to finding a U(1)-bundle ¯L0 on ∂W0 +whose first Chern class θ0 = c1(¯L0) has image of order prime to p under the coboundary +map +H2(∂W0) → H3(W0, ∂W0). +In other words, we need to find suitable U(1)-bundles over each of boundary components +of ∂W0, so that the sum of their first Chern classes is zero (mod p). This required relation +is exactly the condition (1.1) given in the statement of Theorem A. +Let Y denote one of the lens spaces in ∂W0, For convenience, we will identify Z/p ∼= +H2(Y ; Z) by a �→ a · ˆµ, for a ∈ Z/p, where ˆµ ∈ H2(Y ; Z) denotes a standard generator, +Poincar´e dual to the circle fibre class µ ∈ H1(Y ; Z) (introduced in Lemma 6.1). + +FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES +13 +If Y = L(p; ai, bi) then choose the holonomy representation that sends a generator in +π1(Y ) to exp(2πiλi/p). The first Chern class of the associated flat U(1)-bundle is +λi +aibi +[ˆµ] ∈ H2(Yi; Z) ∼= Z/p. +In the case when the π-action on X has only isolated fixed points, the condition for an +extension is that these elements lie in the kernel of δ in H2(∂W0; Z) = � +i H2(Yi; Z) which +is equivalent to the condition � +i +λi +aibi +≡ 0 (mod p). +In the general case when components of the fixed set contain 2-spheres, we need to +consider contributions from the lens spaces Yj = L(pαj; cj, cj). These lens spaces arise +from the free tcj-action on �Yj := ∂ν(Fj) ≈ L(αj; 1). Conisder the induced covering spaces +of Yj by the lens spaces L(paj+1, 1) and L(αj, 1), where αj = pajαj, and note that the +covering maps induce an isomorphism: +(6.5) +H2(Yj; Z) +∼ += +� +∼ += +� +H2(L(paj+1, 1)) ⊕ H2(L(αj, 1)) +∼ += +� +Z/pαj +∼ += +� Z/paj+1 ⊕ Z/αj +Under the two covering maps, the standard cohomology generator ˆµj ∈ H2(Yj; Z) is sent +to the standard generators in H2(L(paj+1, 1)) and H2(L(αj, 1)), respectively, by Lemma +6.3. The maps in the lower sequence are the reductions mod paj+1 and α, after using the +identifications provided by the cohomology generators. +Now the following congruences uniquely determines a Chern class c1(¯Lj) = ℓj +c2 +j +[ˆµj] ∈ +H2(Yj; Z), by choosing: +(6.6) +ℓj +≡ +−λjαj +(mod paj+1), +ℓj +≡ +cjmj +(mod αj). +and hence a U(1)-bundle ¯Lj ց Yj. The minus sign is chosen in the first congruence +because the induced orientation on Yj from ∂W0 is opposite to its orientation as the disk +bundle over S2 with Euler class αj. +By diagram (6.5) and Lemma 6.3, the first Chern class has image +c1(¯Lj) = ℓj +c2 +j +[µj] = −λjαj + cjmj +c2 +j +[µj] ∈ H2(Yj; Z) +with respect to the decomposition H2(Yj; Z) ∼= Z/paj+1 ⊕ Z/αj. After substituting these +expressions into the formula of Theorem 5.1, we see that the sum vanishes mod p, and +hence the required line bundle ¯L0 over W0 exists. +□ +7. Equivariant SU(2) Bundles +In this section we compute a (necessary) congruence relation similar to the previous +section, but for equivariant SU(2)-bundles. +As above, we work over a closed, simply +connected, oriented 4-manifold with a finite homologically trivial cyclic group action. We + +14 +NIMA ANVARI AND IAN HAMBLETON +again use the twisted G-signature formula (with the previously established notation). In +particular, let D denote the signature operator twisted by an equivaraint SU(2)-bundle +E −→ X, then the contribution to the Lefschetz numbers from isolated fixed points is +given by +L(g, D) | pt = (ta + 1) +(ta − 1) +(tb + 1) +(tb − 1)(tλ + t−λ). +We need to compute the contribution from isolated fixed 2-spheres F. +Since E | F = +L ⊕ L−1, we have chg(L ⊕ L−1 | F) = {eλ+z + e−λ−z}[F] and +L(g, D) | F = {2 cot(iθ/2) − csch2(iθ/2)y} chg(L ⊕ L−1 | F)[F] += {2(tc + 1) +(tc − 1) − +4tcy +(tc − 1)2}{eλ(1 + z) + e−λ(1 − z)}[F] += {2(tc + 1) +(tc − 1) − +4tcy +(tc − 1)2}{tλ + t−λ + z(tλ − t−λ)}[F] += − 4tc[F]2 +(tc − 1)2(tλ + t−λ) + 2c1(L)[F](tc + 1) +(tc − 1)(tλ − t−λ). +Also note +χ(1) = ch(E)L(X)[X] = (2 − c2(E))(4 + 1 +3p1) += 2 Sign(X) − 4c2(E). +We now again multiply both sides of the G-signature formula by (t − 1), take Taylor +expansions about t = 1 and reduce coefficients modulo p: +(ta + 1) +(ta − 1) +(tb + 1) +(tb − 1)(t − 1)2(tλ + t−λ) = 8 +ab + 8 +ab(t − 1)+ +2 +3 +(a2 + b2 + 1 + 6λ2) +ab +(t − 1)2 + · · · . +and for the second term, where we let m denote c1(L)[F]: +(t − 1)2{ −4αtc +(tc − 1)2(tλ + t−λ) + 2m(tc + 1) +(tc − 1)(tλ + t−λ)} += −8α +c2 ++ −8α +c2 (t − 1) + 2 +3 +(αc2 − α − 6αλ2 + 12mcλ) +c2 +(t − 1)2 + · · · +Summing over all the fixed sets and simplifying the coefficient of second order term (t−1)2, +we obtain: +2 Sign(X) + +� +i +4λ2 +i +aibi +− +� +j +4αjλ2 +j +c2 +j ++ +� +j +8mjλj +cj +. +Equating this with χ(1) = 2 Sign(X) − 4c2(E) and reducing coefficients modulo p gives +the following congruence relation: + +FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES +15 +Theorem 7.1. Let (E, π) → (X, π) denote an equivariant SU(2)-bundle over a simply +connected, closed 4-manifold with a homologically trivial group action of a finite cyclic +group π = Z/p of odd prime order. Then the following congruence relation holds +� +i +λ2 +i +aibi +− +� +j +αjλ2 +j +c2 +j ++ +� +j +2λj +cj +c1(i∗Lj)[Fj] ≡ −c2(E)[X] +(mod p), +where Lj is a local reduction E | Fj = Lj ⊕ L−1 +j . +Example 7.2 (Linear Models on S4). Let X = S4 with a linear Z/p-action which gives +rotation numbers (a, b) and (a, −b). Let E denote the instanton one equivariant SU(2)- +bundle, i.e. with c2(E) = 1. Then the congruence relation is given by +−c2(E) = λ2 +1 +ab − λ2 +2 +ab +(mod p) +It is elementary to check that for congruence relation is satisfied with the following isotropy +representations +λ1 = b − a +2 +λ2 = a + b +2 +. +over the fibres of the fixed points. +Example 7.3 (Linear Models on CP +2). Let X = CP +2 with a linear Z/p-action with +one isolated fixed point with rotation number (a, −a) for some a (mod p) and a fixed +projective line F with rotation number a (mod p) on the normal bundle. Let E again +denote the instanton one equivariant SU(2)-bundle. The congruence relation gives +−1 ≡ −λ2 +a2 + λ2 +F +a2 + 2mλF +a +where m = c1(i∗L)[F]. There exists two distinct lifts giving rise to equivariant bundles +which admit G-invariant ASD connections. In the case when the equivariant lift comes +from ”bubbling” on the isolated fixed point then m = 0 and +λ ≡ a +(mod p) +λF ≡ 0 +(mod p). +Thus the congruence is satisfied. On the other hand, if we choose the equivariant lift +associated to the fixed 2-sphere (from 3-dimensional fixed connected component in the +moduli space of equivariant ASD connections with c2(E) = 1) then m = −1 and +λ ≡ a/2 +(mod p) +λF ≡ a/2 +(mod p), +again the congruence relation is satisfied. +In the next section we compute the dimension of the moduli space of invariant anti-self +dual connections for a given equivariant SU(2)-bundle. + +16 +NIMA ANVARI AND IAN HAMBLETON +8. Equivariant Index Computation +Let X be a simply connected, closed, smooth negative definite 4-manifold, with a +homologically trivial action of a finite group G. +If E ց X is an SU(2)-bundle with +c2(E) = k, the moduli space M∗ +1(X) of irreducible ASD connections (on an SU(2)-bundle +E with c2(E) = 1) inherits a G-action, and the connected components of the fixed point +set MG +1 (X) correspond to G-invariant ASD connections for certain equivariant lifts of the +G-action on X to E (see [15], [6], [20, Theorem A], [24, §2]). +We want to compute the dimension of the moduli space MG +k (X) of irreducible G- +invariant ASD connections. This is motivated by the following example, for which the +formal dimension dim M∗ +1(X) = 5. +In this case, we expect a dimension formula that +gives 1 and 3-dimensional strata depending on contributions from isolated fixed points +or isolated fixed 2-spheres in X and on the isotropy representations from the equivariant +lift (see [6] and [21] for details). There are similar index calculations in the literature in +various gauge-theoretic settings (for example, see [13, §3], [4]), [28, 29], [35], [1]). +We first very briefly review the dimension calculation in the non-equivariant setting to +set some notation. Let D+ +A = d∗ +A + d+ +A : Ω1(ad E) → Ω0(ad E) ⊕ Ω2 ++(ad E) denote the +anti-self duality operator, and let Mk denote the ASD moduli space with c2(E) = k. Note +that the formal dimension is given by dim Mk = − Ind(D+ +A) and this is given by +(8.1) +Ind(D+ +A) = ˆA(X) ch(S+) ch(adC E)[X] +where S = S+ ⊕S− and ˆA(X) = � +xi/2 +sinh(xi/2) with ch(S) = �(exi/2 + e−xi/2) and ch(S+) − +ch(S−) = �(exi/2 − e−xi/2). Using this we compute +2 ˆA(X) ch(S+) ch(adC E)[X] = (4 + 1 +3p1 + χ)(3 − 4c2(E))[X] += −16c2(E) + 3(p1 +3 + χ). +Thus the index Ind(D+ +A) = −8c2(E) + 3 +2(Sign +χ)(X) and we get the usual expression +dim Mk = 8k − 3/2(χ + Sign)(X) for the dimension of the moduli space. Also note the +following alternative expression for the index: +Ind(D+ +A) = ch(S+ − S−) ch(S+) ch(adC E)Td(TX ⊗ C) +e(X) +[X] += ˆA(X) ch(S+ ⊗ adC E)[X]. +For the equivariant setting E is an equivariant SU(2)-bundle and let D = D+ +A denote the +anti-self duality operator d∗ +A + d+ +A : Ω1(ad E)G → Ω0(ad E)G ⊕ Ω2 ++(ad E)G. We compute +the equivariant index by averaging the Lefschetz numbers as in [13]: +Ind(D) = 1 +p +� +g∈G +L(g, D) + +FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES +17 +Ind(D) = 1 +p{L(1, D) + +� +g̸=1 +L(g, D)} += 1 +p{−8c2(E) + 3 +2(χ + Sign)(X) + +� +g̸=1 +L(g, D)} += 1 +p{−8c2(E) + 3p +2 (χ + Sign)(X/G) − 3 +2(dχ + dσ)(XG) + +� +g̸=1 +L(g, D)} +where pχ(X/G) = χ(X) + dχ with dχ = � +g̸=1 χ(Xg) is the Euler characteristic defect +terms and similarly for the signature defect term: +− 3 +2(dχ + dσ)[pt] = −3 +2(1 − cot(θ1/2) cot(θ2/2)) +− 3 +2(dχ + dσ)[F] = −3 +2(2 + [F]2 csc2(θ/2)), +where (θ1, θ2) are the rotation numbers at an isolated fixed point and θ = cF is the rotation +number on the normal bundle to F. Decomposing the contributions from isolated fixed +points and 2-spheres: +� +g̸=1 +L(g, D)(XG) = +� +g̸=1 +{ +� +i +L(g, D) | (ai,bi) + +� +j +L(g, D) | Fj}. +Now chg(adC E)(pt) = 3 − 4 sin2( πkℓ +p ), with ℓ the isotropy representation on the fiber of E +over the fixed point. The Lefshetz numbers from the fixed sets can be computed directly +from the index formula and are given by: +(8.3) +L(g, D) | pt = −1 +2 [cot(θ1/2) cot(θ2/2) − 1] chg(adC E)[pt] +(8.4) +L(g, D) | F = [−i cot(θ/2) + 1 +2(χ + csc2(θ/2)y)] chg(adC E)[F], +with χ the Euler class of the tangent bundle to F and y is the Euler class of the normal +bundle to F. We first compute the contribution from isolated fixed points. +L(g, D) | pt = −1 +2 [cot(θ1/2) cot(θ2/2) − 1][3 − 4 sin2(πkℓ +p )][pt] += −3 +2[cot(θ1/2) cot(θ2/2) − 1] − 2 sin2(πkℓ +p ) ++ 2 cot(θ1/2) cot(θ2/2) sin2(πkℓ +p ). + +18 +NIMA ANVARI AND IAN HAMBLETON +Summing over all isolated fixed points gives +1 +p +� +g̸=1 +� +i +L(g, D) | (ai,bi) = 3 +2p +� +i +(dχ + dσ)(ai, bi) − 2 +p +� +i +p−1 +� +k=1 +sin(πkℓi +p ) ++ 2 +p +� +i +p−1 +� +k=1 +cot(aiπk +p +) cot(biπk +p ) sin2(πkℓi +p ) += 3 +2p +� +i +(dχ + dσ)(ai, bi) + m + +� +i +ρL(p, ai, bi, ℓi) +where m is the number isolated fixed points with non-trivial representation on the fiber +and ρL(p, a, b, ℓ) is the rho invariant of lens spaces. +We need to compute chg(adC E | F). Since an SU(2) bundle restricted over a fixed 2- +submanifold has a local abelian reduction E | F = L ⊕ L−1 for some L, we have ad E | F = +L2 ⊕ R. We need to compute chg(adC E | F) = chg(L2) + chg(L2) + 1 and this contributes +chg(adC E | F) = (g + gc1(L2)) + (g−1 + g−1c1(L2)) + 1 += (g + g−1 + 1) + c1(L2)(g − g−1) += (3 − 4 sin2(πkℓ +p )) + 2ic1(L2) sin(2πkℓ +p +), +where now ℓ is the isotropy representation on the fibre over the fixed 2-sphere F. Substi- +tuting these terms, the Lesfchetz number L(g, D) | F evaluated on fixed 2-spheres gives: +L(g, D) | F = [−i cot(θ/2) + 1 +2(χ + csc2(θ/2)y)][(3 − 4 sin2(πkℓ +p )) + 2ic1(L2) sin(2πkℓ +p +)][F] += 1 +2[χ + csc2(θ/2)y][3 − 4 sin2(πkℓ +p )] + 2c1(L2) sin(2πkℓ +p +) cot(θ +2). +Let us introduce a kind of rho invariant term for fixed surfaces: +ρF(ℓ) = 2 +p +p−1 +� +k=1 +csc2(πcFk +p +) sin2(πkℓ +p )[F]2 − 4c1(L)[F] +p +p−1 +� +k=1 +sin(2πkℓ +p +) cot(πkcF +p +), +with this notation we have +1 +p +� +g̸=1 +� +j +L(g, D) | Fj = 3 +2p +� +j +(dχ + dσ)[Fj] − 2 +p +� +j +χ(Fj) +p−1 +� +k=1 +sin2(πkℓj +p +) +− +� +j +ρFj(ℓj). + +FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES +19 +Now combining all the terms we obtain: +Ind(DA) = −8 +p c2(E) + 3 +2(χ + Sign)(X/G)) − m + +� +i +ρL(p, ai, bi, ℓi) +− +� +j with ℓj̸=0 +χ(Fj) − +� +j +ρFj(ℓj). +Since dim MG +k (X) = − Ind(DA), the dimension formula is +dim MG +k (X) = 8 +pc2(E) − 3 +2(χ + Sign)(X/G) + m − +� +i +ρL(p, ai, bi, ℓi) ++ +� +j with ℓj̸=0 +χ(Fj) + +� +j +ρFj(ℓj). +Before giving an example we note a few special cases. +When the action on X only +has isolated fixed points, let (ai, bi) denote the rotation numbers and ℓi the isotropy +representation over the points, the formula reduces to the following: +dim MG +k (X) = 8c2(E) +p +− 3 +2(χ + Sign)(X/G) + m − +� +i +ρL(p, ai, bi, ℓi). +For invariant ASD connections on the four-sphere this formula reduces to that of [4, p. +394]. In the case of SO(3)-bundles in the orbifold setting, see Fintushel and Stern [13]. +When the action on X is a smooth involution with fixed 2-sphere and non-trivial action +on fibre cF ≡ ℓ ≡ 1 mod 2 the formula above reduces to: +dim MG +k (X) = 4c2(E) − 3 +2(χ + Sign)(X/G) + χ(F) + [F]2. +which matches with Wang [35, Theorem 18, p. 130]. +We finish this section with an +example. +Example 8.5. Let X = #3CP +2 with a linear Z/p-action with p = 5 that arises from +equivariant connected sums of linear actions in the following way. Take the equivariant +connected sum of two copies of CP +2 along the two dimensional fixed sets which fixes a pro- +jective line and a rotation number of (1, −1) at the isolated fixed points in each copy. Now +at one of the isolated fixed points take the equivariant connected sum with CP +2 that has +a linear action with 3 isolated fixed points with rotation numbers (1, 1), (2, −1), (2, −1). +The result is a smooth, homologically trivial Z/5-action on X that has 3 isolated fixed +points with rotation data {(1, −1), (2, −1), (2, −1)} and a single fixed 2-sphere F with +rotation number cF ≡ 1 (mod p) on the normal bundle and has self intersection −2. +The compactified, equivariant ASD instanton one moduli space M1(X) has dimension +5 with fixed components that are 1 and 3-dimensional which correspond to invariant ASD +connections for a lifted action to the SU(2)-bundle (see [21]). +The boundary of the moduli space is the ”bubbling” of highly concentrated ASD con- +nections which can be identified with a copy of X. The isolated fixed points propagate + +20 +NIMA ANVARI AND IAN HAMBLETON +1-fixed dimensional strata into the moduli space. We will compute the dimension of these +strata using the dimension formula from this section and from the fixed point data. +For example, at the isolated fixed point (2, −1) the highly concentrated instantons +correspond to ASD connections on the 4-sphere, with equivariant lifts matching the linear +models which then pull back to X using the degree 1-map in the formation of the Taubes +boundary. This determines the equivariant lift on X and has isotropy representation tλ1 +over the fixed point (2, −1) with λ1 ≡ −3 (mod p) and tλ2 over all the other fixed point +sets with λ2 ≡ 1 (mod p). The dimension formula gives: +8 +p − ρL(p, 2, −1, −3) − ρL(p, 2, −1, 1) − ρL(p, 1, −1, 1) + χ(F) + ρF(1) = 1. +On the other hand, at a point on the fixed 2-sphere F following the same procedure with +the degree one Taubes map, we can pull-back an equivariant bundle from the linear model +on S4 with a fixed embedded 2-sphere. This time we get an equivariant SU(2)-bundle on +X with c1(L)[F] = −1 in the local reduction E | F = L⊕L−1. The isotropy representation +is tλ over all the fixed point sets with λ ≡ 1 (mod p). We then have: +8 +p − 2ρL(p, 2, −1, 1) − ρL(p, 1, −1, 1) + χ(F) + ρF(1) = 3. +after substituting the data into the dimension formula. +References +[1] N. Anvari, Extending smooth cyclic group actions on the Poincar´e homology sphere, Pacific J. Math. +282 (2016), 9–25. +[2] N. Anvari and I. Hambleton, Cyclic group actions on contractible 4–manifolds, Geometry & Topology +20 (2016), 1127–1155. +[3] +, Cyclic branched coverings of Brieskorn spheres bounding acyclic 4-manifolds, Glasg. Math. +J. 63 (2021), 400–413. +[4] D. M. Austin, SO(3)-instantons on L(p, q) × R, J. Differential Geom. 32 (1990), 383–413. +[5] D. Baraglia and P. Hekmati, Brieskorn spheres, cyclic group actions and the Milnor conjecture, 2022. +[6] P. J. Braam and G. Mati´c, The Smith conjecture in dimension four and equivariant gauge theory, +Forum Math. 5 (1993), 299–311. +[7] W. Chen, Group actions on 4-manifolds: some recent results and open questions, Proceedings of the +G¨okova Geometry-Topology Conference 2009, Int. Press, Somerville, MA, 2010, pp. 1–21. +[8] S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford Mathematical Mono- +graphs, The Clarendon Press Oxford University Press, New York, 1990, Oxford Science Publications. +[9] A. L. Edmonds, Aspects of group actions on four-manifolds, Topology and its Applications 31 (1989), +109–124. +[10] +, A survey of group actions on 4-manifolds, Handbook of group actions. Vol. III, Adv. Lect. +Math. (ALM), vol. 40, Int. Press, Somerville, MA, 2018, pp. 421–460. +[11] A. L. Edmonds and J. H. Ewing, Locally linear group actions on the complex projective plane, Topol- +ogy 28 (1989), 211–223. +[12] +, Realizing forms and fixed point data in dimension four, Amer. J. Math. 114 (1992), 1103– +1126. +[13] R. Fintushel and R. J. Stern, Pseudofree orbifolds, Ann. of Math. (2) 122 (1985), 335–364. +[14] +, Homotopy K3 surfaces containing Σ(2, 3, 7), J. Differential Geom. 34 (1991), 255–265. +[15] M. Furuta, A remark on a fixed point of finite group action on S4, Topology 28 (1989), 35–38. + +FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES +21 +[16] +, Za-invariant SU(2) instantons over the four sphere, Geometry of low-dimensional mani- +folds, 1 (Durham, 1989), London Math. Soc. Lecture Note Ser., vol. 150, Cambridge Univ. Press, +Cambridge, 1990, pp. 161–174. +[17] M. Furuta and Y. Hashimoto, Invariant instantons on S4, J. Fac. Sci. Univ. Tokyo Sect. IA Math. +37 (1990), 585–600. +[18] I. Hambleton and J.-C. Hausmann, Equivariant principal bundles over spheres and cohomogeneity +one manifolds, Proc. London Math. Soc. (3) 86 (2003), 250–272. +[19] +, Equivariant bundles and isotropy representations, Groups Geom. Dyn. 4 (2010), 127–162. +[20] I. Hambleton and R. Lee, Perturbation of equivariant moduli spaces, Math. Ann. 293 (1992), 17–37. +[21] +, Smooth group actions on definite 4-manifolds and moduli spaces, Duke Math. J. 78 (1995), +715–732. +[22] I. Hambleton, R. Lee, and I. Madsen, Rigidity of certain finite group actions on the complex projective +plane, Comment. Math. Helv. 64 (1989), 618–638. +[23] I. Hambleton and S. Pamuk, Rank conditions for finite group actions on 4-manifolds, Canad. J. +Math. 74 (2022), 550–572. +[24] I. Hambleton and M. Tanase, Permutations, isotropy and smooth cyclic group actions on definite +4-manifolds, Geom. Topol. 8 (2004), 475–509. +[25] M. Hedden and P. Kirk, Chern-Simons invariants, SO(3) instantons, and Z/2 homology cobordism, +Chern-Simons gauge theory: 20 years after, AMS/IP Stud. Adv. Math., vol. 50, Amer. Math. Soc., +Providence, RI, 2011, pp. 83–114. +[26] S. Kwasik and T. Lawson, Nonsmoothable Zp actions on contractible 4-manifolds, J. Reine Angew. +Math. 437 (1993), 29–54. +[27] R. K. Lashof, J. P. May, and G. B. Segal, Equivariant bundles with abelian structural group, Pro- +ceedings of the Northwestern Homotopy Theory Conference (Evanston, Ill., 1982), Contemp. Math., +vol. 19, Amer. Math. Soc., Providence, RI, 1983, pp. 167–176. +[28] T. Lawson, Compactness results for orbifold instantons, Math. Z. 200 (1988), 123–140. +[29] +, A note on trigonometric sums arising in gauge theory, Manuscripta Math. 80 (1993), 265– +272. +[30] J. W. Milnor and J. D. Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76, +Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. +[31] D. Ruberman, Involutions on spin 4-manifolds, Proc. Amer. Math. Soc. 123 (1995), 593–596. +[32] N. Saveliev, Floer homology of Brieskorn homology spheres, J. Differential Geom. 53 (1999), 15–87. +[33] +, Invariants for homology 3-spheres, Springer, 2000. +[34] +, Fukumoto-Furuta invariants of plumbed homology 3-spheres, Pacific J. Math. 205 (2002), +465–490. +[35] S. Wang, Moduli spaces over manifolds with involutions, Math. Ann. 296 (1993), 119–138. +[36] D. M. Wilczy´nski, Group actions on the complex projective plane, Trans. Amer. Math. Soc. 303 +(1987), 707–731. +Department of Mathematics & Statistics, McMaster University L8S 4K1, Hamilton, +Ontario, Canada +Email address: anvarin@math.mcmaster.ca +Department of Mathematics & Statistics, McMaster University L8S 4K1, Hamilton, +Ontario, Canada +Email address: hambleton@mcmaster.ca + diff --git a/BtE1T4oBgHgl3EQfVwQu/content/tmp_files/load_file.txt b/BtE1T4oBgHgl3EQfVwQu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..eb2ca728ddda0aa4c74c37e7f00790e37fcaefe3 --- /dev/null +++ b/BtE1T4oBgHgl3EQfVwQu/content/tmp_files/load_file.txt @@ -0,0 +1,626 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf,len=625 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='03105v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='GT] 8 Jan 2023 FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES NIMA ANVARI AND IAN HAMBLETON Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Given a 4-manifold with a homologically trivial and locally-linear cyclic group action, we obtain necessary and sufficient conditions for the existence of equi- variant bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The conditions are derived from the twisted signature formula and are in the form of congruence relations between the fixed point data and the isotropy representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Introduction Finite group actions on 4-manifolds can be studied in various settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We are mainly interested in comparing smooth actions with those which are topological and locally linear, but important examples arise for symplectic 4-manifolds and complex surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Here is a sampling of survey articles and recent work on aspects of this general theme: [1, 2, 3, 5, 6, 7, 10, 15, 22, 20, 21, 23, 26, 31, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We will focus on the existence and classification of equivariant bundles, and their applications in Yang-Mills gauge theory to the study of finite group actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We begin by recalling some standard definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let (X, π) denote a simply-connected, closed 4-manifold X together with a locally linear and homologically trivial action of a cyclic group π = Z/p of prime order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The fixed point set Xπ has Euler characteristic χ(Xπ) = b2(X) + 2, by the Lefschetz fixed point formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' In general Xπ will consist of isolated fixed points and a disjoint union of fixed 2-spheres (see [9, §2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' At each isolated fixed point x ∈ Xπ, the tangent space admits an equi- variant decomposition (TxiX, π) = C(ai) ⊕ C(bi) of complex representation spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let t · (z1, z2) = (ζaiz1, ζbiz2) denote the action for t a fixed generator in the cyclic group π, ζ = e2πi/p, and with integers (ai, bi), both non-zero modulo p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' (i) The integers (ai, bi) are the local tangential rotation data, and are well-defined up to order and simultaneous change in sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' (ii) Similarly, for each point x on a π-fixed 2-sphere Fj, there is a representation C ⊕ C(cj) corresponding to the equivariant splitting TX | Fj = TFj ⊕ N(Fj) where N(Fj) is the normal bundle with rotation ζcj, and cj ̸≡ 0 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Date: January 4, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' This research was partially supported by NSERC Discovery Grant A4000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 1 2 NIMA ANVARI AND IAN HAMBLETON (iii) The total fixed point rotation data is the collection F = {(ai, bi), (cj, αj) | i ∈ I, j ∈ J}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' where I and J index the isolated fixed points and 2-spheres respectively and αj = [Fj]·[Fj] is the self-intersection number of the fixed spheres [Fj] ∈ H2(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' (iv) By an equivariant line bundle (L, π) → (X, π) we mean a principal U(1)-bundle L over X together with a lift of the π-action to the total space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Given such a lift, there exists a set of isotropy representations of the π-action on each fiber over the fixed point set which we denote by L | xi = tλi over isolated fixed points and L | Fj = tλj over a fixed 2-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Denote the collection of these isotropy representations by I = {λi, λj | i ∈ I, j ∈ J}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' With this notation we have our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let (X, π) denote a simply-connected, closed 4-manifold with a locally linear, homologically trivial action of a cyclic group π = Z/p of odd prime order p, with fixed-point rotation data F = {(ai, bi), (cj, αj) | i ∈ I, j ∈ J}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' A collection I of integers {λi, λj | i ∈ I, j ∈ J} can be realized (modulo p) as the isotropy representations of an equivariant line bundle (L, π) → (X, π) if and only if there is a collection of integers {mj | j ∈ J} such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1) � i∈I λi aibi + � j∈J cjmj − λjαj c2 j ≡ 0 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' When a solution exists, the integers mj = c1(i∗L)[Fj] satisfy equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' A special case of this result can be found in [22, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='4], for a cyclic group π of odd order acting locally linearly and semi-freely on the complex projective plane CP 2 with has three isolated fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The necessary condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1) in Theorem A is established by extending the methods of [22, §2] to more general actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The definitions above generalize directly to actions (X, π) where π is any finite group, and the structural group G of the principal bundle is any compact Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' For applications in gauge theory, G = SU(2) is an important example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The details will be left to the reader (see also [18, 19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Some motivating questions Here are some questions related to the general theme (all actions will be assumed to preserve orientation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' More information about some of these directions can be found in the references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Does there exist a smooth Z/p-action on a (homotopy) K3 surface, which induces the identity on integral homology ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' This is a well-known question of Allan Edmonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Note that results of Edmonds and Ewing [12] imply that topological locally linear examples exist for odd primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Does there exist a smooth Z/p-action on a homotopy K3 surface, which contains an invariant embedded Brieskorn homology 3-sphere Σ(2, 3, 7) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES 3 Fintushel and Stern [14] showed that many homotopy K3 surfaces admit embeddings of Σ(2, 3, 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The question concerns the possible existence of an equivariant splitting of the K3 surface along the Brieskorn sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' A Brieskorn homology 3-sphere Σ(a, b, c) admits a free Z/p-action if p ∤ abc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Does there exist a smooth, homologically trivial extension of this action with isolated fixed points to any smooth simply connected negative definite 4-manifold X with boundary Σ(a, b, c) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Anvari [1] proved that the free Z/7 on Σ(2, 3, 5) does not extend in this way over the min- imal negative definite 4-manifold obtained by resolving the link singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The method of proof involves studying equivariant Yang-Mills gauge theory on the non-compact manifold with cylindrical end obtained from X by attaching the end Σ(2, 3, 5)×[0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Information about the Floer homology of Brieskorn spheres is an essential ingredient in tackling this problem (see Saveliev [32, 33, 34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' What sets of rotation numbers can be realized by a smooth, pseudo-free Z/n-action on X = S2 × S2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' A pseudo-free action is one with isolated singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' If the action is semi-free, there are “standard models” with rotation data {(a, b), (c, d), (a, −b), (c, −d)} at the four fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' This question for X = CP 2 was answered in [11, 20], where it turned out that the rotation data was the same for locally linear actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let (X, π) denote a smooth action of a finite group π on a closed, simply connected smooth 4-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Under what conditins does there exist a π-equivariant principal G- bundle over X with prescribed Chern classes, for G = U(1) or G = SU(2) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Some information about this question was provided in [22, 20] for X = CP 2 and π finite cyclic, or more generally for X negative definite (see also [19] for the connection between Chern classes and the isotropy representations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' For X = S4, the existence and classification of such bundles was applied by Austin [4] and Furuta [15, 17, 16] to study group actions via instanton gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The compactification of an equivariant version of Donaldson’s Yang-Mills moduli space [8], [20] involves “bubbling” convergence to equivariant instantons over the 4-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Further applications of equivariant bundles arise in studying the equivariant compactification of moduli spaces over cylindrical end 4-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The G-Signature Formula In this section we review the terms of the G-signature formula that we will need in deriving congruence relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The G-signature of a closed 4-manifold X with an orientation preserving action of a finite group G acting as isometries on X is defined as the virtual representation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1) Sign(X, G) = [H2 +(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' C)] − [H2 −(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' C)] where H2 ±(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' C) are the maximal positive/negative definite G-invariant subspaces of H2(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Taking characters gives the g-signatures (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='2) Sign(g, X) = trg H2 +(X) − trg H2 −(X) 4 NIMA ANVARI AND IAN HAMBLETON which by the G-signature formula can be computed from the fixed point set Xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let D : C∞(Λ+) → C∞(Λ−) denote the signature operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The Lefschetz numbers are computed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='3) L(g, D) = (−1)n(n+1)/2 chg(Λ+ − Λ−)(TX | Xg ⊗ C)Td(TXg ⊗ C) e(TXg) chg(Λ−1Ng ⊗ C) [Xg] Where n = dim Xg and note that TX | Xg = TXg ⊕ Ng and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='4) chg(Λ+ − Λ−)(TX | Xg ⊗ C) = chg(Λ+ − Λ−)(TXg ⊗ C) chg(Λ+ − Λ−)(Ng ⊗ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let F denote a fixed surface, then the contribution to the g-signature is given by L(g, D) | F = (−1)(e−x − ex)(e−y−iθ − ey+iθ) x(1 − ey+iθ)(1 − e−y−iθ) x(−x) (1 − e−x)(1 − ex)[F] = coth �y + iθ 2 � x cot �x 2 � [F] where the following trigonometric identity is used (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='5) coth �x 2 � = e−x − ex (1 − e−x)(1 − ex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' To evaluate on [F], we use the Taylor expansions x coth (x/2) = 2 + 1/6x2 + · · · coth �y + iθ 2 � = coth (iθ/2) − 1 2 csch2 �iθ 2 � y Thus the contribution to the Lefschetz number is given by L(g, D) | F = {2 coth(iθ/2) − csch2(iθ/2)y}[F] = − csch2(iθ/2)[F]2 = csc2(θ/2)[F]2 = −4tcF (tcF − 1)2[F]2 where θ = 2πcF p and cF is the rotation number on the normal fiber of F and t = e2πi/p is a primitive pth root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Similarly we can compute the contribution from isolated fixed points: L(g, D) | pt = (e−iθ1 − eiθ1)(e−iθ2 − eiθ2) (1 − eiθ1)(1 − e−iθ1)(1 − eiθ2)(1 − e−iθ2) = coth(iθ1/2) coth(iθ2/2) = − cot(θ1/2) cot(θ2/2) = (ta + 1)(tb + 1) (ta − 1)(tb − 1) FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES 5 where θ1 = 2πa p and θ2 = 2πb p are the rotation numbers at the fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' By summing over the fixed point set the G-signature formula is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='6) Sign(X) = � i (tai + 1) (tai − 1) (tbi + 1) (tbi − 1) + � j −4αjtcj (tcj − 1)2 where αj denotes the self-intersection [Fj] · [Fj] of the fixed 2-spheres {Fj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' This formula can be viewed as an equation in the cyclotomic field Q[ζ] = Q[t]/Φp(t) where Φp(t) is the cyclotomic polynomial 1 + t + t2 + · · · + tp−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Congruence Relations for the G-Signature Formula In this section we derive congruence relations satisfied by the rotation data F using the G-signature formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We first note that since (ta−1)/(t−1) is a unit in ring of cyclotomic integers Z[ζ] when (a, p) = 1, multiplying both sides of the G-signature formula by (t−1)2 induces an equation in the ring R = Z[ζ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The I-adic expansion of the resulting right- hand side leads to congruence relations relating the rotation data, where I denote the ideal generated by (t − 1) in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Following the method of [22] we lift the equation to Z[t], compute the Taylor expan- sions about t = 1 and reduce the coefficients modulo p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Since the indeterminacy of the coefficients are determined from the expansion of the cyclotomic polynomial Φp(t) (for which p divides the coefficients of its Taylor expansion about t = 1 up to order p − 1) we obtain valid congruence relations by equating coefficients modulo p up to order p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The expansion arising from contributions from isolated fixed points are given by (ta + 1) (ta − 1) (tb + 1) (tb − 1)(t − 1)2 = 4 ab + 4 ab(t − 1) + 1 3 �a2 + b2 + 1 ab � (t − 1)2 − 1 180 �a4 + b4 − 5a2b2 + 3 ab � (t − 1)4 + · · · Similarly the expansion of the second term is given by expressions of the form −4αtc (tc − 1)2(t − 1)2 = −4α c2 + −4α c2 (t − 1) + 1 3 α(c2 − 1) c2 (t − 1)2 − 1 60 α(c − 1)(1 + c + c2 + c3) c2 (t − 1)4 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Equating both sides of the expansion (mod p) from the resulting equation in R we thus obtain the following congruence relations: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' [22, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 625] Let (X, π) denote a simply connected, closed 4-manifold with a homologically trivial, locally-linear group action of a finite cyclic group π = Z/p of odd 6 NIMA ANVARI AND IAN HAMBLETON order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Then the following congruence relations hold (1) � i 1 aibi − � j αj c2 j ≡ 0 (mod p) (2) � i a2 i + b2 i aibi + � j αj ≡ 3 Sign(X) (mod p) (3) � i a4 i + b4 i − 5a2 i b2 i aibi + 3 � j αjc2 j ≡ 0 (mod p) (4) � i 2a6 i − 7a4 i b2 i − 7a2 i b4 i + 2b6 i aibi + 10 � j αjc4 j ≡ 0 (mod p) Higher-order relations are valid up to and including terms of order p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='2 (Linear models on CP 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let G = Z/p with odd prime p act linearly on X = CP 2 by t · [z1 : z2 : z3] = [ζaz1 : z2 : z3] for 0 < a < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The fixed point set consists of one isolated fixed point [1 : 0 : 0] with tangential rotation number (a, a) and a fixed projective line F = {[z1 : z2 : z3] | z1 = 0} with self-intersection +1 and a rotation of cF ≡ a (mod p) on the normal bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Then it is easy to check that the congruence relations are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Similarly, in the case when the action is given by t·[z1 : z2 : z3] = [ζaz1 : ζbz2 : z3] for 0 < a < b < p the action consists of three isolated fixed points [0 : 0 : 1], [1 : 0 : 0], [0 : 1 : 0] with rotation numbers (a, b), (b − a, −a), (a − b, −b) and with some algebra it can be checked that the congruence relations hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Additional examples can be obtained for #nCP 2 by equivariant connected sums along fixed point sets using the linear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Equivariant Line Bundles Let (X, π) denote a simply connected, closed 4-manifold with a homologically trivial action of a finite cyclic group π = Z/p of odd prime p and L → X an equivariant line bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We compute the contribution of a fixed surface F to the twisted G-signature formula: L(g, D) | F = {2 coth(iθ/2) − csch2(iθ/2)y}{chg(i∗L)}[F] = {2 coth(iθ/2) − csch2(iθ/2)y}{ez+iφ}[F] = {2 coth(iθ/2) − csch2(iθ/2)y}{eiφ + eiφz}[F] = {2 coth(iθ/2)eiφz − csch2(iθ/2)yeiφ = 2c1(i∗L)[F](tcF + 1) (tcF − 1)tλ + −4[F]2tcF (tcF − 1)2 tλ where φ = 2πλ p and z = c1(i∗L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Similarly for the contribution to the isolated fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' To summarize, given an equivariant line bundle (L, π) → (X, π), the index of the twisted G-signature operator gives a virtual character given by FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES 7 χ(t) = � i (tai + 1) (tai − 1) (tbi + 1) (tbi − 1)tλi + � j −4αjtcj (tcj − 1)2tλj + � j 2c1(i∗L)[Fj](tcj + 1) (tcj − 1)tλj where {Fj} are fixed 2-spheres of the action on X with αj denoting the self-intersection [Fj] · [Fj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Note that χ(1) = ch(L)L(X)[X] = � 1 + c1(L) + 1 2c1(L)2 � � 4 + p1 3 � [X] (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1) = �p1 3 + 2c1(L)2� [X] = Sign(X) + 2c1(L)2[X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='2) Since χ(t) is a virtual character for G = Z/p we may write χ(t) = �p−1 i=0 aiti for some ai ∈ Z and χ(t)(t − 1)2 = χ(1)(t − 1)2 + higher order terms (mod p) We can then take the Taylor expansion of the right-hand side of the twisted G-signature formula after multiplying by (t−1)2 and equate the first and second order terms to obtain two additional congruence relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The first order term vanishes while the second order term is congruent to χ(1) (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Taking Taylor expansions for these terms in the G-signature formula give: (ta + 1) (ta − 1) (tb + 1) (tb − 1)(t − 1)2tλ = 4 ab + 4(λ + 1) ab (t − 1)+ 1 3 (a2 + b2 + 1 + 6λ2 + 6λ) ab (t − 1)2 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Similarly for the second term: −4αtc (tc − 1)2(t − 1)2tλ = −4α c2 + −4α(1 + λ) c2 (t − 1)+ 1 3 α(c2 − 1 − 6λ − 6λ2) c2 (t − 1)2 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' and for the third term: 2m(tc + 1) (tc − 1)(t − 1)2tλ = 4m c (t − 1) + 2m(2λ + 1) c (t − 1)2 + · · · where m = c1(i∗L)[F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Combining these expressions we obtain the following theorem: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let (L, π) → (X, π) denote an equivariant line bundle over a simply connected, closed 4-manifold with a homologically trivial group action of a finite cyclic 8 NIMA ANVARI AND IAN HAMBLETON group π = Z/p of odd order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Then the following congruence relation holds (i) � i λi aibi − � j λjαj c2 j + � j c1(i∗L)[Fj] cj ≡ 0 (mod p) (ii) � i λ2 i aibi − � j λ2 jαj c2 j + 2 � j λjc1(i∗L)[Fj] cj ≡ c1(L)2[X] (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let π = Z/p act on X = CP 2 preserving an almost complex structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Then the complexified second exterior power of the tangent bundle is an equivariant line bundle see [22, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='3 (Linear models on CP 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let p denote an odd prime and consider X = CP 2 and a linear action t·[z1 : z2 : z3] = [ζaz1 : ζbz2 : z3] for 0 < a < b < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We give an explicit construction of equivariant line bundles over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Consider a finite dimensional complex representation space V = C(λ1) ⊕ C(λ2) ⊕ C(λ3) with action given by ρ ∈ GL3(C): (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='4) ρ = \uf8eb \uf8ed tλ1 tλ2 tλ3 \uf8f6 \uf8f8 : C3 \\ {0} −→ C3 \\ {0} with the λi positive integer weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let S(V ) denote the unit sphere in V then ρ com- mutes with the free S1-action on S(V ) and CP 2 = S(V )/S1 = S5/S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The ρ-action on S(V ) is a lift of the linear action on X if the following system of linear congruences λ1 − λ3 ≡ a, λ2 − λ3 ≡ b λ2 − λ1 ≡ b − a, λ3 − λ1 ≡ −a λ1 − λ2 ≡ a − b, λ3 − λ2 ≡ −b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' has a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' This system has one degree of freedom;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' let λ3 = λ be a fixed parameter, then the isotropy representations over the three isolated fixed points are given by: fixed point p1 = [0 : 0 : 1], rotation number (a, b), with isotropyλ3 ≡ λ fixed point p2 = [1 : 0 : 0], rotation number (b − a, −a), with isotropyλ1 ≡ λ + a fixed point p3 = [0 : 1 : 0], rotation number (−b, a − b)with isotropyλ2 ≡ λ + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The equivariant line bundle L = S(V ) ×S1 C is the canonical bundle over CP 2 and the congruence relations of the theorem are satisfied: � i λi aibi ≡ 0 � i λ2 i aibi ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES 9 In the case when b = 0 the action is given by t · [z1 : z2 : z3] = [ζaz1 : z2 : z3] and has a fixed projective line F = {z1 = 0} with normal rotation number cF ≡ a (mod p) and self-intersection +1, while the isolated fixed point [1 : 0 : 0] has rotation number (a, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The compatibility for a lift of the linear action is the congruence relation: λ − λF ≡ a (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' It is easily seen that the congruence relation of the theorem are satisfied: λ a2 − λF a2 + c1(i∗L)[F] a ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' where we used c1(i∗L)[F] = −1 since the first Chern class of the canonical line bundle over CP 2 is negative of the preferred generator [F] ∈ H2(CP 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) [30, Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='10, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 169].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Similarly, the second relation is λ2 a2 − λ2 F a2 + 2λFc1(i∗L)[F] a ≡ (a + λF)2 − λ2 F a2 + 2λFc1(i∗L)[F] a ≡ 1 ≡ c1(L)2[X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let (X, π) be a homologically trivial of π = Z/p in the setting of Theorem A, with rotation data F = {(ai, bi), (cj, αj) | i ∈ I, j ∈ J}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We say that (X, π) satisfies the condition of Theorem A if there exists a set of isotropy data I = {λi, λj | i ∈ I, j ∈ J}, and a set of integers {mj | j ∈ J} so that the equation given in Theorem A holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' In the next statement, we apply the equivariant connected sum operation to line bun- dles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Suppose that (X, π) satisfies the condition of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' There there is an equivariant connected sum (X♯ CP 2, π) with a linear π-action on CP 2 which also satisfies the condition of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We first suppose that (X, π) contains a fixed 2-sphere Fj with data {αj, cj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let (CP 2, π) be the linear action given by t · [z1 : z2 : z3] = [ζ−cjz1 : z2 : z3] as in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We do the equivariant connected sum (preserving the orientations) along a point in Fj and a point in the fixed 2-sphere of CP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The new data is obtained by (i) adding the data {(−cj, −cj);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' λj} for the newly created isolated fixed point (on CP 2), and (ii) the data {(cj, αj + 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' λj} for the new fixed 2-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' With these choices, it follows that the action (X♯ CP 2, π) satisfies the condition of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The proof in case the action (X, π) has only isolated fixed points is easier, and will be left to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Suppose that (X, π) has data satisfying the condition of Theorem A, and contains a fixed 2-sphere F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let X0 ⊂ X denote the complement of a linear π-invariant 4-ball neighbourhood of a point x ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' If L is an equivariant line bundle over (X♯ CP 2, π), then the restriction of L to X0 extends to an equivariant line bundle over (X, π) realizing the given data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 10 NIMA ANVARI AND IAN HAMBLETON 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The proof of Theorem A The first relation in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1 proves the necessary conditions of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' To prove sufficiency we will need the following lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Note that in a standard lens space Y = L(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' a, b), a generator µ ∈ H1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) is repre- sented by a circle fibre in the fibration S1 → L(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' a, b) → S2 given by the quotient of a free Z/n action on S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The linking paring lk: H1(Y ) × H1(Y ) → Q/Z in the lens space Y = L(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' a, b) is given by lk(µ, µ) = ab p where µ is a generator of H1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) = Z/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' In the usual representation of lens spaces L(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' q) as the quotient of the free Z/n action t · (z1, z2) = (ζz1, ζqz2) on S3, the linking pairing is given by lk(µ, µ) = q n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The diffeomorphism L(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' a∗b) → L(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' a, b) arising from changing the generator in Z/n induces a map on first homology given by multiplication by a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' It follows that the linking pairing on Y is given by lk(aµ, aµ) = a2 · �a∗b n � = ab n ∈ Q/Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' □ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='2 ((See [22, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 621], [25, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 95])).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let φ: π1(Y ) −→ U(1) be the holonomy representation of a flat U(1)-bundle over the lens space Y = L(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' a, b) that sends a generator µ to exp(2πiλ/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Then the Poincar´e dual of the first Chern class PD(c1(L)) is given by λ ab[µ] in H1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The adjoint to the linking form Φ: H1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) → Hom(H1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z), Q/Z) sends m[µ] to lk(mµ, −) which can be identified with the holonomy representation of the flat bundle via : e2πi·lk(mµ,−) : H1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) → U(1) and this maps the generator to exp(2πi · mab/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' It follows that m ≡ λ ab (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' □ If Y is a lens space, we let ˆµ ∈ H2(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) denote a standard cohomology generator: the Poincar´e dual to the circle fibre class µ ∈ H1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let u: Y ′ → Y be a d-fold regular covering of lens space, where H1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) ∼= Z/dn and H1(Y ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) ∼= Z/n, with gcd(d, n) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Then u∗(ˆµ) = ˆµ′ ∈ H2(Y ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' For a d-fold regular covering u: Y ′ → Y of lens spaces, we have u∗[µ′] = d[µ] ∈ H1(Y, Z) and u∗[Y ′] = d[Y ] ∈ H3(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The cohomology generator ˆµ = [Y ] ∩ µ ∈ H2(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) is the Poincar´e dual of µ, and similarly for ˆµ′ ∈ H2(Y ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We have the formula u∗([Y ′] ∩ u∗(ˆµ)) = u∗(µ′) = dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' If H1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) = Z/dn and H1(Y ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) = Z/n, where gcd(d, n) = 1, then u∗(ˆµ) = kˆµ′ implies that k ≡ 1 (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Hence u∗(ˆµ) = ˆµ′ ∈ H2(Y ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' □ FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES 11 Suppose that (X, π) satisfies the assumptions of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let Σ ⊂ X denote the singular set of the action, and let X0 := X − Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Write αj = pajαj, where αj is prime to p (for each fixed 2-sphere Fj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' If the singular set Σ ⊂ X contains an isolated point, then H1(X0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) is a quotient of � Z/αj and has order prime to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let F ⊂ X denote the disjoint union of the fixed 2-spheres, so F = � j Fj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' First note that H1(X0) ∼= H3(X, F) and we have an exact sequence · · → H2(X) → H2(F) → H3(X, F) → H3(X) → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Since H3(X) = 0, and the homology classes of the fixed 2-spheres are linearly independent mod p (by [9, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='6]), it follows that H3(X, F) ∼= H1(X0) is a torsion group of order prime to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Moreover, the exact Mayer-Vietoris sequence 0 → H2(X0) ⊕ H2(F) → H2(X) → H1(∂X0) → H1(X0) → 0 and the equality H1(∂X0) = H1(∂ν(F)) ∼= � Z/αj completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' □ The proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1(i), the indicated formulas hold if (X, π) admits an equivariant line bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' It remains to prove that a solution {λi, λj, mj | i ∈ I, j ∈ J} to the congruence relation � i λi aibi − � j λjαj c2 j + � j mj cj ≡ 0 (mod p) is sufficient for the existence of an equivariant line bundle with {λi, λj} isotropy repre- sentations over the isolated fixed points and 2-spheres respectively, and mj ≡ c1(i∗L | Fj) mod αj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' For simplicity, we will assume that the action (X, π) contains at least one iso- lated fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' This may always be arranged by taking the equivariant connected sum of (X, π) along a fixed 2-sphere with a suitable linear π-action on CP 2 (see Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let X0 = X−N, where N = ν(Σ) is a π-invariant tubular neighbourhood of the singular set Σ ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' More explicitly, X0 is the compact 4-manifold with boundary obtained by removing π-invariant 4-balls around each isolated fixed point and π-invariant tubular neighbourhoods D2 → ν(Fj) → Fj around each π-fixed 2-sphere Fj with rotation tcj on D2-fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Then ν(Fj) is a 2-disk bundle over S2 with Euler class αj[F] ∈ H2(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) ∼= Z, and the lens space ∂ν(Fj) = L(αj, 1) inherits a free Z/p action with rotation number cj on the circle fibre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' If W0 := X0/π denotes the quotient manifold with (regular) covering map q: X0 → W0 classified by u: W0 → Bπ, then the boundary ∂W0 consists of lens spaces Yi = L(p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' ai, bi) and Yj = L(pαj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' cj, cj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Note that H1(W0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) ∼= Z/p ⊕ H1(X0, Z) by the spectral sequence of the covering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='4, H1(X0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) is a quotient of � Z/αj and has order prime to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Recall that π-equivariant line bundles L over (X, π) are classified by an element θ(L) ∈ H2 π(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) = H2(X ×π Eπ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) 12 NIMA ANVARI AND IAN HAMBLETON in the Borel requivariant cohomology of X (see [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Since the π-action on X0 is free, for the restriction L0 ց X0 we have θ(L0) ∈ H2 π(X0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) ∼= H2(W0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z), and θ(L0) = c1(¯L0), where ¯L0 is the line bundle over W0 obtained by dividing out the free π-action on the total space of L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Moreover, in the short exact sequence 0 → H2(Z/p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) c∗ −→ H2(W0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) q∗ −→ H2(X0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) → 0 the pullback q∗(θ(L0)) = c1(q∗(¯L0)) = c1(L0) ∈ H2(X0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The strategy will be to find a suitable element θ(L) ∈ H2 π(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) by studying the Mayer- Vietoris sequence · · → H2 π(X) → H2 π(X0) ⊕ H2 π(N) → H2 π(∂X0) δ−→ H3 π(X) → H3 π(X0) ⊕ H3 π(N) → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' in Borel cohomology associated to the π-equivariant decomposition X = X0 ∪ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We observe the following: (i) The Mayer-Vietoris coboundary map H2 π(∂X0) δ−→ H3 π(X) factors H2 π(∂X0) δ−→ H3 π(X0, ∂X0) ∼= H3 π(X, N) → H3 π(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' (ii) The cokernel of the map H2 π(N) → H2 π(∂X0) has exponent p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' This follows from the commutative diagram of restriction maps H2 π(N) � � H2 π(∂X0) � � � Z/pαj � H2(N) � H2(∂X0) ∼ = � � Z/αj since the map H2 π(N) → H2(N) is surjective (by the Borel spectral sequence) and the map H2(N, ∂N) → H2(N) is adjoint to the (diagonal) intersection form on N, with cokernel H2(∂N) = H2(∂X0), hence determined by the self-intersection numbers {αj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' (iii) We have H3 π(X0, ∂X0) ∼= H3(W0, ∂W0) ∼= H1(W0) ∼= Z/p⊕H1(X0), where H1(X0) is a quotient of � Z/αj and has order prime to p (by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' To complete the proof of Theorem A, it is now enough to produce a class θ0 ∈ H2 π(X0) ∼= H2(W0), which added together with the classes already found in H2 π(N) will have image zero in H2 π(∂X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' By the observations above, this amounts to finding a U(1)-bundle ¯L0 on ∂W0 whose first Chern class θ0 = c1(¯L0) has image of order prime to p under the coboundary map H2(∂W0) → H3(W0, ∂W0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' In other words, we need to find suitable U(1)-bundles over each of boundary components of ∂W0, so that the sum of their first Chern classes is zero (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' This required relation is exactly the condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1) given in the statement of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let Y denote one of the lens spaces in ∂W0, For convenience, we will identify Z/p ∼= H2(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) by a �→ a · ˆµ, for a ∈ Z/p, where ˆµ ∈ H2(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) denotes a standard generator, Poincar´e dual to the circle fibre class µ ∈ H1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) (introduced in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES 13 If Y = L(p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' ai, bi) then choose the holonomy representation that sends a generator in π1(Y ) to exp(2πiλi/p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The first Chern class of the associated flat U(1)-bundle is λi aibi [ˆµ] ∈ H2(Yi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) ∼= Z/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' In the case when the π-action on X has only isolated fixed points, the condition for an extension is that these elements lie in the kernel of δ in H2(∂W0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) = � i H2(Yi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) which is equivalent to the condition � i λi aibi ≡ 0 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' In the general case when components of the fixed set contain 2-spheres, we need to consider contributions from the lens spaces Yj = L(pαj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' cj, cj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' These lens spaces arise from the free tcj-action on �Yj := ∂ν(Fj) ≈ L(αj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Conisder the induced covering spaces of Yj by the lens spaces L(paj+1, 1) and L(αj, 1), where αj = pajαj, and note that the covering maps induce an isomorphism: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='5) H2(Yj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) ∼ = � ∼ = � H2(L(paj+1, 1)) ⊕ H2(L(αj, 1)) ∼ = � Z/pαj ∼ = � Z/paj+1 ⊕ Z/αj Under the two covering maps, the standard cohomology generator ˆµj ∈ H2(Yj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) is sent to the standard generators in H2(L(paj+1, 1)) and H2(L(αj, 1)), respectively, by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The maps in the lower sequence are the reductions mod paj+1 and α, after using the identifications provided by the cohomology generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Now the following congruences uniquely determines a Chern class c1(¯Lj) = ℓj c2 j [ˆµj] ∈ H2(Yj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z), by choosing: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='6) ℓj ≡ −λjαj (mod paj+1), ℓj ≡ cjmj (mod αj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' and hence a U(1)-bundle ¯Lj ց Yj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The minus sign is chosen in the first congruence because the induced orientation on Yj from ∂W0 is opposite to its orientation as the disk bundle over S2 with Euler class αj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' By diagram (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='5) and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='3, the first Chern class has image c1(¯Lj) = ℓj c2 j [µj] = −λjαj + cjmj c2 j [µj] ∈ H2(Yj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) with respect to the decomposition H2(Yj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) ∼= Z/paj+1 ⊕ Z/αj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' After substituting these expressions into the formula of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1, we see that the sum vanishes mod p, and hence the required line bundle ¯L0 over W0 exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Equivariant SU(2) Bundles In this section we compute a (necessary) congruence relation similar to the previous section, but for equivariant SU(2)-bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' As above, we work over a closed, simply connected, oriented 4-manifold with a finite homologically trivial cyclic group action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We 14 NIMA ANVARI AND IAN HAMBLETON again use the twisted G-signature formula (with the previously established notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' In particular, let D denote the signature operator twisted by an equivaraint SU(2)-bundle E −→ X, then the contribution to the Lefschetz numbers from isolated fixed points is given by L(g, D) | pt = (ta + 1) (ta − 1) (tb + 1) (tb − 1)(tλ + t−λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We need to compute the contribution from isolated fixed 2-spheres F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Since E | F = L ⊕ L−1, we have chg(L ⊕ L−1 | F) = {eλ+z + e−λ−z}[F] and L(g, D) | F = {2 cot(iθ/2) − csch2(iθ/2)y} chg(L ⊕ L−1 | F)[F] = {2(tc + 1) (tc − 1) − 4tcy (tc − 1)2}{eλ(1 + z) + e−λ(1 − z)}[F] = {2(tc + 1) (tc − 1) − 4tcy (tc − 1)2}{tλ + t−λ + z(tλ − t−λ)}[F] = − 4tc[F]2 (tc − 1)2(tλ + t−λ) + 2c1(L)[F](tc + 1) (tc − 1)(tλ − t−λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Also note χ(1) = ch(E)L(X)[X] = (2 − c2(E))(4 + 1 3p1) = 2 Sign(X) − 4c2(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We now again multiply both sides of the G-signature formula by (t − 1), take Taylor expansions about t = 1 and reduce coefficients modulo p: (ta + 1) (ta − 1) (tb + 1) (tb − 1)(t − 1)2(tλ + t−λ) = 8 ab + 8 ab(t − 1)+ 2 3 (a2 + b2 + 1 + 6λ2) ab (t − 1)2 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' and for the second term, where we let m denote c1(L)[F]: (t − 1)2{ −4αtc (tc − 1)2(tλ + t−λ) + 2m(tc + 1) (tc − 1)(tλ + t−λ)} = −8α c2 + −8α c2 (t − 1) + 2 3 (αc2 − α − 6αλ2 + 12mcλ) c2 (t − 1)2 + · · · Summing over all the fixed sets and simplifying the coefficient of second order term (t−1)2, we obtain: 2 Sign(X) + � i 4λ2 i aibi − � j 4αjλ2 j c2 j + � j 8mjλj cj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Equating this with χ(1) = 2 Sign(X) − 4c2(E) and reducing coefficients modulo p gives the following congruence relation: FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES 15 Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let (E, π) → (X, π) denote an equivariant SU(2)-bundle over a simply connected, closed 4-manifold with a homologically trivial group action of a finite cyclic group π = Z/p of odd prime order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Then the following congruence relation holds � i λ2 i aibi − � j αjλ2 j c2 j + � j 2λj cj c1(i∗Lj)[Fj] ≡ −c2(E)[X] (mod p), where Lj is a local reduction E | Fj = Lj ⊕ L−1 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='2 (Linear Models on S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let X = S4 with a linear Z/p-action which gives rotation numbers (a, b) and (a, −b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let E denote the instanton one equivariant SU(2)- bundle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' with c2(E) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Then the congruence relation is given by −c2(E) = λ2 1 ab − λ2 2 ab (mod p) It is elementary to check that for congruence relation is satisfied with the following isotropy representations λ1 = b − a 2 λ2 = a + b 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' over the fibres of the fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='3 (Linear Models on CP 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let X = CP 2 with a linear Z/p-action with one isolated fixed point with rotation number (a, −a) for some a (mod p) and a fixed projective line F with rotation number a (mod p) on the normal bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let E again denote the instanton one equivariant SU(2)-bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The congruence relation gives −1 ≡ −λ2 a2 + λ2 F a2 + 2mλF a where m = c1(i∗L)[F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' There exists two distinct lifts giving rise to equivariant bundles which admit G-invariant ASD connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' In the case when the equivariant lift comes from ”bubbling” on the isolated fixed point then m = 0 and λ ≡ a (mod p) λF ≡ 0 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Thus the congruence is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' On the other hand, if we choose the equivariant lift associated to the fixed 2-sphere (from 3-dimensional fixed connected component in the moduli space of equivariant ASD connections with c2(E) = 1) then m = −1 and λ ≡ a/2 (mod p) λF ≡ a/2 (mod p), again the congruence relation is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' In the next section we compute the dimension of the moduli space of invariant anti-self dual connections for a given equivariant SU(2)-bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 16 NIMA ANVARI AND IAN HAMBLETON 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Equivariant Index Computation Let X be a simply connected, closed, smooth negative definite 4-manifold, with a homologically trivial action of a finite group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' If E ց X is an SU(2)-bundle with c2(E) = k, the moduli space M∗ 1(X) of irreducible ASD connections (on an SU(2)-bundle E with c2(E) = 1) inherits a G-action, and the connected components of the fixed point set MG 1 (X) correspond to G-invariant ASD connections for certain equivariant lifts of the G-action on X to E (see [15], [6], [20, Theorem A], [24, §2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We want to compute the dimension of the moduli space MG k (X) of irreducible G- invariant ASD connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' This is motivated by the following example, for which the formal dimension dim M∗ 1(X) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' In this case, we expect a dimension formula that gives 1 and 3-dimensional strata depending on contributions from isolated fixed points or isolated fixed 2-spheres in X and on the isotropy representations from the equivariant lift (see [6] and [21] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' There are similar index calculations in the literature in various gauge-theoretic settings (for example, see [13, §3], [4]), [28, 29], [35], [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We first very briefly review the dimension calculation in the non-equivariant setting to set some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let D+ A = d∗ A + d+ A : Ω1(ad E) → Ω0(ad E) ⊕ Ω2 +(ad E) denote the anti-self duality operator, and let Mk denote the ASD moduli space with c2(E) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Note that the formal dimension is given by dim Mk = − Ind(D+ A) and this is given by (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1) Ind(D+ A) = ˆA(X) ch(S+) ch(adC E)[X] where S = S+ ⊕S− and ˆA(X) = � xi/2 sinh(xi/2) with ch(S) = �(exi/2 + e−xi/2) and ch(S+) − ch(S−) = �(exi/2 − e−xi/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Using this we compute 2 ˆA(X) ch(S+) ch(adC E)[X] = (4 + 1 3p1 + χ)(3 − 4c2(E))[X] = −16c2(E) + 3(p1 3 + χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Thus the index Ind(D+ A) = −8c2(E) + 3 2(Sign +χ)(X) and we get the usual expression dim Mk = 8k − 3/2(χ + Sign)(X) for the dimension of the moduli space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Also note the following alternative expression for the index: Ind(D+ A) = ch(S+ − S−) ch(S+) ch(adC E)Td(TX ⊗ C) e(X) [X] = ˆA(X) ch(S+ ⊗ adC E)[X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' For the equivariant setting E is an equivariant SU(2)-bundle and let D = D+ A denote the anti-self duality operator d∗ A + d+ A : Ω1(ad E)G → Ω0(ad E)G ⊕ Ω2 +(ad E)G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We compute the equivariant index by averaging the Lefschetz numbers as in [13]: Ind(D) = 1 p � g∈G L(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' D) FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES 17 Ind(D) = 1 p{L(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' D) + � g̸=1 L(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' D)} = 1 p{−8c2(E) + 3 2(χ + Sign)(X) + � g̸=1 L(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' D)} = 1 p{−8c2(E) + 3p 2 (χ + Sign)(X/G) − 3 2(dχ + dσ)(XG) + � g̸=1 L(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' D)} where pχ(X/G) = χ(X) + dχ with dχ = � g̸=1 χ(Xg) is the Euler characteristic defect terms and similarly for the signature defect term: − 3 2(dχ + dσ)[pt] = −3 2(1 − cot(θ1/2) cot(θ2/2)) − 3 2(dχ + dσ)[F] = −3 2(2 + [F]2 csc2(θ/2)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' where (θ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' θ2) are the rotation numbers at an isolated fixed point and θ = cF is the rotation number on the normal bundle to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Decomposing the contributions from isolated fixed points and 2-spheres: � g̸=1 L(g, D)(XG) = � g̸=1 { � i L(g, D) | (ai,bi) + � j L(g, D) | Fj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Now chg(adC E)(pt) = 3 − 4 sin2( πkℓ p ), with ℓ the isotropy representation on the fiber of E over the fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The Lefshetz numbers from the fixed sets can be computed directly from the index formula and are given by: (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='3) L(g, D) | pt = −1 2 [cot(θ1/2) cot(θ2/2) − 1] chg(adC E)[pt] (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='4) L(g, D) | F = [−i cot(θ/2) + 1 2(χ + csc2(θ/2)y)] chg(adC E)[F], with χ the Euler class of the tangent bundle to F and y is the Euler class of the normal bundle to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We first compute the contribution from isolated fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' L(g, D) | pt = −1 2 [cot(θ1/2) cot(θ2/2) − 1][3 − 4 sin2(πkℓ p )][pt] = −3 2[cot(θ1/2) cot(θ2/2) − 1] − 2 sin2(πkℓ p ) + 2 cot(θ1/2) cot(θ2/2) sin2(πkℓ p ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 18 NIMA ANVARI AND IAN HAMBLETON Summing over all isolated fixed points gives 1 p � g̸=1 � i L(g, D) | (ai,bi) = 3 2p � i (dχ + dσ)(ai, bi) − 2 p � i p−1 � k=1 sin(πkℓi p ) + 2 p � i p−1 � k=1 cot(aiπk p ) cot(biπk p ) sin2(πkℓi p ) = 3 2p � i (dχ + dσ)(ai, bi) + m + � i ρL(p, ai, bi, ℓi) where m is the number isolated fixed points with non-trivial representation on the fiber and ρL(p, a, b, ℓ) is the rho invariant of lens spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We need to compute chg(adC E | F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Since an SU(2) bundle restricted over a fixed 2- submanifold has a local abelian reduction E | F = L ⊕ L−1 for some L, we have ad E | F = L2 ⊕ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We need to compute chg(adC E | F) = chg(L2) + chg(L2) + 1 and this contributes chg(adC E | F) = (g + gc1(L2)) + (g−1 + g−1c1(L2)) + 1 = (g + g−1 + 1) + c1(L2)(g − g−1) = (3 − 4 sin2(πkℓ p )) + 2ic1(L2) sin(2πkℓ p ), where now ℓ is the isotropy representation on the fibre over the fixed 2-sphere F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Substi- tuting these terms, the Lesfchetz number L(g, D) | F evaluated on fixed 2-spheres gives: L(g, D) | F = [−i cot(θ/2) + 1 2(χ + csc2(θ/2)y)][(3 − 4 sin2(πkℓ p )) + 2ic1(L2) sin(2πkℓ p )][F] = 1 2[χ + csc2(θ/2)y][3 − 4 sin2(πkℓ p )] + 2c1(L2) sin(2πkℓ p ) cot(θ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let us introduce a kind of rho invariant term for fixed surfaces: ρF(ℓ) = 2 p p−1 � k=1 csc2(πcFk p ) sin2(πkℓ p )[F]2 − 4c1(L)[F] p p−1 � k=1 sin(2πkℓ p ) cot(πkcF p ), with this notation we have 1 p � g̸=1 � j L(g, D) | Fj = 3 2p � j (dχ + dσ)[Fj] − 2 p � j χ(Fj) p−1 � k=1 sin2(πkℓj p ) − � j ρFj(ℓj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES 19 Now combining all the terms we obtain: Ind(DA) = −8 p c2(E) + 3 2(χ + Sign)(X/G)) − m + � i ρL(p, ai, bi, ℓi) − � j with ℓj̸=0 χ(Fj) − � j ρFj(ℓj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Since dim MG k (X) = − Ind(DA), the dimension formula is dim MG k (X) = 8 pc2(E) − 3 2(χ + Sign)(X/G) + m − � i ρL(p, ai, bi, ℓi) + � j with ℓj̸=0 χ(Fj) + � j ρFj(ℓj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Before giving an example we note a few special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' When the action on X only has isolated fixed points, let (ai, bi) denote the rotation numbers and ℓi the isotropy representation over the points, the formula reduces to the following: dim MG k (X) = 8c2(E) p − 3 2(χ + Sign)(X/G) + m − � i ρL(p, ai, bi, ℓi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' For invariant ASD connections on the four-sphere this formula reduces to that of [4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 394].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' In the case of SO(3)-bundles in the orbifold setting, see Fintushel and Stern [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' When the action on X is a smooth involution with fixed 2-sphere and non-trivial action on fibre cF ≡ ℓ ≡ 1 mod 2 the formula above reduces to: dim MG k (X) = 4c2(E) − 3 2(χ + Sign)(X/G) + χ(F) + [F]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' which matches with Wang [35, Theorem 18, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 130].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We finish this section with an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let X = #3CP 2 with a linear Z/p-action with p = 5 that arises from equivariant connected sums of linear actions in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Take the equivariant connected sum of two copies of CP 2 along the two dimensional fixed sets which fixes a pro- jective line and a rotation number of (1, −1) at the isolated fixed points in each copy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Now at one of the isolated fixed points take the equivariant connected sum with CP 2 that has a linear action with 3 isolated fixed points with rotation numbers (1, 1), (2, −1), (2, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The result is a smooth, homologically trivial Z/5-action on X that has 3 isolated fixed points with rotation data {(1, −1), (2, −1), (2, −1)} and a single fixed 2-sphere F with rotation number cF ≡ 1 (mod p) on the normal bundle and has self intersection −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The compactified, equivariant ASD instanton one moduli space M1(X) has dimension 5 with fixed components that are 1 and 3-dimensional which correspond to invariant ASD connections for a lifted action to the SU(2)-bundle (see [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The boundary of the moduli space is the ”bubbling” of highly concentrated ASD con- nections which can be identified with a copy of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The isolated fixed points propagate 20 NIMA ANVARI AND IAN HAMBLETON 1-fixed dimensional strata into the moduli space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We will compute the dimension of these strata using the dimension formula from this section and from the fixed point data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' For example, at the isolated fixed point (2, −1) the highly concentrated instantons correspond to ASD connections on the 4-sphere, with equivariant lifts matching the linear models which then pull back to X using the degree 1-map in the formation of the Taubes boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' This determines the equivariant lift on X and has isotropy representation tλ1 over the fixed point (2, −1) with λ1 ≡ −3 (mod p) and tλ2 over all the other fixed point sets with λ2 ≡ 1 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The dimension formula gives: 8 p − ρL(p, 2, −1, −3) − ρL(p, 2, −1, 1) − ρL(p, 1, −1, 1) + χ(F) + ρF(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' On the other hand, at a point on the fixed 2-sphere F following the same procedure with the degree one Taubes map, we can pull-back an equivariant bundle from the linear model on S4 with a fixed embedded 2-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' This time we get an equivariant SU(2)-bundle on X with c1(L)[F] = −1 in the local reduction E | F = L⊕L−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The isotropy representation is tλ over all the fixed point sets with λ ≡ 1 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We then have: 8 p − 2ρL(p, 2, −1, 1) − ρL(p, 1, −1, 1) + χ(F) + ρF(1) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' after substituting the data into the dimension formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' References [1] N.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Department of Mathematics & Statistics, McMaster University L8S 4K1, Hamilton, Ontario, Canada Email address: anvarin@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='mcmaster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='ca Department of Mathematics & Statistics, McMaster University L8S 4K1, Hamilton, Ontario, Canada Email address: hambleton@mcmaster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='ca' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} diff --git a/CNE2T4oBgHgl3EQf9AkT/vector_store/index.faiss 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The United States Government retains +and the publisher, by accepting the article for publication, acknowledges that the United States +Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or +reproduce the published form of this manuscript, or allow others to do so, for the United States +Government purposes. The Department of Energy will provide public access to these results of +federally sponsored research in accordance with the DOE Public Access Plan +(http://energy.gov/downloads/doe-public-access-plan). + + + + + + + + + + + + + + + + + + + + +Discovery of structure-property relations for molecules via hypothesis-driven active +learning over the chemical space + +Ayana Ghosh,1 Sergei V. Kalinin2 and Maxim A. Ziatdinov1,3 + +1 Computational Sciences and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, +TN 37831 USA +2Department of Materials Science and Engineering, University of Knoxville, Knoxville, TN +37996 USA +3Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, TN +37831 USA + + +Discovery of the molecular candidates for applications in drug targets, biomolecular systems, +catalysts, photovoltaics, organic electronics, and batteries, necessitates development of machine +learning algorithms capable of rapid exploration of the chemical spaces targeting the desired +functionalities. Here we introduce a novel approach for the active learning over the chemical +spaces based on hypothesis learning. We construct the hypotheses on the possible relationships +between structures and functionalities of interest based on a small subset of data and introduce +them as (probabilistic) mean functions for the Gaussian process. This approach combines the +elements from the symbolic regression methods such as SISSO and active learning into a single +framework. Here, we demonstrate it for the QM9 dataset, but it can be applied more broadly to +datasets from both domains of molecular and solid-state materials sciences. + + + + + + + + + + + + + +email: ghosha@ornl.gov + + +Introduction +Chemical discovery1-4 is rooted in quantitative structure-activity/property relationships +(QSAR/QSPR).5-12 These efforts primarily rely on finding appropriate representation of molecules +followed by establishing relationships between structure and activity they exhibit. These models +are harnessed to explore chemical space to select molecules of interest 13-15 for drug targets,16-20 +antibiotics,21 catalysts,22-23 photovoltaics,24-27 organic electronics,28 redox-flow batteries.29 In +addition, chemical discovery also includes understanding of chemical processes such as reaction +energy pathways,30-32 optimization of reaction conditions,33 (for e.g., catalytic activity34), +crystallization,35-36 docking,37 and synthesis.38-39 +The QSAR/QSPR techniques have proven to be useful in all (not limited to) such scenarios. +The popularity40-42 remains in their simplicity to incorporate structural information combined with +physicochemical properties, reliability to capture the property landscape, capability to identify +existing chemical patterns, and identify activity cliffs within the data while being computationally +affordable to perform. The descriptors or features can be multi-dimensional descriptors capturing +electronic or topological characteristics. Alternatively, these can be fingerprints that are the +effective representations of molecules via graph-based or string representations (SMILES,43 +SELFIES44). QSAR/QSPR models began its journey almost 60 years ago with the seminal work +lead by Hansch et al.45 in which a few simple descriptors were used to capture a 2D structure- +activity relationship. Since then, this field has seen a steep rise in utilization of variety of traditional +ML algorithms (Naïve Bayes, Support Vector Machine, Random Forest, to name a few) for +property/process prediction followed by validation by experimental synthesis. Its success is also +credited to generation of easily accessible public repositories (e.g., PubChem,46-48 ZINC,49 +ChEMBL,50-51 QM9,52 ANI-1x,53 and QM7-X54) containing structural and physiochemical +properties (computed with quantum mechanical calculations or observed with experiments) on +thousands to millions of molecules. Altogether these have paved the path forward in chemical +design, discovery with day-to-day applications. +If QSAR/QSPR studies have created a revolution in in silico design efforts, applications of +deep neural network (NN) algorithms55-56 have further accelerated this progress. They have +enabled efficient usage of the big data for not only finding molecules of interest but also quantify57- +60 molecular interactions, chemical bonding, inverse design of molecules for targets and gain novel +insights into mechanisms. However, the performance of any of these models is highly dependent +on the quality, quantity, modelability of the datasets. Furthermore, a discovery process necessitates +extrapolation of learned correlative relationships onto the previously unseen regions of chemical +space. +Correspondingly, the task of generation61 of new molecules by going beyond the standard +(manual) design rules or solutions has gained much attention. Several studies62 have been reported +where different NN-based algorithms are explored to accomplish this challenge. Autoencoders are +one of the common models that are being used to encode molecules63 via complex latent +representations to optimize for specific properties and map them back to molecular structures +through decoding. Another set of examples include applications of recurrent NN64 algorithms +where molecule generation is treated as a sequencing task and the algorithm is permitted to + +generate samples at each stage, as informed by the model. There are also studies on using self- +attention driven transformer models65-66 for targeted structure generation. In addition, +reinforcement learning strategies67-68 have been implemented in this context where molecules with +multiple target objectives can be found. The modern molecular generative models have +transformed standard string representations of molecules towards embedded spaces69 with +information on the entire molecular scaffold. +However, the learning approach behind most of the generative models traverse through +latent embeddings. The latter are generally not smooth, precluding direct gradient-based +optimization methods. Once trained over full libraries of molecules, the fraction of the space +occupied with molecules with useful functionalities is typically small, making their discovery +complex. The chemical space is non-differentiable, precluding the gradient-based descent or +simple Gaussian processes (GPs)70-71 based methodologies. ML methods including variational +autoencoders aim to construct a suitable low-dimensional and ideally differentiable latent +embeddings for the chemical space, allowing for the Bayesian optimization (BO)72-74 type +processes. However, these methodologies to date have been based on either construction of the +embedding space for the full library of candidate molecules or finding similar kernel of +representation for these molecules for target explorations. +Historically, Gaussian process (GPs) has been used within the active learning and BO, +making these processes purely data-driven and non-parametric in nature. It interpolates functional +behavior over relatively low-dimensional parameters space. In a classical GP, a kernel function +(such as radial basis function kernel) is utilized to define the degree of correlation across the +parameters space. The kernel parameters are inferred based on the available data, obtained during +exploration-exploitation steps. It does not incorporate any prior information of physical or +chemical behavior of the system in the process. It learns the physics of the system from the data +itself via kernel function with possibilities of leading to suboptimal results. Consequently, the +number of optimization steps necessary to reconstruct functional behavior, even scanning over low +parameters space becomes large. More advanced physics-informed kernel functions75-77 may help +in such cases which is an area of active research. However, proper application of this technique in +the domain of chemical or physical sciences demands going above the usage of conventional GP +in BO framework to model functionalities of interest.78-81 A series of molecular kernels82 such as +fingerprint kernels (e.g., scalar product kernel, Tanimoto kernel), string kernels (e.g., kernels based +on SMILEs, SELFIEs) and graph kernels (typically uses molecular fragments) can be used in GP +framework. Once the best-performing kernel is chosen, BO is performed for applications in real +world scenarios such as optimization of chemical reactions. +In a recent work, Ziatdinov et al.83-85 introduced a physics-augmented algorithm for active +learning and Bayesian optimization. It combines the flexibility of GP models with physical priors +allowing for the hypothesis-driven discovery in ML. To date, it has been applied86 to the +experiments in scanning probe microscopy, providing new insights into the concentration-induced +phase transition and identifying domain growth laws in ferroelectric materials. +Here, we extend the concept of hypothesis learning to molecular discovery. We combine +it with the compressed-sensing methodologies for identifying relevant structural descriptors and + +evaluate multiple automatically generated hypotheses with a reward-driven acquisition (similar to +the reinforcement learning) to select next evaluation points. Specifically, the hypotheses are +selected using the compressed sensing performed on combinations of nonlinear functionalized +features to find a list of the most relevant combinations. This step is followed by balancing +dimensions (i.e., respecting physics constraints) with respect to the target property to formulate +them into feasible equations. Here, we only consider a handful of easily computable features +related to property of interest, to keep the hypotheses in a simple form that is easy to calculate. +Finally, we evaluate the hypotheses over a wide parameters space to predict functionalities of +interest within the active learning loop. We have utilized the QM9 dataset on isolated molecules +as a use case to establish this tail of our work. +A schematic of the generalized framework detailing the active cruise between design and +discovery using hypothesis learning is shown in Figure 1. The results along with estimated +uncertainties show a generalized cost-effective way to approximate structure-property relations, +applicable to a wide variety of material systems. + +Figure 1: Schematic of workflow, from design to discovery. Figure (left panel) shows +commonly used simulation techniques to generate reliable data for various materials systems. The +right panel establishes the active learning loop - combining features to come up with mathematical +formulations as statistically derived scientific hypotheses, to be evaluated for discovery structure- +property relations. + +Coarse-grained MD +Ab-initio MD +Atomistic MD +Quantum +Mathematical +formulations +Selection & +combination of +physical +descriptors +Features +Experiment +Physics-informed +featurization & sparsification +Initialization +Scientific +hypothesis +!"#$!⋯# → &!⋯# +Exploration + +Results and Discussion +General Considerations: +The physics-informed featurization scheme that we designed is built upon the compressed +sensing methodologies utilized by Ghiringhelli et al.87 for features selection, implemented here as +the seed step for the discovery cycle of an active learning process. Similar to the original SISSO +implementation, our physics-informed featurization and sparsification scheme allows for the +selection of the most relevant descriptors which is obtained by using the least absolute shrinkage +and selection operator (LASSO). The LASSO algorithm employed as a part of feature selection +scheme uses the sparsity of the l1 norm to effectively reduce a descriptor set to the most relevant +descriptors (di) contained in full set (D). It selects the non-zero terms of the l1 regularized linear +least squares approximation of the target property (P). The target property is approximated as P(d) += dc, where c is the coefficient (or weight) associated with Ω dimensional descriptor d. The +solution to this equation can be determined by minimizing the argmin (||𝑃 − 𝐷𝑐||! +!) + 𝜆||𝑐||". +The coefficient c is non-zero for all featurized descriptor which is then ranked to determine the +corresponding importance. +The generation of nonlinear combinations of descriptors (di) by applying several mathematical +operators such as, 1/x, √x, x2, x3, log(x), 1/ log(x), and exp(x), on each feature allows to form a +nonlinear mapping between D and P. In addition, the complexity by combining these +functionalized descriptors via summation allows to generate a more effective map between D and +P. We have considered functionalized descriptors utilizing 2 or 3 terms for the purpose of this +study. Here, we note that inclusion of more terms may lead to more accurate correlation to +endpoint. However, it also introduces additional uncertainty carried by each of the terms combined +with mathematical operators. The physics-informed featurization and sparsification method allows +us to combine multiple features in a linear combination to establish direct correlation to the +endpoint target. Within this method, we search over a large combinatorial space, combine features +followed by balancing units/dimensionality (with coefficients) to convert them into feasible +equations. These are the mathematical formulations that are then turned into probabilistic models +(hypotheses) by introducing suitable priors on parameters, applicable to all use cases. + +The second element of the proposed approach is the hypothesis-driven active learning built +upon SISSO-derived functional forms. In the hypothesis learning, we utilize a structured GP (sGP) +as opposed to standard zero mean GP as our surrogate model(s) to insert physics-informed priors +in the GP/BO framework. To illustrate this approach, we note that in the conventional GP/BO +process, GP is defined as + + + + +𝑦 = 𝑓(𝑥) + e, +𝑓 ~ 𝑀𝑉𝑁𝑜𝑟𝑚𝑎𝑙 (𝑚, 𝐾) +( 1) +where MVNormal is a multivariate normal distribution, m is a prior mean function typically set to +0, 𝐾 is a prior covariance functions (kernel), and e is a normally distributed observational noise. + +The training process of GP model involves inferring kernel parameters given the available +set of observations (x, y) using Bayesian inference techniques. Once the training is completed, the + +probabilistic predictions of the function over the unmeasured parameter space can be obtained by +sampling from a distribution: + +𝑓∗ ~ 𝑀𝑉𝑁𝑜𝑟𝑚𝑎𝑙8𝜇$ +%&'(, Σ$ +%&'(< +( 2) +𝜇$ +%&'( = 𝑚(𝑋∗) + 𝐾(𝑋∗, 𝑋|𝜃)𝐾(𝑋, 𝑋|𝜃))"8𝑦 − 𝑚(𝑋)<, +Σ$ +%&'( = 𝐾(𝑋∗, 𝑋∗|𝜃) − 𝐾(𝑋∗, 𝑋|𝜃)(𝑋, 𝑋|𝜃))"𝐾(𝑋, 𝑋∗|𝜃) +( 3) +Here new inputs are denoted by 𝑋∗. We can obtain a posterior predictive distribution for each set +of kernel parameters. The next point to evaluate is then determined by +𝑥*+,( = arg 𝑚𝑎𝑥, +1 +𝑀 D 𝛼F𝜇$! +%&'(, Σ$! +%&'(G +- +./" + +( 4) +Here 𝛼 is a pre-defined acquisition function and M is the number of posterior samples with kernel +parameters. + +Within the sGP, the prior mean function in Equation 1 is substituted by a physics-informed +probabilistic model whose parameters are inferred jointly with the kernel parameters. The posterior +mean function in Equation 3 then becomes +𝜇$!0! +%&'( = 𝑚8𝑋∗|𝜙.< + 𝐾8𝑋∗, 𝑋|𝜃.<𝐾8𝑋, 𝑋|𝜃.< +)" F𝑦 − 𝑚8𝑋|𝜙. 2), we first increase all elements no less than a by one, and +then replace a − 1 with aa − 1. Clearly, the resulting sequence is of the form 1A′ +2aa − 1A′ +1. +In addition, there is a unique way to transform such a sequence into an SRD of the form +0A11A2, i.e., 0a − 1A′ +11A′ +2a. So, there are (m − 1)qn−2,m−1 SRDs lying in this situation. +Analogously, we find there are (n − 2 − m)qn−2,m SRDs of the form 0aA′ +11A′ +2(a − 1). +In summary, for n ≥ 4, the number of SRDs of type 1 and type 2 with m bar-elements +on Γn in the form 0A11A2 is given by +(n − 1)qn−1,m + (n − m − 1)qn−2,m + (m − 1)qn−2,m−1. +case 2: 0A11A2. Consider the induced sequence 1A[r] +1 A[r] +2 first (Recall A[r] +i +denotes the +conjugate-reverse of Ai). Apparently, there are n − m bar-elements in A[r] +1 A[r] +2 . +(i) A[r] +1 = ∅. +In this scenario, A[r] +2 +could essentially be any SRD of length n − 1 with n − m bar- +elements the number of which is given by qn−1,n−m + ¯qn−2,n−m. +(ii) A[r] +1 ̸= ∅. +12 + +When A[r] +2 = ∅, 1A[r] +1 is the conjugate-reverse of A11 thus is an SRD of length n − 1. +Consequently, the number of SRDs in this case is qn−1,n−m. +Suppose A[r] +2 ̸= ∅. Similar to case 1 (ii), there are (n − 2)qn−1,n−m SRDs where there +is no a ∈ [n] such that A[r] +1 +ends with a while A[r] +2 +starts with a − 1 or A[r] +1 +ends with +a − 1 while A[r] +2 starts with a. Suppose otherwise such an a exists. For a fixed a ∈ [n]/[2], +similar to the discussion in case 1 (ii), we claim that +• the sequences of the form 1A[r] +1 +′aa − 1A[r] +2 +′ are in one-to-one correspondence to the +SRDs on the set Γn−1 \ {0, 0} starting with 1 and having n − m − 1 bar-elements +which are counted by (n − m − 1)qn−2,n−m−1; +• the sequences of the form 1A[r] +1 +′(a−1)aA[r] +2 +′ are in one-to-one correspondence to the +SRDs on the set Γn−1 \ {0, 0} starting with 1 and having n − m bar-elements which +are counted by (m − 2)qn−2,n−m. +In summary, for n ≥ 4, the number of SRDs of type 1 and type 2 with m bar-elements +on Γn in the form 0A11A2 is given by +nqn−1,n−m + (m − 1)qn−2,n−m + (n − m − 1)qn−2,n−m−1. +Combining the above two cases together, the theorem follows. +Applying Theorem 14, we have +qn,m =qn,m + qn−1,m +=(n − m − 1)qn−1,m + (n − m − 2)qn−2,m + mqn−1,m + (m − 1)qn−2,m−1 ++ (m − 1)qn−1,m−1 + (m − 1)qn−2,m−1 + (n − m + 1)qn−1,n−m ++ (n − m − 1)qn−2,n−m−1 + (n − m)qn−2,n−m−1 + (n − m − 2)qn−3,n−m−2, +and +qn−1,m−1 =qn−1,m−1 + qn−2,m−1 +=(n − m − 1)qn−2,m−1 + (n − m − 2)qn−3,m−1 + (m − 1)qn−2,m−1 ++ (m−2)qn−3,m−2 + (m−2)qn−2,m−2 + (m−2)qn−3,m−2 + (n−m+1)qn−2,n−m ++ (n − m − 1)qn−3,n−m−1 + (n − m)qn−3,n−m−1 + (n − m − 2)qn−4,n−m−2. +Summing up the above two equations, we can clear all numbers of the form qx,y and arrive +at +qn,m + qn−1,m−1 =nqn−1,m−1 + (n − 1)qn−1,m + (m − 2)qn−2,m−2 + (3n − 5)qn−2,m−1 ++ (n − m − 2)qn−2,m + (2m − 4)qn−3,m−2 + (2n − 2m − 4)qn−3,m−1. +Moving qn−1,m−1 to the right-hand side, we obtain eq. (15) as desired. +13 + +Is it true that there will be more signed relative derangements if we turn more unsigned +elements into signed elements? Put it differently, is it easier to form a relative derangement +if more elements have signs? The answer is apparently negative due to the symmetry of +qn,m. But, how about the cases for m ≤ n/2? This is related to the unimodality of +sequences. The sequence x0, x1, x2, · · · , xn is said to be unimodal if there exists an index +0 ≤ m ≤ n, called the mode of the sequence, such that x0 ≤ · · · ≤ xm−1 ≤ xm ≥ xm+1 ≥ +· · · ≥ xn. +Theorem 15. For any fixed n ≥ 1, the sequence qn,0, qn,1, . . . , qn,n is unimodal. +Proof. Thanks to the symmetry of qn,m, it suffices to prove P(n, m) = qn,m − qn,m−1 ≥ 0 +for m ≤ n/2, where we still make the convention qn,m = 0 if m < 0. We shall prove this +mainly by induction. +First, from the polynomials of QB +n (t) listed in the last section, we observe that for +n = 1, 2, . . . , 9 and m ≤ n/2, P(n, m) ≥ 0. Secondly, we claim +• for any n ≥ 2, P(n, 1) ≥ 0; +• for any n ≥ 4, P(n, 2) ≥ 0. +In order for proving P(n, 1) ≥ 0 in the case of n ≥ 2, we construct an injection from +Qn to QB +n,1 (where QB +n,i denotes the subset containing signed relative derangements with i +bar-elements). For each sequence in Qn, replacing n with n, we obtain a unique sequence +in QB +n,1. Obviously, this is an injection and then P(n, 1) ≥ 0 follows. +Analogously, we construct an injection from QB +n,1 to QB +n,2 for proving P(n, 2) ≥ 0. We +will classify the sequences in QB +n,1 by the largest bar-element. +case 1: If the largest bar-element in QB +n,1 is less than n − 1, then we map it to a +relative derangement obtained by substituting n for n. In this case, the obtained relative +derangements in QB +n,2 have two bar-elements: n and i for some 1 ≤ i < n − 1. +case 2: If the largest bar-element in QB +n,1 is exactly n − 1, and n − 1 is not followed +by n, then we substitute n for n. In the case that n − 1 is followed by n, we replace 1 +with 1 to obtain a sequence in QB +n,2. In this case, the obtained relative derangements in +QB +n,2 have two bar-elements: either n − 1 and n, or n − 1 and 1 with an additional feature +that n − 1 is followed by n. +case 3: Suppose the largest bar-element in QB +n,1 is n. If n−1 is not followed by n, then +we remove the bar of n. Meanwhile, we replace n−1 with n − 1 and 1 with 1. If n follows +n − 1, then we simply replace 1 with 1. In this case, the obtained relative derangements +in QB +n,2 have two bar-elements: either n − 1 and 1 with an additional feature that n − 1 +is not followed by n, or n and 1 with the feature that n follows n − 1. +In the above mapping procedure, relative derangements in QB +n,1 lying in the same +case are clearly mapped to distinct relative derangements in QB +n,2. Moreover, inspecting +the patterns of the contained two bar-elements and the additional features, relative de- +rangements from different cases are mapped to distinct relative derangements in QB +n,2 (for +n ≥ 4) as well. Therefore, the above map is indeed an injection. Hence, P(n, 2) ≥ 0. +14 + +Now suppose for 1 ≤ n ≤ N and any 0 ≤ m ≤ n/2, P(n, m) ≥ 0. Next, we shall show +that P(N + 1, m) ≥ 0 for any 3 ≤ m ≤ (N + 1)/2. Applying Corollary 10, we first have +P(N + 1, m) = qN+1,m − qN+1,m−1 +=N(qN,m − qN,m−2) + (N − m − 1)(qN−1,m − qN−1,m−1) ++ 3(N − 1)(qN−1,m−1 − qN−1,m−2) + (m − 3)(qN−1,m−2 − qN−1,m−3) ++ 2(N − m − 1)(qN−2,m−1 − qN−2,m−2) + 2(m − 3)(qN−2,m−2 − qN−2,m−3). +(22) +We proceed to distinguish two cases. +(i) If 3 ≤ m ≤ (N − 1)/2, we compare the two subscripts of each term qx,y on the RHS +of eq. (22) and find that y ≤ x/2. For instance, since the maximum value of m here is +(N − 1)/2, as to qN−1,m−2, we have m − 2 = (N − 5)/2 which satisfies (N − 1)/2 ≥ m − 2. +Consequently, qN−1,m−2 −qN−1,m−3 ≥ 0 by assumption. Other summands are nonnegative +by the same token. Therefore, P(N + 1, m) ≥ 0 follows. +(ii) If N/2 ≤ m ≤ (N + 1)/2, m equals either N/2 or (N + 1)/2 since m ∈ N. We check +the two subscripts of qx,y and find that y > x/2 in some cases. Therefore, in the following +reasoning, we will make some transformation by the symmetry of qn,m. +When m = N/2, we replace qN−1,m with qN−1,N−m−1 and regroup the terms on the +RHS of eq. (22), and obtain +P(N + 1, m) =N(qN, N +2 − qN, N−4 +2 ) + 3(N − 1)(qN−1, N−2 +2 +− qN−1, N−4 +2 ) ++ N − 6 +2 +(qN−1, N−4 +2 +− qN−1, N−6 +2 ) + (N − 2)(qN−2, N−2 +2 +− qN−2, N−4 +2 ) ++ (N − 6)(qN−2, N−4 +2 +− qN−2, N−6 +2 ). +(23) +Similarly, when m = (N + 1)/2, we replace qN,m with qN,N−m, qN−1,m with qN−1,N−m−1 +and qN−2,m−1 with qN−2,N−m−1 in eq. (22) and regroup the terms to have +P(N + 1, m) =N(qN, N−1 +2 +− qN, N−3 +2 ) + 5N − 3 +2 +(qN−1, N−1 +2 +− qN−1, N−3 +2 ) ++ N − 5 +2 +(qN−1, N−3 +2 +− qN−1, N−5 +2 ) + (N − 5)(qN−2, N−3 +2 +− qN−2, N−5 +2 ). +(24) +Inspecting term by term on the RHS of eq. (23) and eq. (24), they are all nonnegative by +assumption. Therefore, P(N + 1, m) ≥ 0. This completes the proof of the theorem. +Acknowledgements +The authors would like to thank Prof. Yi Wang for pointing out that the roots of QB +n (t)’s +are not necessarily all real. +A +Proof of Corollary 2.13 +In the following, we write ∂F +∂x (x, t) as Fx(x, t) and ∂F +∂t (x, t) as Ft(x, t). Then according to +15 + +the definition of F(x, t), we first have +Fx(x, t) = +� +n≥1 +nQB +n (t)xn−1, Ft(x, t) = +� +n≥1 +QB +n +′(t)xn. +For the terms on right-hand side of eq. (10), multiplying by xn and summing over n ≥ 3, +we respectively obtain +� +n≥3 +(n − 1)tQB +n−1(t)xn = tx2 � +n≥3 +(n − 1)QB +n−1(t)xn−2 += tx2(Fx(x, t) − QB +1 (t)) +� +n≥3 +(n − 1)QB +n−1(t)xn = x2 � +n≥3 +(n − 1)QB +n−1(t)xn−2 += x2(Fx(x, t) − QB +1 (t)) +� +n≥3 +(t3 − t)QB +n−2 +′(t)xn = x2(t3 − t) +� +n≥3 +QB +n−2 +′(t)xn−2 += x2(t3 − t)Ft(x, t) +� +n≥3 +(3n − 5)tQB +n−2(t)xn = t +� � +n≥3 +3nQB +n−2(t)xn − 5 +� +n≥3 +QB +n−2(t)xn� += t +� � +n≥3 +3(n − 2 + 2)QB +n−2(t)xn − 5 +� +n≥3 +QB +n−2(t)xn� += t +� +3 +� +n≥3 +(n − 2)QB +n−2(t)xn + 6 +� +n≥3 +QB +n−2(t)xn − 5 +� +n≥3 +QB +n−2xn� += t +� +3x3 � +n≥3 +(n − 2)QB +n−2(t)xn−3 + x2 � +n≥3 +QB +n−2(t)xn−2� += t +� +3x3Fx(x, t) + x2F(x, t) +� +� +n≥3 +(n − 2)QB +n−2(t)xn = x3 � +n≥3 +(n − 2)QB +n−2(t)xn−3 = x3Fx(x, t) +� +n≥3 +(2t3 − 2t2)QB +n−3 +′(t)xn = (2t3 − 2t2)x3 � +n≥3 +QB +n−3 +′(t)xn−3 += (2t3 − 2t2)x3Ft(x, t) +16 + +� +n≥3 +(2n − 6)tQB +n−3(t)xn = t +� � +n≥3 +2nQB +n−3(t)xn − 6 +� +n≥3 +QB +n−3(t)xn� += t +� +2 +� +n≥3 +(n − 3 + 3)QB +n−3(t)xn − 6 +� +n≥3 +QB +n−3(t)xn� += t +� +2 +� +n≥3 +(n − 3)QB +n−3(t)xn� += t +� +2x4 � +n≥3 +(n − 3)QB +n−3(t)xn−4� += 2tx4Fx(x, t) +According to the computation above, for n ≥ 3, we have +� +n≥3 +QB +n (t)xn =tx2(Fx(x, t) − QB +1 (t)) + x2(Fx(x, t) − QB +1 (t)) + x2(t3 − t)Ft(x, t) ++ t +� +3x3Fx(x, t) + x2F(x, t) +� ++ x3Fx(x, t) ++ (2t3 − 2t2)x3Ft(x, t) + 2tx4Fx(x, t) += +� +(t + 1)x2 + (3t + 1)x3 + 2tx4� +Fx(x, t) ++ +� +(t3 − t)x2 + (2t3 − 2t2)x3� +Ft(x, t) + tx2F(x, t) − (t + 1)2x2. +Then, F(x, t) is given as follows: +F(x, t) =QB +1 (t)x + QB +2 (t)x2 + +� +n≥3 +QB +n (t)xn +=x + tx + t2x2 + 4tx2 + x2 + +� +(t + 1)x2 + (3t + 1)x3 + 2tx4� +Fx(x, t) ++ +� +(t3 − t)x2 + (2t3 − 2t2)x3� +Ft(x, t) + tx2F(x, t) − (t + 1)2x2 += +� +(t + 1)x2 + (3t + 1)x3 + 2tx4� +Fx(x, t) ++ +� +(t3 − t)x2 + (2t3 − 2t2)x3� +Ft(x, t) + tx2F(x, t) + (t + 1)x + 2tx2. +After sorting out the above equations, we eventually obtain +Ft(x, t) + t + 1 + (3t + 1)x + 2tx2 +t(t2 − 1) + 2t2(t − 1)x Fx(x, t) + +tx2 − 1 +t(t2 − 1)x2 + 2t2(t − 1)x3F(x, t) += +−1 − t − 2tx +t(t2 − 1)x + 2t2(t − 1)x2, +completing the proof of Corollary 8. +References +[1] C.-O. Chow. On derangement polynomials of type B. S´em. Lothar. Combin., 55:Art. +B55b, 2006. +17 + +[2] C.-O. Chow. On derangement polynomials of type B. II. J. Combin. Theory Ser. A, +116(4):816–830, 2005. +[3] R.X.F. Chen. Signed relative derangements, reversal distance, and the signed Hult- +man numbers. Ars. Combin., 147:281–288, 2019. +[4] W.Y.C. Chen. The skew, relative, and classical derangements. Discrete Math., +160:235–239, 1996. +[5] W.Y.C. Chen and J.C.Y. Zhang. The skew and relative derangements of type B. +Electron. J. Combin., 14:2147–2153, 2007. +[6] W.Y.C. Chen, R. L. Tang and A.F.Y. Zhao. Derangement polynomials and ex- +cedances of type B. Electron. J. Combin., 16(2):R15, 2009. +[7] L.L. Liu and M. Dong. Cyclic derangement polynomials of the wreath product Cr≀Sn. +Discrete Math., 343(12):112109, 2020. +[8] R.P. Stanley. Enumerative Combinatorics, vol. 1. Cambridge University Press, Cam- +bridge, 1997. +[9] M.L. Wachs. On q-derangement numbers. Proc. Amer. Math. Soc., 106(1):273–278, +1989. +[10] A.F.Y. Zhao. Excedance numbers for the permutations of type B. Electron. J. Com- +bin., 20(2):P28, 2013. +18 + diff --git a/H9AyT4oBgHgl3EQfffh-/content/tmp_files/load_file.txt b/H9AyT4oBgHgl3EQfffh-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..86e080374f32554320d35f6bdb9f1928b1b062aa --- /dev/null +++ b/H9AyT4oBgHgl3EQfffh-/content/tmp_files/load_file.txt @@ -0,0 +1,545 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf,len=544 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='00341v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='CO] 1 Jan 2023 Symmetric polynomials connecting unsigned and signed relative derangements Ricky Xiao-Feng Chen, Yu-Chen Ruan School of Mathematics, Hefei University of Technology Hefei, Anhui 230601, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' China xiaofengchen@hfut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='cn, 1059568476@qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='com Abstract In this paper, we introduce polynomials (in t) of signed relative derangements that track the number of signed elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The polynomials are clearly seen to be in a sence symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Note that relative derangements are those without any signed elements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=', the evaluations of the polynomials at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Also, the numbers of all signed relative derangements are given by the evaluations at t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Then the coefficients of the polynomials connect unsigned and signed relative derangements and show how putting elements with signs affects the formation of derangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' We first prove a recursion satisfied by these polynomials which results in a recursion satisfied by the coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' A combinatorial proof of the latter is provided next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' We also show that the sequences of the coefficients are unimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Moreover, other results are obtained, for instance, a kind of dual of a relation between signed derangements and signed relative derangements previously proved by Chen and Zhang is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Keywords: Derangements, Relative derangements, Symmetric polynomials, Uni- modal Mathematics Subject Classifications: 05C05, 05A19, 05A15 1 Introduction A derangement on a set [n] = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=', n} is a permutation π = π1π2 · · · πn on [n] such that πi ̸= i for all i ∈ [n], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=', a permutation without fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' We use Dn to denote the set of derangements on [n] and Dn to denote the number of derangements on [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The study of derangements may date back to Euler who showed that the probability for a random permutation to be a derangement tends to 1/e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' It is also well known (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=', Stanley [8, Chapter 2]) that Dn = (n − 1)(Dn−1 + Dn−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (1) 1 A relative derangement π = π1π2 · · ·πn on [n] is a permutation such that πi+1 ̸= πi + 1 for 1 ≤ i ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Let Qn denote the set of relative derangements on [n] and Qn = |Qn|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' With the aid of the notion of skew derangements, Chen [4] combinatorially showed that Qn = Dn + Dn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (2) A signed permutation π on [n] can be viewed as a bijection on the set [n] �{1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' , n} such that π(i) = π(i), where j = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Intuitively, a signed permutation on [n] is just an ordinary permutation π = π1π2 · · · πn with some elements associated with a bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For example, 1342 is a signed permutation on {1, 2, 3, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' These elements with a bar are called signed elements or bar-elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The set of signed permutation on [n] is often denoted by Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' A signed derangement [1] on [n] is a signed permutation π = π1π2 · · · πn such that πi ̸= i, for all i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For example, 1342 is a signed derangement in B4, whereas 1342 is not since it has a fixed point 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' A signed relative derangement (or sometimes called relative derangement of type B, see [5]) on [n] is a signed permutation on [n] such that i is not followed by i + 1, and i is not followed by i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For example, 1324 is a signed relative derangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' We denote by DB n and QB n the sets of signed derangements and signed relative derangements on [n], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Let DB n = |DB n | and QB n = |QB n |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Based on the notion of signed skew derangements, Chen and Zhang [5] proved that QB n = DB n + DB n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (3) One of our results in this paper is a kind of dual of this relation, that is, we present a relation expressing DB n in terms of fn that counts an essential subset of sequences in QB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Obviously, the subset of sequences with zero signed elements is Qn and hence Qn ⊂ QB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' It is natural to consider the subset consisting of sequences with m signed elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' As such, a polynomial tracking the number of signed elements is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' While many polynomials or q-analogues associated to derangements have been studied, for instance, the q-enumeration of derangements in Bn by flag major index [1], the excedances of derangements [6,10], the q-enumeration of derangements by major index [9], and the cyclic polynomials of derangements [7], our polynomials here seem to have been overlooked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In addition, our polynomials have a nice property, namely, they are in a sense symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In Section 2, we introduce the symmetric poly- nomials and prove a recursion satisfied by them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Various results are then derived as a consequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Section 3 is devoted to presenting a combinatorial proof of the resulting recursion satisfied by the coefficients as well as proving a unimodality property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' 2 Symmetric polynomials Let b(π) be the number of signed elements in π ∈ QB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The polynomial of signed relative derangements recording the number of signed elements is then given by QB n (t) = � π∈QB n tb(π) = n � m=0 qn,mtm, 2 where qn,m denotes the number of signed relative derangements with exactly m signed elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' It is evident that qn,m = qn,n−m as we can obtain a signed relative derangment with n − m bar-elements by turning a signed element into its unsigned counterpart and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Therefore, the polynomial QB n (t) is in a sense symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Denote by �QB n the set of signed permutations on the set [n] where in each signed permutation two consecutive entries of the form i(i+ 1) or i(i + 1) for some 1 ≤ i < n−1 appears exactly once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For example, 4231 ∈ �QB 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For π ∈ QB n , we denote the resulting sequence from removing n or n whichever appears in π by π↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The following lemma should not be hard to observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For any π ∈ QB n , we have either π↓ ∈ QB n−1 or π↓ ∈ �QB n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Accordingly, we immediately have QB n (t) = � π∈QB n tb(π) = � π∈QB n , π↓∈�QB n−1 tb(π) + � π∈QB n , π↓∈QB n−1 tb(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (4) To obtain a recursion of QB n (t), we next study the two sums on the right-hand side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (4) in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For π = π1π2 · · · πn−1 ∈ QB n−1 and n ≥ 2, denote by S↑(π) the set of sequences in �QB n that result from π by lifting the elements larger than πi (for some 1 ≤ i ≤ n − 1) by one and replacing πi with a length-two sequence πi(πi + 1), where we define the addition for bar-elements by the rule i + 1 = i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Moreover, if an element x appears an entry in π, we write x ∈ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For n ≥ 3 and any π ∈ QB n , we have � π′∈S↑(π) tb(π′) = b(π)tb(π)+1 + (n − b(π))tb(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (5) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For any π = π1π2 · · ·πn ∈ QB n , it has b(π) bar-elements and n − b(π) elements without a bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For any πi ∈ π with a bar, it will generate an additional bar-element after lifting the elements larger than πi (for some 1 ≤ i ≤ n − 1) by one and replacing πi with a length-two sequence πi(πi + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In other words, it will contribute tb(π)+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' However, for any πi ∈ π without a bar, the number of bar-elements in the sequence will not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Therefore, it contributes tb(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Summarizing the two cases gives the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The lemma right below is not difficult to verify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' If π, π′ ∈ QB n−1 and π ̸= π′, then S↑(π) � S↑(π′) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Moreover, �QB n = � π∈QB n−1 S↑(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (6) 3 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For n ≥ 3, we have � π∈QB n , π↓∈�QB n−1 tb(π) = (1 + t) � (t2 − t)QB n−2 ′(t) + (n − 2)QB n−2(t) � , (7) where QB n ′(t) stands for the derivative of QB n (t) with respect to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' First, by construction, there are exactly two signed permutations π, π′ ∈ QB n such that π↓ = π′↓ ∈ �QB n−1, and vice versa, where if n ∈ π then π′ can be obtained by replacing n with n in π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Thus, tb(π↓) = tb(π′↓) = tb(π) = tb(π′)−1 and � π∈QB n , π↓∈�QB n−1 tb(π) = � π′∈�QB n−1 (1 + t)tb(π′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Next, we have � π′∈�QB n−1 tb(π′) = � π′′∈QB n−2 � π′∈S↑(π′′) tb(π′) = � π′′∈QB n−2 � b(π′′) · t + � n − 2 − b(π′′) �� tb(π′′) = � π′′∈QB n−2 � (t − 1)b(π′′)tb(π′′) + (n − 2)tb(π′′)� = (t2 − t)QB n−2 ′(t) + (n − 2)QB n−2(t), where the first two equalities follow from Lemma 3 and Lemma 2, respectively, and then the proof follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For n ≥ 3, we have � π∈QB n , π↓∈QB n−1 tb(π) = (nt + n − 1)QB n−1(t) + (1 − t) � π′∈QB n−1, n−1∈π′ tb(π′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (8) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' A sequence π ∈ QB n where n appears can be clearly obtained by inserting n into a sequence π↓ ∈ QB n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' We distinguish two cases: if n − 1 appears in π↓ ∈ QB n−1, there are n − 1 positions where n can be inserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' if n − 1 appears in π↓ ∈ QB n−1, there are n positions where n can be inserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Note that in both cases, we have b(π) = b(π↓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Thus, � π∈QB n , π↓∈QB n−1, n∈π tb(π) = � π′∈QB n−1, n−1∈π′ (n − 1) · tb(π′) + � π′∈QB n−1, n−1∈π′ n · tb(π′) = (n − 1)QB n−1(t) + � π′∈QB n−1, n−1∈π′ tb(π′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Similarly, the situation of inserting n can be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' We also distinguish two cases: 4 if n − 1 appears in π↓ ∈ QB n−1, there are n positions where n can be inserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' if n − 1 appears in π↓ ∈ QB n−1, there are n − 1 positions where n can be inserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The difference is that in this case, we have b(π) = b(π↓) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Thus, � π∈QB n , π↓∈QB n−1, n∈π tb(π) = � π′∈QB n−1,n−1∈π′ nt · tb(π′) + � π′∈QB n−1,n−1∈π′ (n − 1)t · tb(π′) = ntQB n−1(t) − t � π′∈QB n−1,n−1∈π′ tb(π′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Combining the above two cases, we obtain the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For n ≥ 3, we have � π′∈QB n−1, n−1∈π′ tb(π′) =(n − 1)tQB n−2(t) + t � (t2 − t)QB n−3 ′(t) + (n − 3)QB n−3(t) � − t � π′′∈QB n−2, n−2∈π′′ tb(π′′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (9) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Analogously, we first have � π′∈QB n−1, n−1∈π′ tb(π′) = � π′∈QB n−1, n−1∈π′, π′↓∈QB n−2 tb(π′↓) + � π′∈QB n−1, n−1∈π′, π′↓∈�QB n−2 tb(π′↓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The first sum of the right-hand side has been obtained in Proposition 5 and equals (n − 1)tQB n−2(t) − t � π′′∈QB n−2, n−2∈π′′ tb(π′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Following the proof of Proposition 4, the second sum of the right-hand side equals � π′∈�QB n−2 t · tb(π′) = t � π′′∈QB n−3 � π′∈S↑(π′′) tb(π′) = t � π′′∈QB n−3 � b(π′′) · t + � n − 3 − b(π′′) �� tb(π′′) = t � π′′∈QB n−3 � (t − 1)b(π′′)tb(π′′) + (n − 3)tb(π′′)� = t � (t2 − t)QB n−3 ′(t) + (n − 3)QB n−3(t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The rest is clear and the proof follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Based on Proposition 4–6, we conclude 5 Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For n ≥ 3, the following holds QB n (t) = (n − 1)(t + 1)QB n−1(t) + � (3n − 5)t + (n − 2) � QB n−2(t) + (t3 − t)QB n−2 ′(t) + (2n − 6)tQB n−3(t) + 2t2(t − 1)QB n−3 ′(t), (10) and QB 0 (t) = 0, QB 1 (t) = 1 + t, QB 2 (t) = t2 + 4t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' According to Proposition 4–6, we first obtain QB n (t) = � π∈QB n tb(π) = � π∈QB n , π↓∈�QB n−1 tb(π) + � π∈QB n , π↓∈QB n−1 tb(π) =(1 + t) � (t2 − t)QB n−2 ′(t) + (n − 2)QB n−2(t) � + (nt + n − 1)QB n−1(t) + (1 − t) � π′∈QB n−1 n−1∈π′ tb(π′) =(1 + t) � (t2 − t)QB n−2 ′(t) + (n − 2)QB n−2(t) � + (nt + n − 1)QB n−1(t) + (1 − t) � (n − 1)tQB n−2(t) + t � (t2 − t)QB n−3 ′(t) + (n − 3)QB n−3(t) � − t � π′′∈QB n−2 n−2∈π′↓ tb(π′↓) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Iterating using Proposition 6 and using the fact that � π′∈QB 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' 1∈π′ tb(π′) = t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' we have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='n (t) =(nt + n − 1)QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='n−1(t) + (1 − t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='� n−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='(−1)k+1(n − k)tkQB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='n−k−1(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='+ (−1)n(1 − t)tn−1 + (1 + t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='(t2 − t)QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='n−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='′(t) + (n − 2)QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='n−2(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='+ (1 − t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='�n−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='(−1)k+1tk� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='(t2 − t)QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='n−k−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='′(t) + (n − k − 2)QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='n−k−2(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='=(nt + n − 1)QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='n−1(t) + (2n − 4)QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='n−2(t) + (−1)n(1 − t)tn−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='n−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='(−1)ktk−1� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='(n − k − 1) + (2k + 1 − 2n)t + (n − k)t2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='n−k−1(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='+ (t3 − t)QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='n−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='′(t) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='n−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='(−1)k+1tk+1(2t − 1 − t2)QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='n−k−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (11) Consequently, we have QB n−1(t) = � (n − 1)t + n − 2 � QB n−2(t) + (2n − 6)QB n−3(t) + (−1)n−1(1 − t)tn−2 + n−3 � k=1 (−1)ktk−1� (n − k − 2) + (2k + 3 − 2n)t + (n − k − 1)t2� QB n−k−2(t) + (t3 − t)QB n−3 ′(t) + n−3 � k=1 (−1)k+1tk+1(2t − 1 − t2)QB n−k−3 ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' 6 Then, it is observed that the two sums in the last expression of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (11) equals (−t) � QB n−1(t) − � (n − 1)t + n − 2 � QB n−2(t) − (2n − 6)QB n−3(t) − (−1)n−1(1 − t)tn−2 − (t3 − t)QB n−3 ′(t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Plugging it into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (11) and simplifying completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Based on the obtained recursion eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (10),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' the first few polynomials of QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='n (t) are com- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='puted and listed below: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='1 (t) =t + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='2 (t) =t2 + 4t + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='3 (t) =3t3 + 14t2 + 14t + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='4 (t) =11t4 + 64t3 + 112t2 + 64t + 11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='5 (t) =53t5 + 362t4 + 866t3 + 866t2 + 362t + 53 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='6 (t) =309t6 + 2428t5 + 7252t4 + 10300t3 + 7252t2 + 2428t + 309 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='7 (t) =2119t7 + 18806t6 + 66854t5 + 121838t4 + 121838t3 + 66854t2 + 18806t + 2119 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='8 (t) =16687t8 + 165016t7 + 677656t6 + 1497880t5 + 1937368t4 + 1497880t3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='+ 677656t2 + 165016t + 16687 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='9 (t) =148329t9 + 1616786t8 + 7513658t7 + 19444106t6 + 30752450t5 + 30752450t4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='+ 19444106t3 + 7513658t2 + 1616786t + 148329 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Let F(x, t) = � n≥1 QB n (t)xn be the generating function of QB n (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Then, F(0, t) = 0 and F(x, t) satisfies the following differential equation: ∂F ∂t (x, t) + t + 1 + 3tx + x + 2tx2 t(t2 − 1) + 2t2(t − 1)x ∂F ∂x (x, t) = −1 − t − 2tx t(t2 − 1)x + 2t2(t − 1)x2 − tx2 − 1 t(t2 − 1)x2 + 2t2(t − 1)x3 F(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (12) The proof of Corollary 8 is provided in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Unfortunately, we are unable to solve the differential equation to get explicit formulas for F(x, t) and QB n (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Let π ∈ QB n be chosen uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Then, the expectation and variance of the number of signed elements b(π) are respectively E[b(π)] = n 2, Var[b(π)] = Fn + 2n − n2 4 , where Fn satisfies Fn = � (n − 1)2 + (2n − 2)Fn−1 �QB n−1 QB n + � (3n − 2)(n − 2) + (4n − 3)Fn−2 �QB n−2 QB n + � (2n − 2)(n − 3) + (2n − 2)Fn−3 �QB n−3 QB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Recall that qn,m = qn,n−m, and it is easy to see QB n (1) = n � m=0 qn,m, QB n ′(t) = n � m=0 mqn,mtm−1, QB n ′(1) = n � m=0 mqn,m, QB n ′′(t) = n � m=0 m(m − 1)qn,mtm−2, QB n ′′(1) = n � m=0 m(m − 1)qn,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Consequently, we have E[b(π)] = �n m=0 mqn,m �n m=0 qn,m = �n m=0(m + n − m)qn,m/2 �n m=0 qn,m = QB n ′(1) QB n (1) = n 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' As for the variance, we compute Var[b(π)] = n� m=0 (m − E[b(π)])2qn,m QB n = n� m=0 m2qn,m + n� m=0 E[b(π)]2qn,m − 2 n� m=0 mE[b(π)]qn,m QB n = n� m=0 � m(m − 1) + m � qn,m + n� m=0 E[b(π)]2qn,m − 2 n� m=0 mE[b(π)]qn,m QB n = QB n ′′(1) + QB n ′(1) + E[b(π)]2QB n (1) − 2E[b(π)]QB n ′(1) QB n (1) = QB n ′′(1) QB n (1) + 2n − n2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' From Theorem 7, we next get QB n ′′(1) = (2n − 2)QB n−1 ′(1) + (2n − 2)QB n−1 ′′(1) + (6n − 4)QB n−2 ′(1) + (4n − 3)QB n−2 ′′(1) + (4n − 4)QB n−3 ′(1) + (2n − 2)QB n−3 ′′(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' By dividing both sides by QB n , the following recurrsion of Fn = QB n ′′(1) QB n (1) can be obtained: Fn = � (n − 1)2 + (2n − 2)Fn−1 �QB n−1(1) QB n (1) + � (3n − 2)(n − 2) + (4n − 3)Fn−2 �QB n−2(1) QB n (1) + � (2n − 2)(n − 3) + (2n − 2)Fn−3 �QB n−3(1) QB n (1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' 8 The following corollary follows from Theorem 7 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For n ≥ 3 and m ≥ 0, we have Qn =(n − 1)Qn−1 + (n − 2)Qn−2, (13) QB n =(2n − 1)QB n−1 + (2n − 4)QB n−2, (14) qn,m =(n − 1)qn−1,m−1 + (n − 1)qn−1,m + (m − 2)qn−2,m−2 + (3n − 5)qn−2,m−1 + (n − m − 2)qn−2,m + (2m − 4)qn−3,m−2 + (2n − 2m − 4)qn−3,m−1, (15) where we make the convention that qn,m = 0 if m < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (13) and (14) follow from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (10) by setting t = 0 and t = 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (15) is obtained by comparing the coefficients of tm on both sides of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (10) It is easy to see that the case m = 0 of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (15) agrees with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Of course, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (13) and (14) can be also obtained by making use of the recursions satisfied by Dn, DB n , eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (2) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' We leave the computation to the interested reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In the next section, we will present a direct combinatorial proof of the recursion of qn,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' 3 Recursion and unimodality of qn,m The goal of this section is to first prove the recursion of qn,m combinatorially, and then prove the sequence of qn,m is unimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Before we proceed, we present a connection to the work of the first author [3] using a slight variation of signed relative derangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Recall the definitions there: Let Γn = {(0, −1), (−1, 0), (1, −2), (−2, 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=', (n, −n − 1), (−n − 1, n)} be a set of ordered pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For an ordered pair T = (a, b), the element a is called the left entry of T and denoted by T l = a, while b the right entry of T and denoted by T r = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' A signed relative derangement (SRD) on Γn is a sequence π = T0T1 · · · Tn such that Ti ∈ Γn, each ordered pair appears at most once in π, (a, b) ∈ Γn and (b, a) ∈ Γn cannot be both contained in π, and for 0 ≤ i ≤ n − 1, T r i ̸= −T l i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' This particular form for SRDs was chosen for a reason, as SRDs were also treated as fixed point involutions in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' As such, the first author could provide an upper bound for the number of signed permutations whose reversal distances are maximum possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' An SRD of type 1 on Γn is an SRD π = T0T1T2 · · · Tn such that T0 = (0, −1) and Tn ̸= (n, −n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' An SRD of type 2 on Γn is an SRD π = T0T1T2 · · ·Tn such that T0 = (0, −1) and Tn = (n, −n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Let fn and ˆfn denote the number of SRDs of type 1 and type 2 on Γn, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Clearly, ˆfn = fn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' One of the main results in Chen [3] is the four-term recursion below fn = (2n − 2)fn−1 + (4n − 3)fn−2 + (2n − 2)fn−3, (16) where f1 = 1, f2 = 4, f3 = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' 9 Following [3], we have known that there is a natural bijection for transfroming SRDs on Γn to the signed relative derangements in the classical definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' That is, just view (i, −i−1) as i and (−i−1, i) as i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' But it is worth noting that the condition now becomes that i is not followed by i + 1 and i + 1 is not followed by i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Sometimes it is more convenient to use this definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For instance, let π[r] denote the sequence obtained from π by reading π reversely (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=', right to left) and changing i to i and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Then, if π is an SRD, then π[r] is also an SRD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For example, for an SRD π = 2310, π[r] = 0132 is an SRD too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' We refer to π[r] as the conjugate-reverse of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' This is not true in the classical definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For example, for a signed relative derangement π = 3210, π[r] = 0123 is not a signed relative derangement anymore in the classical definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In the following, we will use the new version of SRDs if not explicitly stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For n ≥ 3, QB n = (fn + fn−1) + (fn−1 + fn−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (17) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The elements π1π2 · · · πn in QB n consist of two classes: π1 = 1 and π1 ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The latter is equivalent to SRDs of type 1 and type 2 and counted by fn + ˆfn = fn + fn−1 as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' As for those starting with 1, the subsequence π2 · · ·πn must not start with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' It is then not hard to see that this class is counted by fn−1 + fn−2, completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In view of Lemma 11, the “core” of QB n is really the subset of sequences not starting with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Also, recall that QB n = DB n + DB n−1 obtained by Chen and Zhang [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Accordingly, it suggests the following relation which can be viewed as a dual of this relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Proposition 12 (Dual of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For n ≥ 2, we have DB n = fn + fn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (18) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' First, we take the opportunity to present a direct combinatorial proof of a recursion of DB n which is an analogue of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Consider signed derangements of length n in DB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' We distinguish the following cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' case 1: If 1 appears, it can be placed at any other n − 1 positions except the first position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Suppose 1 is placed at the k-th position for a fixed 1 < k ≤ n, then we consider the elements k and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' If k is placed at the first position, the remaining n − 2 entries (other than the first and the k-th entries) could essentially form any signed derangement of length n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Then, we have DB n−2 signed derangements in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' If k is not placed at the first position (note that k could still be placed at the first position), viewing k as 1 (and k as 1), the remaining n − 1 entries other than the k-th entry essentially form a signed derangement of length n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Hence, there are DB n−1 signed derangements in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' 10 Since there are n − 1 options for k, we have (n − 1)(DB n−2 + DB n−1) signed derangements where 1 appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' case 2: Consider the case 1 appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Clearly, there are DB n−1 signed derangements where 1 is placed at the first position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' If 1 is not placed at the first position, in analogy with case 1, we have (n−1)(DB n−2+ DB n−1) such signed derangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Summarizing the above discussion, we have DB n = (2n − 1)DB n−1 + (2n − 2)DB n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (19) Next, let Fn = fn + fn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Applying the four-term recurrence eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (16), we have Fn = (2n − 1)fn−1 + (4n − 3)fn−2 + (2n − 2)fn−3 = (2n − 1)Fn−1 + (2n − 2)Fn−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' That is, DB n and Fn satisfy the same recursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Meanwhile, we have DB 2 = F2 = 5, DB 3 = F3 = 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Therefore, DB n and fn + fn−1 also have the same initial values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Thus, it is proved that DB n = fn + fn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' We remark that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (19) can be found in [2], but with a different proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Combining Proposition 12 and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (16), we immediately have an alternative proof of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Now we are in a position to prove the recursion eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Let qn,m denote the number of π = π1π2 · · · πn ∈ QB n with m bar-elements and π1 ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Equivalently, qn,m counts SRDs of type 1 and 2 on Γn that have m bar-elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' We first have the following relation which is an analogue of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' qn,m = qn,m + qn−1,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (20) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For any π ∈ QB n with m bar-elements, π is either in the form π1π2 · · ·πn where π1 ̸= 1 or 1π2 · · ·πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The number of the former is just qn,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' And the number of the latter is equal to the number of π2 · · · πn where π2 ̸= 2, namely qn−1,m, whence the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In the light of Lemma 13, in order for studying qn,m it suffices to study qn,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' To that end, we generalize the idea for proving eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (16) in [3] and obtain Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For n ≥ 4, we have qn,m =(n − 1)qn−1,m + (n − m − 1)qn−2,m + (m − 1)qn−2,m−1 + nqn−1,n−m + (m − 1)qn−2,n−m + (n − m − 1)qn−2,n−m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (21) 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Note that SRDs of type 1 and 2 on Γn with m bar-elements are either in the form 0A11A2 or 0A11A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' We will count SRDs in each case separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' case 1: 0A11A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (i) A2 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In this case, A1 could essentially be any SRD of length n − 1 with m bar-elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' It is easy to see there are qn−1,m + qn−2,m such SRDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (ii) A2 ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Consider the induced sequence 1A2A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' If there exists no a ∈ [n] such that A2 ends with a while A1 starts with a − 1 or A2 ends with a − 1 while A1 starts with a, then the sequence 1A2A1 could be equivalently any SRD of type 1 or 2 of length n−1 and with m bar-elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The latter is counted by qn−1,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Moreover, there are n − 2 ways to transform each such a sequence into sequences of the form A11A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Hence, there are (n − 2)qn−1,m SRDs lying in this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' If otherwise, such an a exists, then by construction a ∈ [n] \\ [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' We claim that for a fixed a ∈ [n] \\ [2], the sequences of the form 1A′ 2aa − 1A′ 1 are in one-to-one correspondence to the SRDs on the set Γn−1 \\ {0, 0} starting with 1 and having m − 1 bar-elements which are counted by qn−2,m−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' the sequences of the form 1A′ 2(a − 1)aA′ 1 are in one-to-one correspondence to the SRDs on the set Γn−1 \\ {0, 0} starting with 1 and having m bar-elements which are counted by qn−2,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The above first case can be seen from replacing aa − 1 with a − 1 and decreasing all other elements greater than a (regardless of if it has a bar) by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In particular, this will lose one bar-element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The second case can be seen analogously, but without losing a bar-element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Conversely, for each of the m − 1 bar-elements in the SRDs on the set Γn−1 \\ {0, 0} starting with 1, say a − 1 (a > 2), we first increase all elements no less than a by one, and then replace a − 1 with aa − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Clearly, the resulting sequence is of the form 1A′ 2aa − 1A′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In addition, there is a unique way to transform such a sequence into an SRD of the form 0A11A2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=', 0a − 1A′ 11A′ 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' So, there are (m − 1)qn−2,m−1 SRDs lying in this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Analogously, we find there are (n − 2 − m)qn−2,m SRDs of the form 0aA′ 11A′ 2(a − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In summary, for n ≥ 4, the number of SRDs of type 1 and type 2 with m bar-elements on Γn in the form 0A11A2 is given by (n − 1)qn−1,m + (n − m − 1)qn−2,m + (m − 1)qn−2,m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' case 2: 0A11A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Consider the induced sequence 1A[r] 1 A[r] 2 first (Recall A[r] i denotes the conjugate-reverse of Ai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Apparently, there are n − m bar-elements in A[r] 1 A[r] 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (i) A[r] 1 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In this scenario, A[r] 2 could essentially be any SRD of length n − 1 with n − m bar- elements the number of which is given by qn−1,n−m + ¯qn−2,n−m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (ii) A[r] 1 ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' 12 When A[r] 2 = ∅, 1A[r] 1 is the conjugate-reverse of A11 thus is an SRD of length n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Consequently, the number of SRDs in this case is qn−1,n−m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Suppose A[r] 2 ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Similar to case 1 (ii), there are (n − 2)qn−1,n−m SRDs where there is no a ∈ [n] such that A[r] 1 ends with a while A[r] 2 starts with a − 1 or A[r] 1 ends with a − 1 while A[r] 2 starts with a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Suppose otherwise such an a exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For a fixed a ∈ [n]/[2], similar to the discussion in case 1 (ii), we claim that the sequences of the form 1A[r] 1 ′aa − 1A[r] 2 ′ are in one-to-one correspondence to the SRDs on the set Γn−1 \\ {0, 0} starting with 1 and having n − m − 1 bar-elements which are counted by (n − m − 1)qn−2,n−m−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' the sequences of the form 1A[r] 1 ′(a−1)aA[r] 2 ′ are in one-to-one correspondence to the SRDs on the set Γn−1 \\ {0, 0} starting with 1 and having n − m bar-elements which are counted by (m − 2)qn−2,n−m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In summary, for n ≥ 4, the number of SRDs of type 1 and type 2 with m bar-elements on Γn in the form 0A11A2 is given by nqn−1,n−m + (m − 1)qn−2,n−m + (n − m − 1)qn−2,n−m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Combining the above two cases together, the theorem follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Applying Theorem 14, we have qn,m =qn,m + qn−1,m =(n − m − 1)qn−1,m + (n − m − 2)qn−2,m + mqn−1,m + (m − 1)qn−2,m−1 + (m − 1)qn−1,m−1 + (m − 1)qn−2,m−1 + (n − m + 1)qn−1,n−m + (n − m − 1)qn−2,n−m−1 + (n − m)qn−2,n−m−1 + (n − m − 2)qn−3,n−m−2, and qn−1,m−1 =qn−1,m−1 + qn−2,m−1 =(n − m − 1)qn−2,m−1 + (n − m − 2)qn−3,m−1 + (m − 1)qn−2,m−1 + (m−2)qn−3,m−2 + (m−2)qn−2,m−2 + (m−2)qn−3,m−2 + (n−m+1)qn−2,n−m + (n − m − 1)qn−3,n−m−1 + (n − m)qn−3,n−m−1 + (n − m − 2)qn−4,n−m−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Summing up the above two equations, we can clear all numbers of the form qx,y and arrive at qn,m + qn−1,m−1 =nqn−1,m−1 + (n − 1)qn−1,m + (m − 2)qn−2,m−2 + (3n − 5)qn−2,m−1 + (n − m − 2)qn−2,m + (2m − 4)qn−3,m−2 + (2n − 2m − 4)qn−3,m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Moving qn−1,m−1 to the right-hand side, we obtain eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (15) as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' 13 Is it true that there will be more signed relative derangements if we turn more unsigned elements into signed elements?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Put it differently, is it easier to form a relative derangement if more elements have signs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The answer is apparently negative due to the symmetry of qn,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' But, how about the cases for m ≤ n/2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' This is related to the unimodality of sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The sequence x0, x1, x2, · · · , xn is said to be unimodal if there exists an index 0 ≤ m ≤ n, called the mode of the sequence, such that x0 ≤ · · · ≤ xm−1 ≤ xm ≥ xm+1 ≥ · · ≥ xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Theorem 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For any fixed n ≥ 1, the sequence qn,0, qn,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' , qn,n is unimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Thanks to the symmetry of qn,m, it suffices to prove P(n, m) = qn,m − qn,m−1 ≥ 0 for m ≤ n/2, where we still make the convention qn,m = 0 if m < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' We shall prove this mainly by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' First, from the polynomials of QB n (t) listed in the last section, we observe that for n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' , 9 and m ≤ n/2, P(n, m) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Secondly, we claim for any n ≥ 2, P(n, 1) ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' for any n ≥ 4, P(n, 2) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In order for proving P(n, 1) ≥ 0 in the case of n ≥ 2, we construct an injection from Qn to QB n,1 (where QB n,i denotes the subset containing signed relative derangements with i bar-elements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For each sequence in Qn, replacing n with n, we obtain a unique sequence in QB n,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Obviously, this is an injection and then P(n, 1) ≥ 0 follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Analogously, we construct an injection from QB n,1 to QB n,2 for proving P(n, 2) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' We will classify the sequences in QB n,1 by the largest bar-element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' case 1: If the largest bar-element in QB n,1 is less than n − 1, then we map it to a relative derangement obtained by substituting n for n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In this case, the obtained relative derangements in QB n,2 have two bar-elements: n and i for some 1 ≤ i < n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' case 2: If the largest bar-element in QB n,1 is exactly n − 1, and n − 1 is not followed by n, then we substitute n for n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In the case that n − 1 is followed by n, we replace 1 with 1 to obtain a sequence in QB n,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In this case, the obtained relative derangements in QB n,2 have two bar-elements: either n − 1 and n, or n − 1 and 1 with an additional feature that n − 1 is followed by n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' case 3: Suppose the largest bar-element in QB n,1 is n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' If n−1 is not followed by n, then we remove the bar of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Meanwhile, we replace n−1 with n − 1 and 1 with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' If n follows n − 1, then we simply replace 1 with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In this case, the obtained relative derangements in QB n,2 have two bar-elements: either n − 1 and 1 with an additional feature that n − 1 is not followed by n, or n and 1 with the feature that n follows n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In the above mapping procedure, relative derangements in QB n,1 lying in the same case are clearly mapped to distinct relative derangements in QB n,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Moreover, inspecting the patterns of the contained two bar-elements and the additional features, relative de- rangements from different cases are mapped to distinct relative derangements in QB n,2 (for n ≥ 4) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Therefore, the above map is indeed an injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Hence, P(n, 2) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' 14 Now suppose for 1 ≤ n ≤ N and any 0 ≤ m ≤ n/2, P(n, m) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Next, we shall show that P(N + 1, m) ≥ 0 for any 3 ≤ m ≤ (N + 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Applying Corollary 10, we first have P(N + 1, m) = qN+1,m − qN+1,m−1 =N(qN,m − qN,m−2) + (N − m − 1)(qN−1,m − qN−1,m−1) + 3(N − 1)(qN−1,m−1 − qN−1,m−2) + (m − 3)(qN−1,m−2 − qN−1,m−3) + 2(N − m − 1)(qN−2,m−1 − qN−2,m−2) + 2(m − 3)(qN−2,m−2 − qN−2,m−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (22) We proceed to distinguish two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (i) If 3 ≤ m ≤ (N − 1)/2, we compare the two subscripts of each term qx,y on the RHS of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (22) and find that y ≤ x/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For instance, since the maximum value of m here is (N − 1)/2, as to qN−1,m−2, we have m − 2 = (N − 5)/2 which satisfies (N − 1)/2 ≥ m − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Consequently, qN−1,m−2 −qN−1,m−3 ≥ 0 by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Other summands are nonnegative by the same token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Therefore, P(N + 1, m) ≥ 0 follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (ii) If N/2 ≤ m ≤ (N + 1)/2, m equals either N/2 or (N + 1)/2 since m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' We check the two subscripts of qx,y and find that y > x/2 in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Therefore, in the following reasoning, we will make some transformation by the symmetry of qn,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' When m = N/2, we replace qN−1,m with qN−1,N−m−1 and regroup the terms on the RHS of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (22), and obtain P(N + 1, m) =N(qN, N 2 − qN, N−4 2 ) + 3(N − 1)(qN−1, N−2 2 − qN−1, N−4 2 ) + N − 6 2 (qN−1, N−4 2 − qN−1, N−6 2 ) + (N − 2)(qN−2, N−2 2 − qN−2, N−4 2 ) + (N − 6)(qN−2, N−4 2 − qN−2, N−6 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (23) Similarly, when m = (N + 1)/2, we replace qN,m with qN,N−m, qN−1,m with qN−1,N−m−1 and qN−2,m−1 with qN−2,N−m−1 in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (22) and regroup the terms to have P(N + 1, m) =N(qN, N−1 2 − qN, N−3 2 ) + 5N − 3 2 (qN−1, N−1 2 − qN−1, N−3 2 ) + N − 5 2 (qN−1, N−3 2 − qN−1, N−5 2 ) + (N − 5)(qN−2, N−3 2 − qN−2, N−5 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (24) Inspecting term by term on the RHS of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (23) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (24), they are all nonnegative by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Therefore, P(N + 1, m) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' This completes the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Acknowledgements The authors would like to thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Yi Wang for pointing out that the roots of QB n (t)’s are not necessarily all real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' A Proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='13 In the following, we write ∂F ∂x (x, t) as Fx(x, t) and ∂F ∂t (x, t) as Ft(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Then according to 15 the definition of F(x, t), we first have Fx(x, t) = � n≥1 nQB n (t)xn−1, Ft(x, t) = � n≥1 QB n ′(t)xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For the terms on right-hand side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (10),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' multiplying by xn and summing over n ≥ 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' we respectively obtain � n≥3 (n − 1)tQB n−1(t)xn = tx2 � n≥3 (n − 1)QB n−1(t)xn−2 = tx2(Fx(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) − QB 1 (t)) � n≥3 (n − 1)QB n−1(t)xn = x2 � n≥3 (n − 1)QB n−1(t)xn−2 = x2(Fx(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) − QB 1 (t)) � n≥3 (t3 − t)QB n−2 ′(t)xn = x2(t3 − t) � n≥3 QB n−2 ′(t)xn−2 = x2(t3 − t)Ft(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) � n≥3 (3n − 5)tQB n−2(t)xn = t � � n≥3 3nQB n−2(t)xn − 5 � n≥3 QB n−2(t)xn� = t � � n≥3 3(n − 2 + 2)QB n−2(t)xn − 5 � n≥3 QB n−2(t)xn� = t � 3 � n≥3 (n − 2)QB n−2(t)xn + 6 � n≥3 QB n−2(t)xn − 5 � n≥3 QB n−2xn� = t � 3x3 � n≥3 (n − 2)QB n−2(t)xn−3 + x2 � n≥3 QB n−2(t)xn−2� = t � 3x3Fx(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) + x2F(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) � � n≥3 (n − 2)QB n−2(t)xn = x3 � n≥3 (n − 2)QB n−2(t)xn−3 = x3Fx(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) � n≥3 (2t3 − 2t2)QB n−3 ′(t)xn = (2t3 − 2t2)x3 � n≥3 QB n−3 ′(t)xn−3 = (2t3 − 2t2)x3Ft(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) 16 � n≥3 (2n − 6)tQB n−3(t)xn = t � � n≥3 2nQB n−3(t)xn − 6 � n≥3 QB n−3(t)xn� = t � 2 � n≥3 (n − 3 + 3)QB n−3(t)xn − 6 � n≥3 QB n−3(t)xn� = t � 2 � n≥3 (n − 3)QB n−3(t)xn� = t � 2x4 � n≥3 (n − 3)QB n−3(t)xn−4� = 2tx4Fx(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) According to the computation above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' for n ≥ 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' we have � n≥3 QB n (t)xn =tx2(Fx(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) − QB 1 (t)) + x2(Fx(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) − QB 1 (t)) + x2(t3 − t)Ft(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) + t � 3x3Fx(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) + x2F(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) � + x3Fx(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) + (2t3 − 2t2)x3Ft(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) + 2tx4Fx(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) = � (t + 1)x2 + (3t + 1)x3 + 2tx4� Fx(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) + � (t3 − t)x2 + (2t3 − 2t2)x3� Ft(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) + tx2F(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) − (t + 1)2x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Then, F(x, t) is given as follows: F(x, t) =QB 1 (t)x + QB 2 (t)x2 + � n≥3 QB n (t)xn =x + tx + t2x2 + 4tx2 + x2 + � (t + 1)x2 + (3t + 1)x3 + 2tx4� Fx(x, t) + � (t3 − t)x2 + (2t3 − 2t2)x3� Ft(x, t) + tx2F(x, t) − (t + 1)2x2 = � (t + 1)x2 + (3t + 1)x3 + 2tx4� Fx(x, t) + � (t3 − t)x2 + (2t3 − 2t2)x3� Ft(x, t) + tx2F(x, t) + (t + 1)x + 2tx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' After sorting out the above equations, we eventually obtain Ft(x, t) + t + 1 + (3t + 1)x + 2tx2 t(t2 − 1) + 2t2(t − 1)x Fx(x, t) + tx2 − 1 t(t2 − 1)x2 + 2t2(t − 1)x3F(x, t) = −1 − t − 2tx t(t2 − 1)x + 2t2(t − 1)x2, completing the proof of Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' References [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='-O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Chow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' On derangement polynomials of type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' S´em.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Lothar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} 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+page_content=' [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Wachs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' On q-derangement numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=', 106(1):273–278, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Zhao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Excedance numbers for the permutations of type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Com- bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=', 20(2):P28, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' 18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} diff --git a/HNE0T4oBgHgl3EQfhgGY/vector_store/index.faiss b/HNE0T4oBgHgl3EQfhgGY/vector_store/index.faiss new file mode 100644 index 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+Abstract. For positive integers a and b, we let Un := Un(a, −b) be the Lucas +sequence {Un}n≥0 of the first kind defined by +U0 = 0, +U1 = 1 +and +Un = aUn−1 + bUn−2 +for n ≥ 2, +and let π(m) := π(a,b)(m) be the period length of {Un}n≥0 modulo the integer +m ≥ 2, where gcd(b, m) = 1. We define an (a, b)-Wall-Sun-Sun prime to be +a prime p such that π(p2) = π(p). When (a, b) = (1, 1), such a prime p is +referred to simply as a Wall-Sun-Sun prime. +We say that a monic polynomial f(x) ∈ Z[x] of degree N is monogenic if +f(x) is irreducible over Q and +{1, θ, θ2, . . . , θN−1} +is a basis for the ring of integers of Q(θ), where f(θ) = 0. +Let f(x) = x2 − ax − b, and let s be a positive integer. Then, with certain +restrictions on a, b and s, we prove that f(xsn) = x2sn −axsn −b is monogenic +for all integers n ≥ 1 if and only if no prime divisor of s is an (a, b)-Wall-Sun- +Sun prime. This result improves and extends previous work of the author in +the special case b = 1. +1. Introduction +Throughout this article, we let (∗) denote the set of conditions: +(∗) + + + + + + + +a and b are positive integers +a ̸≡ 0 +(mod 4) +b is squarefree +D is squarefree, +where +D := +� +a2 + 4b +if a ≡ 1 +(mod 2) +(a/2)2 + b +if a ≡ 0 +(mod 2). +We also let Un := Un(a, −b) denote the nth term of the Lucas sequence {Un}n≥0 +of the first kind defined by +(1.1) +U0 = 0, +U1 = 1 +and +Un = aUn−1 + bUn−2 +for n ≥ 2. +The sequence {Un}n≥0 is well known to be periodic modulo any integer m ≥ 2, +where gcd(b, m) = 1, and we let π(m) := π(a,b)(m) denote the length of the period +of {Un}n≥0 modulo m. +Date: January 16, 2023. +2020 Mathematics Subject Classification. Primary 11R04, 11B39, Secondary 11R09, 12F05. +Key words and phrases. Wall-Sun-Sun prime, monogenic, power-compositional. +1 + +2 +LENNY JONES +Definition 1.1. An (a, b)-Wall-Sun-Sun prime is a prime p with gcd(b, p) = 1, +such that +(1.2) +π(p2) = π(p). +We provide some examples of (a, b)-Wall-Sun-Sun primes in Table 1. +(a, b) +{[p, π(p2)]} +(2, 1) +{[13, 28], [31, 30]} +(3, 26) +{[71, 126]} +(10, 41) +{[29, 120]} +(11, 43) +{[2, 3], [5, 24]} +(15, 14) +{[29, 28]} +(23, 11) +{[2, 3], [3, 3], [71, 35]} +(25, 7) +{[5, 8]} +(27, 22) +{[13, 84]} +Table 1. (a, b)-Wall-Sun-Sun primes p and the corresponding pe- +riod length π(p2) = π(p) +When (a, b) = (1, 1), the sequence {Un}n≥0 is the well-known Fibonacci se- +quence, and the (a, b)-Wall-Sun-Sun primes in this case are known simply as Wall- +Sun-Sun primes [4, 19]. However, at the time this article was written, no Wall- +Sun-Sun primes were known to exist. The existence of Wall-Sun-Sun primes was +first investigated by D. D. Wall [16] in 1960, and subsequently studied by the Sun +brothers [14], who showed a connection with Fermat’s Last Theorem. +When b = 1, primes satisfying (1.2) are also known simply as a-Wall-Sun-Sun +primes [18,19]. We point out that the definition of an a-Wall-Sun-Sun prime given +in [18,19] is a prime p such that +(1.3) +Uπ(p) ≡ 0 +(mod p2). +In the more general situation of (a, b)-Wall-Sun-Sun primes, it is easily seen that +condition (1.2) implies condition (1.3). Although it can be shown that the converse +is true when b = 1 [5], the converse is false in general, as can be seen by the +counterexample (a, b) = (5, 2) with p = 7. In this particular example, we have +π(7) = 48 and U48 ≡ 0 (mod 49), but π(49) = 7π(7) = 336. +Since Wall was +originally concerned with whether there exist any primes p such that (1.2) holds in +the case of (a, b) = (1, 1), we have chosen to use condition (1.2), instead of condition +(1.3), for our definition of the more general (a, b)-Wall-Sun-Sun prime. +Let ∆(f) and ∆(K) denote, respectively, the discriminants over Q of f(x) ∈ Z[x] +and a number field K. We define f(x) ∈ Z[x] to be monogenic if f(x) is monic, +irreducible over Q and {1, θ, θ2, . . . , θdeg(f)−1} is a basis for the ring of integers ZK +of K = Q(θ), where f(θ) = 0. If f(x) is irreducible over Q with f(θ) = 0, then [3] +(1.4) +∆(f) = [ZK : Z[θ]]2 ∆(K). +Observe then, from (1.4), that f(x) is monogenic if and only if ∆(f) = ∆(K). Thus, +if ∆(f) is squarefree, then f(x) is monogenic from (1.4). However, the converse + +GENERALIZED WALL-SUN-SUN PRIMES AND MONOGENIC TRINOMIALS +3 +does not hold in general, and when ∆(f) is not squarefree, it can be quite difficult +to determine whether f(x) is monogenic. +In this article, we establish a connection between (a, b)-Wall-Sun-Sun primes +and the monogenicity of certain power-compositional trinomials. More precisely, +we prove +Theorem 1.2. Let f(x) = x2 − ax − b ∈ Z[x], where a and b satisfy (∗), and let +s ≥ 1 be an integer such that +� +D +p +� += −1 for each prime divisor p ≥ 3 of s, where +� +D +p +� +is the Legendre symbol. Then f(xsn) is monogenic for all integers n ≥ 1 if +and only if no prime divisor of s is an (a, b)-Wall-Sun-Sun prime. +Theorem 1.2 improves and extends previous work of the author on the special +case of b = 1 [9], which was, in part, originally motivated by recent results of +Bouazzaoui [1,2]. Bouazzaoui showed, under certain conditions on the prime p ≥ 3, +that +Q( +√ +d) +is p-rational if and only if +π(a,b)(p2) ̸= π(a,b)(p), +where d > 0 is a fundamental discriminant [17], a = ε + ε and b = −NQ( +√ +d)/Q(ε), +with ε equal to the fundamental unit of Q( +√ +d). We recall, for a prime p ≥ 3, +that a number field K is said to be p-rational if the Galois group of the maximal +pro-p-extension of K which is unramified outside p is a free pro-p-group of rank +r2 + 1, where r2 is the number of pairs of complex embeddings of K. +2. Preliminaries +The formula for the discriminant of an arbitrary monic trinomial, due to Swan +[15], is given in the following theorem. +Theorem 2.1. Let f(x) = xN + AxM + B ∈ Z[x], where 0 < M < N. +Let +r = gcd(N, M), N1 = N/r and M1 = M/r. Then +∆(f) = (−1)N(N−1)/2BM−1Dr, +where +(2.1) +D := N N1BN1−M1 − (−1)N1M M1(N − M)N1−M1AN1. +The next two theorems are due to Capelli [13]. +Theorem 2.2. Let f(x) and h(x) be polynomials in Q[x] with f(x) irreducible. +Suppose that f(α) = 0. Then f(h(x)) is reducible over Q if and only if h(x) − α is +reducible over Q(α). +Theorem 2.3. Let c ∈ Z with c ≥ 2, and let α ∈ C be algebraic. Then xc − α +is reducible over Q(α) if and only if either there is a prime p dividing c such that +α = βp for some β ∈ Q(α) or 4 | c and α = −4β4 for some β ∈ Q(α). +The next proposition follows from Proposition 1 in [20]. +Proposition 2.4. Let b = 1. Then α = (a + +√ +a2 + 4)/2 is the fundamental unit +of Q( +√ +D) with N(α) = −1, where N := NQ(α)/Q denotes the algebraic norm. + +4 +LENNY JONES +In the sequel, for an integer m ≥ 2, we let ordm(∗) denote the order of ∗ modulo +m, and we define (a, b)m := (a (mod m), b (mod m)). For brevity of notation, we +also define +λ := ordp(b2) +and +δp := +�D +p +� +, +where +� +D +p +� +is the Legendre symbol. The following theorem is a compilation of +results from various sources. +Theorem 2.5. Let {Un(a, −b)}n≥0 be the Lucas sequence as defined in (1.1). Let +p be a prime with b ̸≡ 0 (mod p). +(1) π(p) = 2 if and only if (a, b)p = (0, 1). +(2) If p = 2, then +π(2) = +� 2 +if (a, b)4 ∈ {(2, 1), (2, 3)} +3 +if (a, b)4 ∈ {(1, 1), (1, 3), (3, 1), (3, 3)} +and +π(4) = + + + +3 +if (a, b)4 = (3, 3) +4 +if (a, b)4 ∈ {(2, 1), (2, 3)} +6 +if (a, b)4 ∈ {(1, 1), (1, 3), (3, 1)}. +(3) If p ≥ 3, then π(p2) ∈ {π(p), pπ(p)}. +(4) If δp = −1, then 2(p + 1)λ ≡ 0 (mod π(p)). +Proof. Items (1) and (2) follow easily by direct calculation, item (3) can be found +in [11], while item (4) follows from a theorem in [6]. +□ +The following theorem, known as Dedekind’s Index Criterion, or simply Dedekind’s +Criterion if the context is clear, is a standard tool used in determining the mono- +genicity of a polynomial. +Theorem 2.6 (Dedekind [3]). Let K = Q(θ) be a number field, T (x) ∈ Z[x] the +monic minimal polynomial of θ, and ZK the ring of integers of K. Let p be a prime +number and let ∗ denote reduction of ∗ modulo p (in Z, Z[x] or Z[θ]). Let +T(x) = +� +i +τi(x)ei +be the factorization of T (x) modulo p in Fp[x], and set +g(x) = +� +i +τi(x), +where the τi(x) ∈ Z[x] are arbitrary monic lifts of the τi(x). Let h(x) ∈ Z[x] be a +monic lift of T(x)/g(x) and set +F(x) = g(x)h(x) − T (x) +p +∈ Z[x]. +Then +[ZK : Z[θ]] ̸≡ 0 +(mod p) ⇐⇒ gcd +� +F, g, h +� += 1 in Fp[x]. +The next result is essentially an algorithmic adaptation of Theorem 2.6 specifi- +cally for trinomials. + +GENERALIZED WALL-SUN-SUN PRIMES AND MONOGENIC TRINOMIALS +5 +Theorem 2.7. +[8] Let N ≥ 2 be an integer. +Let K = Q(θ) be an algebraic +number field with θ ∈ ZK, the ring of integers of K, having minimal polynomial +f(x) = xN + AxM + B over Q, with gcd(M, N) = r, N1 = N/r and M1 = M/r. +Let D be as defined in (2.1). A prime factor p of ∆(f) does not divide [ZK : Z[θ]] +if and only if p satisfies one of the following items: +(1) when p | A and p | B, then p2 ∤ B; +(2) when p | A and p ∤ B, then +either +p | A2 and p ∤ B1 +or +p ∤ A2 +� +(−B)M1AN1 +2 +− (−B1)N1� +, +where A2 = A/p and B1 = B+(−B)pe +p +with pe || N; +(3) when p ∤ A and p | B, then +either +p | A1 and p ∤ B2 +or +p ∤ A1BM−1 +2 +� +(−A)M1AN1−M1 +1 +− (−B2)N1−M1� +, +where A1 = A+(−A)pj +p +with pj || (N − M), and B2 = B/p; +(4) when p ∤ AB and p | M with N = upm, M = vpm, p ∤ gcd (u, v), then the +polynomials +G(x) : = xN/pm + AxM/pm + B +and +H(x) : = AxM + B + +� +−AxM/pm − B +�pm +p +are coprime modulo p; +(5) when p ∤ ABM, then p2 ∤ D/rN1. +Remark 2.8. We will find both Theorem 2.6 and Theorem 2.7 useful in our inves- +tigations. +The next theorem follows from Corollary (2.10) in [10]. +Theorem 2.9. Let K and L be number fields with K ⊂ L. Then +∆(K)[L:K] �� ∆(L). +3. The Proof of Theorem 1.2 +Throughout this section we let +f(x) = x2 − ax − b ∈ Z[x] +and +α = a + +√ +a2 + 4b +2 +, +where a and b satisfy (∗). We first prove some lemmas. +Lemma 3.1. Let s be a positive integer. Then f(xsn) is irreducible over Q for all +integers n ≥ 1. +Proof. Since D > 1 is squarefree, it follows that f(x) is irreducible over Q, and the +trivial case of s = 1 is true. So, suppose then that s ≥ 2. Note that f(α) = 0. Let +h(x) = xsn and assume, by way of contradiction, that f(h(x)) is reducible. Then, +by Theorems 2.2 and 2.3, we have, for some β ∈ Q(α), that either α = βp for some +prime p dividing s, or α = −4β4 if sn ≡ 0 (mod 4). + +6 +LENNY JONES +If b ≥ 2, then, in either case, we arrive at a contradiction by taking norms, since +N(α) = −b is squarefree but neither N(βp) = N(β)p nor N(−4β4) = 16N(β)4 is +squarefree. Suppose then that b = 1. If α = −4β4, then +−1 = N(α) = N(−4β4) ≡ 0 +(mod 16), +which is impossible. Hence, α = βp for some prime divisor p of s. Then, we see by +taking norms that +N(β)p = N(α) = −1, +which implies that p ≥ 3 and N(β) = −1, since N(β) ∈ Z. Thus, β is a unit, and +therefore β = ±αj for some j ∈ Z, since α is the fundamental unit of Q( +√ +D) by +Proposition 2.4. Consequently, +α = βp = (±1)pαjp, +which implies that (±1)pαjp−1 = 1, contradicting the fact that α has infinite order +in the group of units of the ring of algebraic integers in the real quadratic field +Q( +√ +D). +□ +Remark 3.2. Although here we are assuming that conditions (∗) hold, so that +a ̸≡ 0 (mod 4), the argument given in the proof of Lemma 3.1 for the case of b = 1 +is still valid when a ≡ 0 (mod 4) with the single exception of a = 4 [20] since, +in that case, α = 2 + +√ +5 is not the fundamental unit of Q( +√ +5). However, since +ε = (1 + +√ +5)/2 is the fundamental unit of Q( +√ +5), and α = ε3, Capelli’s theorems +can be used to determine exactly when f(xsn) = x2sn − 4xsn − 1 is reducible and +how f(xsn) factors. +Lemma 3.3. The polynomial f(x) is monogenic. +Proof. By Lemma 3.1, f(x) is irreducible over Q. +Let p be a prime divisor of +∆(f) = a2 + 4b. To examine the monogenicity of f(x), we use Theorem 2.7 with +θ = α. +Suppose first that p | a. Then p | 4b. If p | b, then item (1) of Theorem 2.7 +applies, and we see that [ZK : Z[α]] ̸≡ 0 (mod p) since b is squarefree. Now suppose +that p ∤ b, so that item (2) of Theorem 2.7 applies. Note that p = 2 since p | 4b. +Hence, 2 | a and D = (a/2)2 + b ≡ 1 + b (mod 4) since a ̸≡ 0 (mod 4). Thus, since +D is squarefree and 2 ∤ b, it follows that b ≡ 1 (mod 4) and therefore, +B1 = (−b + b2)/2 = b(b − 1)/2 ≡ 0 +(mod 2). +Also, A2 = −a/2 ≡ 1 (mod 2), since a ̸≡ 0 (mod 4). Thus, +bA2 +2 − (−B1)2 ≡ 1 +(mod 2), +from which we conclude that [[ZK : Z[α]] ̸≡ 0 (mod 2). +Next, suppose that p ∤ a. Then p ∤ 4b since p | (a2 + 4b), and so item (5) of +Theorem 2.7 applies. Since D is squarefree and p ̸= 2, we deduce that p2 ∤ (a2 + 4b) +and consequently, [ZK : Z[α]] ̸≡ 0 (mod p), which completes the proof. +□ +Lemma 3.4. Let p be a prime, with b ̸≡ 0 (mod p). +(1) The prime p = 2 is an (a, b)-Wall-Sun-Sun prime if and only if (a, b)4 = +(3, 3). +(2) If p ≥ 3 and a ≡ 0 (mod p), then p is an (a, b)-Wall-Sun-Sun prime if and +only if ordp2(b) = ordp(b) and a ≡ 0 (mod p2). + +GENERALIZED WALL-SUN-SUN PRIMES AND MONOGENIC TRINOMIALS +7 +Proof. We see that (1) follows from item (2) of Theorem 2.5. +To establish (2), we let {Un}m denote the sequence (1.1) reduced modulo the +integer m ∈ {p, p2}. Since a ≡ 0 (mod p), we can write a = pk, for some positive +integer k. Then, +{Un}p = [0, 1, 0, b, 0, b2, 0, b3, 0, b4, 0, b5, . . .] +and +{Un}p2 = [0, 1, pk, b, 2pkb, b2, 3pkb2, b3, . . . , ordp(b)pkbordp(b)−1, bordp(b), . . .]. +Thus, it follows that p is an (a, b)-Wall-Sun-Sun prime if and only if +π(p2) = π(p) = 2 ordp(b) ⇐⇒ ordp2(b) = ordp(b) +and ordp(b)pkbordp(b)−1 ≡ 0 +(mod p2) +⇐⇒ ordp2(b) = ordp(b) +and a ≡ 0 +(mod p2), +since b ̸≡ 0 (mod p) and ordp(b) ≤ p − 1 ̸≡ 0 (mod p). +□ +Lemma 3.5. Let α = (a− +√ +a2 + 4b)/2, and let p ≥ 3 be a prime such that δp = −1. +Then +(1) ordm(α) = ordm(α) = π(m) for m ∈ {p, p2} and +(2) αp+1 ≡ −b (mod p). +Proof. Note that b ̸≡ 0 (mod p) since δp = −1. It follows from [12] that the order, +modulo an integer m ≥ 3 with gcd(m, b) = 1, of the companion matrix C for the +characteristic polynomial of {Un}n≥0 is π(m). The characteristic polynomial of +{Un}n≥0 is f(x), so that +C = +� +0 +b +1 +a +� +. +Since the eigenvalues of C are α and α, we conclude that +ordm +�� α +0 +0 +α +�� += ordm(C) = π(m), +for m ∈ {p, p2}. +Let z ≥ 1 be an integer, and suppose that αz = a + b +√ +D ∈ Q( +√ +D). +Then +N(αz) = a2 − Db2. But N(αz) = N(α)z = (−b)z, so that a2 − Db2 = (−b)z. Thus, +αz = (−b/α)z = (−b)z/(a + b +√ +D) = (−b)z(a − b +√ +D)/(a2 − Db2) = a − b +√ +D. +Hence, since δp = −1, it follows that +αz ≡ 1 +(mod m) +if and only if +αz ≡ 1 +(mod m) +for m ∈ {p, p2}, which establishes item (1). +By Euler’s criterion, +�� +a2 + 4b +�p+1 += (a2 + 4b)(p−1)/2(a2 + 4b) ≡ δp(a2 + 4b) ≡ −(a2 + 4b) +(mod p), +which implies +�� +a2 + 4b +�p +≡ − +� +a2 + 4b +(mod p). + +8 +LENNY JONES +Hence, +αp+1 = +� +a + +√ +a2 + 4b +2 +� � +a + +√ +a2 + 4b +2 +�p += +� +a + +√ +a2 + 4b +2 +� +p +� +j=0 +�p +j +� �a +2 +�j +�√ +a2 + 4b +2 +�p−j +≡ +� +a + +√ +a2 + 4b +2 +� ��a +2 +�p ++ +�√ +a2 + 4b +2 +�p� +(mod p) +≡ +� +a + +√ +a2 + 4b +2 +� � +a − +√ +a2 + 4b +2 +� +(mod p) +≡ −b +(mod p), +which completes the proof of the lemma. +□ +Lemma 3.6. Let s ≥ 3 and n ≥ 1 be integers. Let p ≥ 3 be a prime such that +p ∤ a, p ∤ b and pm || s with m ≥ 1. Suppose that Fn(x) := f(xsn) and K = Q(θ), +with ZK the ring of integers of K, where Fn(θ) = 0. If δp = −1, then +[ZK : Z[θ]] ≡ 0 +(mod p) +if and only if +α2pmn − aαpmn − b ≡ 0 +(mod p2). +Proof. We apply Theorem 2.6 to T (x) := Fn(x) using the prime p. Let +(3.1) +τ(x) = x2sn/pmn − axsn/pmn − b = f +� +xsn/pmn� +, +and τ(x) = � +i τi(x)ei, where the τi(x) are irreducible in Fp[x]. +Then T(x) = +� +i τi(x)pmnei. Thus, we can let +g(x) = +� +i +τi(x) +and +h(x) = +� +i +τi(x)pmnei−1, +where the τi(x) are monic lifts of the τi(x). Note also that +g(x)h(x) = +� +i +τi(x)pmnei = τ(x) + pw(x), +for some w(x) ∈ Z[x]. Then, in Theorem 2.6, we have that +pF(x) = g(x)h(x) − T (x) += (τ(x) + pw(x))pmn − T (x) += +pmn−1 +� +j=1 +�pmn +j +� +τ(x)j(pw(x))pmn−j + (pw(x))pmn ++ τ(x)pmn − T (x) +≡ τ(x)pmn − T (x) +(mod p2). +(3.2) +Suppose that τ(γ) = 0. Then, we see from (3.1) that +τ(γ)pmn = f(β)pmn = 0, +where β = γsn/pmn, so that γsn = βpmn. Note that f(x) is irreducible in Fp[x] since +δp = −1. Thus, we assume, without loss of generality that γsn = αpmn. Hence, + +GENERALIZED WALL-SUN-SUN PRIMES AND MONOGENIC TRINOMIALS +9 +from (3.2), it follows that +pF(γ) ≡ −T (γ) +(mod p2) +≡ − +� +γ2sn − aγsn − b +� +(mod p2) +≡ − +� +α2pmn − aαpmn − b +� +(mod p2), +which completes the proof of the lemma. +□ +Lemma 3.7. Let p ≥ 3 be a prime such that δp = −1. Then the following condi- +tions are equivalent: +(1) p is an (a, b)-Wall-Sun-Sun prime, +(2) f(αpm) ≡ 0 (mod p2) for all integers m ≥ 1, +(3) f(αpm) ≡ 0 (mod p2) for some integer m ≥ 1. +Proof. First, observe that (2) clearly implies (3). +We show next that (1) implies (2). Because p is an (a, b)-Wall-Sun-Sun prime, +we define +π := π(p2) = π(p). +Since δp = −1, we see from item (4) of Theorem 2.5 that +2(p + 1)λ ≡ 0 +(mod π). +The squares modulo p form a subgroup, of order (p − 1)/2, of the multiplicative +group (Z/pZ)∗. Thus, (p − 1)/2 ≡ 0 (mod λ), so that +2(p + 1)(p − 1)/2 = p2 − 1 ≡ 0 +(mod π). +Consequently, αp2−1 ≡ 1 (mod p2) by item (1) of Lemma 3.5, from which it follows +that +αp2k ≡ α +(mod p2) +and +αp2k+1 ≡ αp +(mod p2), +for every integer k ≥ 1. Hence, +f(αpm) ≡ +� +α2 − aα − b +(mod p2) +if m ≡ 0 +(mod 2), +α2p − aαp − b +(mod p2) +if m ≡ 1 +(mod 2). +Thus, f(αpm) ≡ 0 (mod p2) when m ≡ 0 (mod 2), since α2 − aα − b = 0. Suppose +then that m ≡ 1 (mod 2). Let α = (a− +√ +a2 + 4b)/2. Since p is an (a, b)-Wall-Sun- +Sun prime, and the fact that α = −b/α, we deduce from the Binet-representation +formula for Uπ that +(3.3) +Uπ = απ − απ +α − α += α2π − (−b)π +απ (α − α) ≡ 0 +(mod p2). +Hence, since απ ≡ 1 (mod p2) from item (1) of Lemma 3.5, we conclude from (3.3) +that (−b)π ≡ 1 (mod p2), which implies that +(3.4) +b2(p+1)λ ≡ 1 +(mod p2), +by item (4) of Theorem 2.5. Thus, from (3.4), it follows that +(3.5) +b2(p+1)λ − 1 ≡ (b2λ − 1)B ≡ 0 +(mod p2), +where +B = (b2λ)p + (b2λ)p−1 + · · · + b2λ + 1. + +10 +LENNY JONES +Since b2λ ≡ (b2)λ ≡ 1 (mod p), we see that B ≡ p + 1 ≡ 1 (mod p). Therefore, +(3.6) +b2λ − 1 ≡ 0 +(mod p2), +from (3.5). Also, since δp = −1 and απ ≡ 1 (mod p2), we have from item (4) of +Theorem 2.5 that +(3.7) +α2(p+1)λ − 1 ≡ 0 +(mod p2). +Combining (3.6) and (3.7) yields +(3.8) +α2(p+1)λ − b2λ ≡ +� +αp+1 − b +� � +αp+1 + b +� +C ≡ 0 +(mod p2), +where +C = +� +α2(p+1)�λ−1 ++ +� +α2(p+1)�λ−2 +b2 + · · · + α2(p+1)(b2)λ−2 + (b2)λ−1 +≡ λb2λ+2 +(mod p), +since α2(p+1) ≡ b2 (mod p) from item (2) of Lemma 3.5. Thus, from (3.6) and the +fact that (p − 1)/2 ≡ 0 (mod λ), we deduce that C ≡ λb2 ̸≡ 0 (mod p). Note that +αp+1 − b ̸≡ 0 (mod p) since αp+1 + b ≡ 0 (mod p) and b ̸≡ 0 (mod p). Therefore, +it follows from (3.8) that αp+1 ≡ −b (mod p2). Hence, αp ≡ −bα−1 (mod p2), and +consequently, +f(αpm) ≡ α2p − aαp − b +(mod p2) +≡ +� +−bα−1�2 − a +� +−bα−1� +− b +(mod p2) +≡ −bα−2(α2 − aα − b) +(mod p2) +≡ 0 +(mod p2) +since α2 − aα − b = 0, which completes the proof that (1) implies (2). +Finally, to establish that (3) implies (1), we first note that π(p2) ∈ {π(p), pπ(p)} +by item (3) of Theorem 2.5. Then, in either case, we have that αpπ(p) ≡ 1 (mod p2), +and we conclude from item (4) of Theorem 2.5 that α2p(p+1)λ ≡ 1 (mod p2). Since +(p − 1)/2 ≡ 0 (mod λ), we deduce +α2p(p+1)(p−1)/2 ≡ αp3−p ≡ 1 +(mod p2), +so that αp3 ≡ αp (mod p2). It then follows easily that +(3.9) +αp2k ≡ αp2 +(mod p2) +and +αp2k+1 ≡ αp +(mod p2), +for all integers k ≥ 1. Hence, from (3.9), we have that +(3.10) +f(αpm) ≡ +� +α2p2 − aαp2 − b +(mod p2) +if m ≡ 0 +(mod 2), +α2p − aαp − b +(mod p2) +if m ≡ 1 +(mod 2). +By Hensel, the only zeros of f(x) in (Z/p2Z)[ +√ +D] are α and α = −bα−1. +Suppose that f(αpm) ≡ 0 (mod p2) for some integer m ≡ 1 (mod 2). Then, we +see from (3.10) that +either +αp ≡ α +(mod p2) +or +αp ≡ α +(mod p2). +If αp ≡ α (mod p2), then, from item (2) of Lemma 3.5, we have that +a2 + 4b + a +√ +a2 + 4b +2 += α2 + b ≡ αp+1 + b ≡ 0 +(mod p), + +GENERALIZED WALL-SUN-SUN PRIMES AND MONOGENIC TRINOMIALS +11 +which implies that a2+4b ≡ 0 (mod p), contradicting the fact that δp = −1. Hence, +(3.11) +αp ≡ α ≡ −bα−1 +(mod p2) +or equivalently, +αp+1 ≡ −b +(mod p2). +Since α2p − aαp − b ≡ 0 (mod p2), then α2p − aαp − b ≡ 0 (mod p2) so that +either +αp ≡ α +(mod p2) +or +αp ≡ α +(mod p2). +If αp ≡ α (mod p2), then (−b)p ≡ α2p (mod p2) from (3.11). Hence, +� +a − +√ +a2 + 4b +2 +�2 ++ b ≡ a2 + 4b − a +√ +a2 + 4b +2 +≡ 0 +(mod p), +since αp ≡ α (mod p), which implies that a2 +4b ≡ 0 (mod p), again contradicting +the fact that δp = −1. Therefore, αp ≡ α (mod p2), so that +(−b)p ≡ αp+1 ≡ −b +(mod p2), +from (3.11). Consequently, +(3.12) +(−b)p−1 ≡ 1 +(mod p2). +Combining (3.11) and (3.12) yields +(3.13) +αp2−1 ≡ (αp+1)p−1 ≡ (−b)p−1 ≡ 1 +(mod p2). +Recall that π(p2) ∈ {π(p), pπ(p)} by item (3) of Theorem 2.5. If π(p2) = pπ(p), +then we have by item (1) of Lemma 3.5 and (3.13) that p2 − 1 ≡ 0 (mod p), which +is impossible. Thus, we must have π(p2) = π(p), which completes the proof that (3) +implies (1) when m ≡ 1 (mod 2). Since similar arguments establish this implication +when m ≡ 0 (mod 2), we omit the details. +□ +Proof of Theorem 1.2. For brevity of notation, define +Fn(x) := f(xsn) = x2sn − axsn − b +for n ≥ 0. +Note that F0(x) = f(x). We have that Fn(x) is irreducible for all n ≥ 0 by Lemma +3.1. In the trivial case s = 1, we also see that Fn(x) = f(x) for all n ≥ 0, and +so Fn(x) is monogenic for all n ≥ 0 by Lemma 3.3. So assume that s ≥ 2. By +Theorem 2.1, +(3.14) +∆(Fn) = (−b)sn−1s2nsn(a2 + 4b)sn. +(⇒) Suppose that s has a prime divisor p that is an (a, b)-Wall-Sun-Sun prime. +Note that p ∤ b since b ≡ 1 (mod 2) by item (1) of Lemma 3.4 if p = 2, and δp = −1 +if p ≥ 3. We claim that F1(x) = x2s − axs − b is not monogenic. Let F1(θ) = 0, +K = Q(θ) and ZK be the ring of integers of K. +Suppose first that p = 2, and that 2mv = s, where m ≥ 1 and 2 ∤ v. Then p ∤ a +since (a, b) = (3, 3)4 by item (1) of Lemma 3.4. Applying item (4) of Theorem 2.7 + +12 +LENNY JONES +to F1(x) we see that +G(x) = x2s/2m − axs/2m − b += x2v − axv − b +≡ x2v + xv + 1 +(mod 2) +and +H(x) = −ax2mv − b + (axv + b)2m +2 += +�a2m − a +2 +� +x2mv + +2m−1 +� +j=1 +�2m +j +� +2 +(axv)jb2m−j + b2m − b +2 +≡ +� +x2v + xv + 1 +�2m−1 +(mod 2), +since a2m − a ≡ b2m − b ≡ 2 (mod 4) and [7] +�2m +j +� +≡ +� +0 +(mod 4) +if j ̸= 2m−1 +2 +(mod 4) +if j = 2m. +Thus, G(x) and H(x) are not coprime in F2[x], and therefore, F1(x) is not mono- +genic. +Suppose next that p ≥ 3, and let pm || s with m ≥ 1. If p | a, then p2 | a and +ordp2(b) = ordp(b) by item (2) of Lemma 3.4. Since p ∤ b, we can apply item (2) of +Theorem 2.7 to F1(x) with e = m. Then A2 = −a/p ≡ 0 (mod p) and +B1 = bpm − b +p += b +� +bpm−1 − 1 +� +p +≡ 0 +(mod p), +since ordp2(b) = ordp(b). Thus, [ZK : Z[θ]] ≡ 0 (mod p) and F1(x) is not mono- +genic. +Assume next that p ∤ a. Since δp = −1, we have that +α2pm − aαpm − b ≡ 0 +(mod p2) +by Lemma 3.7. Thus, it follows from Lemma 3.6 that [ZK : Z[θ]] ≡ 0 (mod p), and +therefore, F1(x) is not monogenic, which completes the proof in this direction. +(⇐) Suppose now that no prime divisor of s is an (a, b)-Wall-Sun-Sun prime. By +Lemma 3.3, we see that F0(x) = f(x) is monogenic. Note that F0(α) = 0. For +n ≥ 0, define +αn := α1/sn +and +Kn := Q(αn). +Then α0 = α and, since F0(x) is monogenic, we have that ∆(F0) = ∆(K0). Addi- +tionally, by Lemma 3.1, +Fn(αn) = 0 +and +[Kn+1 : Kn] = s +for all n ≥ 0. We assume that Fn(x) is monogenic, so that ∆(Fn) = ∆(Kn), and we +proceed by induction on n to show that Fn+1(x) is monogenic. Let ZKn+1 denote +the ring of integers of Kn+1. Consequently, by Theorem 2.9, it follows that +∆(Fn)s divides ∆(Kn+1) = +∆(Fn+1) +[ZKn+1 : Z[αn+1]]2 , +which implies that +[ZKn+1 : Z[αn+1]]2 divides ∆(Fn+1) +∆(Fn)s . + +GENERALIZED WALL-SUN-SUN PRIMES AND MONOGENIC TRINOMIALS +13 +We see from (3.14) that +|∆(Fn)s| = (−b)sn+1−ss2nsn+1(a2 + 4b)sn+1 +and +|∆(Fn+1)| = (−b)sn+1−1s2(n+1)sn+1(a2 + 4b)sn+1. +Hence, +���� +∆(Fn+1) +∆(Fn)s +���� = (−b)s−1s2sn+1. +Thus, it is enough to show that gcd(bs, [ZKn+1 : Z[αn+1]]) = 1. +Observe that +gcd(b, s) = 1, since δp = −1 for every prime divisor p of s. +Suppose first that p is a prime divisor of b. If p | a, then it follows that +(3.15) +[ZKn+1 : Z[αn+1]] ̸≡ 0 +(mod p) +by item (1) of Theorem 2.7 since b is squarefree. So, assume that p ∤ a. In this case, +we apply item (3) of Theorem 2.7 to Fn+1(x). Observe that A1 = 0 since p ∤ s, and +B2 = −b/p ̸≡ 0 (mod p) since b is squarefree. Thus, the first condition of item (3) +holds, and therefore once again we have (3.15). +Suppose next that p is a prime divisor of s, with pm || s, and assume first that +p | a. Since p ∤ b, we apply item (2) of Theorem 2.7 to Fn+1(x). If p2 | a, then +ordp2(b) ̸= ordp(b) by item (2) of Lemma 3.4, since p is not an (a, b)-Wall-Sun-Sun +prime. Also, A2 = −a/p ≡ 0 (mod p), and we must show that +B1 = bpm(n+1) − b +p += b +� +bpm(n+1)−1 − 1 +p +� +is not divisible by p. Since ordp2(b) ̸= ordp(b), a straightforward argument shows +that ordp2(b) = p ordp(b). Consequently, +bpm(n+1)−1 ̸≡ 1 +(mod p2), +since pm(n+1) − 1 ̸≡ 0 (mod p). That is, B1 ̸≡ 0 (mod p) and we can conclude +(3.15) in this case. So, suppose then that p || a. Then A2 = −a/p ̸≡ 0 (mod p), +and hence we must show that p does not divide +B := (−B)M1AN1 +2 +− (−B1)N1 = b + + +�a +p +�2 +− b +� +1 − bpm(n+1)−1 +p +�2 + . +Suppose, by way of contradiction, that B ≡ 0 (mod p). If bpm(n+1)−1 ≡ 1 (mod p2), +then b(a/p)2 ≡ 0 (mod p), which is impossible since p || a and p ∤ b. +Thus, +bpm(n+1)−1 ̸≡ 1 (mod p2), from which it follows that b is a square modulo p, con- +tradicting the fact that δp = −1. Hence, B ̸≡ 0 (mod p), and so we have (3.15). +Finally, suppose that p ∤ a. Since p is not an (a, b)-Wall-Sun-Sun prime, we +deduce from Lemma 3.7 that +α2pm(n+1) − aαpm(n+1) − b ̸≡ 0 +(mod p2). +Thus, [ZKn+1 : Z[αn+1]] ̸≡ 0 (mod p) from Lemma 3.6, and therefore, Fn+1(x) is +monogenic, which completes the proof of the theorem. +□ + +14 +LENNY JONES +References +[1] Z. Bouazzaoui, Fibonacci numbers and real quadratic p-rational fields, Period. Math. Hungar. +81 (2020), no. 1, 123–133. +[2] Z. Bouazzaoui, On periods of Fibonacci sequences and real quadratic p-rational fields, Fi- +bonacci Quart. 58 (2020), no. 5, 103–110. +[3] H. Cohen, A Course in Computational Algebraic Number Theory, Springer-Verlag, 2000. +[4] R. Crandall, K. Dilcher and C. 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Mag. +86 (2013), no. 5, 372–380. +[12] D. W. Robinson, A note on linear recurrent sequences modulo m, Amer. Math. Monthly 73 +(1966), 619–621. +[13] A. Schinzel, Polynomials with Special Regard to Reducibility, Encyclopedia of Mathematics +and its Applications, 77, Cambridge University Press, Cambridge, 2000. +[14] Zhi Hong Sun and Zhi Wei Sun, Fibonacci numbers and Fermat’s last theorem, Acta Arith. +60 (1992), no. 4, 371–388. +[15] R. Swan, Factorization of polynomials over finite fields, Pacific J. Math. 12 (1962), 1099– +1106. +[16] D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly 67 (1960), 525–532. +[17] Fundamental Discriminant https://en.wikipedia.org/wiki/Fundamental_discriminant +[18] Wieferich Prime https://en.wikipedia.org/wiki/Wieferich_prime +[19] Wall-Sun-Sun Prime +https://en.wikipedia.org/wiki/Wall%E2%80%93Sun%E2%80%93Sun_prime +[20] H. Yokoi, On real quadratic fields containing units with norm −1, Nagoya Math. J. 33 (1968), +139–152. +Professor Emeritus, Department of Mathematics, Shippensburg University, Shippens- +burg, Pennsylvania 17257, USA +Email address, Lenny Jones: doctorlennyjones@gmail.com + diff --git a/JdE5T4oBgHgl3EQfXg8M/content/tmp_files/load_file.txt b/JdE5T4oBgHgl3EQfXg8M/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2c067104896eeb0854cfdbdf078a1c0d56784691 --- /dev/null +++ b/JdE5T4oBgHgl3EQfXg8M/content/tmp_files/load_file.txt @@ -0,0 +1,510 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf,len=509 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='05566v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='NT] 13 Jan 2023 GENERALIZED WALL-SUN-SUN PRIMES AND MONOGENIC POWER COMPOSITIONAL TRINOMIALS LENNY JONES Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' For positive integers a and b, we let Un := Un(a, −b) be the Lucas sequence {Un}n≥0 of the first kind defined by U0 = 0, U1 = 1 and Un = aUn−1 + bUn−2 for n ≥ 2, and let π(m) := π(a,b)(m) be the period length of {Un}n≥0 modulo the integer m ≥ 2, where gcd(b, m) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' We define an (a, b)-Wall-Sun-Sun prime to be a prime p such that π(p2) = π(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' When (a, b) = (1, 1), such a prime p is referred to simply as a Wall-Sun-Sun prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' We say that a monic polynomial f(x) ∈ Z[x] of degree N is monogenic if f(x) is irreducible over Q and {1, θ, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' , θN−1} is a basis for the ring of integers of Q(θ), where f(θ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Let f(x) = x2 − ax − b, and let s be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Then, with certain restrictions on a, b and s, we prove that f(xsn) = x2sn −axsn −b is monogenic for all integers n ≥ 1 if and only if no prime divisor of s is an (a, b)-Wall-Sun- Sun prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' This result improves and extends previous work of the author in the special case b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Introduction Throughout this article, we let (∗) denote the set of conditions: (∗) \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 a and b are positive integers a ̸≡ 0 (mod 4) b is squarefree D is squarefree, where D := � a2 + 4b if a ≡ 1 (mod 2) (a/2)2 + b if a ≡ 0 (mod 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' We also let Un := Un(a, −b) denote the nth term of the Lucas sequence {Un}n≥0 of the first kind defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='1) U0 = 0, U1 = 1 and Un = aUn−1 + bUn−2 for n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' The sequence {Un}n≥0 is well known to be periodic modulo any integer m ≥ 2, where gcd(b, m) = 1, and we let π(m) := π(a,b)(m) denote the length of the period of {Un}n≥0 modulo m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Date: January 16, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Primary 11R04, 11B39, Secondary 11R09, 12F05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Wall-Sun-Sun prime, monogenic, power-compositional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' 1 2 LENNY JONES Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' An (a, b)-Wall-Sun-Sun prime is a prime p with gcd(b, p) = 1, such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='2) π(p2) = π(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' We provide some examples of (a, b)-Wall-Sun-Sun primes in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' (a, b) {[p, π(p2)]} (2, 1) {[13, 28], [31, 30]} (3, 26) {[71, 126]} (10, 41) {[29, 120]} (11, 43) {[2, 3], [5, 24]} (15, 14) {[29, 28]} (23, 11) {[2, 3], [3, 3], [71, 35]} (25, 7) {[5, 8]} (27, 22) {[13, 84]} Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' (a, b)-Wall-Sun-Sun primes p and the corresponding pe- riod length π(p2) = π(p) When (a, b) = (1, 1), the sequence {Un}n≥0 is the well-known Fibonacci se- quence, and the (a, b)-Wall-Sun-Sun primes in this case are known simply as Wall- Sun-Sun primes [4, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' However, at the time this article was written, no Wall- Sun-Sun primes were known to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' The existence of Wall-Sun-Sun primes was first investigated by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Wall [16] in 1960, and subsequently studied by the Sun brothers [14], who showed a connection with Fermat’s Last Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' When b = 1, primes satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='2) are also known simply as a-Wall-Sun-Sun primes [18,19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' We point out that the definition of an a-Wall-Sun-Sun prime given in [18,19] is a prime p such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='3) Uπ(p) ≡ 0 (mod p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' In the more general situation of (a, b)-Wall-Sun-Sun primes, it is easily seen that condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='2) implies condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Although it can be shown that the converse is true when b = 1 [5], the converse is false in general, as can be seen by the counterexample (a, b) = (5, 2) with p = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' In this particular example, we have π(7) = 48 and U48 ≡ 0 (mod 49), but π(49) = 7π(7) = 336.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Since Wall was originally concerned with whether there exist any primes p such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='2) holds in the case of (a, b) = (1, 1), we have chosen to use condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='2), instead of condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='3), for our definition of the more general (a, b)-Wall-Sun-Sun prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Let ∆(f) and ∆(K) denote, respectively, the discriminants over Q of f(x) ∈ Z[x] and a number field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' We define f(x) ∈ Z[x] to be monogenic if f(x) is monic, irreducible over Q and {1, θ, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' , θdeg(f)−1} is a basis for the ring of integers ZK of K = Q(θ), where f(θ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' If f(x) is irreducible over Q with f(θ) = 0, then [3] (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='4) ∆(f) = [ZK : Z[θ]]2 ∆(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Observe then, from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='4), that f(x) is monogenic if and only if ∆(f) = ∆(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Thus, if ∆(f) is squarefree, then f(x) is monogenic from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' However, the converse GENERALIZED WALL-SUN-SUN PRIMES AND MONOGENIC TRINOMIALS 3 does not hold in general, and when ∆(f) is not squarefree, it can be quite difficult to determine whether f(x) is monogenic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' In this article, we establish a connection between (a, b)-Wall-Sun-Sun primes and the monogenicity of certain power-compositional trinomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' More precisely, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Let f(x) = x2 − ax − b ∈ Z[x], where a and b satisfy (∗), and let s ≥ 1 be an integer such that � D p � = −1 for each prime divisor p ≥ 3 of s, where � D p � is the Legendre symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Then f(xsn) is monogenic for all integers n ≥ 1 if and only if no prime divisor of s is an (a, b)-Wall-Sun-Sun prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='2 improves and extends previous work of the author on the special case of b = 1 [9], which was, in part, originally motivated by recent results of Bouazzaoui [1,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Bouazzaoui showed, under certain conditions on the prime p ≥ 3, that Q( √ d) is p-rational if and only if π(a,b)(p2) ̸= π(a,b)(p), where d > 0 is a fundamental discriminant [17], a = ε + ε and b = −NQ( √ d)/Q(ε), with ε equal to the fundamental unit of Q( √ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' We recall, for a prime p ≥ 3, that a number field K is said to be p-rational if the Galois group of the maximal pro-p-extension of K which is unramified outside p is a free pro-p-group of rank r2 + 1, where r2 is the number of pairs of complex embeddings of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Preliminaries The formula for the discriminant of an arbitrary monic trinomial, due to Swan [15], is given in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Let f(x) = xN + AxM + B ∈ Z[x], where 0 < M < N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Let r = gcd(N, M), N1 = N/r and M1 = M/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Then ∆(f) = (−1)N(N−1)/2BM−1Dr, where (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='1) D := N N1BN1−M1 − (−1)N1M M1(N − M)N1−M1AN1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' The next two theorems are due to Capelli [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Let f(x) and h(x) be polynomials in Q[x] with f(x) irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Suppose that f(α) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Then f(h(x)) is reducible over Q if and only if h(x) − α is reducible over Q(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Let c ∈ Z with c ≥ 2, and let α ∈ C be algebraic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Then xc − α is reducible over Q(α) if and only if either there is a prime p dividing c such that α = βp for some β ∈ Q(α) or 4 | c and α = −4β4 for some β ∈ Q(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' The next proposition follows from Proposition 1 in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Let b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Then α = (a + √ a2 + 4)/2 is the fundamental unit of Q( √ D) with N(α) = −1, where N := NQ(α)/Q denotes the algebraic norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' 4 LENNY JONES In the sequel, for an integer m ≥ 2, we let ordm(∗) denote the order of ∗ modulo m, and we define (a, b)m := (a (mod m), b (mod m)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' For brevity of notation, we also define λ := ordp(b2) and δp := �D p � , where � D p � is the Legendre symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' The following theorem is a compilation of results from various sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Let {Un(a, −b)}n≥0 be the Lucas sequence as defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Let p be a prime with b ̸≡ 0 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' (1) π(p) = 2 if and only if (a, b)p = (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' (2) If p = 2, then π(2) = � 2 if (a, b)4 ∈ {(2, 1), (2, 3)} 3 if (a, b)4 ∈ {(1, 1), (1, 3), (3, 1), (3, 3)} and π(4) = \uf8f1 \uf8f2 \uf8f3 3 if (a, b)4 = (3, 3) 4 if (a, b)4 ∈ {(2, 1), (2, 3)} 6 if (a, b)4 ∈ {(1, 1), (1, 3), (3, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' (3) If p ≥ 3, then π(p2) ∈ {π(p), pπ(p)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' (4) If δp = −1, then 2(p + 1)λ ≡ 0 (mod π(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Items (1) and (2) follow easily by direct calculation, item (3) can be found in [11], while item (4) follows from a theorem in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' □ The following theorem, known as Dedekind’s Index Criterion, or simply Dedekind’s Criterion if the context is clear, is a standard tool used in determining the mono- genicity of a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='6 (Dedekind [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Let K = Q(θ) be a number field, T (x) ∈ Z[x] the monic minimal polynomial of θ, and ZK the ring of integers of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Let p be a prime number and let ∗ denote reduction of ∗ modulo p (in Z, Z[x] or Z[θ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Let T(x) = � i τi(x)ei be the factorization of T (x) modulo p in Fp[x], and set g(x) = � i τi(x), where the τi(x) ∈ Z[x] are arbitrary monic lifts of the τi(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Let h(x) ∈ Z[x] be a monic lift of T(x)/g(x) and set F(x) = g(x)h(x) − T (x) p ∈ Z[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Then [ZK : Z[θ]] ̸≡ 0 (mod p) ⇐⇒ gcd � F, g, h � = 1 in Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' The next result is essentially an algorithmic adaptation of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='6 specifi- cally for trinomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' GENERALIZED WALL-SUN-SUN PRIMES AND MONOGENIC TRINOMIALS 5 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' [8] Let N ≥ 2 be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Let K = Q(θ) be an algebraic number field with θ ∈ ZK, the ring of integers of K, having minimal polynomial f(x) = xN + AxM + B over Q, with gcd(M, N) = r, N1 = N/r and M1 = M/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Let D be as defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' A prime factor p of ∆(f) does not divide [ZK : Z[θ]] if and only if p satisfies one of the following items: (1) when p | A and p | B, then p2 ∤ B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' (2) when p | A and p ∤ B, then either p | A2 and p ∤ B1 or p ∤ A2 � (−B)M1AN1 2 − (−B1)N1� , where A2 = A/p and B1 = B+(−B)pe p with pe || N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' (3) when p ∤ A and p | B, then either p | A1 and p ∤ B2 or p ∤ A1BM−1 2 � (−A)M1AN1−M1 1 − (−B2)N1−M1� , where A1 = A+(−A)pj p with pj || (N − M), and B2 = B/p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' (4) when p ∤ AB and p | M with N = upm, M = vpm, p ∤ gcd (u, v), then the polynomials G(x) : = xN/pm + AxM/pm + B and H(x) : = AxM + B + � −AxM/pm − B �pm p are coprime modulo p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' (5) when p ∤ ABM, then p2 ∤ D/rN1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' We will find both Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='6 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='7 useful in our inves- tigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' The next theorem follows from Corollary (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='10) in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Let K and L be number fields with K ⊂ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Then ∆(K)[L:K] �� ∆(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' The Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='2 Throughout this section we let f(x) = x2 − ax − b ∈ Z[x] and α = a + √ a2 + 4b 2 , where a and b satisfy (∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' We first prove some lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Let s be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Then f(xsn) is irreducible over Q for all integers n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Since D > 1 is squarefree, it follows that f(x) is irreducible over Q, and the trivial case of s = 1 is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' So, suppose then that s ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Note that f(α) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Let h(x) = xsn and assume, by way of contradiction, that f(h(x)) is reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Then, by Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='3, we have, for some β ∈ Q(α), that either α = βp for some prime p dividing s, or α = −4β4 if sn ≡ 0 (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' 6 LENNY JONES If b ≥ 2, then, in either case, we arrive at a contradiction by taking norms, since N(α) = −b is squarefree but neither N(βp) = N(β)p nor N(−4β4) = 16N(β)4 is squarefree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Suppose then that b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' If α = −4β4, then −1 = N(α) = N(−4β4) ≡ 0 (mod 16), which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Hence, α = βp for some prime divisor p of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Then, we see by taking norms that N(β)p = N(α) = −1, which implies that p ≥ 3 and N(β) = −1, since N(β) ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Thus, β is a unit, and therefore β = ±αj for some j ∈ Z, since α is the fundamental unit of Q( √ D) by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Consequently, α = βp = (±1)pαjp, which implies that (±1)pαjp−1 = 1, contradicting the fact that α has infinite order in the group of units of the ring of algebraic integers in the real quadratic field Q( √ D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Although here we are assuming that conditions (∗) hold, so that a ̸≡ 0 (mod 4), the argument given in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='1 for the case of b = 1 is still valid when a ≡ 0 (mod 4) with the single exception of a = 4 [20] since, in that case, α = 2 + √ 5 is not the fundamental unit of Q( √ 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' However, since ε = (1 + √ 5)/2 is the fundamental unit of Q( √ 5), and α = ε3, Capelli’s theorems can be used to determine exactly when f(xsn) = x2sn − 4xsn − 1 is reducible and how f(xsn) factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' The polynomial f(x) is monogenic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='1, f(x) is irreducible over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Let p be a prime divisor of ∆(f) = a2 + 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' To examine the monogenicity of f(x), we use Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='7 with θ = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Suppose first that p | a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Then p | 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' If p | b, then item (1) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='7 applies, and we see that [ZK : Z[α]] ̸≡ 0 (mod p) since b is squarefree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Now suppose that p ∤ b, so that item (2) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='7 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Note that p = 2 since p | 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Hence, 2 | a and D = (a/2)2 + b ≡ 1 + b (mod 4) since a ̸≡ 0 (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Thus, since D is squarefree and 2 ∤ b, it follows that b ≡ 1 (mod 4) and therefore, B1 = (−b + b2)/2 = b(b − 1)/2 ≡ 0 (mod 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Also, A2 = −a/2 ≡ 1 (mod 2), since a ̸≡ 0 (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Thus, bA2 2 − (−B1)2 ≡ 1 (mod 2), from which we conclude that [[ZK : Z[α]] ̸≡ 0 (mod 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Next, suppose that p ∤ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Then p ∤ 4b since p | (a2 + 4b), and so item (5) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='7 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Since D is squarefree and p ̸= 2, we deduce that p2 ∤ (a2 + 4b) and consequently, [ZK : Z[α]] ̸≡ 0 (mod p), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Let p be a prime, with b ̸≡ 0 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' (1) The prime p = 2 is an (a, b)-Wall-Sun-Sun prime if and only if (a, b)4 = (3, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' (2) If p ≥ 3 and a ≡ 0 (mod p), then p is an (a, b)-Wall-Sun-Sun prime if and only if ordp2(b) = ordp(b) and a ≡ 0 (mod p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' GENERALIZED WALL-SUN-SUN PRIMES AND MONOGENIC TRINOMIALS 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' We see that (1) follows from item (2) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' To establish (2), we let {Un}m denote the sequence (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='1) reduced modulo the integer m ∈ {p, p2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Since a ≡ 0 (mod p), we can write a = pk, for some positive integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Then, {Un}p = [0, 1, 0, b, 0, b2, 0, b3, 0, b4, 0, b5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='] and {Un}p2 = [0, 1, pk, b, 2pkb, b2, 3pkb2, b3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' , ordp(b)pkbordp(b)−1, bordp(b), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Thus, it follows that p is an (a, b)-Wall-Sun-Sun prime if and only if π(p2) = π(p) = 2 ordp(b) ⇐⇒ ordp2(b) = ordp(b) and ordp(b)pkbordp(b)−1 ≡ 0 (mod p2) ⇐⇒ ordp2(b) = ordp(b) and a ≡ 0 (mod p2), since b ̸≡ 0 (mod p) and ordp(b) ≤ p − 1 ̸≡ 0 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Let α = (a− √ a2 + 4b)/2, and let p ≥ 3 be a prime such that δp = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Then (1) ordm(α) = ordm(α) = π(m) for m ∈ {p, p2} and (2) αp+1 ≡ −b (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Note that b ̸≡ 0 (mod p) since δp = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' It follows from [12] that the order, modulo an integer m ≥ 3 with gcd(m, b) = 1, of the companion matrix C for the characteristic polynomial of {Un}n≥0 is π(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' The characteristic polynomial of {Un}n≥0 is f(x), so that C = � 0 b 1 a � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Since the eigenvalues of C are α and α, we conclude that ordm �� α 0 0 α �� = ordm(C) = π(m), for m ∈ {p, p2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Let z ≥ 1 be an integer, and suppose that αz = a + b √ D ∈ Q( √ D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Then N(αz) = a2 − Db2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' But N(αz) = N(α)z = (−b)z, so that a2 − Db2 = (−b)z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Thus, αz = (−b/α)z = (−b)z/(a + b √ D) = (−b)z(a − b √ D)/(a2 − Db2) = a − b √ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Hence, since δp = −1, it follows that αz ≡ 1 (mod m) if and only if αz ≡ 1 (mod m) for m ∈ {p, p2}, which establishes item (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' By Euler’s criterion, �� a2 + 4b �p+1 = (a2 + 4b)(p−1)/2(a2 + 4b) ≡ δp(a2 + 4b) ≡ −(a2 + 4b) (mod p), which implies �� a2 + 4b �p ≡ − � a2 + 4b (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' 8 LENNY JONES Hence, αp+1 = � a + √ a2 + 4b 2 � � a + √ a2 + 4b 2 �p = � a + √ a2 + 4b 2 � p � j=0 �p j � �a 2 �j �√ a2 + 4b 2 �p−j ≡ � a + √ a2 + 4b 2 � ��a 2 �p + �√ a2 + 4b 2 �p� (mod p) ≡ � a + √ a2 + 4b 2 � � a − √ a2 + 4b 2 � (mod p) ≡ −b (mod p), which completes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Let s ≥ 3 and n ≥ 1 be integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Let p ≥ 3 be a prime such that p ∤ a, p ∤ b and pm || s with m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Suppose that Fn(x) := f(xsn) and K = Q(θ), with ZK the ring of integers of K, where Fn(θ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' If δp = −1, then [ZK : Z[θ]] ≡ 0 (mod p) if and only if α2pmn − aαpmn − b ≡ 0 (mod p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' We apply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='6 to T (x) := Fn(x) using the prime p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Let (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='1) τ(x) = x2sn/pmn − axsn/pmn − b = f � xsn/pmn� , and τ(x) = � i τi(x)ei, where the τi(x) are irreducible in Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Then T(x) = � i τi(x)pmnei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Thus, we can let g(x) = � i τi(x) and h(x) = � i τi(x)pmnei−1, where the τi(x) are monic lifts of the τi(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Note also that g(x)h(x) = � i τi(x)pmnei = τ(x) + pw(x), for some w(x) ∈ Z[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Then, in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='6, we have that pF(x) = g(x)h(x) − T (x) = (τ(x) + pw(x))pmn − T (x) = pmn−1 � j=1 �pmn j � τ(x)j(pw(x))pmn−j + (pw(x))pmn + τ(x)pmn − T (x) ≡ τ(x)pmn − T (x) (mod p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='2) Suppose that τ(γ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Then, we see from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='1) that τ(γ)pmn = f(β)pmn = 0, where β = γsn/pmn, so that γsn = βpmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Note that f(x) is irreducible in Fp[x] since δp = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Thus, we assume, without loss of generality that γsn = αpmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Hence, GENERALIZED WALL-SUN-SUN PRIMES AND MONOGENIC TRINOMIALS 9 from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='2), it follows that pF(γ) ≡ −T (γ) (mod p2) ≡ − � γ2sn − aγsn − b � (mod p2) ≡ − � α2pmn − aαpmn − b � (mod p2), which completes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Let p ≥ 3 be a prime such that δp = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Then the following condi- tions are equivalent: (1) p is an (a, b)-Wall-Sun-Sun prime, (2) f(αpm) ≡ 0 (mod p2) for all integers m ≥ 1, (3) f(αpm) ≡ 0 (mod p2) for some integer m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' First, observe that (2) clearly implies (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' We show next that (1) implies (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Because p is an (a, b)-Wall-Sun-Sun prime, we define π := π(p2) = π(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Since δp = −1, we see from item (4) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='5 that 2(p + 1)λ ≡ 0 (mod π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' The squares modulo p form a subgroup, of order (p − 1)/2, of the multiplicative group (Z/pZ)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Thus, (p − 1)/2 ≡ 0 (mod λ), so that 2(p + 1)(p − 1)/2 = p2 − 1 ≡ 0 (mod π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Consequently, αp2−1 ≡ 1 (mod p2) by item (1) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='5, from which it follows that αp2k ≡ α (mod p2) and αp2k+1 ≡ αp (mod p2), for every integer k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Hence, f(αpm) ≡ � α2 − aα − b (mod p2) if m ≡ 0 (mod 2), α2p − aαp − b (mod p2) if m ≡ 1 (mod 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Thus, f(αpm) ≡ 0 (mod p2) when m ≡ 0 (mod 2), since α2 − aα − b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Suppose then that m ≡ 1 (mod 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Let α = (a− √ a2 + 4b)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Since p is an (a, b)-Wall-Sun- Sun prime, and the fact that α = −b/α, we deduce from the Binet-representation formula for Uπ that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='3) Uπ = απ − απ α − α = α2π − (−b)π απ (α − α) ≡ 0 (mod p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Hence, since απ ≡ 1 (mod p2) from item (1) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='5, we conclude from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='3) that (−b)π ≡ 1 (mod p2), which implies that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='4) b2(p+1)λ ≡ 1 (mod p2), by item (4) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Thus, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='4), it follows that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='5) b2(p+1)λ − 1 ≡ (b2λ − 1)B ≡ 0 (mod p2), where B = (b2λ)p + (b2λ)p−1 + · · · + b2λ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' 10 LENNY JONES Since b2λ ≡ (b2)λ ≡ 1 (mod p), we see that B ≡ p + 1 ≡ 1 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Therefore, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='6) b2λ − 1 ≡ 0 (mod p2), from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Also, since δp = −1 and απ ≡ 1 (mod p2), we have from item (4) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='5 that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='7) α2(p+1)λ − 1 ≡ 0 (mod p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='7) yields (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='8) α2(p+1)λ − b2λ ≡ � αp+1 − b � � αp+1 + b � C ≡ 0 (mod p2), where C = � α2(p+1)�λ−1 + � α2(p+1)�λ−2 b2 + · · · + α2(p+1)(b2)λ−2 + (b2)λ−1 ≡ λb2λ+2 (mod p), since α2(p+1) ≡ b2 (mod p) from item (2) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Thus, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='6) and the fact that (p − 1)/2 ≡ 0 (mod λ), we deduce that C ≡ λb2 ̸≡ 0 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Note that αp+1 − b ̸≡ 0 (mod p) since αp+1 + b ≡ 0 (mod p) and b ̸≡ 0 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Therefore, it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='8) that αp+1 ≡ −b (mod p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Hence, αp ≡ −bα−1 (mod p2), and consequently, f(αpm) ≡ α2p − aαp − b (mod p2) ≡ � −bα−1�2 − a � −bα−1� − b (mod p2) ≡ −bα−2(α2 − aα − b) (mod p2) ≡ 0 (mod p2) since α2 − aα − b = 0, which completes the proof that (1) implies (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Finally, to establish that (3) implies (1), we first note that π(p2) ∈ {π(p), pπ(p)} by item (3) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Then, in either case, we have that αpπ(p) ≡ 1 (mod p2), and we conclude from item (4) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='5 that α2p(p+1)λ ≡ 1 (mod p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Since (p − 1)/2 ≡ 0 (mod λ), we deduce α2p(p+1)(p−1)/2 ≡ αp3−p ≡ 1 (mod p2), so that αp3 ≡ αp (mod p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' It then follows easily that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='9) αp2k ≡ αp2 (mod p2) and αp2k+1 ≡ αp (mod p2), for all integers k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Hence, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='9), we have that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='10) f(αpm) ≡ � α2p2 − aαp2 − b (mod p2) if m ≡ 0 (mod 2), α2p − aαp − b (mod p2) if m ≡ 1 (mod 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' By Hensel, the only zeros of f(x) in (Z/p2Z)[ √ D] are α and α = −bα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Suppose that f(αpm) ≡ 0 (mod p2) for some integer m ≡ 1 (mod 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Then, we see from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='10) that either αp ≡ α (mod p2) or αp ≡ α (mod p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' If αp ≡ α (mod p2), then, from item (2) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='5, we have that a2 + 4b + a √ a2 + 4b 2 = α2 + b ≡ αp+1 + b ≡ 0 (mod p), GENERALIZED WALL-SUN-SUN PRIMES AND MONOGENIC TRINOMIALS 11 which implies that a2+4b ≡ 0 (mod p), contradicting the fact that δp = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Hence, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='11) αp ≡ α ≡ −bα−1 (mod p2) or equivalently, αp+1 ≡ −b (mod p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Since α2p − aαp − b ≡ 0 (mod p2), then α2p − aαp − b ≡ 0 (mod p2) so that either αp ≡ α (mod p2) or αp ≡ α (mod p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' If αp ≡ α (mod p2), then (−b)p ≡ α2p (mod p2) from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Hence, � a − √ a2 + 4b 2 �2 + b ≡ a2 + 4b − a √ a2 + 4b 2 ≡ 0 (mod p), since αp ≡ α (mod p), which implies that a2 +4b ≡ 0 (mod p), again contradicting the fact that δp = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Therefore, αp ≡ α (mod p2), so that (−b)p ≡ αp+1 ≡ −b (mod p2), from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Consequently, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='12) (−b)p−1 ≡ 1 (mod p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='11) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='12) yields (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='13) αp2−1 ≡ (αp+1)p−1 ≡ (−b)p−1 ≡ 1 (mod p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Recall that π(p2) ∈ {π(p), pπ(p)} by item (3) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' If π(p2) = pπ(p), then we have by item (1) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='5 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='13) that p2 − 1 ≡ 0 (mod p), which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Thus, we must have π(p2) = π(p), which completes the proof that (3) implies (1) when m ≡ 1 (mod 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Since similar arguments establish this implication when m ≡ 0 (mod 2), we omit the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' □ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' For brevity of notation, define Fn(x) := f(xsn) = x2sn − axsn − b for n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Note that F0(x) = f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' We have that Fn(x) is irreducible for all n ≥ 0 by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' In the trivial case s = 1, we also see that Fn(x) = f(x) for all n ≥ 0, and so Fn(x) is monogenic for all n ≥ 0 by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' So assume that s ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='14) ∆(Fn) = (−b)sn−1s2nsn(a2 + 4b)sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' (⇒) Suppose that s has a prime divisor p that is an (a, b)-Wall-Sun-Sun prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Note that p ∤ b since b ≡ 1 (mod 2) by item (1) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='4 if p = 2, and δp = −1 if p ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' We claim that F1(x) = x2s − axs − b is not monogenic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Let F1(θ) = 0, K = Q(θ) and ZK be the ring of integers of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Suppose first that p = 2, and that 2mv = s, where m ≥ 1 and 2 ∤ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Then p ∤ a since (a, b) = (3, 3)4 by item (1) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Applying item (4) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='7 12 LENNY JONES to F1(x) we see that G(x) = x2s/2m − axs/2m − b = x2v − axv − b ≡ x2v + xv + 1 (mod 2) and H(x) = −ax2mv − b + (axv + b)2m 2 = �a2m − a 2 � x2mv + 2m−1 � j=1 �2m j � 2 (axv)jb2m−j + b2m − b 2 ≡ � x2v + xv + 1 �2m−1 (mod 2), since a2m − a ≡ b2m − b ≡ 2 (mod 4) and [7] �2m j � ≡ � 0 (mod 4) if j ̸= 2m−1 2 (mod 4) if j = 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Thus, G(x) and H(x) are not coprime in F2[x], and therefore, F1(x) is not mono- genic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Suppose next that p ≥ 3, and let pm || s with m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' If p | a, then p2 | a and ordp2(b) = ordp(b) by item (2) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Since p ∤ b, we can apply item (2) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='7 to F1(x) with e = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Then A2 = −a/p ≡ 0 (mod p) and B1 = bpm − b p = b � bpm−1 − 1 � p ≡ 0 (mod p), since ordp2(b) = ordp(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Thus, [ZK : Z[θ]] ≡ 0 (mod p) and F1(x) is not mono- genic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Assume next that p ∤ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Since δp = −1, we have that α2pm − aαpm − b ≡ 0 (mod p2) by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Thus, it follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='6 that [ZK : Z[θ]] ≡ 0 (mod p), and therefore, F1(x) is not monogenic, which completes the proof in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' (⇐) Suppose now that no prime divisor of s is an (a, b)-Wall-Sun-Sun prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='3, we see that F0(x) = f(x) is monogenic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Note that F0(α) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' For n ≥ 0, define αn := α1/sn and Kn := Q(αn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Then α0 = α and, since F0(x) is monogenic, we have that ∆(F0) = ∆(K0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Addi- tionally, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='1, Fn(αn) = 0 and [Kn+1 : Kn] = s for all n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' We assume that Fn(x) is monogenic, so that ∆(Fn) = ∆(Kn), and we proceed by induction on n to show that Fn+1(x) is monogenic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Let ZKn+1 denote the ring of integers of Kn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Consequently, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='9, it follows that ∆(Fn)s divides ∆(Kn+1) = ∆(Fn+1) [ZKn+1 : Z[αn+1]]2 , which implies that [ZKn+1 : Z[αn+1]]2 divides ∆(Fn+1) ∆(Fn)s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' GENERALIZED WALL-SUN-SUN PRIMES AND MONOGENIC TRINOMIALS 13 We see from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='14) that |∆(Fn)s| = (−b)sn+1−ss2nsn+1(a2 + 4b)sn+1 and |∆(Fn+1)| = (−b)sn+1−1s2(n+1)sn+1(a2 + 4b)sn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Hence, ���� ∆(Fn+1) ∆(Fn)s ���� = (−b)s−1s2sn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Thus, it is enough to show that gcd(bs, [ZKn+1 : Z[αn+1]]) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Observe that gcd(b, s) = 1, since δp = −1 for every prime divisor p of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Suppose first that p is a prime divisor of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' If p | a, then it follows that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='15) [ZKn+1 : Z[αn+1]] ̸≡ 0 (mod p) by item (1) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='7 since b is squarefree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' So, assume that p ∤ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' In this case, we apply item (3) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='7 to Fn+1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Observe that A1 = 0 since p ∤ s, and B2 = −b/p ̸≡ 0 (mod p) since b is squarefree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Thus, the first condition of item (3) holds, and therefore once again we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Suppose next that p is a prime divisor of s, with pm || s, and assume first that p | a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Since p ∤ b, we apply item (2) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='7 to Fn+1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' If p2 | a, then ordp2(b) ̸= ordp(b) by item (2) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='4, since p is not an (a, b)-Wall-Sun-Sun prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Also, A2 = −a/p ≡ 0 (mod p), and we must show that B1 = bpm(n+1) − b p = b � bpm(n+1)−1 − 1 p � is not divisible by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Since ordp2(b) ̸= ordp(b), a straightforward argument shows that ordp2(b) = p ordp(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Consequently, bpm(n+1)−1 ̸≡ 1 (mod p2), since pm(n+1) − 1 ̸≡ 0 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' That is, B1 ̸≡ 0 (mod p) and we can conclude (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='15) in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' So, suppose then that p || a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Then A2 = −a/p ̸≡ 0 (mod p), and hence we must show that p does not divide B := (−B)M1AN1 2 − (−B1)N1 = b \uf8eb \uf8ed �a p �2 − b � 1 − bpm(n+1)−1 p �2\uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Suppose, by way of contradiction, that B ≡ 0 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' If bpm(n+1)−1 ≡ 1 (mod p2), then b(a/p)2 ≡ 0 (mod p), which is impossible since p || a and p ∤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Thus, bpm(n+1)−1 ̸≡ 1 (mod p2), from which it follows that b is a square modulo p, con- tradicting the fact that δp = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Hence, B ̸≡ 0 (mod p), and so we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Finally, suppose that p ∤ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Since p is not an (a, b)-Wall-Sun-Sun prime, we deduce from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='7 that α2pm(n+1) − aαpm(n+1) − b ̸≡ 0 (mod p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Thus, [ZKn+1 : Z[αn+1]] ̸≡ 0 (mod p) from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='6, and therefore, Fn+1(x) is monogenic, which completes the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' □ 14 LENNY JONES References [1] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Bouazzaoui, Fibonacci numbers and real quadratic p-rational fields, Period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content=' Hungar.' metadata={'source': 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Shippensburg University, Shippens- burg, Pennsylvania 17257, USA Email address, Lenny Jones: doctorlennyjones@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} +page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE5T4oBgHgl3EQfXg8M/content/2301.05566v1.pdf'} diff --git a/JdFOT4oBgHgl3EQfxzTs/vector_store/index.faiss b/JdFOT4oBgHgl3EQfxzTs/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..2630d1400b53e97c94120284e5aca07735ac64f5 --- /dev/null +++ b/JdFOT4oBgHgl3EQfxzTs/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:dad1b1cc9b1a0181f65d1b78ddad9d82526a009c79f4a2acca7c45d0b58019bb +size 2949165 diff --git a/JtFRT4oBgHgl3EQfzzhX/content/tmp_files/2301.13651v1.pdf.txt b/JtFRT4oBgHgl3EQfzzhX/content/tmp_files/2301.13651v1.pdf.txt new file mode 100644 index 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H. Liu,1 C.-Y. Ng,1 and R. Dodson2 +1Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong +2International Centre for Radio Astronomy Research, The University of Western Australia, 35 Stirling Hwy, Crawley, Western Australia +ABSTRACT +PSR B1706−44 is an energetic gamma-ray pulsar located inside supernova remnant (SNR) +G343.1−2.3 and it powers a compact pulsar wind nebula (PWN) that shows torus and jet structure +in X-rays. We present a radio study of the PWN using Australia Telescope Compact Array (ATCA) +observations at 3, 6, 13, and 21 cm. We found an overall arc-like morphology at 3 and 6 cm, and the +“arc” shows two distinct peaks at 6 cm. The radio emission is faint inside the X-ray PWN and only +brightens beyond that. We develop a thick torus model with Doppler boosting effect to explain the +radio PWN structure. The model suggests a bulk flow speed of ∼ 0.2c, which could indicate significant +deceleration of the flow from the X-ray emitting region. Our polarization result reveals a highly ordered +toroidal B-field in the PWN. Its origin is unclear given that the supernova reverse shock should have +interacted with the PWN. At a larger scale, the 13 and 21 cm radio images detected a semi-circular +rim and an east-west ridge of G343.1−2.3. We argue that the latter could possibly be a pulsar tail +rather than a filament of the SNR, as supported by the flat radio spectrum and the alignment between +the magnetic field and its elongation. +Keywords: Pulsar wind nebulae (2215) — Supernova remnants (1667) — Polarimetry (1278) +1. INTRODUCTION +A pulsar is a compact star that is born in a +supernova explosion. It emits periodic signals +and has a strong surface magnetic field. Par- +ticles around a pulsar are accelerated to form +relativistic wind as the pulsar spins down and +loses energy constantly. The relativistic wind +interacts with the ambient medium and forms a +synchrotron nebula, called a pulsar wind nebula +(PWN). A PWN is able to accelerate particles +to very high energies, emitting synchrotron ra- +diation from the radio to hard X-ray bands. +In X-rays, torus-jet features are commonly de- +tected in young PWN systems (Kargaltsev & +Pavlov 2008; Ng & Romani 2004, 2008). Theo- +ries suggest that torus structure is due to the +shocked pulsar wind flowing into the equato- +rial region, and jets are wind in the polar re- +yihanliu@connect.hku.hk +Corresponding author: Y. H. Liu +gions confined by magnetic hoop stress (see +Porth et al. 2017). +In the radio band, how- +ever, these features are rarely seen. +For in- +stance, the Crab, 3C 58, and the PWN inside +G292.0+1.8 are filled with wisps and filamen- +tary structures (Dubner et al. 2017; Bietenholz +2006; Gaensler & Wallace 2003) instead; some +show arc-like structure, such as CTB 87 (Kothes +et al. 2020) and double-lobed morphology, such +as G21.5−0.9, G76.9+1.0, and DA 495 (Bieten- +holz & Bartel 2008; Arzoumanian et al. 2011; +Kothes et al. 2008). +Previous radio polariza- +tion observations also revealed different mag- +netic field configurations in young PWNe. From +theoretical works, the radial component of the +magnetic field decays faster than the toroidal +component (∼r−2 vs. ∼r−1) (Porth et al. 2017). +Therefore, the B-field beyond the termination +shock should be mostly toroidal. +This, how- +ever, is not supported by observations. Only a +few cases, including Vela and Boomerang, show +toroidal B-field (Dodson et al. 2003; Kothes + +2 +et al. 2006), but many others, e.g., the Crab +Nebula, 3C 58, G21.5−0.9, and Dragonfly, have +complex or radial field structure (Reich 2002; +Lai et al. 2022; Jin et al. 2022). The physical +cause of such diverse morphology and magnetic +field structure among radio PWNe is not fully +understood and a larger sample is needed for +further study. +In this work, we present a new radio study +of the PWN powered by the Vela-like pulsar +B1706−44. +It is one of the few γ-ray pul- +sars detected in the early days with EGRET +(McAdam et al. 1993). +It has a characteris- +tic age τc =17.1 kyr and a spin-down power +˙E ≈ 4×1036 erg s−1. A recent study with Chan- +dra found that the pulsar is moving eastward +with a projected velocity of around 130 km s−1 +(de Vries et al. 2021). The association between +the pulsar and the nearby supernova remnant +(SNR) G343.1−2.3 is controversial. The rem- +nant has a circular shell and an east-west ridge +in the southern part (Dodson & Golap 2002). +The pulsar is located at the tip of the ridge, +near the center of the shell. The SNR distance +of ∼ 3.5 kpc estimated from the Σ–D relation- +ship is compatible to the pulsar dispersion mea- +sure distance d ≈ 2.3 kpc (Cordes & Lazio 2002; +Yao et al. 2017; McAdam et al. 1993). +The +High Energy Stereoscopic System (H.E.S.S) de- +tected extended TeV emission west of the pul- +sar, which also has some connection with the +SNR (H. E. S. S. Collaboration et al. 2011). Be- +sides, the pulsar powers an X-ray PWN that has +compact torus and jet structure (Romani et al. +2005). A recent study found diffused emission +around the torus and a long curved outer-jet (de +Vries et al. 2021). +In this study, we aim to perform high reso- +lution radio observations of B1706 PWN for di- +rectly comparing with the compact X-ray struc- +tures in Chandra images and better understand- +ing the PWN magnetic properties. +Previous +observations have detected a radio PWN sur- +rounding the pulsar (Frail et al. 1994; Giacani +et al. 2001; Dodson & Golap 2002; Romani +et al. 2005). However, the pulsar emission was +not clearly distinguished from the PWN. Some +ATCA observations excluded the pulsar emis- +sion, but few observations with high resolution +and sensitivity were included. Besides, there is +no previous study of the magnetic field in the +B1706 PWN. +In this paper, we analyze new and archival +radio observations of the PWN powered by +PSR B1706−44 (hereafter B1706 PWN) and +SNR G343.1−2.3 taken with the Australia Tele- +scope Compact Array (ATCA) at 3, 6, 13, and +21 cm images. We employed new observations +with high resolution aiming to better study the +morphology and polarization information of this +PWN. We describe the observations and data +reduction process in Section 2. Section 3 shows +the results and they are discussed in Section 4. +We summarize our results in Section 5. +2. OBSERVATIONS AND DATA REDUCTION +We carried out new radio observations of +B1706 PWN at 3 and 6 cm bands with ATCA +in 6 km array configurations on 2017 Nov 3 and +2018 Jan 11. We also analyzed archival ATCA +observations taken in the 3, 6, 13, and 21 cm +bands with various array configurations, which +have previously been analyzed by Dodson & Go- +lap (2002); Romani et al. (2005). +All the 3 +and 6 cm band data were taken with the pul- +sar binning mode, providing a high time reso- +lution. +We then only select off-pulse data to +“gate out” the pulsar emission to search for faint +PWN structure in the surrounding. +Table 1 lists the detailed observation param- +eters of all the data. +The 3 and 6 cm obser- +vations were performed simultaneously center- +ing at 8640 MHz and 4800 MHz, as well as at +8997.5 MHz and 5497.5 MHz. Besides, we have +also selected observations in 2003 and 2005 at +8640 MHz and 8384 MHz. Our new observations +in 2017 and 2018 were taken after the Compact +Array Broadband Backend (CABB) upgrade +(Wilson et al. 2011), which increased the band- +width from 128 MHz to 2048 MHz. At 3 cm, the +pre-CABB and post-CABB integration times +are 64.0 hr and 21.2 hr, respectively, with a total +u-v coverage from 0.8 kλ to 197.4 kλ. At 6 cm, +we have 26.7 hr and 21.2 hr pre- and post-CABB + +3 +Table 1. ATCA observations of B1706 PWN used in this study +Obs. Date +Array +Center Freq. +Usable Band- +No. of +Integration +Pulsar +Config. +(MHz) +width (MHz) +Channels +Time (hr) +Binning Mode +3 cm +2002 Jan 06 +750A +8640 +104 +13 +7.7 +Y +2002 Feb 16 +1.5A +8640 +104 +13 +10.2 +Y +2002 Apr 11 +6A +8640 +104 +13 +8.8 +Y +2003 May 18 +1.5C +8384, 8640 +104 +13 +8.7 +Y +2003 Jun 23 +750C +8384, 8640 +104 +13 +10.2 +Y +2003 Aug 02 +6D +8384, 8640 +104 +13 +9.8 +Y +2005 Nov 20 +1.5C +8384, 8640 +104 +13 +4.3 +Y +2005 Dec 27 +6A +8384, 8640 +104 +13 +4.3 +Y +2017 Nov 03 +6A +8997.5 +1728 +433 +10.9 +Y +2018 Jan 11 +6C +8997.5 +1728 +433 +10.3 +Y +6 cm +2002 Jan 06 +750A +4800 +104 +13 +7.7 +Y +2002 Feb 16 +1.5A +4800 +104 +13 +10.2 +Y +2002 Apr 11 +6A +4800 +104 +13 +8.8 +Y +2017 Nov 03 +6A +5497.5 +1728 +433 +10.9 +Y +2018 Jan 11 +6C +5497.5 +1728 +433 +10.3 +Y +13 cm +1998 May 29 +750E +2496 +104 +13 +0.8 +N +1999 Nov 03 +210 +2496 +104 +13 +19.1 +N +21 cm +1998 May 29 +750E +1384 +104 +13 +0.8 +N +1998 Sep 15 +6A +1384 +104 +13 +3.3 +N +1999 Nov 03 +210 +1384 +104 +13 +19.1 +N +2005 Nov 19 +1.5C +1344, 1472 +104 +13 +4.6 +N +2005 Dec 27 +6A +1344, 1472 +104 +13 +1.7 +N +integration time, respectively, covering the u-v +space from 0.85 kλ to 127 kλ. The 13 and 21 cm +datasets with good quality have a total inte- +gration time of 19.9 hr and 29.5 hr, respectively. +The u-v coverage of the observations at 13 cm is +0.2–5.5 kλ and 18–37 kλ, and in the 21 cm band +is 0.1–30 kλ. +We processed the data using the MIRIAD +package (Sault et al. 1995). We first flagged the +edge channels and data affected by severe radio +frequency interference, then followed the stan- +dard procedures to calibrate the flux scale, band +pass, and gains. After calibration, we formed +Stokes I, Q, and U images using multi-frequency +synthesis. We weighted the data inversely pro- +portion to the noise. Since the pre- and post- +CABB data were taken over 15 yr apart, we +formed separated images at 6 cm to check for +any morphological changes. The result shows +no significant variability, we therefore combined + +4 +all off-pulse data at each frequency for a joint +analysis to boost the signal. +We first focused on the region close to the pul- +sar, and generated the 3 cm image with the best +resolution of full width half maximum (FWHM) +6.3′′×3.5′′. +The resulting map has root mean +square (rms) noise of around 0.06 mJy beam−1, +but we did not detect any significant structure +near the pulsar. Then we generated images us- +ing u-v tapering with a larger FWHM to boost +the signal to noise ratio (S/N). Tapering sizes +are 20′′ for the 3 and 6 cm images and 70′′ for +the 13 and 21 cm images. The 3, 6, and 13 cm +images were produced with Brigg’s robust pa- +rameter of 0.5 to suppress side lobes. For the +21cm image, we used natural weighting to max- +imize the sensitivity. +For image decovolution, we first used the +task mossdi to clean strong point sources in +the Stokes I, Q, and U images. +The resid- +ual maps were then cleaned simultaneously us- +ing pmosmem and the models were restored with +beam sizes of 20′′ in 3 and 6 cm and 70′′ in 13 and +21 cm maps. The rms noise is around 0.06, 0.06, +0.7, and 0.8 mJy beam−1 in the Stokes I images +and around 0.05, 0.04, 0.5, and 0.5 mJy beam−1 +in the Stokes Q and U images at 3, 6, 13, and +21 cm, respectively. Finally, we generated po- +larization maps with the task impol. Besides, +we have also applied the same procedure to pro- +duce full Stokes images of the pulsar using the +on pulsed data. +3. RESULTS +3.1. Morphology +Figure 1 shows the total intensity maps of +B1706 PWN during the off-pulse phase in the 3 +and 6 cm bands. The radio PWN is clearly de- +tected. It is elongated in the east-west direction +with a size of ∼ 4′×2′ and wraps PSR B1706−44 +in the north. The eastern part of the PWN is +generally brighter, and the flux density peaks +at 0.7′ and 0.9′ east of the pulsar, reaching +0.60 and 0.85 mJy beam−1 at 3 and 6 cm, re- +spectively. At 3 cm, the nebula has a more uni- +form brightness distribution than at 6 cm, and it +shows arc-like structure overall. We also found +a few protrusions in the PWN, one extends 2′ +north of the pulsar and two others northwest +and southwest from the pulsar extend towards +west 2′ away from the pulsar. +All these pro- +trusions are not thicker than 0.5′. The protru- +sion features should correspond to data with u- +v coverages from ∼ 4.5 to ∼ 15 kλ, which are +included in the 3 cm data. These features are +therefore less likely to result from the missing +flux problem. +We need more observations to +confirm this feature. At 6 cm, the PWN shows +two distinct peaks, resembling two lobes brack- +eting the pulsar. The eastern part has an el- +liptical shape of 2′ in size. It is brighter than +the western part and contains 67% of the flux +density of the PWN. The western lobe is fainter +and more elongated. It has a size of 2′ × 1.5′ +and is oriented along the northeast-southwest +direction. Its surface brightness peaks at ∼ 0.8′ +west of the pulsar and is only about 2/3 of that +of the peak in the eastern lobe. For both radio +images, there is a “bay” feature south of the +pulsar with no detectable radio emission. The +3σ flux density limit is around 0.18 mJy beam−1 +in both the 3 and 6 cm. +The pulsar emis- +sion is clearly detected in the on-pulse data in +both 3 and 6 cm bands with flux densities of +1.0±0.1 mJy beam−1 and 2.9±0.1 mJy beam−1, +respectively. +In Figure 2, we compare the radio images +with a 0.5–7 keV X-ray image obtained with +the Chandra X-ray Observatory. We compared +the X-ray torus/jet feature close to the pul- +sar with the 3 cm radio image with a beam of +6.3′′×3.5′′, and found no counterpart of the X- +ray PWN. The 3 cm radio image has a rms noise +of 0.02 mJy beam−1. We also smoothed the X- +ray image to 20′′, same as that of the radio im- +age. Similarly, there is X-ray emission but no +radio emission in the inner PWN, and the ra- +dio emission only appears in the outer PWN +beyond 10′′ from the pulsar. +Meanwhile, the +X-ray emission fades away in the outer PWN +∼25′′ from the pulsar. The 3 and 6 cm images +also show a linear, jet-like structure extending +west from the end of the northern X-ray jet with +a flux density of 8.2±0.6 mJy at 6 cm. It has a + +5 +!"#$ +% #$ +3 cm +6 cm +Figure 1. Total intensity images of the B1706 PWN at the 3 and 6 cm bands in the off-pulse phase with the pulsar emission +excluded. The gray scale bar on the right is in units of Jy beam−1. The crosses at the center of the images show the position of +PSR B1706−44. The circular beams at the bottom left indicate the beam size of FWHM 20′′ for both images. The rms noise in +both bands is around 60 µJy beam−1. The contours correspond to total intensity levels of 0.18, 0.3, 0.45, and 0.6 mJy beam−1. +1 arcmin +E +N +(a) +(b) +2 arcmin +Radio Features +beyond X-ray Jets +X-ray Jets +Radio PWN +30 arcsec +Torus +Figure 2. (a): Comparison between radio and X-ray emis- +sion of B1706 PWN. The 6 cm radio emission is shown in blue +with the 20′′ restored beam. The Chandra 0.5–7 keV X-ray +image is shown in red, also smoothed to 20′′ resolution. The +inset image is the comparison of the X-ray torus/jet feature +and the 3 cm high resolution radio image, with the X-ray +torus highlighted by the white region +. (b): Same as (a) but both are smoothed to 50′′ resolution. +The cross in white indicates PSR B1706−44. +length of ∼3′ and the width is not clearly re- +solved by the 6 cm observation (see Figure 2a). +We also find similar emission beyond the south- +ern X-ray jet after smoothing the 6 cm intensity +map to 50′′ (see Figure 2b) but it is fainter and +more diffused than the emission in the north. +More data are needed to confirm these features. +Figure 3 shows the total intensity images of +the overall SNR in the 13 and 21 cm bands. +This is the first time that the 13 cm image is +shown, and the SNR shows a similar morphol- +ogy to that at 21 cm: there is a ∼40′ semicircu- +lar rim in the west and a bright east-west ridge +in the south ∼30′ connecting the pulsar to the +western rim. PSR B1706−44 is located at the +tip of the ridge rather than at the center of the +SNR. The pulsar emission is visible in the im- +ages, since no pulsar binning mode was used +in these observations. There is also significant +emission detected at the locations of radio outer +jets at 6 cm. +3.2. Radio spectrum +We measured the flux densities of the over- +all B1706 PWN and each component in differ- +ent bands. +Background subtraction was per- +formed using measurements from nearby source +free regions. The regions selected to measure + +6 +13 cm +21 cm +Figure 3. Total intensity maps of SNR G343.1−2.3 in 13 cm and 21 cm bands. The contours are at levels of 4, 8, 12, and +16 mJy beam−1. The gray scale bars on the right have units of Jy beam−1. The boxes indicate the field of view of Figure 1. Both +images have a beam size of FWHM 70′′, which is shown in the bottom left. The rms noise at 13 cm is around 0.7 mJy beam−1and +at 21 cm is around 0.8 mJy beam−1. +the whole PWN and the eastern and western +lobes in both bands are shown in Figure 9. +The estimated flux densities of the entire +PWN are 18.5±1.0, 21.2±0.7, 39.2±1.4, and +47.4±4.0 mJy in 3 cm, 6 cm, 13 cm, and 21 cm, +respectively. All these are plotted in Figure 4 +and shown in Table 2. The values at 13 and +21 cm have been subtracted for the pulsar flux +density and those at 3 and 6 cm are measured +from the off-pulse phase images. Due to the lack +of short u-v spacing below 0.8 kλ in the 3 and +6 cm observations, the maps have low sensitiv- +ity to structures larger than 4′. +In this case, +Table 2. Flux density of the pulsar, entire PWN, and different +components +Freq. +Total +Eastern +Western +Pulsar +Bands +PWN (mJy) +Lobe (mJy) +Lobe (mJy) +(mJy) +3 cm +18.5±1.0 +8.9±1.0 +10.5±0.8 +1.0±0.1 +6 cm +21.2±0.7 +14.1±0.8 +7.3±0.4 +2.9±0.1 +13 cm +39.2±1.4 +– +– +8.0±0.1 +21 cm +47.4±4.0 +– +– +10.2±0.4 +we separately derived the spectrum from the +two higher and lower frequency bands. +They +both give a similar spectral index α ≈ −0.3 +(Sν ∝ να) (see Figure 4). +We note that our +flux density measurement of the overall PWN +at 6 cm (∼21 mJy) is comparable to the one +measured with the VLA (∼28 mJy), although +slightly lower (Giacani et al. 2001). As the VLA +data has similar u-v coverage as our ATCA ob- +servations. +The discrepancy could be due to +different choices of source and background re- +gions. +The flux densities of the eastern lobe +are 14.1±0.8 mJy and 8.9±1.0 mJy in 6 and +3 cm, respectively; and those of the western +lobe are 7.3±0.4 mJy and 10.5±0.8 mJy, respec- +tively. These give α = −0.97 for the eastern +lobe and α = +0.78 for the western lobe. +We also plotted the 3 and 6 cm images after +filtering data to the same u-v coverages, so that +data in both bands have the same missing flux +problem for a correction of the spectral index. +We obtained a spectral index α ∼ 0 for the +whole PWN and spectral indices of −0.05 and ++0.96 in the eastern lobe and the western lobe, +respectively. The latter is rather unusual and +more observations are needed to confirm this. + + +We also estimated the pulsar flux densities +with extraction region same as the beam size +For measurements at 3 and 6 cm, we generated +images with only on-pulse bins to show the pul +sar emission. The pulsar has flux densities of +1.0±0.1, 2.9±0.1, 8.0±0.1, and 10.2±0.4mJy at +3, 6, 13, and 21 cm, respectively. The results are +B1706 PWN +plotted in Figure 4. Due to low resolution, the +Eastern Lobe +Western Lobe +measurement at 2l cm could be contaminated +PSR B1706-44 +by the PWN emission, but we note that the +ATNF_PSR B1706-44 +result is consistent with the pulsed flux density +8 10 +2 +Frequency (GHz) +obtained from the single dish Parkes Radio Tele- +ATCA +10" +line with the extrapolation of the pulsar spec- +trum (Lyne et al. 1998; Jankowski et al. 2017). +101° +We performed a multiwavelength compari- +son with the Chandra X-ray data. We repro- +cessed all archival data of B1706 PWN using the +CIAO software package (Fruscione et al. 2006), +108 +then extracted the X-ray PWN spectrum using +specextract to fit the background subtracted +107 +spectrum with an absorbed power law. We ob- +1010 +1012 +1016 +1018 +1020 +tained a photon index I ～ 1.53 ±0.07 and a to- +1014 +v(Hz) +tal unabsorbed fux fpun=12.2±0.1x 10-13 erg +ATCA Sensitivity +cm-2 s-1 in the 0.5-7keV range (excluding the +10 °% +pulsar emission). This gives Qx = +-0.53 in the +10° +X-rays. We plot the spectral energy distribution +(SED) of the PWN from radio to X-ray bands +tos +in Figure 4. A comparison with Qradio -0.3 +in the radio band suggests a spectral break of +010 +Aα ～ 0.2. However, we note that the extrap- +olation of the radio and X-ray spectra do not +intersect. This could be due to the X-ray obser- +vations being insensitive to faint emission in the +10 10 +1012 +10 14 +1016 +1018 +outer PWN region. In this case. emission from +v(Hz) +the inner PWN region would dominate and the +Figure 4. Top: radio spectra of the overall B1706 PWN +obtained photon index would be smaller than +and different components. The green dashed line shows the +that of the overall PWN. +extrapolated spectrum of the pulsar emission from 0.4 and +We also did such a comparison in high reso- +1.4GHz data. The flux densities of PSR B1706-44 at 0.4 +and 1.4 GHz are from the ATNF pulsar Catalog (Manchester +lution about the X-ray torus region. For the +et al. 2005) and are shown as triangles with error bars. Mid- +X-ray torus, recent Chandra results show a flux +dle: SED of the PWN from radio to X-ray bands. The black +ftorus=1.26±0.03erg cm-2 s-1 from 0.5 to 7 keV +dots with error bar represent the measurement obtained with +ATCA, and the lines in the top right show the best-fit un- +with a photon index I=-1.46±0.05 (de Vries +absorbed X-ray spectrum obtained from Chandra. Bottom: +et al. 202l). The flux density in radio is esti- +Multiwavelength SED of the X-ray torus. The X-ray spec +mated in a region shown in Figure 2 and show a +trum is extrapolated to 3 cm wavelength (the band in gray), +sensitivity of 0.06 mJy for the torus region. We +and the upper limit in red shows the 3o rms noise of the 3 cm +observations. +compared the radio flux density with the ex-8 +trapolated X-ray spectrum. The SED is shown +in Figure 4. +3.3. Polarization +Figure 5 shows the polarized emission of the +PWN and the SNR. We clipped the 3 and 13 cm +maps where the polarization intensity has a +signal-to-noise ratio (S/N) <3, total intensity +S/N <5, or uncertainty of the position angle +(PA) >10◦. For the 6 and 21 cm maps, we ap- +plied the same clipping criteria for the polar- +ization intensity S/N and PA, but <3 for the +total intensity S/N. The PWN is highly lin- +early polarized in all the bands and the polar- +ized emission generally follows the total inten- +sity. The 3 cm polarization emission is east-west +elongated and the size is ∼ 4′ × 1′. However, it +shows a peak ∼ 0.8′ west of the pulsar, different +from that of the total intensity map. Whereas, +the 6 cm polarization image shows two-lobed +structure resembling the total intensity emis- +sion. +The eastern lobe is brighter and has a +peak flux density of 0.37 mJy beam−1. Polarized +emission was detected in both the northern and +southern radio features beyond X-ray jets, but +the point source in northern one is unpolarized. +The linear polarization fraction in both 3 and +6 cm bands is around 45% for the entire PWN, +and around 85% for the pulsar. The circular po- +larization fraction of the pulsar is around 15% +in both bands. +In the 13 and 21 cm images, we found po- +larized emission on large scales including the +PWN, the ridge, and the SNR shell. +The +PWN is detected at the end of the ridge with +a blob-like structure aligning with the total in- +tensity contours. The PWN polarized emission +is fainter than that of the ridge and the SNR +shell. The polarization fractions of the PWN in +both 13 and 21 cm are around 30%. The shell +structure of the SNR shows significant polarized +emission. The polarization fractions of the en- +tire SNR are around 50% and 40% at 13 cm and +21 cm, respectively. +3.4. Rotation measure and intrinsic magnetic field +orientation +The observed PA of the polarization vectors +are rotated due to Faraday effect in the inter- +stellar medium. The amount of rotation is pro- +portional to the rotation measure (RM) times +square of the wavelength (λ2). We attempted +to derive a high resolution RM map using the +3 and 6 cm data, but it has too large uncer- +tainty to be useful. Therefore, we simply used +the RM of the pulsar (0.7±0.07 rad m−2; John- +ston et al. 2005) to derotate the polarization +vectors at these two bands. To determine the +RM of the SNR, we selected edge channels with +32 MHz bandwidth from the 13 and 21 cm data +to generate Stokes Q and U maps. We used a u- +v taper of 80′′ FWHM to boost the S/N. The im- +ages were then deconvolved using the same pro- +cedure as mentioned above, and restored with a +circular beam of FWHM 80′′, which is the reso- +lution of the lowest frequency band. We formed +four PA maps and applied a linear fit to deter- +mine the RM value at each pixel. The result +is plotted in Figure 6 and the typical uncer- +tainty of the map is ∼1 rad m−2. We found that +the RM of the SNR varies from −90 rad m−2 to ++92 rad m−2 and it is ∼ 0 rad m−2 near the pul- +sar position. The latter is in line with the RM +of PSR B1706−44 (Johnston et al. 2005). The +RM of the PWN and the ridge varies smoothly +compared with that in the SNR rim. +Figure 5 shows the intrinsic magnetic field di- +rection of B1706 PWN at 3 and 6 cm and of +the SNR at 13 and 21 cm after correcting for +the Faraday effect. The magnetic field of the +PWN is highly ordered. It is oriented along the +PWN elongation and wraps around the pulsar +in the north, indicating a toroidal configuration. +For the outer jets, only faint polarized emission +is detected, we are therefore only able to de- +termine the polarization angle in the brightest +regions. +At large scale, the magnetic field of +the ridge well aligns with its elongation. It then +gradually switches to tangential along the rim +of the SNR shell. +4. DISCUSSION +4.1. PWN structure + +9 +3 cm +6 cm +13 cm +21 cm +3 cm +6 cm +13 cm +21 cm +3 cm +6 cm +13 cm +21 cm +Eastern +Lobe +Western +Lobe +6 cm +6 cm +Figure 5. Linear polarized intensity maps of B1706 PWN and SNR G343.1−2.3, overlaid with the total intensity contours and +polarization vectors that show the intrinsic B-field orientation. The contours are at levels of 0.18, 0.3, 0.45, and 0.6 µJy beam−1 +for 3 and 6 cm, and 4, 8, 12, and 16 mJy beam−1 for 13 and 21 cm. The gray scale bars correspond to the polarized intensity +level in units of Jy beam−1. The vector length is proportional to the polarization intensity, with the bars in lower left indicating +polarization intensity of 0.5 mJy beam−1 in 3 and 6 cm maps and 10 mJy beam−1 in 13 and 21 cm maps. +The black cross +represents the position of PSR B1706−44. As shown at the bottom left, the restoring beam sizes are 20′′ FWHM for 3 and 6 cm +images and 70′′ FWHM for 13 and 21 cm images. +Our radio intensity maps of the B1706 PWN +reveal an overall arc-like morphology. +The +emission is bright in the outer region of PWN +but faint in the inner part, which is in con- +trast to the X-ray emission. +Similar X-ray– +radio anti-correlation is also found in a few +other PWNe, including Vela PWN, DA 495, +G76.9+1.0, G319.9−0.7, G327.1−1.1 (Dodson +et al. 2003; Kothes et al. 2008; Arzoumanian +et al. 2011; Kargaltsev et al. 2008; Ng et al. +2010; Ma et al. 2016). The cause is not clearly +understood. +It was suggested that the radio +emission in the inner PWN could be too faint +to detect. As the outflow decelerates, the parti- +cle number density increases outward, resulting +in brighter radio emission. On the other hand, + +10 +Figure 6. +RM map of SNR G343.1−2.3. +The RM values vary from −90 rad m−2 to +92 rad m−2, with the solid squares +representing positive values and hollow boxes for negative. The contours are total intensity levels at 4, 8, 12, and 16 mJy beam−1 +at the 13 cm. +The cross indicates the position of PSR B1706−44 and the box at bottom left corresponds to RM value of +−50 rad m−2. +synchrotron cooling makes the X-ray emission +invisible in the outer PWN (Kargaltsev et al. +2008). +We applied this idea to B1706: +we +extrapolated the X-ray spectrum of the torus +reported by de Vries et al. (2021) down to +the radio band with a simple unbroken power +law. This suggests a flux density from ∼0.01 to +∼0.2 mJy at 3 cm. From our highest resolution +3 cm intensity map, the 3σ limit at this region +is ∼0.06 mJy beam−1, giving a detection limit of +∼0.06 mJy based on the torus area (see Figure +4). In this case, we still cannot firmly rule out +the scenario that the X-ray torus follows a sim- +ple unbroken power-law distribution from radio +to X-rays. A further non-detection with about +3 times better sensitivity is required to reject +such a scenario. Alternatively, we note that the +injected spectrum could have an intrinsic spec- +tral break, if the particle acceleration is due to +magnetic reconnection in the termination shock +or Weibel instability (Lyubarsky & Kirk 2001; +Weibel 1959). In this case, the radio emission +could be a few orders of magnitude fainter, mak- +ing it very difficult to detect. Deeper radio ob- +servations are needed to discriminate between +these. + + +11 +3 +β=0.0 +β=0.1 +β=0.2 +2 +1 +1 +1 +0 +0 +0 +-1 +-1 +-1 +-2 +-2 +-2 +-3 +-2 +0 +2 +-2 +0 +2 +3 +β=0.3 +6cm +3cm +2 +1 +0 +-1 +-2 +-3 +2 +The total intensity images at 6 and 3 cm are shown at the bottom panels for comparison. +Our high resolution radio maps reveal over- +following a similar procedure as Ng & Romani +all arc-like structure for B1706 PWN. which is +(2004, 2008). We built a torus in 3D with cir- +similar to Vela, Boomerang (G106.6+2.9), and +cular cross-section and an outer radius of 2', as- +G76.9+1.0 (Dodson et al. 2003: Kothes et al +suming uniform and isotropic emission inside +2006: Arzoumanian et al. 2011). It was sug +We set the viewing angle between the torus axis +gested that this could be caused by the pas- +and the line of sight to be 53.3 (Romani et al +2005) and the outer boundary radius 6 times +sage of supernova reverse shock and a thick +toroidal model has been developed (Chevalier +of the inner boundary radius. We considered +& Reynolds 2011). We tried to apply the same +Doppler boosting effect following Pelling et al. +model to our 3cm image, but found that it fails +(1987). The apparent intensity I is +to explain the lack of emission in the bay south +I α (1 - n - β)-(1-I) Io, +(1) +of the pulsar. Indeed the model is always sym- +metric and also cannot explain the tongue-like +where n is the unit vector from the observer. +morphology of the Boomerang. It is worthy +B = w/c is the assumed velocity of the radial +post-shock bulk flow, is the photon index in +an asymmetric torus feature close to the pulsar +the rest frame, and Io is the intrinsic intensity of +due to the Doppler boosting effect, such that +synchrotron emission taken to be constant. We +emission is brighter if the particles are moving +projected the model onto the plane of the sky +toward the observer and vice verse. We there- +to generate a 2D brightness map for comparison +fore added Doppler boosting effect to the model +with the data. Figure 7 shows models of a ra-12 +dio B1706 PWN having a thick equatorial torus +and the Doppler boosting effect with different β +values in the bulk flow, as well as the 6 and 3 cm +radio images of the B1706 PWN. In the scenario +of β = 0 (i.e., negligible Doppler boosting), the +model shows two equatorial lobes both having +a brightness peak inside, and a fainter region +between the lobes. We note that this model is +not only horizontally symmetric, but also ver- +tically symmetric. Considering Doppler boost- +ing will result in enhanced brightness in an ap- +proaching bulk flow and reduced brightness in +flows leaving away, such that the upper part is +brighter than the lower part in our model. The +brightness of the upper part becomes compara- +ble to or even overwhelms that of the lobes in +the scenario of β ≥ 0.2. In the latter case, the +model has a kidney-shape feature wrapping the +pulsar with a single peak north of the pulsar +region. +We suggest that the model can cap- +ture characteristic features of the radio PWN +observed, including the overall arc-like PWN +wrapping the pulsar in the north and the faint +bay in the south. Comparisons between these +models and the observations show that a con- +stant value β ∼ 0.2 over the entire torus gives +the best result of the 3 cm PWN, while the 6 cm +PWN can be better described by β ∼ 0.1. +Our 6 cm image is slightly different from the +3 cm image (e.g., the gap between the eastern +and western parts), so that the best fit β are +slightly different. To resolve this, we consider a +spectral gradient across the PWN, as motivated +by the different spectra between the eastern and +western parts (see Figure 4). We fix β = 0.2 and +consider that I0 depends on frequency as +I0,ν = I0,ν0 · ( ν +ν0 +)α, +(2) +where I0,ν is the intrinsic intensity in frequency +ν, and I0,ν0 is the intensity in a reference fre- +quency ν0. We also assume that ν0 = 10 GHz +and +α = 0.9 xEW +Rpwn +, +(3) +where xEW is the position of a point in the east- +west direction with a scale of the PWN radius +Rpwn, indicating α = −0.9 at the eastern end +of the PWN and α = +0.9 at the western end. +We then obtain the model at 3 and 6 cm and +also extrapolate it to around 30 cm. Besides, we +compare all these models with the 3 and 6 cm +ATCA images and 30 cm ASKAP image (Norris +et al. 2019), and all these are shown in Figure +8. Our simulation shows the PWN with a much +brighter eastern part at 30 cm, a larger brighter +eastern lobe and a smaller western lobe at 6 cm, +and lobes connected from the north of the pulsar +at 3 cm. These models can reproduce the main +morphological features of the observed PWN in +these bands. we also tried different values of β +but found that 0.2 gives the best fit result. A +direct comparison with βX = 0.7 in the X-ray +emitting region (Romani et al. 2005) suggests +deceleration along the flow. This also implies +the particle accumulation, which could give rise +to the observed anti-correlation between the ra- +dio and X-ray emission. +Our modeling result shows that B1706 PWN +could have a toroidal structure in 3D, and it +appears as arc-like due to Doppler effect. This +model could be used to explain the arc-like mor- +phology found in other radio PWNe, e.g., Vela +and Boomerang. Finally, we note that toroidal +structure can be resulted from a toroidal B-field +as simulations suggest (Porth et al. 2017). This +is supported by our polarization result, which +reveals a toroidal B-field configuration. +4.2. Equipartition magnetic field +We +estimated +the +equipartition +B-field +strength of the PWN +Beq = [6π(1 + k)c12LsynΦ−1V −1 +pwn]2/7, +(4) +where Vpwn is the emission volume, Lsyn is +the synchrotron luminosity, Φ is a filling factor +for the emission (it is usually taken as 1 even +though not 100% of the volume emits), k is the +ratio between electron energy and the energy of +heavy particles, c12 is a constant related to syn- +chrotron radiation process and weakly depends +on the frequency range (Pacholczyk 1970). We +selected PWN flux density at 6 cm as a reference +and assumed a simple power-law spectrum with + + +13 +30cm +6cm +3cm +ASKAP 30cm +ATCA 6cm +ATCA 3cm +Figure 8. Models of the B1706 PWN in different wavelengths with a gradient in the east-west direction (the top row), as +well as the observation results of B1706 PWN at 30, 6, and 3 cm (the bottom row) taken with ASKAP (Norris et al. 2019) and +ATCA. +a spectral index Q ~ -0.3 from 107 to 1011 Hz to +We also roughly estimated the equapartition +obtain Lsyn = 7.9 × 1030d2.3 erg s-1, where d2.3 +B-field of the linear radio feature beyond the X. +is the source distance in units of 2.3 kpc. To es- +ray jet northwest of the pulsar, assuming it asso- +timate the volume of the PWN, we assumed an +ciated with the PWN. Following the same pro- +oblate spheroid for the emission volume in 3D. +cedure as above and taking the emission volume +The oblate spheroid is 2.6' 4.2' 4.2' in size. +as a cylinder, the flux density measurements at +which gives a volume of Vpwn=2.2 × 1056 cm3. +3 and 6cm gives +These give +Beg = 10.5(1 + k)2/7@-2/7 d2.2/7 μG. +The result is similar if we exclude the point +source, +Taking k=0 and @=1, the B-field is in the order +of 10 uG, slightly lower than 27 μG and 15 μG +Bts* = 59(1 + k)2/7 @-2/7 d,2 +uG +estimated for the inner and outer X-ray PWN, +respectively (de Vries et al. 2021). However, the +This is significantly higher than that of the main +decay in B-field strength is much slower than +radio PWN, but is comparable to that of the +B α 1/r predicted by theory (Porth et al. 2017) +X-ray bright inner PWN. Identification the as- +given that the radio PWN is 5 times larger +sociation between the PWN and this feature +than that in the X-rays. +and requires more following observations and14 +Eastern Lobe +Western Lobe +Shell +Radio Features +beyond X-ray Jets +Ridge +PSR B1706-44 +and the PWN +130 km/s +Figure 9. Total intensity image of SNR G343.1−2.3 at 13 cm. Inset: Zoom in of the PWN at 6 cm. Extraction regions (e.g., +shell, ridge, the whole PWN, and the eastern and western lobes) for spectral measurements are indicated. The circular beams of +20′′ (inset) and 70′′ (SNR) are shown bottom left of the images as scale references. The dashed vector shows the pulsar motion +direction (de Vries et al. 2021). +detailed modeling of the multiwavelength spec- +trum, which is beyond the scope of this work +(e.g., Zhang et al. 2008). +4.3. Nature of the ridge +It is clear from our 13 and 21 cm maps that the +ridge protruding east well aligns with the pul- +sar motion direction (de Vries et al. 2021). We +therefore suggest that the ridge could be a pul- +sar tail instead of SNR structure. In addition, +our polarization maps show a good alignment +between the B-field and the orientation of the +ridge, which is a common feature of pulsar tails +(e.g., G319.9−0.7, G327.1−1.1, and the Mouse; +Ng et al. 2010; Ma et al. 2016; Yusef-Zadeh & +Gaensler 2005). To confirm the nature of the + +15 +ridge, we compared its radio spectrum with that +of the SNR rim, using extraction regions shown +in Figure 9. +We found a significantly flatter +spectral index of −0.3 in the ridge than −1.1 +in the shell, implying that the ridge likely com- +prises of pulsar wind. Moreover, the RM map +shows a comparable RM as that of the pulsar +and no significant variation in the entire ridge, +therefore, supporting this idea. Although the +ridge extends beyond the pulsar birth site sug- +gested by de Vries et al. (2021), it could be +formed by fast outflow or pulsar wind swept +by the SNR reverse shock. The latter scenario +could also explain the TeV emission (H. E. S. S. +Collaboration et al. 2011). Deeper X-ray obser- +vations along the ridge can reveal any spectral +evolution. +Any enhanced synchrotron cooling +could support the interaction with the super- +nova reverse shock. If confirmed, B1706 PWN +could be in the same evolutionary state as Vela +and Boomerang, both are suggested to be af- +ter the passage of reverse shock. We note that +it remains unclear how the toroidal B-field in +all these sources could be retained. It could be +magnetic pressure against compression by the +shock, or the radio PWNe are recently formed +after the shock interaction. Further numerical +simulations are needed to distinguish between +these cases. +As is mentioned, the polarized +emission of the ridge is brighter than that in the +PWN. We suspect that the ridge region should +be thicker than the B1706 PWN locating at the +tip of it. It is also likely that the particles in the +ridge have a larger density due to mixing with +external materials (e.g., SNR ejecta) or particle +slowing down. Therefore, there are more parti- +cles emitting along the line of sight through the +ridge than the PWN. +5. CONCLUSION +We present a radio study of B1706 PWN us- +ing new and archival ATCA observations at 3, +6, 13, and 21 cm bands. Our main results are +summarized below: +• The 3 and 6 cm total intensity images show +an arc-like structure with a scale of ∼ 4′×2′ +wrapping around PSR B1706−44 in the +north. No radio emission is detected at the +X-ray torus and jet location, and the radio +PWN only brightens beyond 10′′ from the +pulsar. We show that the radio PWN mor- +phology can be fit by a thick torus model +with Doppler boosting effect. The result +suggests a bulk flow velocity of 0.2c, lower +than that in the X-ray torus. +• Our polarization results reveal a toroidal +B-field for the PWN and we estimate a +field strength of ∼ 10µG assuming equipar- +tition between particle and magnetic field +energies. This value suggests a slight decay +compared with that of the X-ray bright re- +gion. +• The ridge of the SNR has elongation and +magnetic field well aligned with the pulsar +proper motion direction. It also has a radio +spectrum flatter than the rest of the shell. +All these suggest that it could be a pulsar +tail instead of a filamentary structure of +the SNR. +ACKNOWLEDGMENTS +We thank the anonymous referee for provid- +ing useful suggestions. C.-Y. Ng is supported +by a GRF grant of the Hong Kong Government +under HKU 17301618. 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B., & Fang, J. 2008, ApJ, 676, 1210, +doi: 10.1086/527466 + diff --git a/JtFRT4oBgHgl3EQfzzhX/content/tmp_files/load_file.txt b/JtFRT4oBgHgl3EQfzzhX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3ee20c30e31c5995cee2752c2522c895bd4f206c --- /dev/null +++ b/JtFRT4oBgHgl3EQfzzhX/content/tmp_files/load_file.txt @@ -0,0 +1,1023 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf,len=1022 +page_content='Draft version January 31, 2023 Typeset using LATEX twocolumn style in AASTeX63 Radio Study of the Pulsar Wind Nebula Powered by PSR B1706−44 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Liu,1 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Ng,1 and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Dodson2 1Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong 2International Centre for Radio Astronomy Research, The University of Western Australia, 35 Stirling Hwy, Crawley, Western Australia ABSTRACT PSR B1706−44 is an energetic gamma-ray pulsar located inside supernova remnant (SNR) G343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='1−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='3 and it powers a compact pulsar wind nebula (PWN) that shows torus and jet structure in X-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We present a radio study of the PWN using Australia Telescope Compact Array (ATCA) observations at 3, 6, 13, and 21 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We found an overall arc-like morphology at 3 and 6 cm, and the “arc” shows two distinct peaks at 6 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The radio emission is faint inside the X-ray PWN and only brightens beyond that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We develop a thick torus model with Doppler boosting effect to explain the radio PWN structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The model suggests a bulk flow speed of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='2c, which could indicate significant deceleration of the flow from the X-ray emitting region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Our polarization result reveals a highly ordered toroidal B-field in the PWN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Its origin is unclear given that the supernova reverse shock should have interacted with the PWN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' At a larger scale, the 13 and 21 cm radio images detected a semi-circular rim and an east-west ridge of G343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='1−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We argue that the latter could possibly be a pulsar tail rather than a filament of the SNR, as supported by the flat radio spectrum and the alignment between the magnetic field and its elongation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Keywords: Pulsar wind nebulae (2215) — Supernova remnants (1667) — Polarimetry (1278) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' INTRODUCTION A pulsar is a compact star that is born in a supernova explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' It emits periodic signals and has a strong surface magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Par- ticles around a pulsar are accelerated to form relativistic wind as the pulsar spins down and loses energy constantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The relativistic wind interacts with the ambient medium and forms a synchrotron nebula, called a pulsar wind nebula (PWN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' A PWN is able to accelerate particles to very high energies, emitting synchrotron ra- diation from the radio to hard X-ray bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' In X-rays, torus-jet features are commonly de- tected in young PWN systems (Kargaltsev & Pavlov 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Ng & Romani 2004, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Theo- ries suggest that torus structure is due to the shocked pulsar wind flowing into the equato- rial region, and jets are wind in the polar re- yihanliu@connect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='hku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='hk Corresponding author: Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Liu gions confined by magnetic hoop stress (see Porth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' In the radio band, how- ever, these features are rarely seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' For in- stance, the Crab, 3C 58, and the PWN inside G292.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='0+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='8 are filled with wisps and filamen- tary structures (Dubner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Bietenholz 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Gaensler & Wallace 2003) instead;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' some show arc-like structure, such as CTB 87 (Kothes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2020) and double-lobed morphology, such as G21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='9, G76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='9+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='0, and DA 495 (Bieten- holz & Bartel 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Arzoumanian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Kothes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Previous radio polariza- tion observations also revealed different mag- netic field configurations in young PWNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' From theoretical works, the radial component of the magnetic field decays faster than the toroidal component (∼r−2 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' ∼r−1) (Porth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Therefore, the B-field beyond the termination shock should be mostly toroidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' This, how- ever, is not supported by observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Only a few cases, including Vela and Boomerang, show toroidal B-field (Dodson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Kothes 2 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2006), but many others, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=', the Crab Nebula, 3C 58, G21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='9, and Dragonfly, have complex or radial field structure (Reich 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The physical cause of such diverse morphology and magnetic field structure among radio PWNe is not fully understood and a larger sample is needed for further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' In this work, we present a new radio study of the PWN powered by the Vela-like pulsar B1706−44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' It is one of the few γ-ray pul- sars detected in the early days with EGRET (McAdam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' It has a characteris- tic age τc =17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='1 kyr and a spin-down power ˙E ≈ 4×1036 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' A recent study with Chan- dra found that the pulsar is moving eastward with a projected velocity of around 130 km s−1 (de Vries et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The association between the pulsar and the nearby supernova remnant (SNR) G343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='1−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='3 is controversial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The rem- nant has a circular shell and an east-west ridge in the southern part (Dodson & Golap 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The pulsar is located at the tip of the ridge, near the center of the shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The SNR distance of ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5 kpc estimated from the Σ–D relation- ship is compatible to the pulsar dispersion mea- sure distance d ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='3 kpc (Cordes & Lazio 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' McAdam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The High Energy Stereoscopic System (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='S) de- tected extended TeV emission west of the pul- sar, which also has some connection with the SNR (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Be- sides, the pulsar powers an X-ray PWN that has compact torus and jet structure (Romani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' A recent study found diffused emission around the torus and a long curved outer-jet (de Vries et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' In this study, we aim to perform high reso- lution radio observations of B1706 PWN for di- rectly comparing with the compact X-ray struc- tures in Chandra images and better understand- ing the PWN magnetic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Previous observations have detected a radio PWN sur- rounding the pulsar (Frail et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Giacani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Dodson & Golap 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Romani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' However, the pulsar emission was not clearly distinguished from the PWN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Some ATCA observations excluded the pulsar emis- sion, but few observations with high resolution and sensitivity were included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Besides, there is no previous study of the magnetic field in the B1706 PWN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' In this paper, we analyze new and archival radio observations of the PWN powered by PSR B1706−44 (hereafter B1706 PWN) and SNR G343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='1−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='3 taken with the Australia Tele- scope Compact Array (ATCA) at 3, 6, 13, and 21 cm images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We employed new observations with high resolution aiming to better study the morphology and polarization information of this PWN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We describe the observations and data reduction process in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Section 3 shows the results and they are discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We summarize our results in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' OBSERVATIONS AND DATA REDUCTION We carried out new radio observations of B1706 PWN at 3 and 6 cm bands with ATCA in 6 km array configurations on 2017 Nov 3 and 2018 Jan 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We also analyzed archival ATCA observations taken in the 3, 6, 13, and 21 cm bands with various array configurations, which have previously been analyzed by Dodson & Go- lap (2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Romani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' All the 3 and 6 cm band data were taken with the pul- sar binning mode, providing a high time reso- lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We then only select off-pulse data to “gate out” the pulsar emission to search for faint PWN structure in the surrounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Table 1 lists the detailed observation param- eters of all the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The 3 and 6 cm obser- vations were performed simultaneously center- ing at 8640 MHz and 4800 MHz, as well as at 8997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5 MHz and 5497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Besides, we have also selected observations in 2003 and 2005 at 8640 MHz and 8384 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Our new observations in 2017 and 2018 were taken after the Compact Array Broadband Backend (CABB) upgrade (Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2011), which increased the band- width from 128 MHz to 2048 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' At 3 cm, the pre-CABB and post-CABB integration times are 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='0 hr and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='2 hr, respectively, with a total u-v coverage from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='8 kλ to 197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='4 kλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' At 6 cm, we have 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='7 hr and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='2 hr pre- and post-CABB 3 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' ATCA observations of B1706 PWN used in this study Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Date Array Center Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Usable Band- No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' of Integration Pulsar Config.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' (MHz) width (MHz) Channels Time (hr) Binning Mode 3 cm 2002 Jan 06 750A 8640 104 13 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='7 Y 2002 Feb 16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5A 8640 104 13 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='2 Y 2002 Apr 11 6A 8640 104 13 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='8 Y 2003 May 18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5C 8384, 8640 104 13 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='7 Y 2003 Jun 23 750C 8384, 8640 104 13 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='2 Y 2003 Aug 02 6D 8384, 8640 104 13 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='8 Y 2005 Nov 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5C 8384, 8640 104 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='3 Y 2005 Dec 27 6A 8384, 8640 104 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='3 Y 2017 Nov 03 6A 8997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5 1728 433 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='9 Y 2018 Jan 11 6C 8997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5 1728 433 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='3 Y 6 cm 2002 Jan 06 750A 4800 104 13 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='7 Y 2002 Feb 16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5A 4800 104 13 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='2 Y 2002 Apr 11 6A 4800 104 13 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='8 Y 2017 Nov 03 6A 5497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5 1728 433 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='9 Y 2018 Jan 11 6C 5497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5 1728 433 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='3 Y 13 cm 1998 May 29 750E 2496 104 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='8 N 1999 Nov 03 210 2496 104 13 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='1 N 21 cm 1998 May 29 750E 1384 104 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='8 N 1998 Sep 15 6A 1384 104 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='3 N 1999 Nov 03 210 1384 104 13 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='1 N 2005 Nov 19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5C 1344, 1472 104 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='6 N 2005 Dec 27 6A 1344, 1472 104 13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='7 N integration time, respectively, covering the u-v space from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='85 kλ to 127 kλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The 13 and 21 cm datasets with good quality have a total inte- gration time of 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='9 hr and 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5 hr, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The u-v coverage of the observations at 13 cm is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='2–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5 kλ and 18–37 kλ, and in the 21 cm band is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='1–30 kλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We processed the data using the MIRIAD package (Sault et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We first flagged the edge channels and data affected by severe radio frequency interference, then followed the stan- dard procedures to calibrate the flux scale, band pass, and gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' After calibration, we formed Stokes I, Q, and U images using multi-frequency synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We weighted the data inversely pro- portion to the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Since the pre- and post- CABB data were taken over 15 yr apart, we formed separated images at 6 cm to check for any morphological changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The result shows no significant variability, we therefore combined 4 all off-pulse data at each frequency for a joint analysis to boost the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We first focused on the region close to the pul- sar, and generated the 3 cm image with the best resolution of full width half maximum (FWHM) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='3′′×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The resulting map has root mean square (rms) noise of around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='06 mJy beam−1, but we did not detect any significant structure near the pulsar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Then we generated images us- ing u-v tapering with a larger FWHM to boost the signal to noise ratio (S/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Tapering sizes are 20′′ for the 3 and 6 cm images and 70′′ for the 13 and 21 cm images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The 3, 6, and 13 cm images were produced with Brigg’s robust pa- rameter of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5 to suppress side lobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' For the 21cm image, we used natural weighting to max- imize the sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' For image decovolution, we first used the task mossdi to clean strong point sources in the Stokes I, Q, and U images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The resid- ual maps were then cleaned simultaneously us- ing pmosmem and the models were restored with beam sizes of 20′′ in 3 and 6 cm and 70′′ in 13 and 21 cm maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The rms noise is around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='06, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='06, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='7, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='8 mJy beam−1 in the Stokes I images and around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='04, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5 mJy beam−1 in the Stokes Q and U images at 3, 6, 13, and 21 cm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Finally, we generated po- larization maps with the task impol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Besides, we have also applied the same procedure to pro- duce full Stokes images of the pulsar using the on pulsed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' RESULTS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Morphology Figure 1 shows the total intensity maps of B1706 PWN during the off-pulse phase in the 3 and 6 cm bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The radio PWN is clearly de- tected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' It is elongated in the east-west direction with a size of ∼ 4′×2′ and wraps PSR B1706−44 in the north.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The eastern part of the PWN is generally brighter, and the flux density peaks at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='7′ and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='9′ east of the pulsar, reaching 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='60 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='85 mJy beam−1 at 3 and 6 cm, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' At 3 cm, the nebula has a more uni- form brightness distribution than at 6 cm, and it shows arc-like structure overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We also found a few protrusions in the PWN, one extends 2′ north of the pulsar and two others northwest and southwest from the pulsar extend towards west 2′ away from the pulsar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' All these pro- trusions are not thicker than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The protru- sion features should correspond to data with u- v coverages from ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5 to ∼ 15 kλ, which are included in the 3 cm data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' These features are therefore less likely to result from the missing flux problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We need more observations to confirm this feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' At 6 cm, the PWN shows two distinct peaks, resembling two lobes brack- eting the pulsar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The eastern part has an el- liptical shape of 2′ in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' It is brighter than the western part and contains 67% of the flux density of the PWN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The western lobe is fainter and more elongated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' It has a size of 2′ × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5′ and is oriented along the northeast-southwest direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Its surface brightness peaks at ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='8′ west of the pulsar and is only about 2/3 of that of the peak in the eastern lobe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' For both radio images, there is a “bay” feature south of the pulsar with no detectable radio emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The 3σ flux density limit is around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='18 mJy beam−1 in both the 3 and 6 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The pulsar emis- sion is clearly detected in the on-pulse data in both 3 and 6 cm bands with flux densities of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='1 mJy beam−1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='1 mJy beam−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' In Figure 2, we compare the radio images with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5–7 keV X-ray image obtained with the Chandra X-ray Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We compared the X-ray torus/jet feature close to the pul- sar with the 3 cm radio image with a beam of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='3′′×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5′′, and found no counterpart of the X- ray PWN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The 3 cm radio image has a rms noise of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='02 mJy beam−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We also smoothed the X- ray image to 20′′, same as that of the radio im- age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Similarly, there is X-ray emission but no radio emission in the inner PWN, and the ra- dio emission only appears in the outer PWN beyond 10′′ from the pulsar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Meanwhile, the X-ray emission fades away in the outer PWN ∼25′′ from the pulsar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The 3 and 6 cm images also show a linear, jet-like structure extending west from the end of the northern X-ray jet with a flux density of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='6 mJy at 6 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' It has a 5 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' "#$ % #$ 3 cm 6 cm Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Total intensity images of the B1706 PWN at the 3 and 6 cm bands in the off-pulse phase with the pulsar emission excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The gray scale bar on the right is in units of Jy beam−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The crosses at the center of the images show the position of PSR B1706−44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The circular beams at the bottom left indicate the beam size of FWHM 20′′ for both images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The rms noise in both bands is around 60 µJy beam−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The contours correspond to total intensity levels of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='18, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='45, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='6 mJy beam−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 1 arcmin E N (a) (b) 2 arcmin Radio Features beyond X-ray Jets X-ray Jets Radio PWN 30 arcsec Torus Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' (a): Comparison between radio and X-ray emis- sion of B1706 PWN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The 6 cm radio emission is shown in blue with the 20′′ restored beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The Chandra 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5–7 keV X-ray image is shown in red, also smoothed to 20′′ resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The inset image is the comparison of the X-ray torus/jet feature and the 3 cm high resolution radio image, with the X-ray torus highlighted by the white region .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' (b): Same as (a) but both are smoothed to 50′′ resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The cross in white indicates PSR B1706−44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' length of ∼3′ and the width is not clearly re- solved by the 6 cm observation (see Figure 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We also find similar emission beyond the south- ern X-ray jet after smoothing the 6 cm intensity map to 50′′ (see Figure 2b) but it is fainter and more diffused than the emission in the north.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' More data are needed to confirm these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Figure 3 shows the total intensity images of the overall SNR in the 13 and 21 cm bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' This is the first time that the 13 cm image is shown, and the SNR shows a similar morphol- ogy to that at 21 cm: there is a ∼40′ semicircu- lar rim in the west and a bright east-west ridge in the south ∼30′ connecting the pulsar to the western rim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' PSR B1706−44 is located at the tip of the ridge rather than at the center of the SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The pulsar emission is visible in the im- ages, since no pulsar binning mode was used in these observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' There is also significant emission detected at the locations of radio outer jets at 6 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Radio spectrum We measured the flux densities of the over- all B1706 PWN and each component in differ- ent bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Background subtraction was per- formed using measurements from nearby source free regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The regions selected to measure 6 13 cm 21 cm Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Total intensity maps of SNR G343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='1−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='3 in 13 cm and 21 cm bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The contours are at levels of 4, 8, 12, and 16 mJy beam−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The gray scale bars on the right have units of Jy beam−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The boxes indicate the field of view of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Both images have a beam size of FWHM 70′′, which is shown in the bottom left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The rms noise at 13 cm is around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='7 mJy beam−1and at 21 cm is around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='8 mJy beam−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' the whole PWN and the eastern and western lobes in both bands are shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The estimated flux densities of the entire PWN are 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='0, 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='7, 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='2±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='4, and 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='4±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='0 mJy in 3 cm, 6 cm, 13 cm, and 21 cm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' All these are plotted in Figure 4 and shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The values at 13 and 21 cm have been subtracted for the pulsar flux density and those at 3 and 6 cm are measured from the off-pulse phase images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Due to the lack of short u-v spacing below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='8 kλ in the 3 and 6 cm observations, the maps have low sensitiv- ity to structures larger than 4′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' In this case, Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Flux density of the pulsar, entire PWN, and different components Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Total Eastern Western Pulsar Bands PWN (mJy) Lobe (mJy) Lobe (mJy) (mJy) 3 cm 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='9±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='1 6 cm 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='7 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='1 13 cm 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='2±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='4 – – 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='1 21 cm 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='4±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='0 – – 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='4 we separately derived the spectrum from the two higher and lower frequency bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' They both give a similar spectral index α ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='3 (Sν ∝ να) (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We note that our flux density measurement of the overall PWN at 6 cm (∼21 mJy) is comparable to the one measured with the VLA (∼28 mJy), although slightly lower (Giacani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' As the VLA data has similar u-v coverage as our ATCA ob- servations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The discrepancy could be due to different choices of source and background re- gions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The flux densities of the eastern lobe are 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='8 mJy and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='9±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='0 mJy in 6 and 3 cm, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' and those of the western lobe are 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='4 mJy and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='8 mJy, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' These give α = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='97 for the eastern lobe and α = +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='78 for the western lobe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We also plotted the 3 and 6 cm images after filtering data to the same u-v coverages, so that data in both bands have the same missing flux problem for a correction of the spectral index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We obtained a spectral index α ∼ 0 for the whole PWN and spectral indices of −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='05 and +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='96 in the eastern lobe and the western lobe, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The latter is rather unusual and more observations are needed to confirm this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We also estimated the pulsar flux densities with extraction region same as the beam size For measurements at 3 and 6 cm, we generated images with only on-pulse bins to show the pul sar emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The pulsar has flux densities of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='1, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='1, and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='4mJy at 3, 6, 13, and 21 cm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The results are B1706 PWN plotted in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Due to low resolution, the Eastern Lobe Western Lobe measurement at 2l cm could be contaminated PSR B1706-44 by the PWN emission, but we note that the ATNF_PSR B1706-44 result is consistent with the pulsed flux density 8 10 2 Frequency (GHz) obtained from the single dish Parkes Radio Tele- ATCA 10" line with the extrapolation of the pulsar spec- trum (Lyne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Jankowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 101° We performed a multiwavelength compari- son with the Chandra X-ray data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We repro- cessed all archival data of B1706 PWN using the CIAO software package (Fruscione et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2006), 108 then extracted the X-ray PWN spectrum using specextract to fit the background subtracted 107 spectrum with an absorbed power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We ob- 1010 1012 1016 1018 1020 tained a photon index I ～ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='53 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='07 and a to- 1014 v(Hz) tal unabsorbed fux fpun=12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='1x 10-13 erg ATCA Sensitivity cm-2 s-1 in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5-7keV range (excluding the 10 °% pulsar emission).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' This gives Qx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='53 in the 10° X-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We plot the spectral energy distribution (SED) of the PWN from radio to X-ray bands tos in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' A comparison with Qradio -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='3 in the radio band suggests a spectral break of 010 Aα ～ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' However, we note that the extrap- olation of the radio and X-ray spectra do not intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' This could be due to the X-ray obser- vations being insensitive to faint emission in the 10 10 1012 10 14 1016 1018 outer PWN region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' In this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' emission from v(Hz) the inner PWN region would dominate and the Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Top: radio spectra of the overall B1706 PWN obtained photon index would be smaller than and different components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The green dashed line shows the that of the overall PWN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' extrapolated spectrum of the pulsar emission from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='4 and We also did such a comparison in high reso- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='4GHz data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The flux densities of PSR B1706-44 at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='4 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='4 GHz are from the ATNF pulsar Catalog (Manchester lution about the X-ray torus region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' For the et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2005) and are shown as triangles with error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Mid- X-ray torus, recent Chandra results show a flux dle: SED of the PWN from radio to X-ray bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The black ftorus=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='26±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='03erg cm-2 s-1 from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5 to 7 keV dots with error bar represent the measurement obtained with ATCA, and the lines in the top right show the best-fit un- with a photon index I=-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='46±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='05 (de Vries absorbed X-ray spectrum obtained from Chandra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Bottom: et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 202l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The flux density in radio is esti- Multiwavelength SED of the X-ray torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The X-ray spec mated in a region shown in Figure 2 and show a trum is extrapolated to 3 cm wavelength (the band in gray), sensitivity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='06 mJy for the torus region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We and the upper limit in red shows the 3o rms noise of the 3 cm observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' compared the radio flux density with the ex-8 trapolated X-ray spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The SED is shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Polarization Figure 5 shows the polarized emission of the PWN and the SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We clipped the 3 and 13 cm maps where the polarization intensity has a signal-to-noise ratio (S/N) <3, total intensity S/N <5, or uncertainty of the position angle (PA) >10◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' For the 6 and 21 cm maps, we ap- plied the same clipping criteria for the polar- ization intensity S/N and PA, but <3 for the total intensity S/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The PWN is highly lin- early polarized in all the bands and the polar- ized emission generally follows the total inten- sity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The 3 cm polarization emission is east-west elongated and the size is ∼ 4′ × 1′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' However, it shows a peak ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='8′ west of the pulsar, different from that of the total intensity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Whereas, the 6 cm polarization image shows two-lobed structure resembling the total intensity emis- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The eastern lobe is brighter and has a peak flux density of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='37 mJy beam−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Polarized emission was detected in both the northern and southern radio features beyond X-ray jets, but the point source in northern one is unpolarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The linear polarization fraction in both 3 and 6 cm bands is around 45% for the entire PWN, and around 85% for the pulsar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The circular po- larization fraction of the pulsar is around 15% in both bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' In the 13 and 21 cm images, we found po- larized emission on large scales including the PWN, the ridge, and the SNR shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The PWN is detected at the end of the ridge with a blob-like structure aligning with the total in- tensity contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The PWN polarized emission is fainter than that of the ridge and the SNR shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The polarization fractions of the PWN in both 13 and 21 cm are around 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The shell structure of the SNR shows significant polarized emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The polarization fractions of the en- tire SNR are around 50% and 40% at 13 cm and 21 cm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Rotation measure and intrinsic magnetic field orientation The observed PA of the polarization vectors are rotated due to Faraday effect in the inter- stellar medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The amount of rotation is pro- portional to the rotation measure (RM) times square of the wavelength (λ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We attempted to derive a high resolution RM map using the 3 and 6 cm data, but it has too large uncer- tainty to be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Therefore, we simply used the RM of the pulsar (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='07 rad m−2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' John- ston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2005) to derotate the polarization vectors at these two bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' To determine the RM of the SNR, we selected edge channels with 32 MHz bandwidth from the 13 and 21 cm data to generate Stokes Q and U maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We used a u- v taper of 80′′ FWHM to boost the S/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The im- ages were then deconvolved using the same pro- cedure as mentioned above, and restored with a circular beam of FWHM 80′′, which is the reso- lution of the lowest frequency band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We formed four PA maps and applied a linear fit to deter- mine the RM value at each pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The result is plotted in Figure 6 and the typical uncer- tainty of the map is ∼1 rad m−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We found that the RM of the SNR varies from −90 rad m−2 to +92 rad m−2 and it is ∼ 0 rad m−2 near the pul- sar position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The latter is in line with the RM of PSR B1706−44 (Johnston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The RM of the PWN and the ridge varies smoothly compared with that in the SNR rim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Figure 5 shows the intrinsic magnetic field di- rection of B1706 PWN at 3 and 6 cm and of the SNR at 13 and 21 cm after correcting for the Faraday effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The magnetic field of the PWN is highly ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' It is oriented along the PWN elongation and wraps around the pulsar in the north, indicating a toroidal configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' For the outer jets, only faint polarized emission is detected, we are therefore only able to de- termine the polarization angle in the brightest regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' At large scale, the magnetic field of the ridge well aligns with its elongation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' It then gradually switches to tangential along the rim of the SNR shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' DISCUSSION 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' PWN structure 9 3 cm 6 cm 13 cm 21 cm 3 cm 6 cm 13 cm 21 cm 3 cm 6 cm 13 cm 21 cm Eastern Lobe Western Lobe 6 cm 6 cm Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Linear polarized intensity maps of B1706 PWN and SNR G343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='1−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='3, overlaid with the total intensity contours and polarization vectors that show the intrinsic B-field orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The contours are at levels of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='18, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='45, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='6 µJy beam−1 for 3 and 6 cm, and 4, 8, 12, and 16 mJy beam−1 for 13 and 21 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The gray scale bars correspond to the polarized intensity level in units of Jy beam−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The vector length is proportional to the polarization intensity, with the bars in lower left indicating polarization intensity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5 mJy beam−1 in 3 and 6 cm maps and 10 mJy beam−1 in 13 and 21 cm maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The black cross represents the position of PSR B1706−44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' As shown at the bottom left, the restoring beam sizes are 20′′ FWHM for 3 and 6 cm images and 70′′ FWHM for 13 and 21 cm images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Our radio intensity maps of the B1706 PWN reveal an overall arc-like morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The emission is bright in the outer region of PWN but faint in the inner part, which is in con- trast to the X-ray emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Similar X-ray– radio anti-correlation is also found in a few other PWNe, including Vela PWN, DA 495, G76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='9+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='0, G319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='9−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='7, G327.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='1−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='1 (Dodson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Kothes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Arzoumanian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Kargaltsev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Ng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The cause is not clearly understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' It was suggested that the radio emission in the inner PWN could be too faint to detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' As the outflow decelerates, the parti- cle number density increases outward, resulting in brighter radio emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' On the other hand, 10 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' RM map of SNR G343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='1−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The RM values vary from −90 rad m−2 to +92 rad m−2, with the solid squares representing positive values and hollow boxes for negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The contours are total intensity levels at 4, 8, 12, and 16 mJy beam−1 at the 13 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The cross indicates the position of PSR B1706−44 and the box at bottom left corresponds to RM value of −50 rad m−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' synchrotron cooling makes the X-ray emission invisible in the outer PWN (Kargaltsev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We applied this idea to B1706: we extrapolated the X-ray spectrum of the torus reported by de Vries et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' (2021) down to the radio band with a simple unbroken power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' This suggests a flux density from ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='01 to ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='2 mJy at 3 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' From our highest resolution 3 cm intensity map, the 3σ limit at this region is ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='06 mJy beam−1, giving a detection limit of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='06 mJy based on the torus area (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' In this case, we still cannot firmly rule out the scenario that the X-ray torus follows a sim- ple unbroken power-law distribution from radio to X-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' A further non-detection with about 3 times better sensitivity is required to reject such a scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Alternatively, we note that the injected spectrum could have an intrinsic spec- tral break, if the particle acceleration is due to magnetic reconnection in the termination shock or Weibel instability (Lyubarsky & Kirk 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Weibel 1959).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' In this case, the radio emission could be a few orders of magnitude fainter, mak- ing it very difficult to detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Deeper radio ob- servations are needed to discriminate between these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 11 3 β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='0 β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='1 β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='2 2 1 1 1 0 0 0 1 1 1 2 2 2 3 2 0 2 2 0 2 3 β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='3 6cm 3cm 2 1 0 1 2 3 2 The total intensity images at 6 and 3 cm are shown at the bottom panels for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Our high resolution radio maps reveal over- following a similar procedure as Ng & Romani all arc-like structure for B1706 PWN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' which is (2004, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We built a torus in 3D with cir- similar to Vela, Boomerang (G106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='6+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content="9), and cular cross-section and an outer radius of 2', as- G76." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='9+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='0 (Dodson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2003: Kothes et al suming uniform and isotropic emission inside 2006: Arzoumanian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' It was sug We set the viewing angle between the torus axis gested that this could be caused by the pas- and the line of sight to be 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='3 (Romani et al 2005) and the outer boundary radius 6 times sage of supernova reverse shock and a thick toroidal model has been developed (Chevalier of the inner boundary radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We considered & Reynolds 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We tried to apply the same Doppler boosting effect following Pelling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' model to our 3cm image, but found that it fails (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The apparent intensity I is to explain the lack of emission in the bay south I α (1 - n - β)-(1-I) Io, (1) of the pulsar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Indeed the model is always sym- metric and also cannot explain the tongue-like where n is the unit vector from the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' morphology of the Boomerang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' It is worthy B = w/c is the assumed velocity of the radial post-shock bulk flow, is the photon index in an asymmetric torus feature close to the pulsar the rest frame, and Io is the intrinsic intensity of due to the Doppler boosting effect, such that synchrotron emission taken to be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We emission is brighter if the particles are moving projected the model onto the plane of the sky toward the observer and vice verse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We there- to generate a 2D brightness map for comparison fore added Doppler boosting effect to the model with the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Figure 7 shows models of a ra-12 dio B1706 PWN having a thick equatorial torus and the Doppler boosting effect with different β values in the bulk flow, as well as the 6 and 3 cm radio images of the B1706 PWN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' In the scenario of β = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=', negligible Doppler boosting), the model shows two equatorial lobes both having a brightness peak inside, and a fainter region between the lobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We note that this model is not only horizontally symmetric, but also ver- tically symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Considering Doppler boost- ing will result in enhanced brightness in an ap- proaching bulk flow and reduced brightness in flows leaving away, such that the upper part is brighter than the lower part in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The brightness of the upper part becomes compara- ble to or even overwhelms that of the lobes in the scenario of β ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' In the latter case, the model has a kidney-shape feature wrapping the pulsar with a single peak north of the pulsar region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We suggest that the model can cap- ture characteristic features of the radio PWN observed, including the overall arc-like PWN wrapping the pulsar in the north and the faint bay in the south.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Comparisons between these models and the observations show that a con- stant value β ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='2 over the entire torus gives the best result of the 3 cm PWN, while the 6 cm PWN can be better described by β ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Our 6 cm image is slightly different from the 3 cm image (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=', the gap between the eastern and western parts), so that the best fit β are slightly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' To resolve this, we consider a spectral gradient across the PWN, as motivated by the different spectra between the eastern and western parts (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We fix β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='2 and consider that I0 depends on frequency as I0,ν = I0,ν0 · ( ν ν0 )α, (2) where I0,ν is the intrinsic intensity in frequency ν, and I0,ν0 is the intensity in a reference fre- quency ν0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We also assume that ν0 = 10 GHz and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='9 xEW Rpwn , (3) where xEW is the position of a point in the east- west direction with a scale of the PWN radius Rpwn, indicating α = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='9 at the eastern end of the PWN and α = +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='9 at the western end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We then obtain the model at 3 and 6 cm and also extrapolate it to around 30 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Besides, we compare all these models with the 3 and 6 cm ATCA images and 30 cm ASKAP image (Norris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2019), and all these are shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Our simulation shows the PWN with a much brighter eastern part at 30 cm, a larger brighter eastern lobe and a smaller western lobe at 6 cm, and lobes connected from the north of the pulsar at 3 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' These models can reproduce the main morphological features of the observed PWN in these bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' we also tried different values of β but found that 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='2 gives the best fit result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' A direct comparison with βX = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='7 in the X-ray emitting region (Romani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2005) suggests deceleration along the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' This also implies the particle accumulation, which could give rise to the observed anti-correlation between the ra- dio and X-ray emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Our modeling result shows that B1706 PWN could have a toroidal structure in 3D, and it appears as arc-like due to Doppler effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' This model could be used to explain the arc-like mor- phology found in other radio PWNe, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=', Vela and Boomerang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Finally, we note that toroidal structure can be resulted from a toroidal B-field as simulations suggest (Porth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' This is supported by our polarization result, which reveals a toroidal B-field configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Equipartition magnetic field We estimated the equipartition B-field strength of the PWN Beq = [6π(1 + k)c12LsynΦ−1V −1 pwn]2/7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' (4) where Vpwn is the emission volume,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Lsyn is the synchrotron luminosity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Φ is a filling factor for the emission (it is usually taken as 1 even though not 100% of the volume emits),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' k is the ratio between electron energy and the energy of heavy particles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' c12 is a constant related to syn- chrotron radiation process and weakly depends on the frequency range (Pacholczyk 1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We selected PWN flux density at 6 cm as a reference and assumed a simple power-law spectrum with 13 30cm 6cm 3cm ASKAP 30cm ATCA 6cm ATCA 3cm Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Models of the B1706 PWN in different wavelengths with a gradient in the east-west direction (the top row), as well as the observation results of B1706 PWN at 30, 6, and 3 cm (the bottom row) taken with ASKAP (Norris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2019) and ATCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' a spectral index Q ~ -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='3 from 107 to 1011 Hz to We also roughly estimated the equapartition obtain Lsyn = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='9 × 1030d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='3 erg s-1, where d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='3 B-field of the linear radio feature beyond the X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' is the source distance in units of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='3 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' To es- ray jet northwest of the pulsar, assuming it asso- timate the volume of the PWN, we assumed an ciated with the PWN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Following the same pro- oblate spheroid for the emission volume in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' cedure as above and taking the emission volume The oblate spheroid is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content="6' 4." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content="2' 4." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content="2' in size." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' as a cylinder, the flux density measurements at which gives a volume of Vpwn=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='2 × 1056 cm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 3 and 6cm gives These give Beg = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='5(1 + k)2/7@-2/7 d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='2/7 μG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The result is similar if we exclude the point source, Taking k=0 and @=1, the B-field is in the order of 10 uG, slightly lower than 27 μG and 15 μG Bts* = 59(1 + k)2/7 @-2/7 d,2 uG estimated for the inner and outer X-ray PWN, respectively (de Vries et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' However, the This is significantly higher than that of the main decay in B-field strength is much slower than radio PWN, but is comparable to that of the B α 1/r predicted by theory (Porth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2017) X-ray bright inner PWN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Identification the as- given that the radio PWN is 5 times larger sociation between the PWN and this feature than that in the X-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' and requires more following observations and14 Eastern Lobe Western Lobe Shell Radio Features beyond X-ray Jets Ridge PSR B1706-44 and the PWN 130 km/s Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Total intensity image of SNR G343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='1−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='3 at 13 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Inset: Zoom in of the PWN at 6 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Extraction regions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=', shell, ridge, the whole PWN, and the eastern and western lobes) for spectral measurements are indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The circular beams of 20′′ (inset) and 70′′ (SNR) are shown bottom left of the images as scale references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The dashed vector shows the pulsar motion direction (de Vries et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' detailed modeling of the multiwavelength spec- trum, which is beyond the scope of this work (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=', Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Nature of the ridge It is clear from our 13 and 21 cm maps that the ridge protruding east well aligns with the pul- sar motion direction (de Vries et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We therefore suggest that the ridge could be a pul- sar tail instead of SNR structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' In addition, our polarization maps show a good alignment between the B-field and the orientation of the ridge, which is a common feature of pulsar tails (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=', G319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='9−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='7, G327.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='1−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='1, and the Mouse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Ng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Yusef-Zadeh & Gaensler 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' To confirm the nature of the 15 ridge, we compared its radio spectrum with that of the SNR rim, using extraction regions shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We found a significantly flatter spectral index of −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='3 in the ridge than −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='1 in the shell, implying that the ridge likely com- prises of pulsar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Moreover, the RM map shows a comparable RM as that of the pulsar and no significant variation in the entire ridge, therefore, supporting this idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Although the ridge extends beyond the pulsar birth site sug- gested by de Vries et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' (2021), it could be formed by fast outflow or pulsar wind swept by the SNR reverse shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The latter scenario could also explain the TeV emission (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Deeper X-ray obser- vations along the ridge can reveal any spectral evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Any enhanced synchrotron cooling could support the interaction with the super- nova reverse shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' If confirmed, B1706 PWN could be in the same evolutionary state as Vela and Boomerang, both are suggested to be af- ter the passage of reverse shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We note that it remains unclear how the toroidal B-field in all these sources could be retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' It could be magnetic pressure against compression by the shock, or the radio PWNe are recently formed after the shock interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Further numerical simulations are needed to distinguish between these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' As is mentioned, the polarized emission of the ridge is brighter than that in the PWN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We suspect that the ridge region should be thicker than the B1706 PWN locating at the tip of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' It is also likely that the particles in the ridge have a larger density due to mixing with external materials (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=', SNR ejecta) or particle slowing down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Therefore, there are more parti- cles emitting along the line of sight through the ridge than the PWN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' CONCLUSION We present a radio study of B1706 PWN us- ing new and archival ATCA observations at 3, 6, 13, and 21 cm bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Our main results are summarized below: The 3 and 6 cm total intensity images show an arc-like structure with a scale of ∼ 4′×2′ wrapping around PSR B1706−44 in the north.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' No radio emission is detected at the X-ray torus and jet location, and the radio PWN only brightens beyond 10′′ from the pulsar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' We show that the radio PWN mor- phology can be fit by a thick torus model with Doppler boosting effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The result suggests a bulk flow velocity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content='2c, lower than that in the X-ray torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' Our polarization results reveal a toroidal B-field for the PWN and we estimate a field strength of ∼ 10µG assuming equipar- tition between particle and magnetic field energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' This value suggests a slight decay compared with that of the X-ray bright re- gion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' The ridge of the SNR has elongation and magnetic field well aligned with the pulsar proper motion direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' It also has a radio spectrum flatter than the rest of the shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' All these suggest that it could be a pulsar tail instead of a filamentary structure of the SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFRT4oBgHgl3EQfzzhX/content/2301.13651v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank the anonymous referee for provid- ing useful suggestions.' 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Preprints 2022, 1, 0. +https://doi.org/ +Publisher’s Note: MDPI stays neutral +with regard to jurisdictional claims in +published maps and institutional affil- +iations. +Copyright: +© 2022 by the authors. +Licensee MDPI, Basel, Switzerland. +This article is an open access article +distributed +under +the +terms +and +conditions of the Creative Commons +Attribution (CC BY) license (https:// +creativecommons.org/licenses/by/ +4.0/). +Article +An Inexact Feasible Quantum Interior Point Method for Linearly +Constrained Quadratic Optimization +Zeguan Wu 1 +, Mohammadhossein Mohammadisiahroudi 1 +, Brandon Augustino 1 +, +Xiu Yang 1 +and Tamás Terlaky 1,* +1 +Department of Industrial and Systems Engineering, Lehigh University +* +Correspondence: terlaky@lehigh.edu +Abstract: Quantum linear system algorithms (QLSAs) have the potential to speed up algorithms that rely +on solving linear systems. Interior Point Methods (IPMs) yield a fundamental family of polynomial-time +algorithms for solving optimization problems. IPMs solve a Newton linear system at each iteration to find +the search direction, and thus QLSAs can potentially speed up IPMs. Due to the noise in contemporary +quantum computers, such quantum-assisted IPM (QIPM) only allows an inexact solution for the Newton +linear system. Typically, an inexact search direction leads to an infeasible solution. In our work, we +propose an Inexact-Feasible QIPM (IF-QIPM) and show its advantage in solving linearly constrained +quadratic optimization problems. We also apply the algorithm to ℓ1-norm soft margin support vector +machine (SVM) problems and obtain the best complexity regarding dependence on dimension. This +complexity bound is better than any existing classical or quantum algorithm that produces a classical +solution. +Keywords: Quantum Computing; Interior Point Method; Quadratic Optimization +MSC: 90C20; 90C51; 81P68 +1. Introduction +Linearly constrained quadratic optimization (LCQO) is defined as optimizing a convex +quadratic objective function over a set of linear constraints. This problem reduces to linear opti- +mization when the the quadratic objective function is linear. LCQO has rich theory, algorithms, +and applications. Many machine learning problems are LCQO problems, including variants of +least square problems and variants of support vector machine problems [1,2]. Some important +optimization algorithms also have LCQO subproblems, e.g. sequential quadratic programming +[1]. +The modern age of IPMs launched by Karmarkar’s invention of the projective method for +linear optimization (LO). Since then, a lot of variants of IPMs have been studied for not only +LO problems but also for nonlinear optimization problems, including LCQO problems [3,4]. +Contemporary IPMs look for the optimal solution by moving in a neighbourhood of the +central path. IPMs can be divided into two classes: feasible or infeasible. Feasible IPMs start +with a feasible solution and keep feasibility; infeasible IPMs start with an infeasible interior +solution and so do not require a feasible solution to start with. For LCQO problems with n +variables, some feasible IPMs produce an ϵ-approximate solution in at most O(√n log(1/ϵ)) +IPM iterations, while infeasible IPMs require O(n2 log(1/ϵ)) IPM iterations to generate an +ϵ-approximate solution [5,6]. +At each IPM iteration a linear system needs to be solved to obtain the search direction, +called the Newton direction. Such a Newton linear system is traditionally in the form of +augmented system or the normal equation system. Classically these linear systems can be +solved exactly using Bunch-Parlett factoriztion if the matrices in the systems are symmetric +indefinite [7], or Cholesky factorization if the matrices are symmetric positive definite. The +arXiv:2301.05357v1 [math.OC] 13 Jan 2023 + +BY2 of 21 +complexity of solving the linear systems is O(n3). The linear systems can also be solved +inexactly using some inexact methods, e.g., Krylov subspace methods. Such inexact methods +might take less iterations if the desired accuracy of the solutions to the linear systems is +not high. But such inaccuracy of the solutions to the linear systems, i.e., inaccuracy of the +search directions, might result in infeasibility of the solutions generated by IPMs. To maintain +feasibility of solutions, [8] introduces the so-called orthogonal subspace system (OSS) for LO +problems. A feasible solution can be recovered from an inexact solution to OSS. We extend +their OSS for LO prolems to LCQO problems and provide an efficient method to construct +the OSS. With the OSS, we can obtain an inexact feasible IPM – solving for search direction +inexactly but maintaining the feasibility of solution throughout the process of our IPM. The +feasibility of solution gives better IPM iteration complexity and the bottleneck becomes solving +the linear system, OSS. +With the development of quantum technology, many quantum-assisted algorithms have +been proposed for many optimization problems. Following the invention of quantum algo- +rithms for solving linear systems of equations [9], many researchers are encouraged to study +whether QLSAs would yield quantum speedups in classical algorithms. In particular, QIPMs +have been proposed for for LO problems [10,11] and semidefinite optimization problems [12] +that utilize QLSAs to solve the Newton linear system that arises in each iteration of IPMs. +Similar ideas have also been applied to accelerate the solution of some machine learning +applications, such as linear regression [13] and the support vector machine training problem +[14]. However, linearly constrained quadratic optimization problems, which are fundamental +to both optimization and machine learning, have not been formally studied in the quantum +literature yet. +The remaining part of this paper is organized as follow: in Section 2, we introduce IPMs +for LCQO and the OSS system; in Section 3, we discuss how to use quantum algorithms to find +the Newton directions and analyze the complexity of our IF-QIPM; in Section 4, we apply our +IF-QIPM to support vector machine problem. Discussions are provided in Section 5, and some +technical proofs are moved to the Appendix. +2. Preliminary +2.1. Notations +In this section, we introduce notations we use. Vectors are typically represented by lower- +case letters. For n-dimensional all-zero vector, we represent it with 0n if the dimension is n, +or simply 0 if the dimension is obvious in the context. For n-dimensional all-one vector, we +represent it with en, or simply e if the dimension is obvious in the context. +Matrix are typically represented with upper-case letters. For n-dimensional identity +matrix, we represent it with In×n, or simply I if the dimension is obvious in the context. For +n × m-dimensional all-zero matrix, we represent it with 0n×m, or simply 0 if the dimension if +obvious in the context. For a general n × m-dimensional matrix H, we represent its ith row by +Hi· and jth column by H·j and (i, j) element by Hij or Hi,j. +For real-valued functions f1 and f2 and f3, we write +f1 = O( f2) +if there exits a positive number k4 such that f1 ≤ k4 f2. We write +f1 = ˜O f3( f2) +if there exists a positive number k5 such that f1 ≤ k5 f2 × poly log( f3). + +3 of 21 +2.2. IPMs for LCQO +In this work, LCQO is defined as follow. +Definition 1 (LCQO Problem). For vectors b ∈ Rm, c ∈ Rn, and matrix A ∈ Rm×n with +rank(A) = m ≤ n, and symmetric positive semidefinite matrix Q ∈ Rn×n, we define the primal and +dual LCQO problems as: +(P) +min cTx + 1 +2xTQx, +s.t. Ax = b, +x ≥ 0, +(D) +max bTy − 1 +2xTQx, +s.t. ATy + s − Qx = c, +s ≥ 0, +(1) +where x ∈ Rn is the vector of primal variables, and y ∈ Rm, s ∈ Rn are vectors of the dual variables. +Problem (P) is called the primal problem and (D) is called the dual problem. +The full-row-rankness of matrix A implies that there is no all-zero row in matrix A. We +further make the following assumption on matrix A. +Assumption 1. Matrix A has no all-zero columns. +Remark 1. When matrix A has zero columns, without loss of generality, let us say the nth column is +all-zero, then we can introduce a new variable xn+1 and rewrite the problem into +min +�c +0 +�T� x +xn+1 +� ++ 1 +2 +� x +xn+1 +�T� Q +0n×1 +01×n +0 +�� x +xn+1 +� +, +s.t. +�A·1 +· · · +A·(n−1) +0m×1 +0m×1 +0 +· · · +0 +1 +−1 +�� x +xn+1 +� += +�b +0 +� +, +x ≥ 0, xn+1 ≥ 0. +The new problem is equivalent to the original one. The new problem is still a LCQO problem and has +fewer all-zero columns than the original problem. So we can repeat the procedure to eliminate all the +all-zero columns. In the worst case, we will get a new LCQO problem satisfying Assumption 1 with +2n − m variables and n constraints. +Assumption 2. There exists a solution (x, y, s) such that +Ax = b, x > 0, ATy + s − Qx = c, and s > 0. +The set of primal-dual feasible solutions can be defined as +PD := +� +(x, y, s) ∈ Rn × Rm × Rn : Ax = b, ATy + s − Qx = c, (x, s) ≥ 0 +� +and the set of interior feasible primal-dual solutions can be defined as +PD0 := +� +(x, y, s) ∈ Rn × Rm × Rn : Ax = b, ATy + s − Qx = c, (x, s) > 0 +� +. +According to the strong duality, the set of optimal solutions can be defined as +PD∗ := {(x, y, s) ∈ PD : xs = 0}, + +4 of 21 +where xs denotes the Hadamard, i.e., componentwise product of x and s. Let ϵ > 0, then the +set of ϵ-approximate solutions to Problem (1) can be defined as +PDϵ := +� +(x, y, s) ∈ PD : xTs ≤ nϵ +� +. +(2) +Let X and S be diagonal matrices of x and s, respectively. Under Assumption 2, for all µ > 0, +the perturbed optimality conditions +Ax = b, +ATy + s − Qx = c, +XSe = µe, +(x, s) ≥ 0 +(3) +have a unique solution (x(µ), y(µ), s(µ)) that defines the primal and dual central path +CP := +� +(x, y, s) ∈ PD0|xisi = µ for i ∈ {1, . . . , n}; for µ > 0 +� +. +IPMs apply Newton’s method to solve system (3). At each iteration of infeasible IPMs, a +candidate solution to the primal-dual LCQO pair in (1) is updated by solving the following +linear system to find the Newton direction: +� +� +A +0 +0 +−Q +AT +I +S +0 +X +� +� +� +� +∆x +∆y +∆s +� +� = +� +� +rp +rd +rc +� +�, +(4) +where (rp, rd, rc) are residuals defined as +rp = b − Ax +rd = c − ATy − s +rc = σµe − XSe, +where σ ∈ (0, 1) is the barrier reduction parameter. If rp = 0 and rd = 0, then the solutions +(x, y, s) are primal and dual feasible. Alternatively, we can also define residuals in different +ways as we will show later. Once the Newton direction is found, one can move along the +direction but has to stay in a neighbourhood of the central path, which is defined at the end of +this section. +When the linear system (4) is solved inexactly, that actually leads to inexact infeasible +IPMs. Many researchers have analyzed the performance of inexact infeasible IPMs (II-IPMs). +For LCQO problems, [6] propose an II-IPM using an iterative method to solve the Newton +systems and obtain O(n2 log( 1 +ϵ)) IPM iteration complexity. Here IPM iteration complexity +does not include the complexity contributed by linear system solvers. However, it is known +that feasible IPMs for LCQO problems can achieve O(√n log( 1 +ϵ)) IPM iteration complexity +[15–17]. In [5], the author provides a general inexact feasible IPM for LCQO problems but has +not discussed how to maintain feasibility when inexact linear system solvers are used. In this +work, we will fill the gap by using a method inspired by some QIPM results [8,12] as we shall +discuss later. +In this paper, we consider the following neighborhood of the central path +N2(θ) := +� +(x, y, s) ∈ PD0|∥XSe − µe∥2 ≤ θµ +� +, +(5) + +5 of 21 +where θ ∈ (0, 1). +2.3. Orthogonal Subspaces System +Assuming that (x, y, s) ∈ PD0, to maintain the feasibility of the primal and dual variables, +the first two linear equations in system (4) need to be solved with rp = 0 and rd = 0 exactly, +which can be guaranteed if ∆x lies in the null space of A, denoted as Null(A), and ∆s = +Q∆x − AT∆y. Accordingly, we can rewrite system (4) if we represent ∆x by a basis of Null(A). +To do so, we can partition matrix A to A = +� +AB +AN +� +, where AB is a basis of A. Then we +construct the following matrix +V = +� +A−1 +B AN +−I +� +. +Matrix V has full column rank and satisfies AV = 0, i.e., the columns of V span the null space +of A. Let ∆x = Vλ, where λ ∈ Rn−m is the unknown coefficient vector for ∆x. Subsequently, +we can rewrite system (4) by substituting ∆x and ∆s in the third equation as +SVλ + X +� +QVλ − AT∆y +� += rc ⇔ +� +SV + XQV +−XAT� · +� λ +∆y +� += rc. +(6) +A similar system was proposed and called "Orthogonal Subspaces System" (OSS) in [8,12] and +we use the same name in this work. The matrix in the OSS system (6) is of size n × n, and it is +nonsingular. Even if the OSS system is solved inexactly, primal and dual feasibility is preserved +by computing ∆x = Vλ and ∆s = QVλ − AT∆y. Thus, we can conclude that residual will +only show up in the third equation of (4), i.e., rp = 0 and rd = 0. This nice property of the OSS +system brings much convenience in the analysis of the proposed inexact IPM, and allows to +prove the to-date best iteration complexity. +3. Inexact Feasible IPM with QLSAs +In this section, we propose our IF-QIPM for LCQO problems. We start with the IF-IPM +structure introduced by [5] and describe how to convert it into an IF-QIPM. Then we analyze +the construction of the OSS system, and finally, we analyze the complexity for our IF-QIPM. +3.1. IF-IPM for LCQO +In [5], the author studies a general conceptual form IF-IPM for QCLO problems by +assuming the feasibility of primal and dual variables, which induces the following system +� +� +A +0 +0 +−Q +AT +I +S +0 +X +� +� +� +� +∆x +∆y +∆s +� +� = +� +� +0 +0 +rc +� +�, +(7) +where rc = σµe − XSe with σ ∈ (0, 1) being the reduction factor of the central path parameter +µ, i.e., µnew = σµ. When system (7) is solved with rc = σµe − XSe inexactly yielding an +error r, if ∥r∥2 ≤ δ∥rc∥2 for some δ ∈ (0, 1), then the inexact IPM produces an ϵ-approximate +solution to Problem (1) in O(√n log(1/ϵ)) iterations. The author of [5] does not specify how +to solve system (7) inexactly, how to preserve primal and dual feasibility, and how to satisfy +the convergence conditions described in [5]. Specifically, the convergence conditions are posed +on the right-hand-side and the inexactness error of the system (7). + +6 of 21 +Now we present a general procedure how to solve system (7) inexactly, while the inexact- +ness error occurs only in the third equation of system (7). Let (λ, ∆y) be an inexact solution for +system (6) and r be the corresponding inexactness error, so we have +� +SV + XQV +−XAT� · +� λ +∆y +� += rc + r. +The corresponding Newton step +∆x = Vλ +∆s = Q∆x − AT∆y +satisfies +� +� +A +0 +0 +−Q +AT +I +S +0 +X +� +� · +� +� +∆x +∆y +∆s +� +� = +� +� +0 +0 +rc + r +� +�. +Recall that once (λ, ∆y) is determined, then (∆x, ∆s) is also determined. An interesting property +is that, if (λ, ∆y) and (∆x, ∆y, ∆s) can be deduced from each other, then the OSS system and +system (7) yield the same error term r. Hence the convergence conditions built upon system (7) +can be directly examined using the residual rc and error r of the OSS system. Let ϵOSS be the +target accuracy of the OSS system (6), i.e., +∥(λ − λ∗, ∆y − ∆y∗∥2 ≤ ϵOSS, +where (λ∗, ∆y∗) is the accurate solution. To make the IF-IPM converge, according to [5], we +need +∥r∥2 = +���� +� +SV + XQV +−XAT� · +� λ +∆y +� +− rc +���� +2 +≤ +��� +SV + XQV +−XAT��� +2ϵOSS +≤ δ∥rc∥2. +So +ϵOSS ≤ δ +∥rc∥2 +��� +SV + XQV +−XAT��� +2 +is sufficient for the IF-IPM to converge. We present the IF-IPM in Algorithm 1. +Algorithm 1 Short-step IF-IPM +1: Choose ϵ > 0, δ ∈ (0, 1), θ ∈ (0, 1), β ∈ (0, 1) and σ = (1 − +β +√n). +2: k ← 0 +3: Choose initial feasible interior solution (x0, y0, s0) ∈ N (θ) +4: while (xk, yk, sk) /∈ PDϵ do +5: +µk ← (xk)Tsk +n +6: +ϵk +OSS ← δ∥rk +c∥2/ +��� +SkV + XkQVk +−XkAT��� +2 +7: +(λk, ∆yk) ← solve system (6) with accuracy ϵk +OSS +8: +∆xk = Vλk and ∆sk = −AT∆yk +9: +(xk+1, yk+1, sk+1) ← (xk, yk, sk) + (∆xk, ∆yk, ∆sk) +10: +k ← k + 1 +11: end while +12: return (xk, yk, sk) + +7 of 21 +In the quantum-assisted IF-IPM, or IF-QIPM, we are proposing to accelerate Step 7 using +quantum algorithms. In the next sections, we investigate how to use quantum algorithms to +build and solve the OSS system and get the Newton direction. +3.2. IF-QIPM for LCQO +The pseudocode of our IF-QIPM is presented in Algorithm 2. At each iteration of the +IF-QIPM, we construct and solve system (6) and compute the Newton direction using quantum +algorithms. +Algorithm 2 Short-step IF-QIPM +1: Choose ϵ > 0, δ ∈ (0, 1), θ ∈ (0, θ0), β ∈ (0, 1) and σ = (1 − +β +√n). +2: k ← 0 +3: Choose initial feasible interior solution (x0, y0, s0) ∈ N (θ) +4: while (xk, yk, sk) /∈ PDϵ do +5: +µk ← (xk)Tsk +n +6: +ϵk +OSS ← δ∥rk +c∥2/ +��� +SkV + XkQVk +−XkAT��� +2 +7: +(λk, ∆yk) ← solve system (6) with accuracy ϵk +OSS quantumly +8: +∆xk = Vλk and ∆sk = −AT∆yk +9: +(xk+1, yk+1, sk+1) ← (xk, yk, sk) + (∆xk, ∆yk, ∆sk) +10: +k ← k + 1 +11: end while +12: return (xk, yk, sk) +Here θ0 < 1 and its value will be discussed later. First, we introduce some notations to +simplify the OSS system. In the kth iteration of Algorithm 2, let +Mk = +� +SkV + XkQV +−XkAT� +, zk = +� λk +∆yk +� +. +Then the OSS system can be rewritten as +Mkzk = rk +c. +As discussed in [8], to solve the OSS system (6) using quantum algorithms, we need to first +rewrite it as the normalized Hermitian OSS system +1 +√ +2 +��Mk�� +F +� +0 +Mk +(Mk)T +0 +� +· +� 0 +zk +� += +1 +√ +2 +��Mk�� +F +. +�rk +c +0 +� +. +(8) +To use the QLSAs mentioned earlier, we need to turn the linear system (8) into a quantum +linear system using the block-encoding introduced in [18]. To this end, we first decompose the +coefficiennt matrix in linear system (8) as +1 +√ +2 +��Mk�� +F +� +0 +Mk +(Mk)T +0 +� += +1 +√ +2 +��Mk�� +F +� +0 +0 +(Mk)T +0 +� ++ +1 +√ +2 +��Mk�� +F +�0 +Mk +0 +0 +� +, +(9) + +8 of 21 +where +� +0 +0 +(Mk)T +0 +� += +� +� +0n×n +0n×n +0n×n +0(n−m)×n +VT +0(n−m)×n +0m×n +0m×n +−A +� +�× +� +� +� +� +0n×n +0n×n +Sk +0n×n +0n×n +0n×n +� +� + +� +� +0n×n +0n×n +0n×n +0n×n +Q +0n×n +0n×n +0n×n +In×n +� +� +� +� +0n×n +0n×n +Xk +0n×n +Xk +0n×n +� +� +� +�. +(10) +To compute matrix V, we need to find a basis matrix AB of matrix A and we need to compute +the inverse matrix A−1 +B . Both steps are nontrivial and can be expensive. However, we can +reformulate the LCQO problem as follows +min cTx + 1 +2xTQx +s.t. +�I +0 +A +0 +I +−A +�� +� +x′ +x′′ +x +� +� = +� b +−b +� +x ≥ 0, x′ ≥ 0, x′′ ≥ 0. +In this case, we have an obvious basis +AB = +�I +0 +0 +I +� +and matrix V can be constructed efficiently +V = +� +A−1 +B AN +−I +� += +� +� +�I +0 +0 +I +�� A +−A +� +−I +� +� = +� +� +A +−A +−I +� +�. +Since matrix A has no all-zero rows, matrix V has no all-zero rows either. This property of the +reformulation is useful in the analysis of the proposed IF-QIPM but we do not want to build +the complexity analysis on the reformulated problem. So without loss of the generality we may +make the following assumption. +Assumption 3. Matrix A is of the form A = +� +I +AN +� +. +To simplify the analysis, we further assume the input data are integers. +Assumption 4. The input data of Problem (1) are integers. +Following from the two assumptions above, we have the following lemma. +Lemma 1. Matrix V equals to +V = +�AN +−I +� +and +min +i=1,...,n{∥Vi·∥2 +2} = min{1, +min +i=1,...,m ∥(AN)i·∥2 +2} = 1, +where Vi· and (AN)i· are the ith row of V and AN, respectively. + +9 of 21 +Now we are ready to give θ0 in our definnition of central path neighbourhood, see (5). We +set +θ0 = min +� +1 +3√n, +1 +4∥QVVT∥F + 1 +� +. +(11) +We also define ωk as the maximum of the values of primal variables and dual slack variables in +the kth iteration. +Definition 2. Let (xk, yk, sk) be the a candidate solution for Problem (1), then +ωk = +max +i∈{1,...,n}{xk +i , sk +i }. +In this work, we assume access to quantum random access memory, QRAM. Then Step +7 of Algorithm 2 consists of three parts: 1.) use block-encoding to build system (8); 2.) use +QLSAs to solve system (8); 3.) use quantum tomography algorithms (QTAs) to extract classical +solution. We use the block-encoding methods introduced in [18] to block-encode linear system +(8). +Proposition 1. In the kth iteration of Algorithm 2, use the block-encoding methods introduced in [18] +and the decomposition described in equations (9) and (10), a +�� +∥V∥2 +F + ∥A∥2 +F +√ +2ωk +∥Mk∥F +( +√ +2∥Q∥F + +√ +2 + 1), O(poly log(n)), ϵQLSA +κ3 +Mk +� +-block-encoding of the matrix in the system (8) can be implemented efficiently and the complexity will be +dominated by the complexity of the QLSA step. Here ϵQLSA is the accuracy required for the QLSA step +and κMk is the condition number of matrix Mk. +Proof. See Appendix A for proof. +The complexity contributed by block-encoding is negligible compared with the complexity +contributed by QLSAs and QTAs so we ignore it here. To establish the total complexity +contributed by QLSAs and QTAs, we first need to analyze the accuracy of QLSA characterized +by ϵQLSA and the accuracy of QTA characterized by ϵQTA and their relationship. +In each iteration, we use a QLSA to solve the block-encoded version of system (8) and get +an ϵQLSA-approximate solution. Then we use a QTA to extract an ϵQTA-approximate solution +from the quantum machine. Here, for QLSA and QTA, ˜z is an ϵ-approximate solution of z +means +���� +˜z +∥˜z∥2 +− +z +∥z∥2 +���� +2 +≤ ϵ, +which is different from the concept of ϵ-approximate solutions defined in (2). +Similar to [11], the QLSA we use is proposed by [19] and the QTA we use is proposed by +[20]. Following the argument in Section 2 in [11], we can set the relationship among ϵQLSA, +ϵQTA, and ϵk +OSS as +ϵQLSA = ϵQTA = 1 +2 · +√ +2∥Mk∥F +∥rkc∥2 +ϵk +OSS, +(12) + +10 of 21 +where ϵk +OSS is defined as the ℓ2 norm of the residual when solving system (8) inexactly in the kth +iteration. Here we did not add superscript for ϵQLSA and ϵQTA and the reason shall be revealed +later. Let +�˜0k +˜zk +� +be an inexact solution for system (8) in the kth iteration. Then the norm of residual of system (8), +which is ϵk +OSS, and the norm of residual of system (6), which is ∥Mk ˜zk − rk +c∥2, satisfies +ϵk +OSS = +����� +1 +√ +2∥Mk∥F +� +0 +Mk +(Mk)T +0 +��˜0k +˜zk +� +− +1 +√ +2∥Mk∥F +�rk +c +0 +������ +2 += +����� +1 +√ +2∥Mk∥F +� Mk ˜zk +(Mk)T ˜0k +� +− +1 +√ +2∥Mk∥F +�rk +c +0 +������ +2 +≥ +����� +1 +√ +2∥Mk∥F +Mk ˜zk − +1 +√ +2∥Mk∥F +rk +c +����� +2 +≥ +1 +√ +2∥Mk∥F +∥Mk ˜zk − rk +c∥2. +Recall that the error arising from the OSS system (6) is the same as the error in the full Newton +system (7), then we can directly use the convergence condition provided in Gondzio’s analysis +to the IF-IPM scheme in [5], i.e., +∥Mk ˜zk − rk +c∥2 ≤ δ∥rk +c∥2, +where δ ∈ (0, 1) is a constant parameter. We can require +∥Mk ˜zk − rk +c∥2 ≤ +√ +2∥Mk∥Fϵk +OSS ≤ δ∥rk +c∥2 +and it follows that +ϵk +OSS ≤ +δ∥rk +c∥2 +√ +2 +��Mk�� +F +. +Then choosing +ϵQLSA = ϵQTA = ∥Mk∥Fϵk +OSS +√ +2∥rkc∥2 +≤ δ +2 +ensures the convergence of the IF-QIPM. The complexities for each step are also available now. +Using the QLSA from [19] and QTA from [20], we have the complexity for QLSA and QTA +TQLSA = ˜On, ¯ω, 1 +ϵ +� +κMk +ωk +∥Mk∥F +� +, +TQTA = ˜On(n). +Note that the complexity of the block-encoding procedure is dominated by that of QLSA +and QTA and thus we ignore the complexity contributed by block-encoding. In Step 8, the +complexity contributed by computing Newton step from OSS solution is O(n2). The total +complexity for the kth iteration of IF-QIPM will be +˜On, ¯ω, 1 +ϵ +� +nωkκMk +∥Mk∥F ++ n2 +� +. +(13) + +11 of 21 +3.2.1. Bound for ωk/∥Mk∥F +In this section, all the quantities we consider are from the kth iteration. For simplicity, we +ignore superscript k in this section unless we need it. Using the property of trace, we have +∥M∥2 +F = tr(MTM) += tr +� +(SV + XQV)(SV + XQV)T + XATAX +� += tr +� +(SV + XQV)(SV + XQV)T� ++ tr +� +XATAX +� += tr +� +SVVTS +� ++ tr +� +XQVVTS +� ++ tr +� +SVVTQX +� ++ tr +� +XQVVTQX +� ++ tr +� +XATAX +� +. +For the non-symmetric term, due to cyclic invariant property of trace, we have +tr +� +XQVVTS +� += tr +� +SXQVVT� +. +Recall the central path neighbourhood we defined in (5), we define a matrix E such that +E = 1 +µθ (XS − µI). +(14) +It is obvious that E is a diagonal matrix and satisfies +∥Ee∥2 < 1, +which leads to +| tr(E)| ≤ ∥Ee∥1 ≤ √ +n∥E∥F = √ +n∥Ee∥2 < √ +n and I − E ≻ 0 and I + E ≻ 0. +With this, we can have +tr +� +XQVVTS +� += tr +� +SXQVVT� += tr +� +(θµE + µI)QVVT� += tr +� +θµEQVVT� ++ tr +� +µQVVT� +. +For the second term, we know Q and VTQV are both positive semidefinite. So we can have +tr +� +QVVT� += tr +� +VTQV +� +≥ 0 +because of the cyclic invariant property of trace. According to the Cauchy–Schwarz inequality, +we have +tr +� +EQVVT�2 +≤ ∥E∥2 +F∥QVVT∥2 +F. +So we have +tr +� +EQVVT� +≥ −∥QVVT∥F. + +12 of 21 +Thus, we have +tr +� +XQVVTS +� += tr +� +θµEQVVT� ++ tr +� +µQVVT� +≥ µ +� +tr +� +QVVT� +− θ∥QVVT∥F +� +≥ −θµ∥QVVT∥F +≥ −µ +4 , +where the last inequality holds due to condition (11). So we can bound ∥M∥F by +∥M∥2 +F = tr +� +SVVTS +� ++ tr +� +XQVVTS +� ++ tr +� +SVVTQX +� ++ tr +� +XQVVTQX +� ++ tr +� +XATAX +� +≥ tr +� +SVVTS +� ++ tr +� +XQVVTQX +� ++ tr +� +XATAX +� +− µ +2 . +Since XQVVTQX ⪰ 0, we have +∥M∥2 +F ≥ tr +� +SVVTS +� ++ tr +� +XATAX +� +− µ +2 . +Since X and S are both positive diagonal matrices, we have +∥M∥2 +F ≥ tr +� +SVVTS +� ++ tr +� +XATAX +� +− µ +2 += ∑ +i +s2 +i (VVT)ii + ∑ +i +x2 +i (ATA)ii − µ +2 +≥ ω2 − µ +2 . +As we said in the very beginning of this section, at each iteration ω is indeed ωk but the +superscript is ignored here. Now we are going to find a bound for µ so we can further bound +∥M∥2 +F. Since ω is the upper bound for the magnitude of the primal and dual slack variables, +we have +ω2 ≥ xisi. +Recall the definition of matrix E, see (14). So we have +ω2 ≥ xisi = µ + θµEii ≥ µ − θµ = (1 − θ)µ. +So +∥M∥2 +F ≥ ω2 − µ +2 ≥ ω2 − 1 +2 +ω2 +1 − θ ≥ ω2 − 1 +2 +ω2 +1 − 1/3 = ω2 +4 , +where the last inequality follows from the bound for θ, see (11). So we have +ω +∥M∥F +≤ 2 = O(1). +3.2.2. Bound for κMk +Similar to the previous section, we ignore the supercript k unless we need it. We will start +with a general result and then work on the matrix Mk. The following lemma is a well-known +result regarding condition numbers of matrices and can be proven using Courant-Fischer-Weyl +Min-Max principle [21]. + +13 of 21 +Lemma 2. For any full row rank matrix P ∈ Rm×n and symmetric positive definite matrix D ∈ Rn×n, +their condition number satisfies +κ(PDPT) ≤ κ(D)κ(PPT). +Next, we analyze the matrix in the OSS system (8). Specifically, we focus on MTM since +we are interested in the spectral property of the OSS system (8). Using the matrix E defined +in (14), we have the following decomposition +MTM = +�VT(S + XQ)T(S + XQ)V +−VT(S + XQ)TXAT +−AX(S + XQ)V +AX2AT +� += +�VT(S + XQ)T(S + XQ)V +−VTµ(θE)AT − VTQTX2AT +−Aµ(θE)V − AX2QV +AX2AT +� += +�VT +0 +0 +A +��(S + XQ)T(S + XQ) +−µθE − QX2 +−µθE − X2Q +X2 +��VT +0 +0 +A +�T +. +The second equality holds because +−VTSXAT − VTQTX2AT = −VTµ(I + θE)AT − VTQTX2AT += −VTµ(θE)AT − VTQX2AT. +Here we used that AV = 0 and Q is symmetric. Then, plugging (14) into the first diagonal +block of the decomposition we obtained earlier, we have +MTM = +�VT +0 +0 +A +���S2 + 2µQ + µθ(EQ + QE) + QX2Q +−µθE − QX2 +−µθE − X2Q +X2 +���VT +0 +0 +A +�T += +�VT +0 +0 +A +���S2 + 2µQ + µθ(EQ + QE) +−µθE +−µθE +0 +� ++ +�QX2Q +−QX2 +−X2Q +X2 +���VT +0 +0 +A +�T += +�VT +0 +0 +A +���I +−Q +0 +I +��S2 + 2µQ +−µθE +−µθE +0 +�� I +0 +−Q +I +� ++ +�I +−Q +0 +I +��0 +0 +0 +X2 +�� I +0 +−Q +I +���VT +0 +0 +A +�T += +�VT +0 +0 +A +��I +−Q +0 +I +��S2 + 2µQ +−µθE +−µθE +X2 +�� I +0 +−Q +I +��VT +0 +0 +A +�T +. +(15) +The first two matrices are nonsingular, so we can apply the Lemma 2 and thus we only need to +study the middle matrix. Denote the middle matrix by Ψ. Observe that Ψ is almost the same +as its counterpart in [8]. Subsequently we have the following result regarding the spectral +property of Mk. +Lemma 3. When (x, y, s) ∈ N (θ) and θ ∈ +� +0, min +� +1 +3√n, +1 +4∥QVVT∥F+1 +�� +, the condition number of +matrix Mk satisfies +κMk = O +� +(ωk)2 + µkσmax(Q) +µk +κVAQ +� +, +where κVAQ is the condition number of the matrix +�VT +0 +0 +A +��I +−Q +0 +I +� +. +Proof. The proof is in Appendix B. +Putting all these together, we have the complexity for our IF-QIPM for LCQO problems. + +14 of 21 +Theorem 1. The IF-QIPM for LCQO problems stops with final duality gap less than ϵ in at most +O +�√n log(1/ϵ) +� +IPM iterations and in each IPM iteration, the Newton direction can be obtained with +complexity ˜On, ¯ω, 1 +ϵ +� +n +� +¯ω2 +ϵ + σmax(Q) +� +κVAQ + n2� +, where ¯ω = maxk ωk. +Proof. The complexity bound for the IPM iterations comes from the result in [5]. According to +(13), the complexity for obtaining the Newton direction is +˜On, ¯ω, 1 +ϵ +� +nωkκMk +∥Mk∥F ++ n2 +� +. +(16) +Combining this with the result in Sec. 3.2.1, the bound in Lemma 3, and µk ≥ ϵ, we have +˜On, ¯ω, 1 +ϵ +� +nωkκMk +∥Mk∥F ++ n2 +� += ˜On, ¯ω, 1 +ϵ +� +n +� ¯ω2 +ϵ + σmax(Q) +� +κVAQ + n2 +� +. +(17) +4. Application in Support Vector Machine Problems +In this section, we discuss how to use our IF-QIPM to solve SVM problems. We show that +our algorithm can solve l1-norm soft margin SVM problems with best complexity compared +with any existing classical or quantum algorithms. +The ordinary SVM problem works on a linearly separable dataset, in which the data points +have binary labels. The ordinary SVM aims to find a hyperplane correctly separating the data +points with maximum margin. However, in practice the data points are not necessarily linearly +separable. To allow mislabelling, the concept of soft margin SVM was introduced in [22]. Let +{(φi, ζi) ∈ Rm × {−1, +1}|i = 1, . . . , n} be the set of data points, Φ be a matrix with ith column +being φi, and Z be a diagonal matrix with ith diagonal element being ζi. The SVM problem +with l1-norm soft margin can be formulated as below. +min +(ξ,w,t)∈Rn×Rm×R +1 +2∥w∥2 +2 + C∥ξ∥1 +ζi(⟨w, φi⟩ + t) ≥ 1 − ξi, i = 1, . . . , n +ξi ≥ 0, i = 1, . . . , n. +(18) +Here (w, t) determines a hyperplane and C is a penalty parameter. In [14], the authors rewrote +the SVM problem as a second order conic optimization (SOCO) problem and use the quantum +algorithm they proposed to solve the resulting SOCO problem. They claim the complexity of +their algorithm has O(n2) dependence on the dimension, which is better than any classical +algorithm. However, the algorithm in [14] is invalid. Their algorithm is an Inexact Infeasible- +QIPM (II-QIPM) while they used the IPM complexity for Feasible-QIPM, which ignores at least +O(n1.5) dependence on n. They also missed the symmetrization of the Newton step, which is +necessary for SOCO problems and makes their Newton step invalid. +Aside from [14], some pure quantum algorithms for SVM problems are also proposed. +In [23], the authors propose a pure quantum algorithm for SVM problems. They claim the +complexity is O(κ3 +effϵ−3 log(mn)), where κeff is the condition number of a matrix involving the +kernel matrix and ϵ is the accuracy. In the worst case, κeff = O(m). Their complexity is worse +than ours regarding the dependence of dimension and accuracy. In addition, their algorithm +does not provide classical solutions. Namely, the solution is in the quantum machine and we +can not read or use it in a classical computer. However, our algorithm produces a classical +solution. + +15 of 21 +To convert the problem into standard form LCQO, we introduce (w+, w−) ∈ Rm+ × Rm+, +(t+, t−) ∈ R+ × R+, and a slack variable ρ ∈ Rn+. Then we can get the following formulation +min +w+,w−,t+,t−,ξ,ρ +1 +2∥w+ − w−∥2 +2 + C∥ξ∥1 +ζi(⟨w+ − w−, φi⟩ + t+ − t−) + ξi − ρi = 1, i = 1, . . . , n +(ξ, w+, w−, t+, t−, ρ) ≥ 0. +It is a standard form LCQO problem with nonnegative variables (w+, w−, t+, t−, ξ, ρ) ∈ Rm × +Rm × R × R × Rn × Rn and parameters +c = +� +� +02m+2 +Cen +0n +� +� +Q = +� +� +Im×m +−Im×m +0m×(2+2n) +−Im×m +Im×m +0m×(2+2n) +0(2+2n)×m +0(2+2n)×m +0(2+2n)×(2+2n) +� +� +A = +� +ZΦT +−ZΦT +Z +−Z +In×n +−In×n +� +b = e. +So we can use the proposed IF-QIPM for LCQO problems to solve the ℓ1-norm soft margin +SVM problems and get an ϵ-approximate solution with complexity +˜On, ¯ω, 1 +ϵ +� +n1.5 +� ¯ω2 +ϵ + σmax(Q) +� +κVAQ + n2.5 +� +. +This dependence on dimension is better than any existing quantum or classical algorithm. +5. Discussion +In this work, we present an IF-QIPM for LCQO problems by combining the IF-IPM frame- +work proposed in [5] and the OSS system introduced in [8]. Our algorithm has n1.5 dependence +on n, which is better than any existing algorithms for LCQO problems. The dependence on the +accuracy is polynomial, which is worse than classic IPMs. Iterative refinement method might +help improve the dependence on the accuracy but that could be another work. +Author Contributions: Conceptualization, Zeguan Wu and Tamás Terlaky; Methodology, Zeguan Wu; +Supervision, Xiu Yang and Tamás Terlaky; Validation, Zeguan Wu, Mohammadhossein Mohammadisi- +ahroudi, Brandon Augustino, Xiu Yang and Tamás Terlaky; Writing – original draft, Zeguan Wu; Writing +– review & editing, Zeguan Wu, Mohammadhossein Mohammadisiahroudi, Brandon Augustino, Xiu +Yang and Tamás Terlaky. +Funding: This work was supported by Defense Advanced Research Projects Agency as part of the project +W911NF2010022: The Quantum Computing Revolution and Optimization: Challenges and Opportunities. +Institutional Review Board Statement: Not applicable . +Informed Consent Statement: Not applicable . +Data Availability Statement: Not applicable. +Conflicts of Interest: The funder had no role in the design of the study; in the writing of the manuscript; +or in the decision to publish the results. + +16 of 21 +Abbreviations +The following abbreviations are used in this manuscript: +IF-IPM +Inexact Feasible Interior Point Method +IF-QIPM +INexact Feasible Quantum Interior Point Methods +IPM +Interior Point Method +LCQO +Linearly Constrained Quadratic Optimization +LO +Linear Optimization +OSS +Orthogonal Subspace System +QIPM +Quantum Interior Point Method +QLSA +Quantum Linear System Algorithm +QTA +Quantum Tomography Algorithm +SOCO +Second Order Conic Optimization +SVM +Support Vector Machine +Appendix A. Block-encoding of the OSS system +In this section, we ignore the superscript k for simplicity. As described in Eq. (9), we first +block encode each of the matrices involved in (10). With V, A, S and X given and are stored in +a quantum accessible data structure (we ignore the complexity to store the classical information +into the quantum machine). For the first matrix +M1 = +� +� +0n×n +0n×n +0n×n +0(n−m)×n +VT +0(n−m)×n +0m×n +0m×n +−A +� +�, +a +�� +∥V∥2 +F + ∥A∥2 +F, O(poly log(n)), ϵ1 +� +-block-encoding of M1 can be implemented according to Lemma 50 from [18] efficiently. +The second matrix +M2 = +� +� +0n×n +0n×n +S +0n×n +0n×n +0n×n +� +� +is both 1-row-sparse and 1-column-sparse. By the definition of ω, each element of M2/ω has +absolute value at most 1. According to Lemma 48 in [18], a +(1, O(poly log(n)), ϵ2) +-block-encoding of M2/ω can be implemented efficiently. +The third matrix +M3 = +� +� +0n×n +0n×n +0n×n +0n×n +Q +0n×n +0n×n +0n×n +In×n +� +� +can be decomposed into +M3 = +� +� +0n×n +0n×n +0n×n +0n×n +Q +0n×n +0n×n +0n×n +0n×n +� +� + +� +� +0n×n +0n×n +0n×n +0n×n +0n×n +0n×n +0n×n +0n×n +In×n +� +�. +Then we can block-encode the two matrices first, and then apply linear combination to obtain +M3. In fact, a +(∥Q∥F, O(poly log(n)), ϵ3) + +17 of 21 +-block-encoding of the left matrix can be implemented according to Lemma 50 from [18] +efficiently and a +(1, O(poly log(n)), ϵ3) +-block-encoding of the right matrix can be implemented efficiently according to Lemma 48 in +[18]. With the state-preparation cost of the linear combination coefficient vector (1, 1) neglected, +a +(∥Q∥F + 1, O(poly log(n)), (∥Q∥F + 1)ϵ3) +-block-encoding of M3 can be implemented efficiently according to Lemma 52 from [18]. +The fourth matrix +M4 = +� +� +0n×n +0n×n +X +0n×n +X +0n×n +� +� +is 1-row-sparse and 2-column-sparse. After being scaled by 1 +ω, each element of M4/ω has +absolute value at most 1. According to Lemma 48 in [18], a +�√ +2, O(poly log(n)), ϵ4 +� +-block-encoding of M4/ω can be implemented efficiently. +For the matrix multiplication M3M4/ω, a +�√ +2∥Q∥F + +√ +2, O(poly log(n)), (∥Q∥F + 1)( +√ +2ϵ3 + ϵ4) +� +-block-encoding can be implemented efficiently according to Lemma 53 from [18]. +For the linear combination M2/ω + M3M4/ω, the cost for the state-preparation of the +coefficient vector (1, 1) is negligible and thus a +�√ +2∥Q∥F + +√ +2 + 1, O(poly log(n)), ( +√ +2∥Q∥F + +√ +2 + 1)(ϵ3 + 1 +√ +2 +ϵ4) +� +-block-encoding can be implemented efficiently according to Lemma 52 from [18]. +For the matrix multiplication of M1(M2/ω + M3M4/ω), a +�� +∥V∥2 +F + ∥A∥2 +F( +√ +2∥Q∥F + +√ +2 + 1), +O(poly log(n)), +� +∥V∥2 +F + ∥A∥2 +F( +√ +2∥Q∥F + +√ +2 + 1)(ϵ3 + 1 +√ +2 +ϵ4) + ( +√ +2∥Q∥F + +√ +2 + 1)ϵ1 +� +-block-encoding can be implemented efficiently according to Lemma 53 from [18]. +Finally, considering that the complexity of state-preparation of the vector +( +ω +√ +2∥M∥F +, +ω +√ +2∥M∥F +) + +18 of 21 +can be neglected, a +�� +∥V∥2 +F + ∥A∥2 +F +√ +2ω +∥M∥F +( +√ +2∥Q∥F + +√ +2 + 1), +O(poly log(n)), +� +∥V∥2 +F + ∥A∥2 +F +√ +2ω +∥M∥F +( +√ +2∥Q∥F + +√ +2 + 1)2 +�� +∥V∥2 +F + ∥A∥2 +F(ϵ3 + 1 +√ +2 +ϵ4) + ϵ1 +�� +-block-encoding of the coefficient matrix of system (8) can be implemented efficiently according +to Lemma 52 from [18]. We can choose +ϵ1 = ϵQLSA +κ3 +M +1 +2K +ϵ2 = +ϵ1 +2 +� +∥V∥2 +F + ∥A∥2 +F +ϵ3 = ϵ2 +ϵ4 = +√ +2ϵ2, +where K depends on the initial data +K = +√ +2 +� +∥V∥2 +F + ∥A∥2 +F( +√ +2∥Q∥F + +√ +2 + 1)2. +Now, considering that the complexity for all the block-encoding algorithms we have used so far +have poly-logarithmic dependence on the dimension and accuracy, and that, for i = 1, 2, 3, 4 +O +� +poly log( 1 +ϵi +) +� += O(poly log(κM)), +the complexity for block-encoding will be dominated by the complexity for QLSA because +QLSA has linear dependence on κM. So we can ignore the complexity of block-encoding. +Appendix B. Spectral Analysis for Matrix Ψ +In this section, we provide the spectral analysis for the matrix +Ψ = +�S2 + 2µQ +−µθE +−µθE +X2 +� +. +(A1) +Just like in the previous section, for simplicity, we ignore the superscript k. We can do the +following decomposition +�S2 + 2µQ +−µθE +−µθE +X2 +� += +� +S2 +−µθE +−µθE +X2 +� ++ +�2µQ +0 +0 +0 +� +. +Let us use the following notation +Ψ1 = +� +S2 +−µθE +−µθE +X2 +� +Ψ2 = +�2µQ +0 +0 +0 +� +. + +19 of 21 +It can be proven that Ψ1 is positive definite. The majority of the proof of this conclusion comes +from the paper [8]. For the reader’s convenience, we provide the complete proof here. +Matrix Ψ1 is a block diagonal matrix, with all the four blocks being diagonal matrices. So +we can easily compute the eigenvalues using the characteristic polynomial +det(Ψ1 − qI) = det +�� +X2 − qI +�� +S2 − qI +� − θ2µ2E2� += +n +∏ +i=1 +�� +x2 +i − q +�� +s2 +i − q +� − θ2µ2E2 +ii +� +. +Clearly, det(Ψ1 − qI) = 0 gives n quadratic equations and each quadratic equation gives two +eigenvalues. The two eigenvalues from the ith quadratic equation are +qi+ = 1 +2 +� +(x2 +i + s2 +i ) + +� +(x2 +i + s2 +i )2 − 4x2 +i s2 +i + 4θ2µ2E2 +ii +� +and +qi− = 1 +2 +� +(x2 +i + s2 +i ) − +� +(x2 +i + s2 +i )2 − 4x2 +i s2 +i + 4θ2µ2E2 +ii +� +. +Recalling the definition of E in (14), we can write +qi− = 1 +2 +� +(x2 +i + s2 +i ) − +� +(x2 +i + s2 +i )2 − 4x2 +i s2 +i + 4(xisi − µ)2 +� += 1 +2 +� +(x2 +i + s2 +i ) − +� +(x2 +i + s2 +i )2 + 4(xisi − µ + xisi)(xisi − µ − xisi) +� += 1 +2 +� +(x2 +i + s2 +i ) − +� +(x2 +i + s2 +i )2 − 4µ(2xisi − µ) +� += 1 +2 +� +(x2 +i + s2 +i ) − +� +(x2 +i + s2 +i )2 − 4µ(2θµEii + µ) +� +. +One can verify that the square root always exists because +(x2 +i + s2 +i )2 − 4µ(2xisi − µ) ≥ 4(xisi)2 − 4µ(2xisi) + 4µ2 += 4(xisi − µ)2 +≥ 0. + +20 of 21 +With θ ∈ +� +0, min +� +1 +3√n, +1 +4∥QVVT∥F+1 +�� +, we have +qi− ≥ 1 +2 +� +(x2 +i + s2 +i ) − +� +(x2 +i + s2 +i )2 − 4µ(2θµEii + µ) +� +≥ 1 +2 +� +(x2 +i + s2 +i ) − +� +(x2 +i + s2 +i )2 − 4µ(−2µ +1 +3√n + µ) +� += 1 +2 +� +(x2 +i + s2 +i ) − +� +(x2 +i + s2 +i )2 − 4 +3µ2 +� += 1 +2 +4 +3µ2 +(x2 +i + s2 +i ) + +� +(x2 +i + s2 +i )2 − 4 +3µ2 +≥ 1 +2 +4 +3µ2 +(x2 +i + s2 +i ) + +� +(x2 +i + s2 +i )2 += +µ2 +3(x2 +i + s2 +i ) +> 0. +This means that matrix Ψ1 is positive definite and its eigenvalues coincide with its singular +values because Ψ1 is also real and symmetric. Analogously, we have +qi+ = 1 +2 +� +(x2 +i + s2 +i ) + +� +(x2 +i + s2 +i )2 − 4µ(2θµEii + µ) +� +≤ 1 +2 +� +(x2 +i + s2 +i ) + (x2 +i + s2 +i ) + 2µ +� +(2θEii + 1) +� +≤ 1 +2 +� +(x2 +i + s2 +i ) + (x2 +i + s2 +i ) + 2µ +√ +2 +� += (x2 +i + s2 +i ) + +√ +2µ. +So the condition number of Ψ satisfies +κ(Ψ) ≤ σmax(Ψ1) + σmax(Ψ2) +σmin(Ψ1) + σmin(Ψ2) += maxi qi+ + σmax(Ψ2) +minj qj− + σmin(Ψ2) +≤ maxi{x2 +i + s2 +i } + +√ +2µ + 2µσmax(Q) +minj +µ2 +3(x2 +i +s2 +i ) += +3 maxi{x2 +i + s2 +i } +� +maxi{x2 +i + s2 +i } + +√ +2µ + 2µσmax(Q) +� +µ2 +≤ +3ω2� +ω2 + +√ +2µ + 2µσmax(Q) +� +µ2 +, + +21 of 21 +where the last inequality comes from the definition of ω. Since ω2 ≥ xisi ≥ (1 − θ)µ, we have +κ(Ψ) = O +�ω2(ω2 + µσmax(Q)) +µ2 +� +. +Using Lemma 2, we can also bound the condition number of matrix M by +κM = +� +κ(MTM) +≤ +� +κ(Ψ)κVAQ +≤ O +�(ω2 + µσmax(Q)) +µ +κVAQ +� +. +References +1. +Nocedal, J.; Wright, S.J. 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Physical Review Letters 2014, +113, 130503. + diff --git a/ONE4T4oBgHgl3EQf9w4Q/content/tmp_files/load_file.txt b/ONE4T4oBgHgl3EQf9w4Q/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5c944c96b9114d0ef5105df20641d657c0803c72 --- /dev/null +++ b/ONE4T4oBgHgl3EQf9w4Q/content/tmp_files/load_file.txt @@ -0,0 +1,653 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf,len=652 +page_content='Citation: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Preprints 2022, 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='org/ Publisher’s Note: MDPI stays neutral with regard to 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Article An Inexact Feasible Quantum Interior Point Method for Linearly Constrained Quadratic Optimization Zeguan Wu 1 , Mohammadhossein Mohammadisiahroudi 1 , Brandon Augustino 1 , Xiu Yang 1 and Tamás Terlaky 1,* 1 Department of Industrial and Systems Engineering, Lehigh University Correspondence: terlaky@lehigh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='edu Abstract: Quantum linear system algorithms (QLSAs) have the potential to speed up algorithms that rely on solving linear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Interior Point Methods (IPMs) yield a fundamental family of polynomial-time algorithms for solving optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' IPMs solve a Newton linear system at each iteration to find the search direction, and thus QLSAs can potentially speed up IPMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Due to the noise in contemporary quantum computers, such quantum-assisted IPM (QIPM) only allows an inexact solution for the Newton linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Typically, an inexact search direction leads to an infeasible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' In our work, we propose an Inexact-Feasible QIPM (IF-QIPM) and show its advantage in solving linearly constrained quadratic optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' We also apply the algorithm to ℓ1-norm soft margin support vector machine (SVM) problems and obtain the best complexity regarding dependence on dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' This complexity bound is better than any existing classical or quantum algorithm that produces a classical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Keywords: Quantum Computing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Interior Point Method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Quadratic Optimization MSC: 90C20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' 90C51;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' 81P68 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Introduction Linearly constrained quadratic optimization (LCQO) is defined as optimizing a convex quadratic objective function over a set of linear constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' This problem reduces to linear opti- mization when the the quadratic objective function is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' LCQO has rich theory, algorithms, and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Many machine learning problems are LCQO problems, including variants of least square problems and variants of support vector machine problems [1,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Some important optimization algorithms also have LCQO subproblems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' sequential quadratic programming [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' The modern age of IPMs launched by Karmarkar’s invention of the projective method for linear optimization (LO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Since then, a lot of variants of IPMs have been studied for not only LO problems but also for nonlinear optimization problems, including LCQO problems [3,4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Contemporary IPMs look for the optimal solution by moving in a neighbourhood of the central path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' IPMs can be divided into two classes: feasible or infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Feasible IPMs start with a feasible solution and keep feasibility;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' infeasible IPMs start with an infeasible interior solution and so do not require a feasible solution to start with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' For LCQO problems with n variables, some feasible IPMs produce an ϵ-approximate solution in at most O(√n log(1/ϵ)) IPM iterations, while infeasible IPMs require O(n2 log(1/ϵ)) IPM iterations to generate an ϵ-approximate solution [5,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' At each IPM iteration a linear system needs to be solved to obtain the search direction, called the Newton direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Such a Newton linear system is traditionally in the form of augmented system or the normal equation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Classically these linear systems can be solved exactly using Bunch-Parlett factoriztion if the matrices in the systems are symmetric indefinite [7], or Cholesky factorization if the matrices are symmetric positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' The arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='05357v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='OC] 13 Jan 2023 BY2 of 21 complexity of solving the linear systems is O(n3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' The linear systems can also be solved inexactly using some inexact methods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=', Krylov subspace methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Such inexact methods might take less iterations if the desired accuracy of the solutions to the linear systems is not high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' But such inaccuracy of the solutions to the linear systems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=', inaccuracy of the search directions, might result in infeasibility of the solutions generated by IPMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' To maintain feasibility of solutions, [8] introduces the so-called orthogonal subspace system (OSS) for LO problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' A feasible solution can be recovered from an inexact solution to OSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' We extend their OSS for LO prolems to LCQO problems and provide an efficient method to construct the OSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' With the OSS, we can obtain an inexact feasible IPM – solving for search direction inexactly but maintaining the feasibility of solution throughout the process of our IPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' The feasibility of solution gives better IPM iteration complexity and the bottleneck becomes solving the linear system, OSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' With the development of quantum technology, many quantum-assisted algorithms have been proposed for many optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Following the invention of quantum algo- rithms for solving linear systems of equations [9], many researchers are encouraged to study whether QLSAs would yield quantum speedups in classical algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' In particular, QIPMs have been proposed for for LO problems [10,11] and semidefinite optimization problems [12] that utilize QLSAs to solve the Newton linear system that arises in each iteration of IPMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Similar ideas have also been applied to accelerate the solution of some machine learning applications, such as linear regression [13] and the support vector machine training problem [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' However, linearly constrained quadratic optimization problems, which are fundamental to both optimization and machine learning, have not been formally studied in the quantum literature yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' The remaining part of this paper is organized as follow: in Section 2, we introduce IPMs for LCQO and the OSS system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' in Section 3, we discuss how to use quantum algorithms to find the Newton directions and analyze the complexity of our IF-QIPM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' in Section 4, we apply our IF-QIPM to support vector machine problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Discussions are provided in Section 5, and some technical proofs are moved to the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Preliminary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Notations In this section, we introduce notations we use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Vectors are typically represented by lower- case letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' For n-dimensional all-zero vector, we represent it with 0n if the dimension is n, or simply 0 if the dimension is obvious in the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' For n-dimensional all-one vector, we represent it with en, or simply e if the dimension is obvious in the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Matrix are typically represented with upper-case letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' For n-dimensional identity matrix, we represent it with In×n, or simply I if the dimension is obvious in the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' For n × m-dimensional all-zero matrix, we represent it with 0n×m, or simply 0 if the dimension if obvious in the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' For a general n × m-dimensional matrix H, we represent its ith row by Hi· and jth column by H·j and (i, j) element by Hij or Hi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' For real-valued functions f1 and f2 and f3, we write f1 = O( f2) if there exits a positive number k4 such that f1 ≤ k4 f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' We write f1 = ˜O f3( f2) if there exists a positive number k5 such that f1 ≤ k5 f2 × poly log( f3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' 3 of 21 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' IPMs for LCQO In this work, LCQO is defined as follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Definition 1 (LCQO Problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' For vectors b ∈ Rm, c ∈ Rn, and matrix A ∈ Rm×n with rank(A) = m ≤ n, and symmetric positive semidefinite matrix Q ∈ Rn×n, we define the primal and dual LCQO problems as: (P) min cTx + 1 2xTQx, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Ax = b, x ≥ 0, (D) max bTy − 1 2xTQx, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' ATy + s − Qx = c, s ≥ 0, (1) where x ∈ Rn is the vector of primal variables, and y ∈ Rm, s ∈ Rn are vectors of the dual variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Problem (P) is called the primal problem and (D) is called the dual problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' The full-row-rankness of matrix A implies that there is no all-zero row in matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' We further make the following assumption on matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Matrix A has no all-zero columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' When matrix A has zero columns, without loss of generality, let us say the nth column is all-zero, then we can introduce a new variable xn+1 and rewrite the problem into min �c 0 �T� x xn+1 � + 1 2 � x xn+1 �T� Q 0n×1 01×n 0 �� x xn+1 � , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' �A·1 · · A·(n−1) 0m×1 0m×1 0 · · 0 1 −1 �� x xn+1 � = �b 0 � , x ≥ 0, xn+1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' The new problem is equivalent to the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' The new problem is still a LCQO problem and has fewer all-zero columns than the original problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' So we can repeat the procedure to eliminate all the all-zero columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' In the worst case, we will get a new LCQO problem satisfying Assumption 1 with 2n − m variables and n constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' There exists a solution (x, y, s) such that Ax = b, x > 0, ATy + s − Qx = c, and s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' The set of primal-dual feasible solutions can be defined as PD := � (x, y, s) ∈ Rn × Rm × Rn : Ax = b, ATy + s − Qx = c, (x, s) ≥ 0 � and the set of interior feasible primal-dual solutions can be defined as PD0 := � (x, y, s) ∈ Rn × Rm × Rn : Ax = b, ATy + s − Qx = c, (x, s) > 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' According to the strong duality, the set of optimal solutions can be defined as PD∗ := {(x, y, s) ∈ PD : xs = 0}, 4 of 21 where xs denotes the Hadamard, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=', componentwise product of x and s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Let ϵ > 0, then the set of ϵ-approximate solutions to Problem (1) can be defined as PDϵ := � (x, y, s) ∈ PD : xTs ≤ nϵ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' (2) Let X and S be diagonal matrices of x and s, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Under Assumption 2, for all µ > 0, the perturbed optimality conditions Ax = b, ATy + s − Qx = c, XSe = µe, (x, s) ≥ 0 (3) have a unique solution (x(µ), y(µ), s(µ)) that defines the primal and dual central path CP := � (x, y, s) ∈ PD0|xisi = µ for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' , n};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' for µ > 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' IPMs apply Newton’s method to solve system (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' At each iteration of infeasible IPMs, a candidate solution to the primal-dual LCQO pair in (1) is updated by solving the following linear system to find the Newton direction: � � A 0 0 −Q AT I S 0 X � � � � ∆x ∆y ∆s � � = � � rp rd rc � �, (4) where (rp, rd, rc) are residuals defined as rp = b − Ax rd = c − ATy − s rc = σµe − XSe, where σ ∈ (0, 1) is the barrier reduction parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' If rp = 0 and rd = 0, then the solutions (x, y, s) are primal and dual feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Alternatively, we can also define residuals in different ways as we will show later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Once the Newton direction is found, one can move along the direction but has to stay in a neighbourhood of the central path, which is defined at the end of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' When the linear system (4) is solved inexactly, that actually leads to inexact infeasible IPMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Many researchers have analyzed the performance of inexact infeasible IPMs (II-IPMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' For LCQO problems, [6] propose an II-IPM using an iterative method to solve the Newton systems and obtain O(n2 log( 1 ϵ)) IPM iteration complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Here IPM iteration complexity does not include the complexity contributed by linear system solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' However, it is known that feasible IPMs for LCQO problems can achieve O(√n log( 1 ϵ)) IPM iteration complexity [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' In [5], the author provides a general inexact feasible IPM for LCQO problems but has not discussed how to maintain feasibility when inexact linear system solvers are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' In this work, we will fill the gap by using a method inspired by some QIPM results [8,12] as we shall discuss later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' In this paper, we consider the following neighborhood of the central path N2(θ) := � (x, y, s) ∈ PD0|∥XSe − µe∥2 ≤ θµ � , (5) 5 of 21 where θ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Orthogonal Subspaces System Assuming that (x, y, s) ∈ PD0, to maintain the feasibility of the primal and dual variables, the first two linear equations in system (4) need to be solved with rp = 0 and rd = 0 exactly, which can be guaranteed if ∆x lies in the null space of A, denoted as Null(A), and ∆s = Q∆x − AT∆y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Accordingly, we can rewrite system (4) if we represent ∆x by a basis of Null(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' To do so, we can partition matrix A to A = � AB AN � , where AB is a basis of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Then we construct the following matrix V = � A−1 B AN −I � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Matrix V has full column rank and satisfies AV = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=', the columns of V span the null space of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Let ∆x = Vλ, where λ ∈ Rn−m is the unknown coefficient vector for ∆x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Subsequently, we can rewrite system (4) by substituting ∆x and ∆s in the third equation as SVλ + X � QVλ − AT∆y � = rc ⇔ � SV + XQV −XAT� · � λ ∆y � = rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' (6) A similar system was proposed and called "Orthogonal Subspaces System" (OSS) in [8,12] and we use the same name in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' The matrix in the OSS system (6) is of size n × n, and it is nonsingular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Even if the OSS system is solved inexactly, primal and dual feasibility is preserved by computing ∆x = Vλ and ∆s = QVλ − AT∆y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Thus, we can conclude that residual will only show up in the third equation of (4), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=', rp = 0 and rd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' This nice property of the OSS system brings much convenience in the analysis of the proposed inexact IPM, and allows to prove the to-date best iteration complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Inexact Feasible IPM with QLSAs In this section, we propose our IF-QIPM for LCQO problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' We start with the IF-IPM structure introduced by [5] and describe how to convert it into an IF-QIPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Then we analyze the construction of the OSS system, and finally, we analyze the complexity for our IF-QIPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' IF-IPM for LCQO In [5], the author studies a general conceptual form IF-IPM for QCLO problems by assuming the feasibility of primal and dual variables, which induces the following system � � A 0 0 −Q AT I S 0 X � � � � ∆x ∆y ∆s � � = � � 0 0 rc � �, (7) where rc = σµe − XSe with σ ∈ (0, 1) being the reduction factor of the central path parameter µ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=', µnew = σµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' When system (7) is solved with rc = σµe − XSe inexactly yielding an error r, if ∥r∥2 ≤ δ∥rc∥2 for some δ ∈ (0, 1), then the inexact IPM produces an ϵ-approximate solution to Problem (1) in O(√n log(1/ϵ)) iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' The author of [5] does not specify how to solve system (7) inexactly, how to preserve primal and dual feasibility, and how to satisfy the convergence conditions described in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Specifically, the convergence conditions are posed on the right-hand-side and the inexactness error of the system (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' 6 of 21 Now we present a general procedure how to solve system (7) inexactly, while the inexact- ness error occurs only in the third equation of system (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Let (λ, ∆y) be an inexact solution for system (6) and r be the corresponding inexactness error, so we have � SV + XQV −XAT� · � λ ∆y � = rc + r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' The corresponding Newton step ∆x = Vλ ∆s = Q∆x − AT∆y satisfies � � A 0 0 −Q AT I S 0 X � � · � � ∆x ∆y ∆s � � = � � 0 0 rc + r � �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Recall that once (λ, ∆y) is determined, then (∆x, ∆s) is also determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' An interesting property is that, if (λ, ∆y) and (∆x, ∆y, ∆s) can be deduced from each other, then the OSS system and system (7) yield the same error term r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Hence the convergence conditions built upon system (7) can be directly examined using the residual rc and error r of the OSS system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Let ϵOSS be the target accuracy of the OSS system (6), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=', ∥(λ − λ∗, ∆y − ∆y∗∥2 ≤ ϵOSS, where (λ∗, ∆y∗) is the accurate solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' To make the IF-IPM converge, according to [5], we need ∥r∥2 = ���� � SV + XQV −XAT� · � λ ∆y � − rc ���� 2 ≤ ��� SV + XQV −XAT��� 2ϵOSS ≤ δ∥rc∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' So ϵOSS ≤ δ ∥rc∥2 ��� SV + XQV −XAT��� 2 is sufficient for the IF-IPM to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' We present the IF-IPM in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Algorithm 1 Short-step IF-IPM 1: Choose ϵ > 0, δ ∈ (0, 1), θ ∈ (0, 1), β ∈ (0, 1) and σ = (1 − β √n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' 2: k ← 0 3: Choose initial feasible interior solution (x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' y0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' s0) ∈ N (θ) 4: while (xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' yk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' sk) /∈ PDϵ do 5: µk ← (xk)Tsk n 6: ϵk OSS ← δ∥rk c∥2/ ��� SkV + XkQVk −XkAT��� 2 7: (λk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' ∆yk) ← solve system (6) with accuracy ϵk OSS 8: ∆xk = Vλk and ∆sk = −AT∆yk 9: (xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' yk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' sk+1) ← (xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' yk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' sk) + (∆xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' ∆yk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' ∆sk) 10: k ← k + 1 11: end while 12: return (xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' yk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' sk) 7 of 21 In the quantum-assisted IF-IPM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' or IF-QIPM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' we are proposing to accelerate Step 7 using quantum algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' In the next sections, we investigate how to use quantum algorithms to build and solve the OSS system and get the Newton direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' IF-QIPM for LCQO The pseudocode of our IF-QIPM is presented in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' At each iteration of the IF-QIPM, we construct and solve system (6) and compute the Newton direction using quantum algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Algorithm 2 Short-step IF-QIPM 1: Choose ϵ > 0, δ ∈ (0, 1), θ ∈ (0, θ0), β ∈ (0, 1) and σ = (1 − β √n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' 2: k ← 0 3: Choose initial feasible interior solution (x0, y0, s0) ∈ N (θ) 4: while (xk, yk, sk) /∈ PDϵ do 5: µk ← (xk)Tsk n 6: ϵk OSS ← δ∥rk c∥2/ ��� SkV + XkQVk −XkAT��� 2 7: (λk, ∆yk) ← solve system (6) with accuracy ϵk OSS quantumly 8: ∆xk = Vλk and ∆sk = −AT∆yk 9: (xk+1, yk+1, sk+1) ← (xk, yk, sk) + (∆xk, ∆yk, ∆sk) 10: k ← k + 1 11: end while 12: return (xk, yk, sk) Here θ0 < 1 and its value will be discussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' First, we introduce some notations to simplify the OSS system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' In the kth iteration of Algorithm 2, let Mk = � SkV + XkQV −XkAT� , zk = � λk ∆yk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Then the OSS system can be rewritten as Mkzk = rk c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' As discussed in [8], to solve the OSS system (6) using quantum algorithms, we need to first rewrite it as the normalized Hermitian OSS system 1 √ 2 ��Mk�� F � 0 Mk (Mk)T 0 � � 0 zk � = 1 √ 2 ��Mk�� F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' �rk c 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' (8) To use the QLSAs mentioned earlier, we need to turn the linear system (8) into a quantum linear system using the block-encoding introduced in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' To this end, we first decompose the coefficiennt matrix in linear system (8) as 1 √ 2 ��Mk�� F � 0 Mk (Mk)T 0 � = 1 √ 2 ��Mk�� F � 0 0 (Mk)T 0 � + 1 √ 2 ��Mk�� F �0 Mk 0 0 � , (9) 8 of 21 where � 0 0 (Mk)T 0 � = � � 0n×n 0n×n 0n×n 0(n−m)×n VT 0(n−m)×n 0m×n 0m×n −A � �× � � � � 0n×n 0n×n Sk 0n×n 0n×n 0n×n � � + � � 0n×n 0n×n 0n×n 0n×n Q 0n×n 0n×n 0n×n In×n � � � � 0n×n 0n×n Xk 0n×n Xk 0n×n � � � �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' (10) To compute matrix V, we need to find a basis matrix AB of matrix A and we need to compute the inverse matrix A−1 B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Both steps are nontrivial and can be expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' However, we can reformulate the LCQO problem as follows min cTx + 1 2xTQx s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' �I 0 A 0 I −A �� � x′ x′′ x � � = � b −b � x ≥ 0, x′ ≥ 0, x′′ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' In this case, we have an obvious basis AB = �I 0 0 I � and matrix V can be constructed efficiently V = � A−1 B AN −I � = � � �I 0 0 I �� A −A � −I � � = � � A −A −I � �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Since matrix A has no all-zero rows, matrix V has no all-zero rows either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' This property of the reformulation is useful in the analysis of the proposed IF-QIPM but we do not want to build the complexity analysis on the reformulated problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' So without loss of the generality we may make the following assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Matrix A is of the form A = � I AN � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' To simplify the analysis, we further assume the input data are integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' The input data of Problem (1) are integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Following from the two assumptions above, we have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Matrix V equals to V = �AN −I � and min i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=',n{∥Vi·∥2 2} = min{1, min i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=',m ∥(AN)i·∥2 2} = 1, where Vi· and (AN)i· are the ith row of V and AN, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' 9 of 21 Now we are ready to give θ0 in our definnition of central path neighbourhood, see (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' We set θ0 = min � 1 3√n, 1 4∥QVVT∥F + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' (11) We also define ωk as the maximum of the values of primal variables and dual slack variables in the kth iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Let (xk, yk, sk) be the a candidate solution for Problem (1), then ωk = max i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=',n}{xk i , sk i }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' In this work, we assume access to quantum random access memory, QRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Then Step 7 of Algorithm 2 consists of three parts: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=') use block-encoding to build system (8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=') use QLSAs to solve system (8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=') use quantum tomography algorithms (QTAs) to extract classical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' We use the block-encoding methods introduced in [18] to block-encode linear system (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' In the kth iteration of Algorithm 2, use the block-encoding methods introduced in [18] and the decomposition described in equations (9) and (10), a �� ∥V∥2 F + ∥A∥2 F √ 2ωk ∥Mk∥F ( √ 2∥Q∥F + √ 2 + 1), O(poly log(n)), ϵQLSA κ3 Mk � block-encoding of the matrix in the system (8) can be implemented efficiently and the complexity will be dominated by the complexity of the QLSA step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Here ϵQLSA is the accuracy required for the QLSA step and κMk is the condition number of matrix Mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' See Appendix A for proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' The complexity contributed by block-encoding is negligible compared with the complexity contributed by QLSAs and QTAs so we ignore it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' To establish the total complexity contributed by QLSAs and QTAs, we first need to analyze the accuracy of QLSA characterized by ϵQLSA and the accuracy of QTA characterized by ϵQTA and their relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' In each iteration, we use a QLSA to solve the block-encoded version of system (8) and get an ϵQLSA-approximate solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Then we use a QTA to extract an ϵQTA-approximate solution from the quantum machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Here, for QLSA and QTA, ˜z is an ϵ-approximate solution of z means ���� ˜z ∥˜z∥2 − z ∥z∥2 ���� 2 ≤ ϵ, which is different from the concept of ϵ-approximate solutions defined in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Similar to [11], the QLSA we use is proposed by [19] and the QTA we use is proposed by [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Following the argument in Section 2 in [11], we can set the relationship among ϵQLSA, ϵQTA, and ϵk OSS as ϵQLSA = ϵQTA = 1 2 · √ 2∥Mk∥F ∥rkc∥2 ϵk OSS, (12) 10 of 21 where ϵk OSS is defined as the ℓ2 norm of the residual when solving system (8) inexactly in the kth iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Here we did not add superscript for ϵQLSA and ϵQTA and the reason shall be revealed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Let �˜0k ˜zk � be an inexact solution for system (8) in the kth iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Then the norm of residual of system (8), which is ϵk OSS, and the norm of residual of system (6), which is ∥Mk ˜zk − rk c∥2, satisfies ϵk OSS = ����� 1 √ 2∥Mk∥F � 0 Mk (Mk)T 0 ��˜0k ˜zk � − 1 √ 2∥Mk∥F �rk c 0 ������ 2 = ����� 1 √ 2∥Mk∥F � Mk ˜zk (Mk)T ˜0k � − 1 √ 2∥Mk∥F �rk c 0 ������ 2 ≥ ����� 1 √ 2∥Mk∥F Mk ˜zk − 1 √ 2∥Mk∥F rk c ����� 2 ≥ 1 √ 2∥Mk∥F ∥Mk ˜zk − rk c∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Recall that the error arising from the OSS system (6) is the same as the error in the full Newton system (7), then we can directly use the convergence condition provided in Gondzio’s analysis to the IF-IPM scheme in [5], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=', ∥Mk ˜zk − rk c∥2 ≤ δ∥rk c∥2, where δ ∈ (0, 1) is a constant parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' We can require ∥Mk ˜zk − rk c∥2 ≤ √ 2∥Mk∥Fϵk OSS ≤ δ∥rk c∥2 and it follows that ϵk OSS ≤ δ∥rk c∥2 √ 2 ��Mk�� F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Then choosing ϵQLSA = ϵQTA = ∥Mk∥Fϵk OSS √ 2∥rkc∥2 ≤ δ 2 ensures the convergence of the IF-QIPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' The complexities for each step are also available now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Using the QLSA from [19] and QTA from [20], we have the complexity for QLSA and QTA TQLSA = ˜On, ¯ω, 1 ϵ � κMk ωk ∥Mk∥F � , TQTA = ˜On(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Note that the complexity of the block-encoding procedure is dominated by that of QLSA and QTA and thus we ignore the complexity contributed by block-encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' In Step 8, the complexity contributed by computing Newton step from OSS solution is O(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' The total complexity for the kth iteration of IF-QIPM will be ˜On, ¯ω, 1 ϵ � nωkκMk ∥Mk∥F + n2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' (13) 11 of 21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Bound for ωk/∥Mk∥F In this section, all the quantities we consider are from the kth iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' For simplicity, we ignore superscript k in this section unless we need it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Using the property of trace, we have ∥M∥2 F = tr(MTM) = tr � (SV + XQV)(SV + XQV)T + XATAX � = tr � (SV + XQV)(SV + XQV)T� + tr � XATAX � = tr � SVVTS � + tr � XQVVTS � + tr � SVVTQX � + tr � XQVVTQX � + tr � XATAX � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' For the non-symmetric term, due to cyclic invariant property of trace, we have tr � XQVVTS � = tr � SXQVVT� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Recall the central path neighbourhood we defined in (5), we define a matrix E such that E = 1 µθ (XS − µI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' (14) It is obvious that E is a diagonal matrix and satisfies ∥Ee∥2 < 1, which leads to | tr(E)| ≤ ∥Ee∥1 ≤ √ n∥E∥F = √ n∥Ee∥2 < √ n and I − E ≻ 0 and I + E ≻ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' With this, we can have tr � XQVVTS � = tr � SXQVVT� = tr � (θµE + µI)QVVT� = tr � θµEQVVT� + tr � µQVVT� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' For the second term, we know Q and VTQV are both positive semidefinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' So we can have tr � QVVT� = tr � VTQV � ≥ 0 because of the cyclic invariant property of trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' According to the Cauchy–Schwarz inequality, we have tr � EQVVT�2 ≤ ∥E∥2 F∥QVVT∥2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' So we have tr � EQVVT� ≥ −∥QVVT∥F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' 12 of 21 Thus, we have tr � XQVVTS � = tr � θµEQVVT� + tr � µQVVT� ≥ µ � tr � QVVT� − θ∥QVVT∥F � ≥ −θµ∥QVVT∥F ≥ −µ 4 , where the last inequality holds due to condition (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' So we can bound ∥M∥F by ∥M∥2 F = tr � SVVTS � + tr � XQVVTS � + tr � SVVTQX � + tr � XQVVTQX � + tr � XATAX � ≥ tr � SVVTS � + tr � XQVVTQX � + tr � XATAX � − µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Since XQVVTQX ⪰ 0, we have ∥M∥2 F ≥ tr � SVVTS � + tr � XATAX � − µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Since X and S are both positive diagonal matrices, we have ∥M∥2 F ≥ tr � SVVTS � + tr � XATAX � − µ 2 = ∑ i s2 i (VVT)ii + ∑ i x2 i (ATA)ii − µ 2 ≥ ω2 − µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' As we said in the very beginning of this section, at each iteration ω is indeed ωk but the superscript is ignored here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Now we are going to find a bound for µ so we can further bound ∥M∥2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Since ω is the upper bound for the magnitude of the primal and dual slack variables, we have ω2 ≥ xisi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Recall the definition of matrix E, see (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' So we have ω2 ≥ xisi = µ + θµEii ≥ µ − θµ = (1 − θ)µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' So ∥M∥2 F ≥ ω2 − µ 2 ≥ ω2 − 1 2 ω2 1 − θ ≥ ω2 − 1 2 ω2 1 − 1/3 = ω2 4 , where the last inequality follows from the bound for θ, see (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' So we have ω ∥M∥F ≤ 2 = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Bound for κMk Similar to the previous section, we ignore the supercript k unless we need it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' We will start with a general result and then work on the matrix Mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' The following lemma is a well-known result regarding condition numbers of matrices and can be proven using Courant-Fischer-Weyl Min-Max principle [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' 13 of 21 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' For any full row rank matrix P ∈ Rm×n and symmetric positive definite matrix D ∈ Rn×n, their condition number satisfies κ(PDPT) ≤ κ(D)κ(PPT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Next, we analyze the matrix in the OSS system (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Specifically, we focus on MTM since we are interested in the spectral property of the OSS system (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Using the matrix E defined in (14), we have the following decomposition MTM = �VT(S + XQ)T(S + XQ)V −VT(S + XQ)TXAT −AX(S + XQ)V AX2AT � = �VT(S + XQ)T(S + XQ)V −VTµ(θE)AT − VTQTX2AT −Aµ(θE)V − AX2QV AX2AT � = �VT 0 0 A ��(S + XQ)T(S + XQ) −µθE − QX2 −µθE − X2Q X2 ��VT 0 0 A �T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' The second equality holds because −VTSXAT − VTQTX2AT = −VTµ(I + θE)AT − VTQTX2AT = −VTµ(θE)AT − VTQX2AT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Here we used that AV = 0 and Q is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' plugging (14) into the first diagonal block of the decomposition we obtained earlier,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' we have MTM = �VT 0 0 A ���S2 + 2µQ + µθ(EQ + QE) + QX2Q −µθE − QX2 −µθE − X2Q X2 ���VT 0 0 A �T = �VT 0 0 A ���S2 + 2µQ + µθ(EQ + QE) −µθE −µθE 0 � + �QX2Q −QX2 −X2Q X2 ���VT 0 0 A �T = �VT 0 0 A ���I −Q 0 I ��S2 + 2µQ −µθE −µθE 0 �� I 0 −Q I � + �I −Q 0 I ��0 0 0 X2 �� I 0 −Q I ���VT 0 0 A �T = �VT 0 0 A ��I −Q 0 I ��S2 + 2µQ −µθE −µθE X2 �� I 0 −Q I ��VT 0 0 A �T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' (15) The first two matrices are nonsingular, so we can apply the Lemma 2 and thus we only need to study the middle matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Denote the middle matrix by Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Observe that Ψ is almost the same as its counterpart in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Subsequently we have the following result regarding the spectral property of Mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' When (x, y, s) ∈ N (θ) and θ ∈ � 0, min � 1 3√n, 1 4∥QVVT∥F+1 �� , the condition number of matrix Mk satisfies κMk = O � (ωk)2 + µkσmax(Q) µk κVAQ � , where κVAQ is the condition number of the matrix �VT 0 0 A ��I −Q 0 I � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' The proof is in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Putting all these together, we have the complexity for our IF-QIPM for LCQO problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' 14 of 21 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' The IF-QIPM for LCQO problems stops with final duality gap less than ϵ in at most O �√n log(1/ϵ) � IPM iterations and in each IPM iteration, the Newton direction can be obtained with complexity ˜On, ¯ω, 1 ϵ � n � ¯ω2 ϵ + σmax(Q) � κVAQ + n2� , where ¯ω = maxk ωk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' The complexity bound for the IPM iterations comes from the result in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' According to (13), the complexity for obtaining the Newton direction is ˜On, ¯ω, 1 ϵ � nωkκMk ∥Mk∥F + n2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' (16) Combining this with the result in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='1, the bound in Lemma 3, and µk ≥ ϵ, we have ˜On, ¯ω, 1 ϵ � nωkκMk ∥Mk∥F + n2 � = ˜On, ¯ω, 1 ϵ � n � ¯ω2 ϵ + σmax(Q) � κVAQ + n2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' (17) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Application in Support Vector Machine Problems In this section, we discuss how to use our IF-QIPM to solve SVM problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' We show that our algorithm can solve l1-norm soft margin SVM problems with best complexity compared with any existing classical or quantum algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' The ordinary SVM problem works on a linearly separable dataset, in which the data points have binary labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' The ordinary SVM aims to find a hyperplane correctly separating the data points with maximum margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' However, in practice the data points are not necessarily linearly separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' To allow mislabelling, the concept of soft margin SVM was introduced in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Let {(φi, ζi) ∈ Rm × {−1, +1}|i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' , n} be the set of data points, Φ be a matrix with ith column being φi, and Z be a diagonal matrix with ith diagonal element being ζi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' The SVM problem with l1-norm soft margin can be formulated as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' min (ξ,w,t)∈Rn×Rm×R 1 2∥w∥2 2 + C∥ξ∥1 ζi(⟨w, φi⟩ + t) ≥ 1 − ξi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' , n ξi ≥ 0, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' (18) Here (w, t) determines a hyperplane and C is a penalty parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' In [14], the authors rewrote the SVM problem as a second order conic optimization (SOCO) problem and use the quantum algorithm they proposed to solve the resulting SOCO problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' They claim the complexity of their algorithm has O(n2) dependence on the dimension, which is better than any classical algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' However, the algorithm in [14] is invalid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Their algorithm is an Inexact Infeasible- QIPM (II-QIPM) while they used the IPM complexity for Feasible-QIPM, which ignores at least O(n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='5) dependence on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' They also missed the symmetrization of the Newton step, which is necessary for SOCO problems and makes their Newton step invalid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Aside from [14], some pure quantum algorithms for SVM problems are also proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' In [23], the authors propose a pure quantum algorithm for SVM problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' They claim the complexity is O(κ3 effϵ−3 log(mn)), where κeff is the condition number of a matrix involving the kernel matrix and ϵ is the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' In the worst case, κeff = O(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Their complexity is worse than ours regarding the dependence of dimension and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' In addition, their algorithm does not provide classical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Namely, the solution is in the quantum machine and we can not read or use it in a classical computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' However, our algorithm produces a classical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' 15 of 21 To convert the problem into standard form LCQO, we introduce (w+, w−) ∈ Rm+ × Rm+, (t+, t−) ∈ R+ × R+, and a slack variable ρ ∈ Rn+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Then we can get the following formulation min w+,w−,t+,t−,ξ,ρ 1 2∥w+ − w−∥2 2 + C∥ξ∥1 ζi(⟨w+ − w−, φi⟩ + t+ − t−) + ξi − ρi = 1, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' , n (ξ, w+, w−, t+, t−, ρ) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' It is a standard form LCQO problem with nonnegative variables (w+, w−, t+, t−, ξ, ρ) ∈ Rm × Rm × R × R × Rn × Rn and parameters c = � � 02m+2 Cen 0n � � Q = � � Im×m −Im×m 0m×(2+2n) −Im×m Im×m 0m×(2+2n) 0(2+2n)×m 0(2+2n)×m 0(2+2n)×(2+2n) � � A = � ZΦT −ZΦT Z −Z In×n −In×n � b = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' So we can use the proposed IF-QIPM for LCQO problems to solve the ℓ1-norm soft margin SVM problems and get an ϵ-approximate solution with complexity ˜On, ¯ω, 1 ϵ � n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='5 � ¯ω2 ϵ + σmax(Q) � κVAQ + n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='5 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' This dependence on dimension is better than any existing quantum or classical algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Discussion In this work, we present an IF-QIPM for LCQO problems by combining the IF-IPM frame- work proposed in [5] and the OSS system introduced in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Our algorithm has n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='5 dependence on n, which is better than any existing algorithms for LCQO problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' The dependence on the accuracy is polynomial, which is worse than classic IPMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Iterative refinement method might help improve the dependence on the accuracy but that could be another work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Author Contributions: Conceptualization, Zeguan Wu and Tamás Terlaky;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Methodology, Zeguan Wu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Supervision, Xiu Yang and Tamás Terlaky;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Validation, Zeguan Wu, Mohammadhossein Mohammadisi- ahroudi, Brandon Augustino, Xiu Yang and Tamás Terlaky;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Writing – original draft, Zeguan Wu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Writing – review & editing, Zeguan Wu, Mohammadhossein Mohammadisiahroudi, Brandon Augustino, Xiu Yang and Tamás Terlaky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Funding: This work was supported by Defense Advanced Research Projects Agency as part of the project W911NF2010022: The Quantum Computing Revolution and Optimization: Challenges and Opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Institutional Review Board Statement: Not applicable .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Informed Consent Statement: Not applicable .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Data Availability Statement: Not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Conflicts of Interest: The funder had no role in the design of the study;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' in the writing of the manuscript;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' or in the decision to publish the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='16 of 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='Abbreviations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='The following abbreviations are used in this manuscript: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='IF-IPM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='Inexact Feasible Interior Point Method ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='IF-QIPM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='INexact Feasible Quantum Interior Point Methods ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='IPM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='Interior Point Method ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='LCQO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='Linearly Constrained Quadratic Optimization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='LO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='Linear Optimization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='OSS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='Orthogonal Subspace System ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='QIPM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='Quantum Interior Point Method ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='QLSA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='Quantum Linear System Algorithm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='QTA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='Quantum Tomography Algorithm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='SOCO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='Second Order Conic Optimization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='SVM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='Support Vector Machine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Block-encoding of the OSS system In this section, we ignore the superscript k for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' As described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' (9), we first block encode each of the matrices involved in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' With V, A, S and X given and are stored in a quantum accessible data structure (we ignore the complexity to store the classical information into the quantum machine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' For the first matrix M1 = � � 0n×n 0n×n 0n×n 0(n−m)×n VT 0(n−m)×n 0m×n 0m×n −A � �, a �� ∥V∥2 F + ∥A∥2 F, O(poly log(n)), ϵ1 � block-encoding of M1 can be implemented according to Lemma 50 from [18] efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' The second matrix M2 = � � 0n×n 0n×n S 0n×n 0n×n 0n×n � � is both 1-row-sparse and 1-column-sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' By the definition of ω, each element of M2/ω has absolute value at most 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' According to Lemma 48 in [18], a (1, O(poly log(n)), ϵ2) block-encoding of M2/ω can be implemented efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' The third matrix M3 = � � 0n×n 0n×n 0n×n 0n×n Q 0n×n 0n×n 0n×n In×n � � can be decomposed into M3 = � � 0n×n 0n×n 0n×n 0n×n Q 0n×n 0n×n 0n×n 0n×n � � + � � 0n×n 0n×n 0n×n 0n×n 0n×n 0n×n 0n×n 0n×n In×n � �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Then we can block-encode the two matrices first, and then apply linear combination to obtain M3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' In fact, a (∥Q∥F, O(poly log(n)), ϵ3) 17 of 21 block-encoding of the left matrix can be implemented according to Lemma 50 from [18] efficiently and a (1, O(poly log(n)), ϵ3) block-encoding of the right matrix can be implemented efficiently according to Lemma 48 in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' With the state-preparation cost of the linear combination coefficient vector (1, 1) neglected, a (∥Q∥F + 1, O(poly log(n)), (∥Q∥F + 1)ϵ3) block-encoding of M3 can be implemented efficiently according to Lemma 52 from [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' The fourth matrix M4 = � � 0n×n 0n×n X 0n×n X 0n×n � � is 1-row-sparse and 2-column-sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' After being scaled by 1 ω, each element of M4/ω has absolute value at most 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' According to Lemma 48 in [18], a �√ 2, O(poly log(n)), ϵ4 � block-encoding of M4/ω can be implemented efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' For the matrix multiplication M3M4/ω, a �√ 2∥Q∥F + √ 2, O(poly log(n)), (∥Q∥F + 1)( √ 2ϵ3 + ϵ4) � block-encoding can be implemented efficiently according to Lemma 53 from [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' For the linear combination M2/ω + M3M4/ω, the cost for the state-preparation of the coefficient vector (1, 1) is negligible and thus a �√ 2∥Q∥F + √ 2 + 1, O(poly log(n)), ( √ 2∥Q∥F + √ 2 + 1)(ϵ3 + 1 √ 2 ϵ4) � block-encoding can be implemented efficiently according to Lemma 52 from [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' For the matrix multiplication of M1(M2/ω + M3M4/ω), a �� ∥V∥2 F + ∥A∥2 F( √ 2∥Q∥F + √ 2 + 1), O(poly log(n)), � ∥V∥2 F + ∥A∥2 F( √ 2∥Q∥F + √ 2 + 1)(ϵ3 + 1 √ 2 ϵ4) + ( √ 2∥Q∥F + √ 2 + 1)ϵ1 � block-encoding can be implemented efficiently according to Lemma 53 from [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Finally, considering that the complexity of state-preparation of the vector ( ω √ 2∥M∥F , ω √ 2∥M∥F ) 18 of 21 can be neglected, a �� ∥V∥2 F + ∥A∥2 F √ 2ω ∥M∥F ( √ 2∥Q∥F + √ 2 + 1), O(poly log(n)), � ∥V∥2 F + ∥A∥2 F √ 2ω ∥M∥F ( √ 2∥Q∥F + √ 2 + 1)2 �� ∥V∥2 F + ∥A∥2 F(ϵ3 + 1 √ 2 ϵ4) + ϵ1 �� block-encoding of the coefficient matrix of system (8) can be implemented efficiently according to Lemma 52 from [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' We can choose ϵ1 = ϵQLSA κ3 M 1 2K ϵ2 = ϵ1 2 � ∥V∥2 F + ∥A∥2 F ϵ3 = ϵ2 ϵ4 = √ 2ϵ2, where K depends on the initial data K = √ 2 � ∥V∥2 F + ∥A∥2 F( √ 2∥Q∥F + √ 2 + 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Now, considering that the complexity for all the block-encoding algorithms we have used so far have poly-logarithmic dependence on the dimension and accuracy, and that, for i = 1, 2, 3, 4 O � poly log( 1 ϵi ) � = O(poly log(κM)), the complexity for block-encoding will be dominated by the complexity for QLSA because QLSA has linear dependence on κM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' So we can ignore the complexity of block-encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Spectral Analysis for Matrix Ψ In this section, we provide the spectral analysis for the matrix Ψ = �S2 + 2µQ −µθE −µθE X2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' (A1) Just like in the previous section, for simplicity, we ignore the superscript k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' We can do the following decomposition �S2 + 2µQ −µθE −µθE X2 � = � S2 −µθE −µθE X2 � + �2µQ 0 0 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Let us use the following notation Ψ1 = � S2 −µθE −µθE X2 � Ψ2 = �2µQ 0 0 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' 19 of 21 It can be proven that Ψ1 is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' The majority of the proof of this conclusion comes from the paper [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' For the reader’s convenience, we provide the complete proof here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Matrix Ψ1 is a block diagonal matrix, with all the four blocks being diagonal matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' So we can easily compute the eigenvalues using the characteristic polynomial det(Ψ1 − qI) = det �� X2 − qI �� S2 − qI � − θ2µ2E2� = n ∏ i=1 �� x2 i − q �� s2 i − q � − θ2µ2E2 ii � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Clearly, det(Ψ1 − qI) = 0 gives n quadratic equations and each quadratic equation gives two eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' The two eigenvalues from the ith quadratic equation are qi+ = 1 2 � (x2 i + s2 i ) + � (x2 i + s2 i )2 − 4x2 i s2 i + 4θ2µ2E2 ii � and qi− = 1 2 � (x2 i + s2 i ) − � (x2 i + s2 i )2 − 4x2 i s2 i + 4θ2µ2E2 ii � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Recalling the definition of E in (14), we can write qi− = 1 2 � (x2 i + s2 i ) − � (x2 i + s2 i )2 − 4x2 i s2 i + 4(xisi − µ)2 � = 1 2 � (x2 i + s2 i ) − � (x2 i + s2 i )2 + 4(xisi − µ + xisi)(xisi − µ − xisi) � = 1 2 � (x2 i + s2 i ) − � (x2 i + s2 i )2 − 4µ(2xisi − µ) � = 1 2 � (x2 i + s2 i ) − � (x2 i + s2 i )2 − 4µ(2θµEii + µ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' One can verify that the square root always exists because (x2 i + s2 i )2 − 4µ(2xisi − µ) ≥ 4(xisi)2 − 4µ(2xisi) + 4µ2 = 4(xisi − µ)2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' 20 of 21 With θ ∈ � 0, min � 1 3√n, 1 4∥QVVT∥F+1 �� , we have qi− ≥ 1 2 � (x2 i + s2 i ) − � (x2 i + s2 i )2 − 4µ(2θµEii + µ) � ≥ 1 2 � (x2 i + s2 i ) − � (x2 i + s2 i )2 − 4µ(−2µ 1 3√n + µ) � = 1 2 � (x2 i + s2 i ) − � (x2 i + s2 i )2 − 4 3µ2 � = 1 2 4 3µ2 (x2 i + s2 i ) + � (x2 i + s2 i )2 − 4 3µ2 ≥ 1 2 4 3µ2 (x2 i + s2 i ) + � (x2 i + s2 i )2 = µ2 3(x2 i + s2 i ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' This means that matrix Ψ1 is positive definite and its eigenvalues coincide with its singular values because Ψ1 is also real and symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Analogously, we have qi+ = 1 2 � (x2 i + s2 i ) + � (x2 i + s2 i )2 − 4µ(2θµEii + µ) � ≤ 1 2 � (x2 i + s2 i ) + (x2 i + s2 i ) + 2µ � (2θEii + 1) � ≤ 1 2 � (x2 i + s2 i ) + (x2 i + s2 i ) + 2µ √ 2 � = (x2 i + s2 i ) + √ 2µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' So the condition number of Ψ satisfies κ(Ψ) ≤ σmax(Ψ1) + σmax(Ψ2) σmin(Ψ1) + σmin(Ψ2) = maxi qi+ + σmax(Ψ2) minj qj− + σmin(Ψ2) ≤ maxi{x2 i + s2 i } + √ 2µ + 2µσmax(Q) minj µ2 3(x2 i +s2 i ) = 3 maxi{x2 i + s2 i } � maxi{x2 i + s2 i } + √ 2µ + 2µσmax(Q) � µ2 ≤ 3ω2� ω2 + √ 2µ + 2µσmax(Q) � µ2 , 21 of 21 where the last inequality comes from the definition of ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Since ω2 ≥ xisi ≥ (1 − θ)µ, we have κ(Ψ) = O �ω2(ω2 + µσmax(Q)) µ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Using Lemma 2, we can also bound the condition number of matrix M by κM = � κ(MTM) ≤ � κ(Ψ)κVAQ ≤ O �(ω2 + µσmax(Q)) µ κVAQ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Nocedal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Horn, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Johnson, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Matrix analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Cambridge university press, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Cortes, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Vapnik, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Support-vector networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Machine Learning 1995, 20, 273–297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Rebentrost, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Mohseni, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Lloyd, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Quantum support vector machine for big data classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} +page_content=' Physical Review Letters 2014, 113, 130503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE4T4oBgHgl3EQf9w4Q/content/2301.05357v1.pdf'} diff --git a/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf b/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..51387e9b5f8c1e132af88bc6b9a80e9b8f4603cc --- /dev/null +++ b/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7de72999e5695687ebfbd43256b16b988d5fa5d0ed862362002b17b755fc0ef4 +size 202842 diff --git a/OtE1T4oBgHgl3EQfuAVi/vector_store/index.faiss b/OtE1T4oBgHgl3EQfuAVi/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..25c83fc75bbf712063cdaf0664831e2ba019de35 --- /dev/null +++ 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[physics.optics] 20 Jan 2023 +Inhibited spontaneous emission of quantum dots weakly coupled to off resonant silver +nanoplatelets and silver nanowires +Harshavardhan R. Kalluru1,∗ Binita Tongbram2, Ashish Biswas3, and Jaydeep K. Basu4 +Department of Physics, Indian Institute of Science Bengaluru, 560012, India.1,2,3,4 +(Dated: January 23, 2023) +Spontaneous emission (SE) rate of any light emitters directly scales with the locally available +modes for photons. The emission rate can be modified, by changing the dielectric environment of +light emitters. Generally cavities with modes in resonance to light emission frequency, are used to +amplify the light emission rate. The Fermi golden rule predicts that if the cavity modes are off- +resonant to the emission frequency, then the SE rate is suppressed. In this study, we demonstrate +that the SE of colloidal alloyed quantum dots is inhibited by coupling them to chemically synthe- +sized Silver nanowires and Silver nanoplatelet systems. The silver nanoplatelet and silver nanowire +plasmonic resonance modes are in ultraviolet and infrared regions of the electromagnetic spectrum. +The quantum dots emit in visible region of light. This off-resonant weak coupling of emitters and +cavities results in emission rate suppression and is quantified by time resolved photoluminescence +(TRPL) measurements. TRPL decay profiles show that the emission rate can be suppressed by +coupling self assembled quantum dot monolayers to a single silver nanoplatelet and a single silver +nanowire respectively. +I. +INTRODUCTION +The spontaneous emission (SE) rate changes with the +available cavity modes in the vicinity of emitters. This +was first reported by E.M. Purcell.[1] The SE rate is di- +rectly proportional to the local photonic density of states +(PDOS). So in presence of cavity modes which are reso- +nant to emission frequency, the SE rate is enhanced rel- +ative to vacuum emission rate.[2] If the cavity modes are +off resonant, then the SE rate is inhibited relative to vac- +uum emission rate.[3] The PDOS can be quantified and +measured as the ratio of the spontaneous radiaitive emis- +sion rate near a cavity (Γc) relative to spontaneous ra- +diative emission rate in vacuum (Γo). The ratio is known +as Purcell factor. The Purcell factor (FP),[4] which es- +sentially accounts for the available cavity mode density +is given by, +FP = 3λ3 +o +4π · +�Q +V +� +(1) +where Q is the cavity quality factor, V is cavity mode +volume and λo is the cavity mode resonance wavelength. +The Purcell factor has two components, one is radia- +tive (Fr +P) and the other one is non-radiative (Fnr +P ). As +the radiative emission rate is the reciprocal of the spon- +taneous life time (τ) and the radiative part of the Purcell +factor can be experimentally estimated by time resolved +photoluminescence (TRPL) measurements. +FP = Fr +P + Fnr +P +(2) +For +silver +plasmonic +cavities, +the +absorption +of +light +through +non-radiative +decay +occurs +via +four +mechanisms.[5] They are classified into phonon mediated +∗ kallurureddy@iisc.ac.in +absorption, absorption mediated by electron-electron +scattering, direct surface plasmon scattering mediated +absorption and inter-band absorption processes. The di- +rect experimental measurement of non-radiative part of +Purcell effect, involves accounting absorption for each +of these four processes and is beyond the scope this +manuscript. So, only the radiative part of Purcell effect +is discussed. +Fr +P = Γc +Γo += τo +τc +(3) +Here Γo and Γc are the radiative decay rate of emitters +in vacuum and near the cavity. τo and τc are the corre- +sponding lifetimes of the emitters in vacuum and near +the cavity. If the cavity mode is resonant to the emis- +sion, then the Fr +P > 1, which means the radiative emis- +sion rate is enhanced . If the cavity mode is off-resonant, +then the Fr +P < 1, which means the radiative emission rate +is inhibited. The resonant and off-resonant emission rate +modification is a consequence of weak coupling between +the emitter and cavity modes. +In this study, CdSe-ZnS alloyed quantum dots (AQDs) +are used as emitters. +The plasmonic cavities used are +Silver nanowires (AgNWs) and Silver nanoplatelets (Ag- +NPLs). The self-assembled AQD layers are transferred +onto the cavities and the coupled systems are studied. +The position of plasmonic dipolar modes (DM) of the +AgNPLs and AgNWs, depends upon the aspect ratio (a) +of these systems. The aspect ratio[6] is defined as the +ratio of lateral size to thickness of a nanostructure. In +the case of AgNPLs, the ratio of triangular edge length +to AgNPL thickness is considered as aspect ratio. In the +case of AgNWs, the ratio of wire length to wire thick- +ness is considered as the aspect ratio. Typically for the +synthesized AgNWs in this study, aspect ratio is 241. +AgNPLs are synthesized in three aspect ratios of 3, 261 +and 551. + +2 +AgNWs have localized surface plasmon resonances +along longitudinal and transverse directions of the +wire.[7] Similarly triangular AgNPLs have in-plane ori- +ented plasmonic dipolar modes (DM) and quadrupolar +modes (QM).[8–10] It is well established that for silver +nanowires, the longitudinal surface plasmon resonance +(LSPR), moves from visible to infra red region with in- +creasing aspect ratio.[11, 12] The absorbance peaks of +LSPR peaks are in infra red region and can be measured +by electron energy loss spectroscopy (EELS).[13] The Ag- +NPL dipolar plasmonic resonance also shifts to higher +wavelengths with increasing aspect ratio.[14, 15] So con- +sidering the large aspect ratios of silver nanoplatelets and +silver nanowires in this study, all the plasmonic mode res- +onances are off resonant relative to the AQD emission. +II. +EXPERIMENTAL METHODS +The AQDs are synthesized by protocol mentioned +in[16]. +The AgNPLs and AgNWs are synthesized +and cleaned by protocols mentioned in appendices A +and B respectively. The synthesized nanoplatelets and +nanowires are imaged and characterized by scanning +transmission electron microscopy (STEM), energy dis- +persive spectroscopy (EDS), high angle annual dark field +(STEM-HAADF) spatial mapping. (Appendices A and +B) The AgNWs and AgNPLs are transferred by dip- +coating onto Silicon substrates with 300 nm oxide, using a +Kibron dipper. Subsequently langmuir-schaefer (LS) self- +assembled monolayers are prepared and transferred using +a KSV langmuir setup. (Appendix C) The transferred +nanostructures are imaged by atomic force microscopy +(AFM). +The absorption spectra of AgNPLs are measured in a +Perkin-Elmer lambda-35 solution uv-vis absorption spec- +trometer. +The Photoluminescence (PL) and time re- +solved photoluminescence (TRPL) are measured with a +Witec alpha 300 confocal microscope equipped with a +peltier cooled CCD spectrometer and a Picoquant SPAD +detector. +The PL spectra are excited with a 532 nm +diode laser and exciting laser line is cut-off with 532 nm +edge long pass filter. The PL integration time is set at +5 s and averaged over such 4 accumulation cycles. The +TRPL spectra are excited with a 405 nm pulsed laser and +cutoff with a set of 405 nm and 488 nm edge long pass +filters. The TRPL integration time is set as 5 s and 10 +accumulation cycles. +III. +RESULTS AND DISCUSSION +The TEM images of the chemically synthesized Silver +nano-platelets (AgNPLs) and Silver nanowires (AgNWs) +are shown in Fig. 1. The chemically synthesized nanos- +tructures generally are defective. The defects are gener- +ally grouped into ridge and groove defects.[17] Such nm +sized defects, strongly scatter the surface plasmons.[18] +FIG. 1. +(a) and (b) show the TEM images of Silver +nanoplatelets (AgNPLs) and silver nanowires (AgNWs) of as- +pect ratios of 3 and 241 respectively. (c) and (d) show the +high resolution TEM images of Silver nanoplatelets and Silver +nanowires. The encircled regions in red and green, show the +ridge and groove type defects respectively. +The ridge and groove defects scatter the surface plasmons +and are can be used as hot spots[19] for surface enhanced +raman scattering. Such defects can also create shoulders +in scattering spectra. +In subsection A, the coupling data of AgNPLs with +AQDs is discussed and AgNW-AQD coupling data is dis- +cussed in the subsection B. +A. +AgNPL-AQD coupling +The AgNPLs are deposited on Silicon substrates and +imaged by AFM. Then 2 monolayers of AQDs are de- +posited on the AgNPLs directly. +The AgNPLs aspect +ratio can be controlled by varying the precursor concen- +tration used in synthesis. Three sets of AgNPLs with +aspect ratios a1=400, a2=261 and a3=3 are synthesized. +Only the AgNPLs with aspect ratios a1 and a2 are used +for coupling to AQDs. The synthesized triangular Ag- +NPLs edges are blunted. This is known as snipping.[20] +With large aspect ratios[21] and snipping of AgNPLs, +the in-plane dipolar modes of AgNPLs move to higher +wavelengths, typically to infra red region of light. The +position of DM modes of AgNPLs of aspect ratio a3=3 is +at 1027 nm, as shown in Fig. 2. The Absorption spectra +of AgNPLs with higher aspect ratios are beyond the de- +tection range of our instrument facilities, so they are not +shown. AgNPLs of two aspect ratios a1 and a2 are dip- + +(a) +(b) +50nm +2 +um +(c) +(p) +10nm +200nm3 +FIG. 2. shows the AgNPL (a3=3) absorbance and AQD PL +superimposed over each other. +FIG. 3. (a) and (b) show the AFM images of a single AgNPL +and a single AgNPL coated with 2L-AQDs, respectively. (c) +and (d) show the corresponding AFM height profiles of single +AgNPL and a single AgNPL coated with 2L-AQDs, respec- +tively +coated on to Silicon substrates. Then two consecutive +AQDs monolayers are deposited on the AgNPLs. The +AFM image and topography profile of a bare AgNPL +and 2 AQD monolayers deposited AgNPL are shown in +Fig. 3, respectively. The AFM images of the bare Ag- +NPLs and AQD deposited AgNPLs, are shown in Figs. +4(a) and 4(b). The AFM images indicate that the AgN- +PLs are completely covered with AQDs. The PL spatial +map of the coupled AQDs-AgNPLs is shown in Fig. 4 +(c). The bright spots in PL spatial map are on AgNPLs. +FIG. 4. +(a) and (b) show the AFM images of bare AgN- +PLs on Silicon and 2 AQD monolayer coated AgNPLs on Sil- +icon respectively.(c) shows the PL emission spatial map of +2L-AQDs-AgNPLs system. (d) shows the typical PL spectra +of of 2L-AQDs and 2L-AQDs-coupled to a single AgNPL. +A typical spectra of AQDs coupled to a single AgNPL +is shown in Fig. 4(d) along with control PL spectra of 2 +monolayer AQDs on Silicon. Similarly the TRPL spectra +is collected for control AQDs and AQDs coupled to single +AgNPL. The PL spectra is not deformed and shows no +extra features as shoulders. The TRPL spectra show that +the AQD SE rate is inhibited on the AgNPL and is AQD +emission rate faster in case of the control sample. This +FIG. 5. (a) and (b) show the respective TRPL decay profiles +of 2L-AQDs on Silicon and 2L-AQDs on single AgNPLs with +aspect ratios a1 and a2. +confirms that the AQDs are weakly coupled to AgNPL +and the coupled system shows suppression of SE.[22, 23] +To quantify, the SE rate inhibition of AQDs, the TRPL +intensity (I) is fitted with the following function, where +the A1 and A1 are the amplitudes of the decay functions. +τ1 and τ2 are the respective decay lifetimes. The fitting + +(a) +(b) +AQD-contro +AQD-control +A +AQD-2L-AgNP-a1 +AQD-2L-AgNP-a2 +1.00 +twoexponentialfitAQDcontrol +intensity +1.00 +twoexponentialfitAQDcontro +twoexponentialfitAQD-2L-AGNP-a1 +twoexponentialfitAQD-2L-AGNP-a2 +0.75 +0.75 +TRPL +Normalized +0.50 +0.50 +lormalized +DO +40 +0.25 +0.25 +0.00 +0.00 +0 +10 +20 +30 +40 +50 +0 +10 +20 +30 +40 +50 +time(ns) +time (ns)(a) +(b) +nm +nm +200 +100 +10 μm +10 μm +75 +100 +50 +0 +25 +0 +-100 +(c) +(d) +10000 +-O-AQD-control-PL +-0-AQD-2L-AgNP-PL +1400 +(a.u.) +8000 +1200 +6000 +intensity +4000 +1000 +2000 +800 +600 +540 +560 +580 +600 +620 +640 +660 +680 +700 +wavelength(nm) +40 μm(a) +(c) +nm +6 +1 μm +BareAgNP +4 +4 +Height (nm) +2 +0 +0 +-4 +-2 +0 +1 +2 +3 +4 +5 +-8 +distance (μm) +(q) +(d) +nm +5 +AgNP-2L-AQD +80 +Height (nm) +-5 +40 +-10 +-15 +0 +-20 +-25 +0 +1 +2 +3 +-40 +1 μm +distance (μm)0.30 +-△- AgNP in water +1.00 +-0- QD-PL +DM +0.25 +0.75 +Absorbance +0.20 +QD-PL +QM +0.50 +0.15 +0.25 +0.10 +0.05 +500 +600 +700 +800 +900 +1000 +wavelength (nm)4 +TABLE I. The table shows the fitting components of lifetime +of AQDs coupled to AgNPL +Sample +A1 τ1 (ns) A2 τ2 (ns) +τavg (ns) +AQD +0.13 +1.06 +0.66 +10.37 +8.86 ± 1.46 +AQD-AgNPL-a1 0.37 +4.86 +0.54 +16.74 +11.89 ± 1.36 +AQD-AgNPL-a2 0.24 +2.68 +0.68 +12.25 +9.78 ± 0.34 +parameters are shown in Table I. +I = A1·e− t +τ1 + A2·e− t +τ2 +(2) +The weighted average lifetime is given by +τavg = A1 · τ1 + A2 · τ2 +A1 + A2 +(3) +The Purcell factor is calculated from equation (1), us- +ing weighted average lifetime from fitting TRPL spectra. +The Purcell factors (FP) for single AgNPL of aspect ra- +tio a1 and a2 are 0.75 ± 0.21 and 0.96 ± 0.18. As a1 > +a2, the DM wavelength of AgNPL a1 is larger than that +of AgNPL a2. +So the Purcell factor of AQDs coupled +to AgNPL a1 is smaller, which is indicator of increased +inhibition of AQD SE. So with increasing aspect ratio of +AgNPLs, Purcell inhibition of SE also increases. +B. +AgNW-AQD coupling +The control sample is bare Silver nanowires (AgNWs) +deposited on the Silicon. The control sample is imaged +by TEM and AFM. The AFM imaging indicates that the +AgNW thickness is approximately 36.1 nm, as shown in +Fig. 2(c). The typical aspect ratio of AgNWs is about +240. On a separate control sample, 10 nm of spacer layer +is deposited onto the nanowires by spin-coating 5 mg/ml +polymethyl methacrylate (PMMA) solution in toluene. +(Appendix C) The one monolayer (1L) of AQDs is trans- +ferred by the LS method on top of the spacer and the +AQD monolayer is about 5.6 nm thick.[16] The AFM +height profile of the AQD deposited sample with spacer, +is 55.8 nm as shown in Fig. 2(d). +The PL spectra of the coupled AQD-AgNPL system +are measured. The optical image of the system is shown +in Fig. 3(a) and the spatial PL emission map is shown in +Fig. 3(b). The PL spectra of control sample: monolayer +AQD on spacer and single AgNW-spacer-AQD mono- +layer are shown in Fig. +3(c). +The PL spectra of the +coupled system shows a shoulder feature. The PL spec- +tra is fitted with two Gaussian functions and the peak +to peak separation is 160 meV. The amplitude and full +width half maximum (FWHM) of Gaussian functions 1 +and 2 are represented by A1, A2, w1, w2 respectively. +I(λ) = Io + A1·e +− +� +λ−λ1 +2w2 +1 +� ++ A2·e +− +� +λ−λ2 +2w2 +2 +� +(4) +FIG. 6. (a) and (b) show the AFM images of bare AgNWs +and 1L AQDs coated AgNW with 10 nm polymer spacer, re- +spectively.(c) and (d) show the AFM profiles of a bare AgNW +and 1L AQDs coated AgNW, with 10 nm polymer spacer re- +spectively. +TABLE II. The table shows the fitting components of life- +time of AQDs on spacer and AQDs coupled to spacer coated +AgNWs +Sample +A1 +w1 +A2 +w2 +wavg +1L-AQD-AgNW 0.21 22.19 0.95 47.22 42.67 ± 0.24 +5L-AQD-AgNW 0.18 23.91 0.94 56.40 51.27 ± 0.38 +wavg = A1 · w1 + A2 · w2 +A1 + A2 +(5) +The PL fitting parameters are indicated in Table II. Here +λ1 and λ2 are the central positions of the constituent +gaussian functions 1 and 2. The λ1 and λ2 positions for +multiple AgNWs are 554.55 nm ± 0.11 nm and 597.97 +nm ± 0.04 nm. For single AgNW, they are 553.60 nm +± 0.08 nm and 596.45 nm ± 0.03 nm. Such shoulders in +quantum dot PL spectra might be observed due to Rabi +splitting of strongly coupled systems.[24] For N emitter- +strong coupling, the Rabi splitting scales with number +of emitters.[25] The coupling coefficient (g) of N dipolar +emitters with a cavity mode field (E) is given by[26, 27] +ℏg = +√ +N +� +⃗µ · ⃗E +� +(6) +Here µ is transition dipole moment of the emitter.[28] +The Rabi splitting magnitude is 2g. So the Rabi split- +ting, i.e., the peak to peak separation should scale with +number of emitters. So if the number of emitters is in- +creased by 5 times, then Rabi splitting magnitude should + +(a) +(b) +nm +nm +150 +40 +1 μm +100 +20 +50 +0 +0 +-20 +50 +Tum +40 +(c) +(p) +BareAgNW +50 +AgNW-spacer-1L-AQD +20 +40 +10 +Height (nm) +30 +Height (nm) +0 +20 +10 +10 +0 +-20 +-10 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Distance (μum) +Distance (μm)5 +increase 2.2 fold. 5 monolayers of AQDs are transferred +FIG. 7. (a) shows the optical image of the 1L AQDs deposited +on 10 nm spacer coated AgNWs on Silicon. A typical single +nanowire is shown by the blue circle. Multiple nanowire clus- +ters are shown by the red circle. (b) shows the corresponding +PL spatial map of the 1L AQDs deposited on 10 nm spacer +coated AgNWs on Silicon. (c) shows the PL spectral deforma- +tion of 1L-AQDs-coupled to AgNWs relative to 1L-AQDs.(d) +shows the fitting of PL spectra of 1L-AQDs-coupled to Ag- +NWs. +on an AgNW with spacer and compared with 1 mono- +layer AQDs on a AgNW. The PL spectra of both the +samples are measured. The shoulder separation (λ2−λ1) +for 1L AQDs coupled to AgNW is 42.85 nm. The sep- +aration should scale to 95.82 nm for 5L AQDs coupled +to AgNW, in case of true strong coupling. Instead the +shoulder separation of the 5L AQDs coupled to AgNW is +measured as 43.42 nm, which is with in 1% change from +the initial value for 1L AQDs coupled to AgNW. This +null result confirms that there is no emitter number de- +pendence. So, strong coupling is conclusively ruled out. +The shoulder is attributed to scattering by AgNW. +Also, as the AQD concentration changes from 1L to +5L, the weighted average spectral line width (wavg) in- +creases from 42.67 nm to 51.27 nm, i.e. by 8.60 nm ± 0.62 +nm. The PL spectral broadening with increased emitter +concentration is indicative of weak coupling. To find out +more about the nature of coupling of AQDs to AgNWs +radiative lifetime of AQDs is measured. It is evident from +the raw data in Fig. 8(b), that the SE rate of 1L-AQDs +on AgNWs is slower compared to free space decay rate of +1L-AQDs, which confirms the coupled system is in weak +coupling. Similar fitting approach of AgNPLs is adapted +for AgNWs. The fitting parameters are shown in Table +III. +FIG. 8. +(a) shows the PL spectra of 1L-AQDs-coupled to +AgNWs and 5L-AQDs-coupled to AgNWs. +(b) shows the +normalized raw SE rate profiles of 1L-AQDs on spacer coated +Silicon, 1L-AQDs-spacer-single AgNW and 1L-AQDs-spacer- +multiple AgNWs respectively.(c) and (d) show the inhibition +of SE rate of 1L-AQDs for single and multiple AgNWs respec- +tively. +TABLE III. The table shows the fitting components of lifetime +of AQDs on spacer and AQDs coupled to spacer coated single +and multiple AgNWs +Sample +A1 τ1 (ns) A2 τ2 (ns) +τavg (ns) +AQD-1L +0.35 +1.77 +0.57 +10.78 +7.35 ± 0.07 +AQD-1L-AgNW 0.29 +4.05 +0.65 +14.12 +10.99 ± 0.35 +AQD-1L-AgNWs 0.31 +4.57 +0.63 +16.02 +12.24 ± 0.84 +The Purcell factor (FP) is calculated from the ratio of +of AQD lifetime to the weighted average life time (τavg). +For Single AgNWs, the Purcell factor is 0.67 ± 0.06 and +for multiple AgNWs it is 0.60 ± 0.12. +The SE inhibition of AQDs emitting in the visible re- +gion of light is expected, as the plasmonic resonances +of Silver nanowires are either in ultraviolet or infrared +regions.[29, 30] The lifetime decay profiles reaffirm that +the absence of resonant cavity modes inhibits the SE of +AQDs.[31] +IV. +CONCLUSION +The Silver nanowire and Silver nanoplatelet plasmonic +resonances are off resonant to alloyed quantum dot spon- +taneous emission frequency. This results in inhibition of +spontaneous emission of quantum dots coupled to Sil- +ver nanowires and Silver nanoplatelets. +The sponta- + +(a) +(q) +1100 +1.0 +AQD-1L-spacer-control +。=QD-1L-AgNW-PL +AQD-1L-spacer-single-AgNW +--QD-5L-AgNW-PL +intensity +1000 +AQD-1L-spacer-multiple-AgNWs +0.8 +(a.u.) +intensity +900 +Normalized TRPL +0.6 +0.4 +800 +PL +0.2 +700 +0.0 +540 +560 +580 +600 +620 +640 +660 +680 +700 +0 +10 +20 +30 +40 +50 +(c) +wavelength (nm) +(d) +time (ns) +1.0 +AQD-1L-control +1.0 +AQD-1L-control +AQD-1L-spacer-single-AgNW +AQD-1L-spacer-multiple-AgNWs +A +intensi +0.8 +twoexponentialfit-AQD-control +intens +0.8 +twoexponentialfit-AQD-contro +twoexponentialfit-AQD-AgNw +twoexponentialfit-AQD-AgNWs +A +0.6 +0.6 +V +TRPI +0.4 +A +0.4 +Normalized +口 +lormalized +0.2 +A +0.2 +A +0.0 +0.0 +0 +10 +20 +30 +40 +50 +0 +10 +20 +30 +40 +50 +time (ns) +time (ns)(a) +(q) +10 μm +10 μm +(c) +(d) +AQD-1L-PL +AQD-1L-AgNW-PL +1.0 +1.0 +AQD-1L-AgNW-PL +FitPeak1 +sity +FitPeak2 +0.8 +0.8 +inten +Cumualtive Fit +0.6 +0.6 +PL +0.4 +0.4 +0.2 +0.2 +0.0 +0.0 +540 +560 +580 +600 +620 +640 +660 +680 +700 +540 +560 +580 +600 +620 +640 +660 +680 +700 +wavelength (nm) +wavelength(nm)6 +neous emission inhibition is quantified in terms of Pur- +cell factor. Purcell factors of 0.67 and 0.75 are observed +for quantum dots coupled to silver nanowire and silver +nanoplatelet respectively. With increasing aspect ratio of +AgNPL and with increasing number of silver nanowires, +the spontaneous emission rate is increasingly inhibited. +ACKNOWLEDGMENTS +B. Tongbram thanks the Department of science and +technology (DST), Inspire faculty programme for fellow- +ship. H.R. Kalluru thanks the Micro and nano character- +ization facility (MNCF-CeNSE), IISc for access to titan +themis 300 kV TEM facility. +Appendix A: AgNPL synthesis method and +Characterization +The Silver Nanoplates (AgNPLs) are synthesized in hy- +drophilic phase, with capping agent Polyvinylpyrrolidone +(PVP), as per the reported protocol.[32] The glassware +used for the synthesis is cleaned following the RCA SC- +I protocol, subsequently rinsed thrice in DI water and +dried. Silver nitrate (AgNO3-99%), sodium borohydride +(NaBH4-99.99%), sodium tri-citrate dihydrate (TSCDH- +99%) and polyvinylpyrrolidone-40K are procured from +Merck. +30% w/W hydrogen peroxide solution is pur- +chased from SDFCL. +A 100 ml borosilicate conical glass flask and 46.68 ml +DI water is added to the flask. Then 120 µL of hydrogen +peroxide solution is added to the conical flask. 140 mg +of PVP is dissolved in 1 ml DI water and the whole PVP +solution is added to the conical flask. 22.3 mg of TSCDH +is dissolved in 1 ml DI water and the solution is added +to the flask. Now the flask is placed on magnetic stirrer +setup. The stirrer was turned on and set at 800 rpm for +rigorous mixing of precursors, at room temperature (300 +K). +A glass vial is placed in an ice bath and 4 ml DI wa- +ter is added to it. The ice cold water is kept ready for +dissolving NaBH4. 8.6 mg silver nitrate is dissolved in 1 +ml DI water and 0.2 ml of the solution is added to the +flask. Immediately the solution colour turned to pale yel- +low. 15.1 mg of NaBH4 is added to ice cold water and +mixed thoroughly using a suction pipette. NaBH4 solu- +tion needs to be immediately used after preparation, as +it degrades with time. 1 ml of NaBH4 solution is added +to the mixture of precursors. The reaction mixture im- +mediately turned deep brownish yellow. +After approximately 30 minutes of continuous stirring, +the solution turns dark red and finally to deep brown. +The whole colour change process happens with in 1-2 +minutes. The colour change of solution is due to shifting +of localised surface plasmon resonance peak shift due to +lateral growth of AgNPLs in the reaction mixture. The +reaction mixture is cleaned by centrifuging at 10000 rpm +for 10 minutes. The precipitate is dark in colour and the +supernatant is brownish. The supernatant is discarded +and precipitate is again dispersed in DI water. Then the +centrifuging process is repeated further twice, by select- +ing precipitate and discarding supernatant. After third +centrifuging, the precipitate is dispersed in ethanol or DI +water. The resultant solution is purple and is used for +further characterization and measurements. +For TEM measurements, the AgNPL solution (20 +µg/ml) in ethanol is drop casted on to a copper transmis- +sion electron microscopy (TEM) grid and dried in a dessi- +cator under vacuum for 12 hours. The dried TEM grid +is cleaned with argon plasma (chamber vacuum 5x10-4 +Torr and incident power 22 W) for 40 seconds. STEM- +HAADF mapping of the AgNPLs is shown in Fig. 9. The +Silver characterstic X-ray intensity follows the contours +of the AgNPL volume. This indicates that the AgNPLs +are made of Silver. The Carbon X-ray intensity map in- +dicates the distribution of PVP ligand over the AgNPL. +FIG. 9. (a) shows the typical dark field STEM-HAADF im- +ages of AgNPLs on a TEM grid. (b) and (c) show the silver +and carbon atomic distribution on AgNPLs of aspect ratio +a3. +(d) shows the superimposed silver atomic distribution +and darkfield STEM-HAADF image. +The AgNPLs of three aspect ratios are synthesized by +making the following changes. The AgNPLs of aspect +ratio a1 are synthesized by following above mentioned +procedure. For synthesizing the AgNPLs with aspect ra- +tio a2, the volumes of the precursors (AgNO3 ; NaBH4) +dropped in the conical flask are changed from (0.2 ml +; 1 ml) to (0.1 ml; 0.5 ml) respectively. To synthesize +the AgNPLs of aspect ratio of a3, the reaction volume of +precursors required for AgNPLs of aspect ratio of a2 is +reduced by half, such that the reaction mixture volume +is 25 ml. + +HAADF +Ag +(a) +(q) +50nm +50nm +HAADF +Ag +(c) +(d) +50nm +50nm7 +FIG. 10. (a) and (b) show the typical magnified and large area +view of the TEM images of drop casted AgNPLs of aspect +ratio a3 on TEM grid.(c) and (d) show the distribution in +lateral size and thickness of AgNPLs of aspect ratio a3 +FIG. 11. (a) shows the typical isothermal compression and +compression cycle for transfer of AQDs with a Langmuir- +Schaefer setup. (b)shows the AFM image of AgNPL of typical +aspect ratio a2=261 +Appendix B: AgNW synthesis method and +Characterization +The Silver nanowires (AgNWs) are synthesized in hy- +drophobic phase with capping agent Oleyl amine (OAm), +as per the reported protocol.[33] The glassware used for +the synthesis is cleaned by the RCA SC-I protocol and +rinsed thrice in DI water and dried. +Silver bromide +(AgBr-99%), copper chloride (CuCl2-99.5%), n-Hexane +(99%) and oleyl amine (70%) are procured from Merck. +0.3 gL-1 CuCl2-OAm solution is prepared by dissolving +3 mg of CuCl2 in 10 mL of OAm at 60o C. The temper- +ature is maintained for 10 min and then the solution is +cooled to ambient temperature. +TABLE IV. The table shows the elemental analysis obtained +from the EDS spectrum of AgNWs. +Element Atomic fraction (%) Error (%) +Carbon +95.79 +4.05 +Nitrogen +1.39 +0.28 +Copper +0.83 +0.11 +Silver +1.98 +0.24 +In a round bottom borosilicate glass flask 0.1g AgBr, +34 µL of CuCl2-OAm solution and 5 ml OAm is added. +The flask is heated to 160o C and maintained at the same +temperature for 6 hours. Then the heating element is +turned off and reaction mixture is allowed to reach am- +bient temperature naturally. The final reaction mixture +colour is dark gray, indicating formation of AgNWs. +The reaction mixture is dispersed in 15 ml n-hexane +and centrifuged at 6000 rpm for 10 minutes. The precip- +itate is then dispersed in hexane and centrifuged for two +more cycles. The precipitate after 3 cycles is dispersed +in n-hexane and stored in dark for characterization and +measurements. +FIG. 12. shows the AgNW EDS spectra with characteristic X- +ray peaks of constituent atoms. The inlay shows the zoomed +peaks of the EDS spectra of AgNWs. +For TEM measurements, the AgNW solution (10 +µg/ml) in hexane is drop casted on to a copper trans- +mission electron microscopy (TEM) grid and processed +identical manner of AgNPLs on TEM grid. +The EDS +spectra of the AgNWs is measured and shown in Fig. +10. The elemental distribution is shown in table IV. The +carbon and nitrogen content is attributed to oleyl amine +ligands. The Copper content is attributed to both Cop- +per TEM grid and CuCl2 seeding process of AgNWs. The +Silver content is attributed to AgNWs. + +60000 +AgNW EDS spectrum +8000 +50000 +Cu-Kα +7000 +Intensity (Counts) +Intensity (Counts) +6000 +40000 +5000 +4000 +30000 +N +3000 +Cu-Kβ +Ag-Lβ +Cu-L +20000 +2000 +1000 +10000 +0 +2 +4 +9 +8 +10 +Energy (KeV) +0 +0 +2 +4 +9 +8 +10 +Energy (KeV)nm +(a) +42 +(b) +40 +Surface pressure (mN.m*1) +三 +38 +40 +34 +20 +32 +30 +0 +28 +26 +24 +-20 +22 +4000 +6000 +8000 +10000 +12000 +14000 +1 μm +Trough area (mm3)(a) +(b) +50 nm +500nm +(c) +(d) +25 +20 +5 +4 +15 +count +3 +C +10 +2 +5 +1 +0 +0 +10 +20 +30 +40 +50 +60 +8 +10 +12 +14 +16 +18 +20 +Lateral size (nm) +Thickness (nm)8 +TABLE V. The table shows the synthesized nanostructure +specifications. +Sample +Lateral size (µm) thickness (nm) Aspect ratio +AgNW +8.435 +35 +241 +AgNP-a1 +2.243 +5.6 +400 +AgNP-a2 +1.176 +4.5 +261 +AgNP-a3 +0.043 +14 +3 +Appendix C: Sample preparation procedure +The synthesized AgNPLs are hydrophilic and AgNWs +are hydrophobic in nature. +AgNPLs are transferred +onto Silicon substrate by dip-coating at water and hex- +ane interface, by following the procedure mentioned in +report.[34] The self assembly of hydrophilic particles at +water-hexane interface reduces clustering of AgNPLs and +is proffered for studying properties of single AgNPLs. +Typically in a large glass petridish, the substrate is +placed and is attached to a motorized dipper (KSV make) +and 200 ml deionized (DI) water is poured over the sub- +strate, till the substrate gets completely immersed. +Then 200 ml hexane is poured over DI water, which +floats over water. +Then 3 ml of 2 mg/ml solution of +AgNPLs in DI water is dispersed in the DI water phase +gently below Hexane. The system is allowed to stabilize +for 15 minutes. There are two interfaces in this system. +First interface is at the hexane-air boundary. The other +interface is the hexane-water boundary. +Then the substrate is brought up from water phase +to the hexane-water interface, at a rate of 10 mm/min. +Once the substrate touches water-hexane interface, the +AgNPLs at the hexane-water interface gets transferred +onto the substrate. The substrate is then moved up to +hexane-air interface and the hexane is allowed to dry at +ambient conditions. +AgNWs are transferred onto Silicon substrate by dip- +coating at hexane air interface in a similar procedure fol- +lowed for AgNPLs, with a major difference. The amount +of hexane poured over DI water is limited to 10 ml, so +that a thin Hexane phase floats over DI water. Then 5 ml +of 2.2 mg/ml AgNW solution in hexane is dispersed in DI +water phase. AgNWs are hydrophobic and move immedi- +ately to hexane phase. The system is allowed to stabilize +for 15 minutes and then the substrate is brought up to- +wards the hexane-water interface at a rate of 10 mm/min. +As soon as the substrate crosses hexane-air interface, the +AgNWs get transferred on the substrate. The substrate +is then dried at ambient conditions. +Poly-methyl-methacryalate (PMMA), 350K molecular +weight is procured from Merck and 5 mg/ml solution is +prepared in Toluene. The solution is spin-coated on a +clean Silicon substrate at 3000 rpm and for 60 s. The +resultant film thickness is characterized by X-ray reflec- +tivity (XRR) measurement. The XRR fringe separation +is a measure of film thickness (h). 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Hens, +Langmuir 26, 7732 (2010). + diff --git a/QdFAT4oBgHgl3EQf0B72/content/tmp_files/load_file.txt b/QdFAT4oBgHgl3EQf0B72/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b21ae36cfc670251f22a06e13c3f6fabaa08d402 --- /dev/null +++ b/QdFAT4oBgHgl3EQf0B72/content/tmp_files/load_file.txt @@ -0,0 +1,850 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf,len=849 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='08702v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='optics] 20 Jan 2023 Inhibited spontaneous emission of quantum dots weakly coupled to off resonant silver nanoplatelets and silver nanowires Harshavardhan R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Kalluru1,∗ Binita Tongbram2, Ashish Biswas3, and Jaydeep K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Basu4 Department of Physics, Indian Institute of Science Bengaluru, 560012, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='1,2,3,4 (Dated: January 23, 2023) Spontaneous emission (SE) rate of any light emitters directly scales with the locally available modes for photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The emission rate can be modified, by changing the dielectric environment of light emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Generally cavities with modes in resonance to light emission frequency, are used to amplify the light emission rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The Fermi golden rule predicts that if the cavity modes are off- resonant to the emission frequency, then the SE rate is suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' In this study, we demonstrate that the SE of colloidal alloyed quantum dots is inhibited by coupling them to chemically synthe- sized Silver nanowires and Silver nanoplatelet systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The silver nanoplatelet and silver nanowire plasmonic resonance modes are in ultraviolet and infrared regions of the electromagnetic spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The quantum dots emit in visible region of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' This off-resonant weak coupling of emitters and cavities results in emission rate suppression and is quantified by time resolved photoluminescence (TRPL) measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' TRPL decay profiles show that the emission rate can be suppressed by coupling self assembled quantum dot monolayers to a single silver nanoplatelet and a single silver nanowire respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' INTRODUCTION The spontaneous emission (SE) rate changes with the available cavity modes in the vicinity of emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' This was first reported by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Purcell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' [1] The SE rate is di- rectly proportional to the local photonic density of states (PDOS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' So in presence of cavity modes which are reso- nant to emission frequency, the SE rate is enhanced rel- ative to vacuum emission rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' [2] If the cavity modes are off resonant, then the SE rate is inhibited relative to vac- uum emission rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' [3] The PDOS can be quantified and measured as the ratio of the spontaneous radiaitive emis- sion rate near a cavity (Γc) relative to spontaneous ra- diative emission rate in vacuum (Γo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The ratio is known as Purcell factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The Purcell factor (FP),[4] which es- sentially accounts for the available cavity mode density is given by, FP = 3λ3 o 4π · �Q V � (1) where Q is the cavity quality factor, V is cavity mode volume and λo is the cavity mode resonance wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The Purcell factor has two components, one is radia- tive (Fr P) and the other one is non-radiative (Fnr P ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' As the radiative emission rate is the reciprocal of the spon- taneous life time (τ) and the radiative part of the Purcell factor can be experimentally estimated by time resolved photoluminescence (TRPL) measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' FP = Fr P + Fnr P (2) For silver plasmonic cavities, the absorption of light through non-radiative decay occurs via four mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' [5] They are classified into phonon mediated ∗ kallurureddy@iisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='in absorption, absorption mediated by electron-electron scattering, direct surface plasmon scattering mediated absorption and inter-band absorption processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The di- rect experimental measurement of non-radiative part of Purcell effect, involves accounting absorption for each of these four processes and is beyond the scope this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' So, only the radiative part of Purcell effect is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Fr P = Γc Γo = τo τc (3) Here Γo and Γc are the radiative decay rate of emitters in vacuum and near the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' τo and τc are the corre- sponding lifetimes of the emitters in vacuum and near the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' If the cavity mode is resonant to the emis- sion, then the Fr P > 1, which means the radiative emis- sion rate is enhanced .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' If the cavity mode is off-resonant, then the Fr P < 1, which means the radiative emission rate is inhibited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The resonant and off-resonant emission rate modification is a consequence of weak coupling between the emitter and cavity modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' In this study, CdSe-ZnS alloyed quantum dots (AQDs) are used as emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The plasmonic cavities used are Silver nanowires (AgNWs) and Silver nanoplatelets (Ag- NPLs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The self-assembled AQD layers are transferred onto the cavities and the coupled systems are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The position of plasmonic dipolar modes (DM) of the AgNPLs and AgNWs, depends upon the aspect ratio (a) of these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The aspect ratio[6] is defined as the ratio of lateral size to thickness of a nanostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' In the case of AgNPLs, the ratio of triangular edge length to AgNPL thickness is considered as aspect ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' In the case of AgNWs, the ratio of wire length to wire thick- ness is considered as the aspect ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Typically for the synthesized AgNWs in this study, aspect ratio is 241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' AgNPLs are synthesized in three aspect ratios of 3, 261 and 551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 2 AgNWs have localized surface plasmon resonances along longitudinal and transverse directions of the wire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' [7] Similarly triangular AgNPLs have in-plane ori- ented plasmonic dipolar modes (DM) and quadrupolar modes (QM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' [8–10] It is well established that for silver nanowires, the longitudinal surface plasmon resonance (LSPR), moves from visible to infra red region with in- creasing aspect ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' [11, 12] The absorbance peaks of LSPR peaks are in infra red region and can be measured by electron energy loss spectroscopy (EELS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' [13] The Ag- NPL dipolar plasmonic resonance also shifts to higher wavelengths with increasing aspect ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' [14, 15] So con- sidering the large aspect ratios of silver nanoplatelets and silver nanowires in this study, all the plasmonic mode res- onances are off resonant relative to the AQD emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' EXPERIMENTAL METHODS The AQDs are synthesized by protocol mentioned in[16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The AgNPLs and AgNWs are synthesized and cleaned by protocols mentioned in appendices A and B respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The synthesized nanoplatelets and nanowires are imaged and characterized by scanning transmission electron microscopy (STEM), energy dis- persive spectroscopy (EDS), high angle annual dark field (STEM-HAADF) spatial mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' (Appendices A and B) The AgNWs and AgNPLs are transferred by dip- coating onto Silicon substrates with 300 nm oxide, using a Kibron dipper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Subsequently langmuir-schaefer (LS) self- assembled monolayers are prepared and transferred using a KSV langmuir setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' (Appendix C) The transferred nanostructures are imaged by atomic force microscopy (AFM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The absorption spectra of AgNPLs are measured in a Perkin-Elmer lambda-35 solution uv-vis absorption spec- trometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The Photoluminescence (PL) and time re- solved photoluminescence (TRPL) are measured with a Witec alpha 300 confocal microscope equipped with a peltier cooled CCD spectrometer and a Picoquant SPAD detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The PL spectra are excited with a 532 nm diode laser and exciting laser line is cut-off with 532 nm edge long pass filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The PL integration time is set at 5 s and averaged over such 4 accumulation cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The TRPL spectra are excited with a 405 nm pulsed laser and cutoff with a set of 405 nm and 488 nm edge long pass filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The TRPL integration time is set as 5 s and 10 accumulation cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' RESULTS AND DISCUSSION The TEM images of the chemically synthesized Silver nano-platelets (AgNPLs) and Silver nanowires (AgNWs) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The chemically synthesized nanos- tructures generally are defective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The defects are gener- ally grouped into ridge and groove defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' [17] Such nm sized defects, strongly scatter the surface plasmons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' [18] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' (a) and (b) show the TEM images of Silver nanoplatelets (AgNPLs) and silver nanowires (AgNWs) of as- pect ratios of 3 and 241 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' (c) and (d) show the high resolution TEM images of Silver nanoplatelets and Silver nanowires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The encircled regions in red and green, show the ridge and groove type defects respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The ridge and groove defects scatter the surface plasmons and are can be used as hot spots[19] for surface enhanced raman scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Such defects can also create shoulders in scattering spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' In subsection A, the coupling data of AgNPLs with AQDs is discussed and AgNW-AQD coupling data is dis- cussed in the subsection B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' AgNPL-AQD coupling The AgNPLs are deposited on Silicon substrates and imaged by AFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Then 2 monolayers of AQDs are de- posited on the AgNPLs directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The AgNPLs aspect ratio can be controlled by varying the precursor concen- tration used in synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Three sets of AgNPLs with aspect ratios a1=400, a2=261 and a3=3 are synthesized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Only the AgNPLs with aspect ratios a1 and a2 are used for coupling to AQDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The synthesized triangular Ag- NPLs edges are blunted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' This is known as snipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' [20] With large aspect ratios[21] and snipping of AgNPLs, the in-plane dipolar modes of AgNPLs move to higher wavelengths, typically to infra red region of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The position of DM modes of AgNPLs of aspect ratio a3=3 is at 1027 nm, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The Absorption spectra of AgNPLs with higher aspect ratios are beyond the de- tection range of our instrument facilities, so they are not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' AgNPLs of two aspect ratios a1 and a2 are dip- (a) (b) 50nm 2 um (c) (p) 10nm 200nm3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' shows the AgNPL (a3=3) absorbance and AQD PL superimposed over each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' (a) and (b) show the AFM images of a single AgNPL and a single AgNPL coated with 2L-AQDs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' (c) and (d) show the corresponding AFM height profiles of single AgNPL and a single AgNPL coated with 2L-AQDs, respec- tively coated on to Silicon substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Then two consecutive AQDs monolayers are deposited on the AgNPLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The AFM image and topography profile of a bare AgNPL and 2 AQD monolayers deposited AgNPL are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The AFM images of the bare Ag- NPLs and AQD deposited AgNPLs, are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 4(a) and 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The AFM images indicate that the AgN- PLs are completely covered with AQDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The PL spatial map of the coupled AQDs-AgNPLs is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 4 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The bright spots in PL spatial map are on AgNPLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' (a) and (b) show the AFM images of bare AgN- PLs on Silicon and 2 AQD monolayer coated AgNPLs on Sil- icon respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' (c) shows the PL emission spatial map of 2L-AQDs-AgNPLs system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' (d) shows the typical PL spectra of of 2L-AQDs and 2L-AQDs-coupled to a single AgNPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' A typical spectra of AQDs coupled to a single AgNPL is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 4(d) along with control PL spectra of 2 monolayer AQDs on Silicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Similarly the TRPL spectra is collected for control AQDs and AQDs coupled to single AgNPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The PL spectra is not deformed and shows no extra features as shoulders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The TRPL spectra show that the AQD SE rate is inhibited on the AgNPL and is AQD emission rate faster in case of the control sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' This FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' (a) and (b) show the respective TRPL decay profiles of 2L-AQDs on Silicon and 2L-AQDs on single AgNPLs with aspect ratios a1 and a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' confirms that the AQDs are weakly coupled to AgNPL and the coupled system shows suppression of SE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' [22, 23] To quantify, the SE rate inhibition of AQDs, the TRPL intensity (I) is fitted with the following function, where the A1 and A1 are the amplitudes of the decay functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' τ1 and τ2 are the respective decay lifetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The fitting (a) (b) AQD-contro AQD-control A AQD-2L-AgNP-a1 AQD-2L-AgNP-a2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='00 twoexponentialfitAQDcontrol intensity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='00 twoexponentialfitAQDcontro twoexponentialfitAQD-2L-AGNP-a1 twoexponentialfitAQD-2L-AGNP-a2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='75 TRPL Normalized 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='50 lormalized DO 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='00 0 10 20 30 40 50 0 10 20 30 40 50 time(ns) time (ns)(a) (b) nm nm 200 100 10 μm 10 μm 75 100 50 0 25 0 100 (c) (d) 10000 O-AQD-control-PL 0-AQD-2L-AgNP-PL 1400 (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=') 8000 1200 6000 intensity 4000 1000 2000 800 600 540 560 580 600 620 640 660 680 700 wavelength(nm) 40 μm(a) (c) nm 6 1 μm BareAgNP 4 4 Height (nm) 2 0 0 4 2 0 1 2 3 4 5 8 distance (μm) (q) (d) nm 5 AgNP-2L-AQD 80 Height (nm) 5 40 10 15 0 20 25 0 1 2 3 40 1 μm distance (μm)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='30 △- AgNP in water 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='00 0- QD-PL DM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='75 Absorbance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='20 QD-PL QM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='05 500 600 700 800 900 1000 wavelength (nm)4 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The table shows the fitting components of lifetime of AQDs coupled to AgNPL Sample A1 τ1 (ns) A2 τ2 (ns) τavg (ns) AQD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='66 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='37 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='86 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='46 AQD-AgNPL-a1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='37 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='54 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='74 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='89 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='36 AQD-AgNPL-a2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='68 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='25 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='34 parameters are shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' I = A1·e− t τ1 + A2·e− t τ2 (2) The weighted average lifetime is given by τavg = A1 · τ1 + A2 · τ2 A1 + A2 (3) The Purcell factor is calculated from equation (1), us- ing weighted average lifetime from fitting TRPL spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The Purcell factors (FP) for single AgNPL of aspect ra- tio a1 and a2 are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='21 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' As a1 > a2, the DM wavelength of AgNPL a1 is larger than that of AgNPL a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' So the Purcell factor of AQDs coupled to AgNPL a1 is smaller, which is indicator of increased inhibition of AQD SE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' So with increasing aspect ratio of AgNPLs, Purcell inhibition of SE also increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' AgNW-AQD coupling The control sample is bare Silver nanowires (AgNWs) deposited on the Silicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The control sample is imaged by TEM and AFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The AFM imaging indicates that the AgNW thickness is approximately 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='1 nm, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The typical aspect ratio of AgNWs is about 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' On a separate control sample, 10 nm of spacer layer is deposited onto the nanowires by spin-coating 5 mg/ml polymethyl methacrylate (PMMA) solution in toluene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' (Appendix C) The one monolayer (1L) of AQDs is trans- ferred by the LS method on top of the spacer and the AQD monolayer is about 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='6 nm thick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' [16] The AFM height profile of the AQD deposited sample with spacer, is 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='8 nm as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The PL spectra of the coupled AQD-AgNPL system are measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The optical image of the system is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 3(a) and the spatial PL emission map is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The PL spectra of control sample: monolayer AQD on spacer and single AgNW-spacer-AQD mono- layer are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The PL spectra of the coupled system shows a shoulder feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The PL spec- tra is fitted with two Gaussian functions and the peak to peak separation is 160 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The amplitude and full width half maximum (FWHM) of Gaussian functions 1 and 2 are represented by A1, A2, w1, w2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' I(λ) = Io + A1·e − � λ−λ1 2w2 1 � + A2·e − � λ−λ2 2w2 2 � (4) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' (a) and (b) show the AFM images of bare AgNWs and 1L AQDs coated AgNW with 10 nm polymer spacer, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' (c) and (d) show the AFM profiles of a bare AgNW and 1L AQDs coated AgNW, with 10 nm polymer spacer re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The table shows the fitting components of life- time of AQDs on spacer and AQDs coupled to spacer coated AgNWs Sample A1 w1 A2 w2 wavg 1L-AQD-AgNW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='21 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='95 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='22 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='24 5L-AQD-AgNW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='18 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='94 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='40 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='38 wavg = A1 · w1 + A2 · w2 A1 + A2 (5) The PL fitting parameters are indicated in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Here λ1 and λ2 are the central positions of the constituent gaussian functions 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The λ1 and λ2 positions for multiple AgNWs are 554.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='55 nm ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='11 nm and 597.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='97 nm ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='04 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' For single AgNW, they are 553.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='60 nm ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='08 nm and 596.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='45 nm ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='03 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Such shoulders in quantum dot PL spectra might be observed due to Rabi splitting of strongly coupled systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' [24] For N emitter- strong coupling, the Rabi splitting scales with number of emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' [25] The coupling coefficient (g) of N dipolar emitters with a cavity mode field (E) is given by[26, 27] ℏg = √ N � ⃗µ · ⃗E � (6) Here µ is transition dipole moment of the emitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' [28] The Rabi splitting magnitude is 2g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' So the Rabi split- ting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=', the peak to peak separation should scale with number of emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' So if the number of emitters is in- creased by 5 times, then Rabi splitting magnitude should (a) (b) nm nm 150 40 1 μm 100 20 50 0 0 20 50 Tum 40 (c) (p) BareAgNW 50 AgNW-spacer-1L-AQD 20 40 10 Height (nm) 30 Height (nm) 0 20 10 10 0 20 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='0 Distance (μum) Distance (μm)5 increase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='2 fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 5 monolayers of AQDs are transferred FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' (a) shows the optical image of the 1L AQDs deposited on 10 nm spacer coated AgNWs on Silicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' A typical single nanowire is shown by the blue circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Multiple nanowire clus- ters are shown by the red circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' (b) shows the corresponding PL spatial map of the 1L AQDs deposited on 10 nm spacer coated AgNWs on Silicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' (c) shows the PL spectral deforma- tion of 1L-AQDs-coupled to AgNWs relative to 1L-AQDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' (d) shows the fitting of PL spectra of 1L-AQDs-coupled to Ag- NWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' on an AgNW with spacer and compared with 1 mono- layer AQDs on a AgNW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The PL spectra of both the samples are measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The shoulder separation (λ2−λ1) for 1L AQDs coupled to AgNW is 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='85 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The sep- aration should scale to 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='82 nm for 5L AQDs coupled to AgNW, in case of true strong coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Instead the shoulder separation of the 5L AQDs coupled to AgNW is measured as 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='42 nm, which is with in 1% change from the initial value for 1L AQDs coupled to AgNW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' This null result confirms that there is no emitter number de- pendence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' So, strong coupling is conclusively ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The shoulder is attributed to scattering by AgNW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Also, as the AQD concentration changes from 1L to 5L, the weighted average spectral line width (wavg) in- creases from 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='67 nm to 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='27 nm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' by 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='60 nm ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='62 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The PL spectral broadening with increased emitter concentration is indicative of weak coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' To find out more about the nature of coupling of AQDs to AgNWs radiative lifetime of AQDs is measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' It is evident from the raw data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 8(b), that the SE rate of 1L-AQDs on AgNWs is slower compared to free space decay rate of 1L-AQDs, which confirms the coupled system is in weak coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Similar fitting approach of AgNPLs is adapted for AgNWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The fitting parameters are shown in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' (a) shows the PL spectra of 1L-AQDs-coupled to AgNWs and 5L-AQDs-coupled to AgNWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' (b) shows the normalized raw SE rate profiles of 1L-AQDs on spacer coated Silicon, 1L-AQDs-spacer-single AgNW and 1L-AQDs-spacer- multiple AgNWs respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' (c) and (d) show the inhibition of SE rate of 1L-AQDs for single and multiple AgNWs respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The table shows the fitting components of lifetime of AQDs on spacer and AQDs coupled to spacer coated single and multiple AgNWs Sample A1 τ1 (ns) A2 τ2 (ns) τavg (ns) AQD-1L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='57 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='78 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='07 AQD-1L-AgNW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='29 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='65 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='12 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='35 AQD-1L-AgNWs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='31 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='63 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='02 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='84 The Purcell factor (FP) is calculated from the ratio of of AQD lifetime to the weighted average life time (τavg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' For Single AgNWs, the Purcell factor is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='06 and for multiple AgNWs it is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The SE inhibition of AQDs emitting in the visible re- gion of light is expected, as the plasmonic resonances of Silver nanowires are either in ultraviolet or infrared regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' [29, 30] The lifetime decay profiles reaffirm that the absence of resonant cavity modes inhibits the SE of AQDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' [31] IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' CONCLUSION The Silver nanowire and Silver nanoplatelet plasmonic resonances are off resonant to alloyed quantum dot spon- taneous emission frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' This results in inhibition of spontaneous emission of quantum dots coupled to Sil- ver nanowires and Silver nanoplatelets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The sponta- (a) (q) 1100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='0 AQD-1L-spacer-control 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='=QD-1L-AgNW-PL AQD-1L-spacer-single-AgNW --QD-5L-AgNW-PL intensity 1000 AQD-1L-spacer-multiple-AgNWs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='8 (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=') intensity 900 Normalized TRPL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='4 800 PL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='2 700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='0 540 560 580 600 620 640 660 680 700 0 10 20 30 40 50 (c) wavelength (nm) (d) time (ns) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='0 AQD-1L-control 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='0 AQD-1L-control AQD-1L-spacer-single-AgNW AQD-1L-spacer-multiple-AgNWs A intensi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='8 twoexponentialfit-AQD-control intens 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='8 twoexponentialfit-AQD-contro twoexponentialfit-AQD-AgNw twoexponentialfit-AQD-AgNWs A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='6 V TRPI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='4 A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='4 Normalized 口 lormalized 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='2 A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='2 A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='0 0 10 20 30 40 50 0 10 20 30 40 50 time (ns) time (ns)(a) (q) 10 μm 10 μm (c) (d) AQD-1L-PL AQD-1L-AgNW-PL 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='0 AQD-1L-AgNW-PL FitPeak1 sity FitPeak2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='8 inten Cumualtive Fit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='6 PL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='0 540 560 580 600 620 640 660 680 700 540 560 580 600 620 640 660 680 700 wavelength (nm) wavelength(nm)6 neous emission inhibition is quantified in terms of Pur- cell factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Purcell factors of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='67 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='75 are observed for quantum dots coupled to silver nanowire and silver nanoplatelet respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' With increasing aspect ratio of AgNPL and with increasing number of silver nanowires, the spontaneous emission rate is increasingly inhibited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' ACKNOWLEDGMENTS B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Tongbram thanks the Department of science and technology (DST), Inspire faculty programme for fellow- ship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Kalluru thanks the Micro and nano character- ization facility (MNCF-CeNSE), IISc for access to titan themis 300 kV TEM facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Appendix A: AgNPL synthesis method and Characterization The Silver Nanoplates (AgNPLs) are synthesized in hy- drophilic phase, with capping agent Polyvinylpyrrolidone (PVP), as per the reported protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' [32] The glassware used for the synthesis is cleaned following the RCA SC- I protocol, subsequently rinsed thrice in DI water and dried.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Silver nitrate (AgNO3-99%), sodium borohydride (NaBH4-99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='99%), sodium tri-citrate dihydrate (TSCDH- 99%) and polyvinylpyrrolidone-40K are procured from Merck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 30% w/W hydrogen peroxide solution is pur- chased from SDFCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' A 100 ml borosilicate conical glass flask and 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='68 ml DI water is added to the flask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Then 120 µL of hydrogen peroxide solution is added to the conical flask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 140 mg of PVP is dissolved in 1 ml DI water and the whole PVP solution is added to the conical flask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='3 mg of TSCDH is dissolved in 1 ml DI water and the solution is added to the flask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Now the flask is placed on magnetic stirrer setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The stirrer was turned on and set at 800 rpm for rigorous mixing of precursors, at room temperature (300 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' A glass vial is placed in an ice bath and 4 ml DI wa- ter is added to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The ice cold water is kept ready for dissolving NaBH4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='6 mg silver nitrate is dissolved in 1 ml DI water and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='2 ml of the solution is added to the flask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Immediately the solution colour turned to pale yel- low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='1 mg of NaBH4 is added to ice cold water and mixed thoroughly using a suction pipette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' NaBH4 solu- tion needs to be immediately used after preparation, as it degrades with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 1 ml of NaBH4 solution is added to the mixture of precursors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The reaction mixture im- mediately turned deep brownish yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' After approximately 30 minutes of continuous stirring, the solution turns dark red and finally to deep brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The whole colour change process happens with in 1-2 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The colour change of solution is due to shifting of localised surface plasmon resonance peak shift due to lateral growth of AgNPLs in the reaction mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The reaction mixture is cleaned by centrifuging at 10000 rpm for 10 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The precipitate is dark in colour and the supernatant is brownish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The supernatant is discarded and precipitate is again dispersed in DI water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Then the centrifuging process is repeated further twice, by select- ing precipitate and discarding supernatant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' After third centrifuging, the precipitate is dispersed in ethanol or DI water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The resultant solution is purple and is used for further characterization and measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' For TEM measurements, the AgNPL solution (20 µg/ml) in ethanol is drop casted on to a copper transmis- sion electron microscopy (TEM) grid and dried in a dessi- cator under vacuum for 12 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The dried TEM grid is cleaned with argon plasma (chamber vacuum 5x10-4 Torr and incident power 22 W) for 40 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' STEM- HAADF mapping of the AgNPLs is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The Silver characterstic X-ray intensity follows the contours of the AgNPL volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' This indicates that the AgNPLs are made of Silver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The Carbon X-ray intensity map in- dicates the distribution of PVP ligand over the AgNPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' (a) shows the typical dark field STEM-HAADF im- ages of AgNPLs on a TEM grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' (b) and (c) show the silver and carbon atomic distribution on AgNPLs of aspect ratio a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' (d) shows the superimposed silver atomic distribution and darkfield STEM-HAADF image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The AgNPLs of three aspect ratios are synthesized by making the following changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The AgNPLs of aspect ratio a1 are synthesized by following above mentioned procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' For synthesizing the AgNPLs with aspect ra- tio a2, the volumes of the precursors (AgNO3 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' NaBH4) dropped in the conical flask are changed from (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='2 ml ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 1 ml) to (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='1 ml;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='5 ml) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' To synthesize the AgNPLs of aspect ratio of a3, the reaction volume of precursors required for AgNPLs of aspect ratio of a2 is reduced by half, such that the reaction mixture volume is 25 ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' HAADF Ag (a) (q) 50nm 50nm HAADF Ag (c) (d) 50nm 50nm7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' (a) and (b) show the typical magnified and large area view of the TEM images of drop casted AgNPLs of aspect ratio a3 on TEM grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' (c) and (d) show the distribution in lateral size and thickness of AgNPLs of aspect ratio a3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' (a) shows the typical isothermal compression and compression cycle for transfer of AQDs with a Langmuir- Schaefer setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' (b)shows the AFM image of AgNPL of typical aspect ratio a2=261 Appendix B: AgNW synthesis method and Characterization The Silver nanowires (AgNWs) are synthesized in hy- drophobic phase with capping agent Oleyl amine (OAm), as per the reported protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' [33] The glassware used for the synthesis is cleaned by the RCA SC-I protocol and rinsed thrice in DI water and dried.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Silver bromide (AgBr-99%), copper chloride (CuCl2-99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='5%), n-Hexane (99%) and oleyl amine (70%) are procured from Merck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='3 gL-1 CuCl2-OAm solution is prepared by dissolving 3 mg of CuCl2 in 10 mL of OAm at 60o C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The temper- ature is maintained for 10 min and then the solution is cooled to ambient temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The table shows the elemental analysis obtained from the EDS spectrum of AgNWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Element Atomic fraction (%) Error (%) Carbon 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='79 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='05 Nitrogen 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='28 Copper 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='11 Silver 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='24 In a round bottom borosilicate glass flask 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='1g AgBr, 34 µL of CuCl2-OAm solution and 5 ml OAm is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The flask is heated to 160o C and maintained at the same temperature for 6 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Then the heating element is turned off and reaction mixture is allowed to reach am- bient temperature naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The final reaction mixture colour is dark gray, indicating formation of AgNWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The reaction mixture is dispersed in 15 ml n-hexane and centrifuged at 6000 rpm for 10 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The precip- itate is then dispersed in hexane and centrifuged for two more cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The precipitate after 3 cycles is dispersed in n-hexane and stored in dark for characterization and measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' shows the AgNW EDS spectra with characteristic X- ray peaks of constituent atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The inlay shows the zoomed peaks of the EDS spectra of AgNWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' For TEM measurements, the AgNW solution (10 µg/ml) in hexane is drop casted on to a copper trans- mission electron microscopy (TEM) grid and processed identical manner of AgNPLs on TEM grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The EDS spectra of the AgNWs is measured and shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The elemental distribution is shown in table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The carbon and nitrogen content is attributed to oleyl amine ligands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The Copper content is attributed to both Cop- per TEM grid and CuCl2 seeding process of AgNWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The Silver content is attributed to AgNWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 60000 AgNW EDS spectrum 8000 50000 Cu-Kα 7000 Intensity (Counts) Intensity (Counts) 6000 40000 5000 4000 30000 N 3000 Cu-Kβ Ag-Lβ Cu-L 20000 2000 1000 10000 0 2 4 9 8 10 Energy (KeV) 0 0 2 4 9 8 10 Energy (KeV)nm (a) 42 (b) 40 Surface pressure (mN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='m*1) 三 38 40 34 20 32 30 0 28 26 24 20 22 4000 6000 8000 10000 12000 14000 1 μm Trough area (mm3)(a) (b) 50 nm 500nm (c) (d) 25 20 5 4 15 count 3 C 10 2 5 1 0 0 10 20 30 40 50 60 8 10 12 14 16 18 20 Lateral size (nm) Thickness (nm)8 TABLE V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The table shows the synthesized nanostructure specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Sample Lateral size (µm) thickness (nm) Aspect ratio AgNW 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='435 35 241 AgNP-a1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='243 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='6 400 AgNP-a2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='176 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='5 261 AgNP-a3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='043 14 3 Appendix C: Sample preparation procedure The synthesized AgNPLs are hydrophilic and AgNWs are hydrophobic in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' AgNPLs are transferred onto Silicon substrate by dip-coating at water and hex- ane interface, by following the procedure mentioned in report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' [34] The self assembly of hydrophilic particles at water-hexane interface reduces clustering of AgNPLs and is proffered for studying properties of single AgNPLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Typically in a large glass petridish, the substrate is placed and is attached to a motorized dipper (KSV make) and 200 ml deionized (DI) water is poured over the sub- strate, till the substrate gets completely immersed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Then 200 ml hexane is poured over DI water, which floats over water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Then 3 ml of 2 mg/ml solution of AgNPLs in DI water is dispersed in the DI water phase gently below Hexane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The system is allowed to stabilize for 15 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' There are two interfaces in this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' First interface is at the hexane-air boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The other interface is the hexane-water boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Then the substrate is brought up from water phase to the hexane-water interface, at a rate of 10 mm/min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Once the substrate touches water-hexane interface, the AgNPLs at the hexane-water interface gets transferred onto the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The substrate is then moved up to hexane-air interface and the hexane is allowed to dry at ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' AgNWs are transferred onto Silicon substrate by dip- coating at hexane air interface in a similar procedure fol- lowed for AgNPLs, with a major difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The amount of hexane poured over DI water is limited to 10 ml, so that a thin Hexane phase floats over DI water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Then 5 ml of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='2 mg/ml AgNW solution in hexane is dispersed in DI water phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' AgNWs are hydrophobic and move immedi- ately to hexane phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The system is allowed to stabilize for 15 minutes and then the substrate is brought up to- wards the hexane-water interface at a rate of 10 mm/min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' As soon as the substrate crosses hexane-air interface, the AgNWs get transferred on the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The substrate is then dried at ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Poly-methyl-methacryalate (PMMA), 350K molecular weight is procured from Merck and 5 mg/ml solution is prepared in Toluene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The solution is spin-coated on a clean Silicon substrate at 3000 rpm and for 60 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The resultant film thickness is characterized by X-ray reflec- tivity (XRR) measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The XRR fringe separation is a measure of film thickness (h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The film thickness is calculated as per procedure mentioned in[35] The film thickness turns out as 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content='45 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Mono- layers of AQDs are synthesized as per the reported procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' [36, 37] The AQDs are transferred by self- assembly via Langmuir-Schaefer (LS) method,[38] as per the procedure mentioned in the report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' [16, 39] The mono- layer formation in a typical LS method is indicated by saturation of surface pressure in isothermal compression cycle of the AQDs[40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The typical compression and expansion cycle of LS isotherm is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' The formed monolayer is transferred on to AgNWs with spacer, by lowering and stamping the substrate onto the self assembled monolayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' Once the stamping is done, the substrate is retracted and dried in ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} +page_content=' [1] E.' metadata={'source': 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26, 7732 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFAT4oBgHgl3EQf0B72/content/2301.08702v1.pdf'} diff --git a/RtE0T4oBgHgl3EQfkgHa/content/tmp_files/2301.02475v1.pdf.txt b/RtE0T4oBgHgl3EQfkgHa/content/tmp_files/2301.02475v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..31bcfa3f183990d002112ae990cc9387893313dc --- /dev/null +++ b/RtE0T4oBgHgl3EQfkgHa/content/tmp_files/2301.02475v1.pdf.txt @@ -0,0 +1,1465 @@ +HEARABLES: FEASIBILITY OF RECORDING CARDIAC RHYTHMS FROM SINGLE EAR +LOCATIONS +Metin Yarici, Wilhelm Von Rosenberg, Ghena Hammour, Harry Davies, Pierluigi Amadori, Nico Ling, +Yiannis Demiris, Danilo P. Mandic +Department of Electrical and Electronic Engineering, Imperial College London, SW7 2AZ, UK +E-mails: {metin.yarici16, d.mandic}@imperial.ac.uk +ABSTRACT +Wearable technologies are envisaged to provide critical support to future healthcare systems. Hearables - devices worn in the ear +- are of particular interest due to their ability to provide health monitoring in an efficient, reliable and unobtrusive way. Despite +the considerable potential of these devices, the ECG signal that can be acquired through a hearable device worn on a single +ear is still relatively unexplored. Biophysics modelling of ECG volume conduction was used to establish principles behind the +single ear ECG signal, and measurements of cardiac rhythms from 10 subjects were found to be in good correspondence with +simulated equivalents. Additionally, the viability of the single ear ECG in real-world environments was determined through +one hour duration measurements during a simulated driving task on 5 subjects. Results demonstrated that the single ear ECG +resembles the Lead I signal, the most widely used ECG signal in the identification of heart conditions such as myocardial +infarction and atrial fibrillation, and was robust against real-world measurement noise, even after prolonged measurements. +This study conclusively demonstrates that hearables can enable continuous monitoring of vital signs in an unobtrusive and +seamless way, with the potential for reliable identification and management of heart conditions such as myocardial infarction +and atrial fibrillation. +Index Terms— wearable health, electrocardiogram, ear ECG +1. INTRODUCTION +The use of wearable technologies for monitoring vital signs has become increasingly widespread in the society, both for +recreational and medical purposes. Most often, these devices are integrated into wearable garments and accessories, and +are concealed and miniaturised for convenience of the user. The most common choices for wearable vital signs monitoring +technologies are smart watches and chest straps. Smart watches typically utilise the photoplethysmogram (PPG) to provide +continuous monitoring of pulse and respiration, while chest straps may also record the electrocardiogram (ECG). However, +PPGs lack the information necessary to understand the functioning of the heart [1], while chest-worn devices are not suitable for +everyday use due to their obtrusive nature. Consequently, alternative solutions that provide ECG measurements in a convenient +and user-friendly manner have attracted significant research interest. +One such solution is the ’hearable’ device - a wearable that fits in the ear and can serve both as an audio accessory and +a platform for health monitoring [2, 3]. For hearables, the stability of the head relative to the vital organs during sitting, +sleeping, walking, and eating results in superior monitoring capability in real-world scenarios, relative to devices attached to +the limbs [4]. For this reason, wearable devices that are placed on the regions of the skin surface surrounding the ear are also +gaining popularity. Da He et al. [5] and Casson et al. [6] demonstrated the effectiveness of behind-the-ear and scalp based +recordings of ECG, ballistocardiogram (BCG), and PPG for heart rate monitoring, while Celik et al. [7, 8] provided evidence +of similar results when measurements were taken from electrodes placed on the ear and neck. Despite the success of these +proof-of-concept studies, the difference in ECG potential from the various head locations is not yet well understood. +To that end, our previous work investigated the ECG potential at different head positions from inside a helmet [9]. An +assessment of the performance of the channels under consideration was conducted with regard to the successful application of +an R-peak detection algorithm [10]. However, through experimentation alone, it is difficult to clarify the effect of the manifold +factors that can influence the sensitivity of the ECG sensors [11]. For example, variations in user movements, muscle activity, +brain activity, and electrode-skin contact can all adversely affect the quality of the ECG signal [12, 13], so that the current +arXiv:2301.02475v1 [physics.med-ph] 6 Jan 2023 + +studies are limited by either a lack empirical evidence or support from theoretical models +(a) +(b) +Helix +Concha +Scalp (upper) +Scalp (lower) +Neck (upper) +Neck (lower) +Left arm +below | z [mm], pz +(0.12mAm) | above +-40 +0 +80 0 60 +-50 +-30 +-10 +10 +30 +0 +20 +-20 +-40 +-20 +20 +40 +60 +right | x [mm], px (0.12mAm) | left +T +R +P +Q +heart vector +S + y [mm], + py (0.12mAm) +(c) +Fig. 1: Biophysics model of ECG propagation. (a) 3D Model structure: Organs and tissues in the torso and head are encased +in a skin surface structure. (b) Heart vector: The orientation and magnitude of the current dipole p in px, py, and pz; the heart +vector at one point in time is shown in blue, and the trace of the tip of the heart vector from the start of the cycle until the current +position (axes in 0.12 mAm) is shown in orange; the heart muscle is shown in pink in the background (axes in millimetres). (c) +Sensor set-up: For the purpose of mapping the ECG on the left side of the head, sensors were placed on scalp, ear, and neck +positions. Potential difference between the upper and lower scalp (scalp ECG), helix and concha (ear ECG), and upper and +lower neck electrodes (neck ECG) was extracted. +A comprehensive theoretical approach to mapping ECG potentials can be achieved through forward modelling [14, 15]. +This process involves applying Maxwell’s equations to a dielectric model of the body in order to estimate the propagation of +cardiac potentials from the heart to the head-positions of interest. This offers the advantage of examining the ECG in a way +that is isolated from other sources of electrical activity, such as brain and muscle activity. Furthermore, such modelling can +enable the evaluation of the characteristic ECG - that is, the timings and shape of the waves of the cardiac cyle - available +from various wearable ECG channels. Early ECG modelling was based on simplified representations of the human body (for +example, a homogeneous conductive sphere [16]), however more realistic models have since been produced which include +separate shells representing the heart and surrounding body [17], and realistically shaped geometries for specific organs in the +chest cavity [18, 19, 20, 21]. A model which enables the simulation of ECG propagation to the head and ear locations was +first introduced in our previous work [11]. A whole-body model incorporating tissue from the torso and head was employed +to provide a rigorous theoretical basis for the possibility of high-quality head-ECG. In addition to simulations, a systematic +analysis of the ECG surface potential from head and in-ear channels was conducted, measuring the ECG potential difference +between the left and right side of the head, as well as between the left and right ears. Moreover, the characteristic timing and +shape of the ECG from the different channels was evaluated through both measurements and simulations, opening new avenues +for wearable head-based cardiac monitoring that provides more than just heart rate detection. +Ideally, wearable vital signs measurements should be conducted from a device worn on a single ear. However, as previ- +ously discussed, the ECG potential available on the single ear, or the surrounding locations on the scalp and neck, is not well +understood. To this end, we follow the approach outlined in [11] and assess the ECG potential on a single side of the head, ear, +and neck. +This study seeks to determine the amplitude, shape, and timing of ECG waveforms from specific sites through both real-life +recordings and biophysics modelling. Once the possibility of recording ECG potentials from each location was established, a +second experiment was conducted in a more dynamic, real-world environment, which allowed for the recordings to be evaluated +over an extended period of driving in a simulator. This enabled further assessment of the impacts of varying levels of brain +and muscle activity, as well as the user’s physical movement on the signal quality, and with greater confidence. Results from +both experimentation and simulations have established the feasibility of recording ear ECG, and neck ECG signals that possess +characteristics similar to those present in the standard Lead I ECG signal. Comprehensive evaluation of performance of the +wearable-ECG under consideration relative to Lead I has also been presented. Key findings reveal differences in performance +that arise with different recording lengths from each recording channel, thereby revealing the suitability of a given channel for + +Z +×y-0.16 +-0.18 +-0.2 +-0.22 +-0.24 +-0.26 +-0.28 +Z +-0.3 +y +0.32Z +x0.4 +0.3 +0.2 +0.1 +0 +-0.1 +-0.2 +-0.3 +Z +x +-0.4(b) +30 +(0.12 mAm) I above ) +20 +10 +0- +heart vector +-10- +-20 +R +-30 +-40 +-50 +40 +-40 +-20 +0 +20 +40 +60 +80. 0 +(0.12 mAm) 1 back-various applications in cardiac monitoring. Overall, our results indicate that the single ear ECG channel and the cross ear ECG +channel can provide robust ECG monitoring during prolonged real-world tasks. +Our findings demonstrate the potential of ear ECG for capturing cardiac rhythms with shapes that are very similar to Lead +I from the standard limb leads, even during challenging, real-world recordings. This framework holds promise for examining +heart conditions that are visible in multiple consecutive cardiac cycles in Lead I, including but not limited to myocardial infarc- +tion (the ST segment is elevated), first-degree atrioventricular block (the PR interval is longer than 200 ms), atrial fibrillation +(absence of the P-wave, found in 2% to 3% of the population in Europe and the USA [22]), sinus tachycardia (increased heart +rate and a shortened P-T duration) and atrial flutter (an increased frequency of the P-wave relative to the QRS complex as a +result of rapid atria contraction) [23, 24, 25]. Moreover, the proposed ear ECG framework offers the possibility of 24/7 continu- +ous and unobtrusive cardiac monitoring and can alert the user to the presence of universal signatures of heart malfunction, such +as absent P-waves and elongated ST-segments. Such a new perspective can be beneficial to many existing applications, such as +remote monitoring for patients with cardiac conditions such as arrhythmias, or heart failure [26], sports and fitness monitoring +of athletes [27], and assessment of the effect of physical strain and stress in workplace environments [28, 29]. +2. METHODS +In the present study, the feasibility of recording cardiac rhythms from a single ear is established through real-world measure- +ments and biophysics simulations. First, theoretical grounding for the measurement of single ear ECG is provided through +biophysics modelling of ECG propagation from the heart to various positions of interest around the head and ears, through the +use of an accurate volume conductor model of the human body that was first presented in [11]. +Model geometry +The geometry within the presented model is based on the VHP-Female Computational Phantom v. 2.1 and v. 2.2 [30] - a +data-set consisting of three-dimensional geometry for major organs and tissues in the human body. The phantom was edited +in instances where computational problems were encountered with the mesh of the structures in the model in the modelling +software (COMSOL Multiphysics [31]). The model comprised geometric representations for major structures surrounding the +heart and in the head, and a complete whole-body volume enveloped by an outer skin structure; the top-half of the model is +shown in Figure 1a). For the purpose of realistic modelling of electric fields within the body, a large sphere or radius r = 3.3 m, +filled with air, surrounded the body and provided an electrical ground in the model. The complete mesh consisted of 560.630 +domain elements and 72.286 boundary elements, and the average edge length was 6 mm. +Electric properties of the body and cardiac current sources +The relevant electric properties of the modelled human tissues (conductivity and relative permittivity at an excitation frequency +of 15 Hz) were extracted from the database in [32], and are provided in the supplemental material (Table S1). Previous research +has revealed that isotropic and anisotropic treatments of the heart structure in simulations of ECG yield similar results [33]. +Therefore, an isotropic treatment of the heart structure was adopted within the present model. Multi-source modelling of +the cardiac potential has been shown to perform best for prediction of the ECG potential on the torso [34], however, owing +to the relatively large distance between the head and heart, the present model employs single source modelling. The heart +source dynamics over the course of a single cardiac cycle are shown in Figure 1b; a comprehensive description of the electrical +dynamics of the cardiac cycle is provided in [35]. The source dipole in the presented model is a superposition of three orthogonal +vectors representing the projection of the cardiac potential over the course of a single cardiac cycle at a normal to the sagittal, +transversal, and coronal planes. The heart vectors were acquired from real-world measurement from a subject with no known +cardiac abnormalities. The cardiac cycle was taken to start and end 200 ms before and 400 ms after the maximum of the R-wave. +Experiment A: ECG mapping on the single ear +ECG potential on candidate positions for head-based wearable ECG monitoring platforms was assessed through measurements +on ten subjects with GRASS Ag/AGCl cup electrodes. single ear ECG from the left ear was recorded through one electrode +placed on the helix and another on the concha; this reflects two locations that are available to stand-alone, single ear hearable +devices. The electrode positioning for the remaining head-ECG channels was devised such that the ear, scalp, and neck channels +each spanned equal distances across the head-surface; in this way, differences in ECG potential in each of these regions could + +be assessed. For the scalp channel, the electrodes were positioned according to the following methodology: the lower scalp +electrode was placed at a distance of L/2 above the helix-electrode, where L equals the separation between the helix- and +concha-electrodes. The upper scalp electrode was placed at a distance of L above the lower scalp electrode. An equivalent +placing system was employed for the neck channel, whereby neck electrodes were placed below the concha-electrode. The ear, +scalp, and neck electrodes were all positioned along a vertical line which intersected the points at which each subject’s helix- +and concha-electrodes were positioned (see Figure 1c)). A standard limb-ECG channel was also created between the left and +right wrist, whereby electrodes were positioned on the left and right volar central zones (just below the palm) [11, 36]. Prior +to the application of the electrodes, the skin at each location was prepared through cleaning with medical wipes and abrasion +with NuPrep gel. A layer of 10-20 conductive paste was also applied to the electrodes prior to placement, in order to improve +conduction between the skin surface and the electrodes. Measurements were conducted via a custom bio-amplifier programmed +to digitise the potential difference between the upper and lower scalp electrodes (scalp ECG), the helix and concha electrodes +(single ear ear ECG), the upper and lower scalp electrodes (neck ECG) and the left and right wrist electrodes (wrist ECG). A +sampling rate of 500 Hz was used and the helix-electrode served as a bias. During the measurements, subjects were seated and +instructed to close their eyes while minimising eye saccades, and movement of the head, neck, and arms. The recordings were +performed under the Imperial College London ethics committee approval JRCO 20IC6414, and all subjects gave full informed +consent. +Experiment B: real-world feasibility +Once the cardiac potential on the scalp, ear and neck regions had been established for subjects while at rest, the feasibility of +recording cardiac rhythms in a real-world scenario was investigated. Single ear measurements on both the left and right ears +were conducted, in addition to cross-ear measurements between the left and right ears, for which we previously established the +feasibility of recordings on 6 subjects at rest [11, 37, 38]. Five of the ten subjects from Experiment A were instructed to drive in +a virtual reality (VR) driving simulator environment while measurements were taken from the described ear ECG channels and +a reference-ECG channel from the wrists. Data was acquired via a g.tec g.USBamp (2011) bio-amplifier at a sampling rate of +1.200 Hz. In order to ensure comfortable ear ECG measurements for the duration of the driving task, custom fabric electrodes +were used to record the ECG signal from the concha [39]. The fabric electrodes employed within this study were constructed +from silver-coated thread that is interwoven with elastic fibres. In [2], electrodes of this type were mounted on visco-elastic +earpieces and then placed in the ear canal of five subjects over the course of a normal working day. Measurements were shown +to be stable after prolonged periods of unrestrained activity which included walking, eating, and talking. While the ear canal is +an ideal position to record the ECG, as outlined in [11, 37], in the current experiment, a PPG sensor was placed in the position +of the ear canal for measurement of ear-SpO2, and is the subject of ongoing analysis [40]. Therefore, the concha - another +convenient location to record physiological signals from - was used. In the present study, the electrodes were secured to the +concha using soft, malleable non-allergenic silicone, shaped to each person’s ear. Prior to application of the electrodes, the skin +was prepared through cleaning with medical wipes and abrasion with NuPrep gel. A thin layer of Signa Gel conductive gel was +applied to the surface of the fabric electrode in order to help establish good conduction at the skin-electrode interface. For the +left and right ear single ear ECG measurements, a reference electrode was positioned on a second location which is suitable for +monitoring through Hearable devices - the helix. As with the previous electrode positions, the ipsi-lateral helix was prepared +with medical wipes and NuPrep gel prior to electrode connection. The GRASS Ag/AgCl electrodes were used to record the +signal from the helix. A layer of 10-20 conductive paste was applied to aid conduction at the skin-electrode interface. As a bias +electrode, a second Ag/AgCl electrode was placed on the ipsi-lateral earlobe, with the same skin preparation. For the cross-ear +ECG measurement, the left concha electrode was referenced to the right helix electrode, with the left ear lobe electrode serving +as the bias. For the reference ECG measurement, for Subjects 1-3, a wrist ECG measurement was conducted, whereby the +left wrist electrode was referenced to the right wrist electrode. For Subjects 4 and 5, a reference ECG signal was obtained +by referencing the left wrist electrode to the right helix electrode. All reference channels were biased with the left ear lobe +electrode. Vertical electro-oculogram (VEOG) channels were also recorded via one electrode below and another above each +eye. The previously described skin preparation and conductive paste was also applied for the VEOG measurements. +2.1. Signal processing +All signal processing was performed in MatLab. The processing steps for the recorded ECG data are described in Algorithm1, +and consisted of the following procedures. The ECG data from both Experiment A and B were first bandpass-filtered between +0.5 Hz and 95 Hz using a third-order Butterworth filter and notch-filtered with a second-order IIR-filter with a centre frequency +of fc = 50 Hz and a band-width of w = 5 Hz. For data collected in Experiment B (during the driving task), an additional + +Algorithm 1: Signal processing steps +1 Record electric potential differences raw_ECG with one reference channel (wrist ECG, Lead I) and multiple +head-ECG channels. +2 For data that was recorded as part of Experiment B, perform threshold- and blink-related artifact rejection on +raw_ECG. +3 Bandpass- and notch-filter the reference channel, respectively, using a third-order Butterworth filter with a lower +cut-off frequency of fmin = 1 Hz and an upper cut-off frequency of fmax = 95 Hz, and a second-order IIR-filter with +a centre frequency of fc = 50 Hz and a band-width of w = 5 Hz, to give filtered_reference. +4 Perform R-wave detection on filtered_reference according to [41, 42]. +5 Bandpass-filter the signals in all channels in raw_ECG using a third-order Butterworth filter with a lower cut-off +frequency of fmin = 1 Hz and an upper cut-off frequency of fmax = 30 Hz, to give filtered_ECG. +6 Extract cardiac cycles from filtered_ECG around the identified R-waves (within a −200 ms to +400 ms window) +using the R-wave timings obtained in Step 3 from the wrist ECG. +7 Find the median the cardiac rhythms for different lengths of data - ranging from 2 − 540 cardiac cycles. +8 Calculate four metrics for the quality assessment of the median cardiac cycles from individual channels: (i) correlation +between cardiac rhythms in a given channel and Lead I, (ii) ratios of the amplitude of P-, Q-, S-, and T- waves relative +to the R-wave in a given channel, (iii) timing of P-, Q-, S-, and T-waves relative to the timing of the R-wave in a given +channel, and (iv) normalised variance in the channels — the root-mean-square error (RMSE) of the differences +between the individual cardiac rhythms and the grand-median cardiac rhythm, divided by the standard deviation of +the grand-median cardiac rhythm, in the channel under consideration. +cleaning procedure which removed blink artifacts and large amplitude deflections, such as those caused by motion and jaw- +clenching, was applied. Details of the artifact rejection are provided in the supplemental material. +Averaging over multiple consecutive cardiac cycles was conducted in order to extract full cardiac rhythms. In [11], a +mutlimodal in-ear sensor was shown to provide a stand-alone solution for the reliable measurement and identification of cardiac +cycles from ear ECG; a collocated MEMS sensor placed beneath the surface of the fabric electrodes was used to detect pressure +waves in the ear canal surface associated with the heart beating (e.g., the ballistocardiogram). This served as a reference for the +timing of the cardiac cycles in the ear ECG signal detected by the fabric electrodes and, in conjunction with a matched-filtering +technique [9, 10], enabled reliable identification of cardiac cycle timing in the low SNR ear ECG signal. Such techniques +were not the subject of the present study, therefore timings of the R-peaks that were extracted from the reference channels +were used here. However, it is feasible that future developments of the single ear ECG presented in this study will incorporate +the described multimodal sensing capabilities. The use of the reference channel in this way is made possible by virtue of the +ECG signals from all locations on the body sharing a common electrical source. The R-peak timings were identified using +the Pan-Tompkins algorithm implementation in MatLab [42, 41]. From within the ear ECG signals, segments of 600 ms +length containing the cardiac rhythms were extracted (200 ms prior to and 400 ms after the timing of the R-peak), enabling the +identification of all the key components (P-, Q-, R-, S-, and T-waves) in the signal. +Median cardiac rhythms were obtained for different numbers of cardiac cycles N, ranging from N = 2 to N = 540. Since +the success of the extraction of cardiac rhythms is dependant on the number of averages taken, and the number of cardiac +cycles varies greatly during activities such as driving, different extraction procedures were categorised in terms of the number +of cardiac cycles included in the segment, as opposed to the length of time in which the duration took place. During normal +heart function (roughly 60 bpm - 80 bpm), conducting averages over the values of N ranging from 2 to 540 produced results +that provide an indication of the performance of the ear ECG ranging from recording duration of a few seconds up to roughly +9 minutes. For each value of N tested, the maximum number of median cardiac rhythms were obtained. For example, for a +recording spanning M = 10 complete cardiac cycles, an N = 2 median cardiac rhythm could be extracted 8 times (M − N). +A grand-median cardiac rhythm was extracted from the reference-ECG signal of each subject to serve as a benchmark during +analysis of ear ECG cardiac rhythm quality. For rigour, four performance metrics were calculated for each individual ear ECG +cardiac rhythm. The four calculated metrics were as follows: +i) The Pearson correlation coefficient between the cardiac rhythms from a given channel and a gran-median from the wrist +ECG (Lead I); + +ii) Root mean square (RMS) of the ratios between the R-wave and remaining wave amplitudes in a given channel, bench- +marked against equivalent ratios in Lead I; +iii) Root-mean-square error (RMSE) of the timings of the P-, Q-, S- and T-waves relative to the R-wave in a given channel, +relative to timings in lead I. Details of the wave-timing calculation are provided in the supplemental material; +iv) RMSE between the grand-median cardiac rhythm of a given channel and all individual cardiac rhythms recorded in +that same channel, where the RMSE was normalised by dividing by the standard deviation of the grand-median cardiac +rhythm. +(b) +0.4 +0.3 +0.2 +0.1 +0 +-0.1 +-0.2 +-0.3 +-0.4 +Surface potential [mV] +(a) +Time [s] +Scalp +Ear +Neck +5µV +ECG measurements +Reference R-peak timing +Fig. 2: Simulated and recorded cardiac electric potentials. (a) Simulated electric potential on the head surface and upper torso +at the time of the R-wave peak (in milliVolts) in the biophysics model. (b) Recorded ECG traces from the scalp, ear, and neck +ECG channels from a single subject. The ECG potential at the instance of the R-peaks in the reference (Lead I) channel are +indicated (red circle). +3. RESULTS +In order to provide rigorous theoretical support to the ear ECG measurements, simulations of ECG propagation throughout +the body were conducted. A single ECG cycle was simulated, ranging from the time t1 = −200 ms prior to the timing of +the maximum of the R-wave, to t2 = 400 ms after (600 ms in total), at a temporal resolution of TR = 2 ms. A snap-shot of +the so-simulated cardiac cycle at the timing of the maximum of the R-wave is shown in Figure 6a). Notice the large potential +difference between the left and right sides of the torso - positions where conventional Lead I electrodes are placed. Observe the +reduction in potential difference across the surface of the head (where the considered ear ECG channels are based) relative to +the torso. Scalp, ear, and neck ECG cardiac rhythms extracted from virtual sensor positions on a single side of the head (Figure +1c)) and are displayed in Figure 3a) (blue traces). +3.1. Correspondence between simulation and measurement: Experiment A +The narrow neck structure impedes ECG potential as it propagates from the heart to the head surface. It is therefore intuitive +that the gradient in the ECG potential will be highest at the base of the neck, where large potential differences begin to occur, +and lowest towards the top of the head. Measurements (from Subject 6) in Figure 2 demonstrate such a topography over the +neck and head surface; note that the P-,Q-,R-,S-, and T-waves waves in the Lead I cardiac rhythm are identifiable in the neck +ECG channel, demonstrating a large potential difference at the position of the neck. On the other hand, in the ear ECG trace, +R-peaks are intermittently identifiable, and in the scalp trace, there is no evidence of ECG visible without further processing. +Averaging multiple cardiac cycles (e.g., as in Algorithm 1) enables extraction of the cardiac rhythms from the lower SNR ECG +measurements. + +0.4 +0.3 +0.2 +0.1 +0 +-0.1 +-0.2 +-0.3 +Z +x +-0.40.4 +0.3 +0.2 +0.1 +0 +-0.1 +-0.2 +-0.3 +Z +x +-0.40.4 +0.3 +0.2 +0.1 +0 +-0.1 +-0.2 +-0.3 +Z +x +-0.4Time [s] +Simulation +Potential [µV] +(a) +Potential [µV] +(b) +Lead I +Neck +Ear +Scalp +-0.2 +0 +0.2 +0.4 +-200 +0 +200 +400 +600 +-166 ms +-34 ms +0 ms +32 ms +268 ms +-0.2 +0 +0.2 +0.4 +-10 +0 +10 +20 +30 +-166 ms +-36 ms +-2 ms +26 ms +268 ms +-0.2 +0 +0.2 +0.4 +-2 +0 +2 +4 +-164 ms +-38 ms +-4 ms +24 ms +264 ms +-0.2 +0 +0.2 +0.4 +-2 +0 +2 +4 +-158 ms +-34 ms +-4 ms +20 ms +260 ms +-400 +-200 +0 +200 +400 +600 +800 +-120 ms +-32 ms +2 ms +32 ms +244 ms +-5 +0 +5 +10 +15 +-116 ms +-32 ms +2 ms +30 ms +242 ms +-2 +0 +2 +4 +-116 ms +-32 ms +2 ms +32 ms +242 ms +-2 +0 +2 +4 +-116 ms +-34 ms +2 ms +30 ms +242 ms +Measurement +Fig. 3: Simulated and measured cardiac rhythms in scalp, ear, and neck ECG. (a) Simulations. (b) Measurements - cardiac +rhythms based on averaging over a ten minute recording. ECG potentials as the timings of the P-, Q-, R-, S-, and T-waves (from +left to right) are circled. The timing of each eave is indicated in ms. +The model predictions in Figure 3a) (blue traces) reveal the characteristic shape and timings of the ECG signal at each +location. If there were no other sources (e.g., brain activity and muscle activity) which contributed to the potential difference +at these sites, these are the ECG signals that we would expect to obtain. As expected, the amplitude of the ECG on the head +decreases the higher up the head the channels are placed, in both the measured and simulated channels. Overall, in both +measurements and simulations, the scalp, ear and neck signals closely match the Lead I signal. This can be attributed to the fact +that the channels under consideration and Lead I form a similar projection plane for the heart vector (Figure 1b). Therefore, in +this study, the obtained scalp, ear, and neck rhythms were bench-marked against the Lead I equivalent wrist ECG. The measured +traces in Figure 3b) were obtained by averaging over the entire set of cardiac cycles in the ten minute recording, according to +Algorithm 1. In this way, although high SNR and accurate measurements of cardiac rhythms were extracted, the performance +of the channel is not well understood at shorter measurement lengths, when a lower number of cardiac cycles are available. +3.2. Ear ECG cardiac rhythms: Experiment A +The scalp, ear, and neck traces in Figure 6 were based on averaging over 540 cardiac cycles (roughly ten minute recordings), +and help establish the potential difference available to the scalp, ear, and neck ECG channels, due to their high SNR. However, +in practice, shorter recording lengths are desirable; cardiac rhythms obtained through averaging over the first 240 recorded +cardiac cycles (roughly 4 minutes of data) are shown in Figure 4 for 5 subjects. The cardiac rhythms are plotted in black for +the reference-channel and blue for the scalp, ear, and neck ECG channels. While the neck ECG rhythms were consistent across +subjects, more variation was observed for the ear and scalp channels. We have included examples of both good and bad ECG, +from Subjects 1-3, and Subject 4-5, respectively, which reflect the level of variation observed across all subjects. Waveforms +from the remaining 5 subjects are provided in the supplemental material. The scalp ECG results are considerably worse than +the ear and neck ECG results, which primarily stems from i) poorer electrode skin contact through hair on the scalp, ii) the +lower amplitude ECG signal at the location of the scalp relative to the ear and neck, and iii) the large amplitude background +noise (EEG and temporal muscle-EMG) on the temporal scalp. Median cardiac cycles from the first N = 600 cycles are also +plotted in green. Minimal difference is observed for the high quality neck and ear ECG channels, however, little change is also +observed for the lower quality scalp ECG for Subject 4 and 5; this indicates that a compact device attached on this position on +the scalp might not be suitable for ECG recordings. +In instances where urgent care may be required following the diagnosis of an ECG abnormality (such as myocardial infarc- +tion following ST elevation), variations in data requirements for the identification of the abnormality are crucial. The reliability +of ECG extraction from each channel will be dependant on the number of cycles used to acquire the averaged ECG rhythm, +since the averaging process improves the SNR for noise that is uncorrelated with the ECG. Therefore, performance metrics + +Lead I +Neck +Ear +Scalp +S1 +S2 +S3 +S4 +S5 +Time [s] +Potential [µV] +(a) +Potential [µV] +(b) +Potential [µV] +(c) +Potential [µV] +(d) +Potential [µV] +(e) +-200 +0 +200 +400 +-124 ms +-32 ms +2 ms +30 ms +224 ms +-4 +-2 +0 +2 +4 +6 +-118 ms +-42 ms +-4 ms +24 ms +220 ms +-1 +0 +1 +2 +-120 ms +-44 ms +0 ms +28 ms +192 ms +-0.5 +0 +0.5 +1 +-114 ms +-32 ms +2 ms +32 ms +216 ms +-200 +0 +200 +400 +600 +-166 ms +-34 ms +0 ms +32 ms +270 ms +-10 +0 +10 +20 +-168 ms +-36 ms +-2 ms +26 ms +268 ms +-2 +0 +2 +-162 ms +-40 ms +-4 ms +28 ms +270 ms +-1 +0 +1 +2 +-198 ms +-32 ms +-4 ms +18 ms +226 ms +-400 +-200 +0 +200 +400 +600 +-162 ms +-30 ms +4 ms +34 ms +264 ms +-5 +0 +5 +-164 ms +-34 ms +2 ms +32 ms +254 ms +-1 +0 +1 +-160 ms +-30 ms +4 ms +32 ms +242 ms +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +-182 ms +-32 ms +4 ms +34 ms +222 ms +-100 +0 +100 +200 +-178 ms +-34 ms +2 ms +32 ms +222 ms +-4 +-2 +0 +2 +4 +6 +-174 ms +-36 ms +0 ms +28 ms +218 ms +-0.5 +0 +0.5 -188 ms +-46 ms +-2 ms +30 ms +186 ms +-0.5 +0 +0.5 +-178 ms +-36 ms +-4 ms +42 ms +164 ms +-200 +0 +200 +400 +-114 ms +-30 ms +4 ms +32 ms +258 ms +-10 +0 +10 +20 +-114 ms +-32 ms +2 ms +32 ms +262 ms +-1 +0 +1 +-100 ms +-32 ms +2 ms +32 ms +256 ms +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 -182 ms +-8 ms +24 ms +42 ms +230 ms +-0.2 +0 +0.2 +0.4 +-0.2 +0 +0.2 +0.4 +-0.2 +0 +0.2 +0.4 +-0.2 +0 +0.2 +0.4 +Fig. 4: Cardiac rhythms from wrist, scalp, ear, and neck ECG of 5 subjects ((a) - (d)). Reference lead I cardiac cycles are +displayed along the first column. ECG acquired after N = 240 cycles (blue) and N = 540 cycles (green). ECG potentials +at timings of P-, Q-, R-, S-, and T-waves in each channel are circled for the N = 240 cycle ECG. Timings of the waves are +indicated in ms next to each wave. +were calculated for averages formed from different numbers of cardiac cycles, ranging from N = 2 to N = 240 cardiac cycles +(Figure 5). The superiority of the neck ECG is evident, providing high fidelity Lead I ECG after averaging over a low number of +cycles, while the scalp ECG performed the worst over the three evaluations. Representative performance metrics after averaging +over N = 240 cycles - a moderate data requirement - are provided in Table 1. +3.3. Correspondence between simulation and measurement: Experiment B +While the performance of the ear ECG at rest has been demonstrated through the analysis of data collected in Experiment A, the +feasibility of recording theses signals in real-world environments must also be established. To that end, measurements during +a driving task were conducted on five subjects over the course of one hour. Ear ECG data was collected from both the left and +right ears (in a single ear, stand-alone fashion, and from between the ears, as was demonstrated in [11, 37]). +In order to help provide a better understanding of the ECG potential available at single and cross ear locations, in Figure 2a), + +102 +0 +10 +20 +30 +0.2 +0.4 +0.6 +0.8 +1 +1 +2 +3 +4 +102 +Pearson corr. coeff. [r] +102 +1 +10 +1 +10 +Amplitude ratio +Wave timing [ms] +Cardiac cycles [N] +(a) +(b) +(c) +Scalp +Ear +Neck +Performance metrics +1 +10 +r=0.56 +r=0.86 +r=0.97 +y=2.1 +y=1.5 +y=2.3 +3 ms +10 ms +20 ms +N=240 +N=240 +N=240 +Fig. 5: Performance metrics for scalp, ear, and neck ECG after varying levels of averaging (N = cycles). (a) correlation of +the cardiac rhythms with the grand-median lead I cardiac rhythm, (b) RMS amplitude ratio between the R-peak and P-,Q-,S-, +and T- peaks for a given channel, normalised by the values from lead I (c) RMSE of the timings of the P-,Q-,S-, and T- waves +relative to the R-wave between a given channel and the lead I channel. A vertical line indicates the values at N = 240 cycles. +Values for the scalp ECG and neck ECG at N = 240 cycles are displayed in Table 1. +Right ear [mV] +(a) +Concha +Helix +Left ear [mV] +(b) +-0.32 +-0.33 +-0.34 +-0.35 +Reference R-peak timing +Left +Right +Cross-ear +Ear-ECG measurements +Time [s] +(c) +0.02mV +Fig. 6: Simulated and recorded cardiac electric potentials. Simulations of (a) left ear and (b) right ear surfaces on the model at +the timing of the R-peak. The outline of the mesh of the model is shown in black to improve the clarity of the shape of the ear +surface. (c) Recorded ECG traces from left ear, right ear, and cross ear ECG channels from a single subject. ECG potentials in +each channel at the timings of the R-peaks in the reference (lead I) channel are circled. +a magnified view of the simulated potential over the ears is provided. The range of potentials spanning the left and right ears +(i.e., the maximum available potential from any configuration of ear ECG electrodes) is considerably lower ( 0.03 mv) than that +on the torso ( 0.8 mv). Moreover, the potential over the surface of a single ear (i.e., the maximum available potential from any +configuration of single ear ECG electrodes) is even further reduced to ( 0.001 mv). Measurements from Subject 4 exemplify the +lower amplitude of single ear ECG relative to the cross ear channel; note that the R-peaks are in the bandpass-filtered waveform +of the cross ear ECG channel, however, for the single ear ECG, the SNR is too low enable such identification. The reader should +note that in the single ear data from Experiment A, R-peaks were identifiable in the bandpass-filtered signal. The difference +observed between these waveforms is likely down to the fact that the recordings during Experiment B were conducted in the +presence of larger amplitude noise arising due to muscle activity, and after periods of motion of the user, during which the +electrode-skin contact can be compromised. +Figure 7 shows measurements and simulations revealing the characteristic shape and timing of the cardiac rhythms from the +single ear and cross ear ECG channels. In both measurements and simulations, two salient features are apparent. The first is the + +-0.16 +-0.18 +-0.2 +-0.22 +-0.24 +-0.26 +-0.28 +Z +-0.3 +y +0.32-0.21 +-0.22 +-0.23 +-0.24 +-0.25 +-0.26 +Z +-0.27 +y +-0.28elevated amplitude in the cross ear signal relative to the single ear signals, explained clearly by the surface potentials in Figure +2a). Next, the good correspondence between the Lead I signal and the cross ear signal is evident (as demonstrated in [11, 37], as +well as the correspondence between the Lead I signal and both single ear signals. However, there is a contradiction between the +simulated and measured R-peak amplitudes in the left and right ears, whereby the amplitude of the right ear R-peak is observed +to be marginally higher than that in the left ear signal, whereas the measurements across all subjects indicate that the left ear +R-peak is the higher (mean = 0.4 µV). This apparent discrepancy should be attributed to the high sensitivity of the model +predictions when measuring potential differences across a small distance, whereby small changes in electrode positioning can +lead to amplitude changes that are of the same magnitude as the potential difference being measured. Based on the fact that +the right ear is slightly further away from the heart than the left ear, one would expect the left ear signal to be slightly higher +in amplitude than the right ear signal (as reflected in the measurements), however, further investigation would be needed to +establish this difference, for example, through precisely positioned measurements on multiple subjects. +Potential [µV] +(a) +Potential [µV] +(b) +Time [s] +Simulation +Lead I +Cross ear +Right ear +Left ear +-500 +0 +500 +-120 ms +-30 ms +4 ms +34 ms +246 ms +-10 +0 +10 +-116 ms +-30 ms +4 ms +38 ms +246 ms +-4 +-2 +0 +2 +4 +6 +-116 ms +-30 ms +4 ms +34 ms +244 ms +-4 +-2 +0 +2 +4 +6 +-114 ms +-30 ms +4 ms +66 ms +248 ms +Measurement +-0.2 +0 +0.2 +0.4 +0 +500 +1000 +-137 ms +-37 ms +0 ms +30 ms +247 ms +-0.2 +0 +0.2 +0.4 +-10 +0 +10 +20 +-137 ms +-37 ms +0 ms +27 ms +247 ms +-0.2 +0 +0.2 +0.4 +-2 +0 +2 +-137 ms +-37 ms +0 ms +30 ms +250 ms +-0.2 +0 +0.2 +0.4 +-2 +0 +2 +-137 ms +-37 ms +0 ms +27 ms +240 ms +Fig. 7: Simulated and measured cardiac rhythms in wrist, left ear, right ear, and cross ear ECG. (a) Simulations. (b) Measure- +ments - cardiac rhythms based on averaging over a sixty minute recording. ECG potentials as the timings of the P-, Q-, R-, S-, +and T-waves (from left to right) are circled. The timing of each eave is indicated in ms. +3.4. Ear ECG cardiac rhythms: Experiment B +Cardiac rhythms extracted from N = 240 cycles for all five subjects are shown in Figure 8. The displayed cardiac rhythms +were taken from the last 240 cycles recorded during the experiment - such that the impact of real-world recording scenarios +(e.g., the presence of muscle movement and physical movement of the user) would be present. Out of the ear ECG channels, +the cross ear ECG was more robust, and faithfully retained the Lead I information in each example provided, demonstrating the +suitability of this channel to real world recording scenarios. For example, the inter-subject variations in the shape of the Lead I +signals are reflected well in the cross ear channel for all five subjects. For the single ear measurements, the Lead I characteristics +were still retained, however more distortion in the signal is evident (for Subjects 4 and 5 in particular). Note that, for Subjects 4 +and 5, the Lead I amplitude was relatively low compared with Subjects 1-3, indicating that the amplitude of the ECG was low +across the whole body for these participants. The implication of this result is that the single ear ECG quality might be highly +dependant on the amplitude of electrical activity generated by the heart. For regular ECG, and indeed, even cross-ear ECG, +smaller amplitude heart potentials do not severely effect the ECG signal quality, which provides an explanation for the higher +ECG quality. Despite the higher distortion evident in the single ear traces in Figure 8, for the most part, the multiple waves were +still readily discernible. Observe that, as previously discussed, the right ear amplitude was lower than the left ear amplitude. +This is also reflected in more distortion visible in the right ear channels. +Over the course of long-term measurements in real-world scenarios, such as driving, regular movement of the user will +induce motion and EMG artifacts in the signal, while also likely compromising the skin-electrode contacts of the channel. In +turn, this will all increase the number of cardiac cycles required in order to obtain faithful ECG. Therefore, we have provided an +analysis of the performance of the channels with varying data lengths over the course of a one hour driving trial on five subjects. + +Lead I +Cross ear +Left ear +Right ear +-500 +0 +500 +-130 ms +-30 ms +3 ms +27 ms +253 ms +-10 +0 +10 +-133 ms +-33 ms +3 ms +27 ms +260 ms +-2 +0 +2 +-123 ms +-33 ms +3 ms +30 ms +250 ms +-1 +0 +1 +2 +-157 ms +-33 ms +3 ms +27 ms +253 ms +-500 +0 +500 +-107 ms +-27 ms +3 ms +27 ms +210 ms +-10 +0 +10 +-110 ms +-27 ms +3 ms +27 ms +210 ms +-2 +0 +2 +-107 ms +-27 ms +3 ms +27 ms +227 ms +-0.5 +0 +0.5 +1 +1.5 +-107 ms +-27 ms +7 ms +30 ms +193 ms +0 +500 +1000 +-140 ms +-33 ms +3 ms +27 ms +250 ms +0 +10 +20 +-137 ms +-33 ms +0 ms +27 ms +243 ms +-1 +0 +1 +2 +3 +-137 ms +-33 ms +0 ms +27 ms +260 ms +-1 +0 +1 +2 +-137 ms +-33 ms +0 ms +27 ms +223 ms +-200 +-100 +0 +100 +-137 ms +-27 ms +3 ms +30 ms +270 ms +-10 +-5 +0 +5 +-123 ms +-17 ms +13 ms +37 ms +260 ms +-1 +0 +1 -150 ms +-20 ms +10 ms +40 ms +297 ms +-0.5 +0 +0.5 -127 ms +-17 ms +17 ms +43 ms +273 ms +-200 +0 +200 +-123 ms +-40 ms +7 ms +33 ms +247 ms +-5 +0 +5 +-193 ms +-57 ms +-17 ms +10 ms +233 ms +-1 +0 +1 +-200 ms +-50 ms +-13 ms +17 ms +223 ms +-0.5 +0 +0.5 -157 ms +-53 ms +-23 ms +7 ms +227 ms +S1 +S2 +S3 +S4 +S5 +Potential [µV] +(a) +Potential [µV] +(b) +Potential [µV] +(c) +Potential [µV] +(d) +Potential [µV] +(e) +-0.2 +0 +0.2 +0.4 +-0.2 +0 +0.2 +0.4 +-0.2 +0 +0.2 +0.4 +-0.2 +0 +0.2 +0.4 +Time [s] +Fig. 8: Cardiac rhythms from wrist, left ear, right ear, and cross ear ECG of 5 subjects ((a) - (d)). Reference lead I cardiac +cycles are displayed along the first column. ECG acquired after N = 240 cycles (blue) and N = 600 cycles (green). ECG +potentials at timings of P-, Q-, R-, S-, and T-waves in each channel are circled for the N = 240 cycle ECG. Timings of the +waves are indicated in ms. +Performance metrics were calculated on all cardiac cycles recorded during the one hour trial from each subject. Moreover, +the performance was evaluated for different levels of averaging (expressed as the number cardiac cycles that were averaged +over). Figure 5a) displays the Pearson correlation between the ear ECG cardiac rhythms and Lead I. Correlation results in +[11] (r = 0.96 for 4 minutes of cross-ear ECG) from a different cohort of subjects (N = 6) are in good agreement with the +results in this study (r = 0.93 for N = 240 cycles - or roughly 4 minutes of data) from five subjects, demonstrating that the +performance of the cross-ear ECG can be robust to real-world measurement noise. For single ear ECG, the performance during +real world measurements was also stable, with the left single-ear channels performing almost identically in both experiments. +For example, the correlation at N = 240 cycles in experiment A and B, respectively, for the left ear single ear channel were 0.85 +and 0.86, while the respective timing errors were 10 ms and 7 ms. Relative to the cross-ear ECG, observe the lower performance +of the single ear channel, particularly at smaller values of N. However, at higher values of N, the performance of the single ear +channels were similar to the cross-ear channel, indicating that the single ear ECG is also a viable option for cardiac monitoring + +in real-world scenarios. +The observation of a slightly reduced amplitude of the right ear signal relative to that in the left ear is further supported by +the performance metrics in Figure 9. However, it is also likely that poorer electrode skin contact would have contributed to the +lower performance of the right ear channel. +Table 1 provides performance metrics for the channels under consideration from both Experiment A and B. The values +represent the performance of each channel at a value of N = 240, which was laso used t obtain the average cardiac ryhtms +displayed in Figures 4 and 8. The performance metrics that are probided in Table 1 represent the mean across all cardiac +rhtyhms from within each recording. Performance metrics for the left ear ECG channel were evaluated for data collected in +both Experiment A and B. Note that the the performance of the left ear ECG channel (with respect to the timing error and +amplitude ratio metrics) was worse in Experiment A relative to Experiment B. This result might seem counter-intuitive, since +the data from Experiment A was collected during ideal conditions (with the subject at rest), whereas the data from Experiment +B was collected in noise-prone conditions (driving in a simulator environment). However, this discrepancy was possibly caused +by the different recording duration over which the performance metrics were evaluated in the two Experiments. For example, at +N = 240 cycles, in recordings from experiment A (10 min duration), a total of 400 cardiac rhythms were evaluated, whereas +in recordings from experiment B (60 min duration), roughly 3400 cardiac rhythms were evaluated. +102 +0 +10 +20 +30 +102 +0 +0.5 +1 +102 +1 +1.5 +2 +2.5 +Pearson corr. coeff. [r] +1 +10 +1 +10 +1 +10 +Amplitude ratio +Wave timing [ms] +Cardiac cycles (N) +(a) +(b) +(c) +Performance metrics +Left ear +Right ear +Cross-ear +r=0.76 +r=0.94 +r=0.85 +y=1.4 +y=1.2 +y=1.4 +5 ms +7 ms +9 ms +N=240 +N=240 +N=240 +Fig. 9: Performance metrics for left ear, right ear, and cross ear ECG after varying levels of averaging (N = cycles). (a) +correlation of the cardiac rhythms with the grand-median lead I cardiac rhythm, (b) RMS amplitude ratio between the R-peak +and P-,Q-,S-, and T- peaks for a given channel, normalised by the values from lead I (c) RMSE of the timings of the P-,Q-,S-, +and T- waves relative to the R-wave between a given channel and the lead I channel. A vertical line indicates the values at +N = 240 cycles, displayed in Table 1. +i) Correlation +ii) Amplitude difference +ii) Time difference +ECG Channel +meas. +sim. +meas. +sim. +meas. (ms) +sim. (ms) +iii) var. +Wrist - rest +1 +1 +1 +1 +0 +0 +1 +Neck - rest +0.97 +0.99 +1.5 +1 +3 +1 +1 +cross ear - driving +0.94 +0.99 +1 +1.1 +5 +2 +3 +left ear - rest +0.86 +0.99 +2.1 +1.1 +10 +2 +6 +left ear - driving +0.85 +0.99 +1.1 +1.1 +7 +2 +6 +right ear - driving +0.76 +0.99 +1.3 +1.2 +9 +2 +9 +Scalp - rest +0.56 +0.97 +2.3 +1.5 +20 +3 +9 +Table 1: Mean performance for cardiac rhythms (N = 240 cycles) for scalp, cross ear, left ear, right ear, and neck ECG +channels. For the left ear channel, data (i) correlation of the cardiac rhythms with the grand-median lead I cardiac rhythm, +(ii) RMS amplitude ratio between the R-peak and P-,Q-,S-, and T- peaks for a given channel, normalise dby the values from +lead I iii) RMSE of the timings of the P-,Q-,S-, and T- waves relative to the R-wave between a given channel and the lead I +channel, and (iv) normalised variance. The column heading ‘meas.’ denotes results for measured cardiac rhythms, whereas +’sim.’ denotes results for the simulated rhythms. Values for left ear, right ear, and cross ear ECG were calculated with data from +Experiment B, while values for scalp, and neck ECG were calculated wit data from Experiment A. + +4. CONCLUSION +The ability to monitor ECG will be a key feature in future wearable health systems. A primary position to record physiological +signals from is the ear as a result of its stable location relative to the vital organs during everyday activities and its ability to +house commonplace accessories such as earbuds. However, the ECG signal that is available over the surface of a single ear +had not yet been established. First, the difference in ECG potential on the ear and in surrounding regions of the neck and scalp +was investigated. These results can support both existing and prospective wearable ECG platforms that utilise scalp, ear, and +neck locations. Measurements of single ear ECG on ten subjects, during an ideal scenario of resting while sitting helped to +demonstrate, for the first time, the characteristic timing and shape of the ECG signal available at the single ear location. Further +measurements, including both single ear and cross ear ECG, on five subjects during a one hour driving task demonstrated +real-world feasibility. Both the cross ear and single ear ECG were demonstrated to be robust to real world environments over +prolonged recording periods, providing valuable evidence for the use of such technology in society. Future work will consider +the integration of additional cardiac monitoring sensors into the ear worn platform, such as the PPG and BCG, and consider +various cardiac conditions. +Acknowledgment +This work was supported by the Racing Foundation grant 285/2018 and the MURI/EPSRC grant EP/P008461. +5. REFERENCES +[1] J. 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Mandic, “In-ear spo2: A tool for wearable, unobtrusive monitoring of +core blood oxygen saturation,” Sensors, vol. 20, no. 17, p. 4879, 2020. +[41] J. Pan and W. J. Tompkins, “A real-time QRS detection algorithm,” IEEE Transactions on Biomedical Engineering, no. 3, +pp. 230–236, 1985. +[42] H. Sedghamiz, “Matlab implementation of Pan Tompkins ECG QRS detector,” Code Available at the File Exchange Site +of MathWorks, 2014. + diff --git a/RtE0T4oBgHgl3EQfkgHa/content/tmp_files/load_file.txt b/RtE0T4oBgHgl3EQfkgHa/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2ff17b619fe7da6eea5a3510c556d52e5ad4fddb --- /dev/null +++ b/RtE0T4oBgHgl3EQfkgHa/content/tmp_files/load_file.txt @@ -0,0 +1,918 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf,len=917 +page_content='HEARABLES: FEASIBILITY OF RECORDING CARDIAC RHYTHMS FROM SINGLE EAR LOCATIONS Metin Yarici, Wilhelm Von Rosenberg, Ghena Hammour, Harry Davies, Pierluigi Amadori, Nico Ling, Yiannis Demiris, Danilo P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Mandic Department of Electrical and Electronic Engineering, Imperial College London, SW7 2AZ, UK E-mails: {metin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='yarici16, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='mandic}@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='uk ABSTRACT Wearable technologies are envisaged to provide critical support to future healthcare systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Hearables - devices worn in the ear are of particular interest due to their ability to provide health monitoring in an efficient, reliable and unobtrusive way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Despite the considerable potential of these devices, the ECG signal that can be acquired through a hearable device worn on a single ear is still relatively unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Biophysics modelling of ECG volume conduction was used to establish principles behind the single ear ECG signal, and measurements of cardiac rhythms from 10 subjects were found to be in good correspondence with simulated equivalents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Additionally, the viability of the single ear ECG in real-world environments was determined through one hour duration measurements during a simulated driving task on 5 subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Results demonstrated that the single ear ECG resembles the Lead I signal, the most widely used ECG signal in the identification of heart conditions such as myocardial infarction and atrial fibrillation, and was robust against real-world measurement noise, even after prolonged measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' This study conclusively demonstrates that hearables can enable continuous monitoring of vital signs in an unobtrusive and seamless way, with the potential for reliable identification and management of heart conditions such as myocardial infarction and atrial fibrillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Index Terms— wearable health, electrocardiogram, ear ECG 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' INTRODUCTION The use of wearable technologies for monitoring vital signs has become increasingly widespread in the society, both for recreational and medical purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Most often, these devices are integrated into wearable garments and accessories, and are concealed and miniaturised for convenience of the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The most common choices for wearable vital signs monitoring technologies are smart watches and chest straps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Smart watches typically utilise the photoplethysmogram (PPG) to provide continuous monitoring of pulse and respiration, while chest straps may also record the electrocardiogram (ECG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' However, PPGs lack the information necessary to understand the functioning of the heart [1], while chest-worn devices are not suitable for everyday use due to their obtrusive nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Consequently, alternative solutions that provide ECG measurements in a convenient and user-friendly manner have attracted significant research interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' One such solution is the ’hearable’ device - a wearable that fits in the ear and can serve both as an audio accessory and a platform for health monitoring [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' For hearables, the stability of the head relative to the vital organs during sitting, sleeping, walking, and eating results in superior monitoring capability in real-world scenarios, relative to devices attached to the limbs [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' For this reason, wearable devices that are placed on the regions of the skin surface surrounding the ear are also gaining popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Da He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' [5] and Casson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' [6] demonstrated the effectiveness of behind-the-ear and scalp based recordings of ECG, ballistocardiogram (BCG), and PPG for heart rate monitoring, while Celik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' [7, 8] provided evidence of similar results when measurements were taken from electrodes placed on the ear and neck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Despite the success of these proof-of-concept studies, the difference in ECG potential from the various head locations is not yet well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' To that end, our previous work investigated the ECG potential at different head positions from inside a helmet [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' An assessment of the performance of the channels under consideration was conducted with regard to the successful application of an R-peak detection algorithm [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' However, through experimentation alone, it is difficult to clarify the effect of the manifold factors that can influence the sensitivity of the ECG sensors [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' For example, variations in user movements, muscle activity, brain activity, and electrode-skin contact can all adversely affect the quality of the ECG signal [12, 13], so that the current arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='02475v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='med-ph] 6 Jan 2023 studies are limited by either a lack empirical evidence or support from theoretical models (a) (b) Helix Concha Scalp (upper) Scalp (lower) Neck (upper) Neck (lower) Left arm below | z [mm], pz (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='12mAm) | above 40 0 80 0 60 50 30 10 10 30 0 20 20 40 20 20 40 60 right | x [mm], px (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='12mAm) | left T R P Q heart vector S y [mm], py (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='12mAm) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' 1: Biophysics model of ECG propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' (a) 3D Model structure: Organs and tissues in the torso and head are encased in a skin surface structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' (b) Heart vector: The orientation and magnitude of the current dipole p in px, py, and pz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' the heart vector at one point in time is shown in blue, and the trace of the tip of the heart vector from the start of the cycle until the current position (axes in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='12 mAm) is shown in orange;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' the heart muscle is shown in pink in the background (axes in millimetres).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' (c) Sensor set-up: For the purpose of mapping the ECG on the left side of the head, sensors were placed on scalp, ear, and neck positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Potential difference between the upper and lower scalp (scalp ECG), helix and concha (ear ECG), and upper and lower neck electrodes (neck ECG) was extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' A comprehensive theoretical approach to mapping ECG potentials can be achieved through forward modelling [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' This process involves applying Maxwell’s equations to a dielectric model of the body in order to estimate the propagation of cardiac potentials from the heart to the head-positions of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' This offers the advantage of examining the ECG in a way that is isolated from other sources of electrical activity, such as brain and muscle activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Furthermore, such modelling can enable the evaluation of the characteristic ECG - that is, the timings and shape of the waves of the cardiac cyle - available from various wearable ECG channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Early ECG modelling was based on simplified representations of the human body (for example, a homogeneous conductive sphere [16]), however more realistic models have since been produced which include separate shells representing the heart and surrounding body [17], and realistically shaped geometries for specific organs in the chest cavity [18, 19, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' A model which enables the simulation of ECG propagation to the head and ear locations was first introduced in our previous work [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' A whole-body model incorporating tissue from the torso and head was employed to provide a rigorous theoretical basis for the possibility of high-quality head-ECG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' In addition to simulations, a systematic analysis of the ECG surface potential from head and in-ear channels was conducted, measuring the ECG potential difference between the left and right side of the head, as well as between the left and right ears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Moreover, the characteristic timing and shape of the ECG from the different channels was evaluated through both measurements and simulations, opening new avenues for wearable head-based cardiac monitoring that provides more than just heart rate detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Ideally, wearable vital signs measurements should be conducted from a device worn on a single ear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' However, as previ- ously discussed, the ECG potential available on the single ear, or the surrounding locations on the scalp and neck, is not well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' To this end, we follow the approach outlined in [11] and assess the ECG potential on a single side of the head, ear, and neck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' This study seeks to determine the amplitude, shape, and timing of ECG waveforms from specific sites through both real-life recordings and biophysics modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Once the possibility of recording ECG potentials from each location was established, a second experiment was conducted in a more dynamic, real-world environment, which allowed for the recordings to be evaluated over an extended period of driving in a simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' This enabled further assessment of the impacts of varying levels of brain and muscle activity, as well as the user’s physical movement on the signal quality, and with greater confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Results from both experimentation and simulations have established the feasibility of recording ear ECG, and neck ECG signals that possess characteristics similar to those present in the standard Lead I ECG signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Comprehensive evaluation of performance of the wearable-ECG under consideration relative to Lead I has also been presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Key findings reveal differences in performance that arise with different recording lengths from each recording channel, thereby revealing the suitability of a given channel for Z ×y-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='28 Z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='3 y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='32Z x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='3 Z x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4(b) 30 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='12 mAm) I above ) 20 10 0- heart vector 10- 20 R 30 40 50 40 40 20 0 20 40 60 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' 0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='12 mAm) 1 back-various applications in cardiac monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Overall, our results indicate that the single ear ECG channel and the cross ear ECG channel can provide robust ECG monitoring during prolonged real-world tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Our findings demonstrate the potential of ear ECG for capturing cardiac rhythms with shapes that are very similar to Lead I from the standard limb leads, even during challenging, real-world recordings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' This framework holds promise for examining heart conditions that are visible in multiple consecutive cardiac cycles in Lead I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' including but not limited to myocardial infarc- tion (the ST segment is elevated),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' first-degree atrioventricular block (the PR interval is longer than 200 ms),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' atrial fibrillation (absence of the P-wave,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' found in 2% to 3% of the population in Europe and the USA [22]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' sinus tachycardia (increased heart rate and a shortened P-T duration) and atrial flutter (an increased frequency of the P-wave relative to the QRS complex as a result of rapid atria contraction) [23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' 24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Moreover, the proposed ear ECG framework offers the possibility of 24/7 continu- ous and unobtrusive cardiac monitoring and can alert the user to the presence of universal signatures of heart malfunction, such as absent P-waves and elongated ST-segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Such a new perspective can be beneficial to many existing applications, such as remote monitoring for patients with cardiac conditions such as arrhythmias, or heart failure [26], sports and fitness monitoring of athletes [27], and assessment of the effect of physical strain and stress in workplace environments [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' METHODS In the present study, the feasibility of recording cardiac rhythms from a single ear is established through real-world measure- ments and biophysics simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' First, theoretical grounding for the measurement of single ear ECG is provided through biophysics modelling of ECG propagation from the heart to various positions of interest around the head and ears, through the use of an accurate volume conductor model of the human body that was first presented in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Model geometry The geometry within the presented model is based on the VHP-Female Computational Phantom v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='1 and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 [30] - a data-set consisting of three-dimensional geometry for major organs and tissues in the human body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The phantom was edited in instances where computational problems were encountered with the mesh of the structures in the model in the modelling software (COMSOL Multiphysics [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The model comprised geometric representations for major structures surrounding the heart and in the head, and a complete whole-body volume enveloped by an outer skin structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' the top-half of the model is shown in Figure 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' For the purpose of realistic modelling of electric fields within the body, a large sphere or radius r = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='3 m, filled with air, surrounded the body and provided an electrical ground in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The complete mesh consisted of 560.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='630 domain elements and 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='286 boundary elements, and the average edge length was 6 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Electric properties of the body and cardiac current sources The relevant electric properties of the modelled human tissues (conductivity and relative permittivity at an excitation frequency of 15 Hz) were extracted from the database in [32], and are provided in the supplemental material (Table S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Previous research has revealed that isotropic and anisotropic treatments of the heart structure in simulations of ECG yield similar results [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Therefore, an isotropic treatment of the heart structure was adopted within the present model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Multi-source modelling of the cardiac potential has been shown to perform best for prediction of the ECG potential on the torso [34], however, owing to the relatively large distance between the head and heart, the present model employs single source modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The heart source dynamics over the course of a single cardiac cycle are shown in Figure 1b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' a comprehensive description of the electrical dynamics of the cardiac cycle is provided in [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The source dipole in the presented model is a superposition of three orthogonal vectors representing the projection of the cardiac potential over the course of a single cardiac cycle at a normal to the sagittal, transversal, and coronal planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The heart vectors were acquired from real-world measurement from a subject with no known cardiac abnormalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The cardiac cycle was taken to start and end 200 ms before and 400 ms after the maximum of the R-wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Experiment A: ECG mapping on the single ear ECG potential on candidate positions for head-based wearable ECG monitoring platforms was assessed through measurements on ten subjects with GRASS Ag/AGCl cup electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' single ear ECG from the left ear was recorded through one electrode placed on the helix and another on the concha;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' this reflects two locations that are available to stand-alone, single ear hearable devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The electrode positioning for the remaining head-ECG channels was devised such that the ear, scalp, and neck channels each spanned equal distances across the head-surface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' in this way, differences in ECG potential in each of these regions could be assessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' For the scalp channel, the electrodes were positioned according to the following methodology: the lower scalp electrode was placed at a distance of L/2 above the helix-electrode, where L equals the separation between the helix- and concha-electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The upper scalp electrode was placed at a distance of L above the lower scalp electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' An equivalent placing system was employed for the neck channel, whereby neck electrodes were placed below the concha-electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The ear, scalp, and neck electrodes were all positioned along a vertical line which intersected the points at which each subject’s helix- and concha-electrodes were positioned (see Figure 1c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' A standard limb-ECG channel was also created between the left and right wrist, whereby electrodes were positioned on the left and right volar central zones (just below the palm) [11, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Prior to the application of the electrodes, the skin at each location was prepared through cleaning with medical wipes and abrasion with NuPrep gel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' A layer of 10-20 conductive paste was also applied to the electrodes prior to placement, in order to improve conduction between the skin surface and the electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Measurements were conducted via a custom bio-amplifier programmed to digitise the potential difference between the upper and lower scalp electrodes (scalp ECG), the helix and concha electrodes (single ear ear ECG), the upper and lower scalp electrodes (neck ECG) and the left and right wrist electrodes (wrist ECG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' A sampling rate of 500 Hz was used and the helix-electrode served as a bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' During the measurements, subjects were seated and instructed to close their eyes while minimising eye saccades, and movement of the head, neck, and arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The recordings were performed under the Imperial College London ethics committee approval JRCO 20IC6414, and all subjects gave full informed consent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Experiment B: real-world feasibility Once the cardiac potential on the scalp, ear and neck regions had been established for subjects while at rest, the feasibility of recording cardiac rhythms in a real-world scenario was investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Single ear measurements on both the left and right ears were conducted, in addition to cross-ear measurements between the left and right ears, for which we previously established the feasibility of recordings on 6 subjects at rest [11, 37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Five of the ten subjects from Experiment A were instructed to drive in a virtual reality (VR) driving simulator environment while measurements were taken from the described ear ECG channels and a reference-ECG channel from the wrists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Data was acquired via a g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='tec g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='USBamp (2011) bio-amplifier at a sampling rate of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='200 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' In order to ensure comfortable ear ECG measurements for the duration of the driving task, custom fabric electrodes were used to record the ECG signal from the concha [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The fabric electrodes employed within this study were constructed from silver-coated thread that is interwoven with elastic fibres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' In [2], electrodes of this type were mounted on visco-elastic earpieces and then placed in the ear canal of five subjects over the course of a normal working day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Measurements were shown to be stable after prolonged periods of unrestrained activity which included walking, eating, and talking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' While the ear canal is an ideal position to record the ECG, as outlined in [11, 37], in the current experiment, a PPG sensor was placed in the position of the ear canal for measurement of ear-SpO2, and is the subject of ongoing analysis [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Therefore, the concha - another convenient location to record physiological signals from - was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' In the present study, the electrodes were secured to the concha using soft, malleable non-allergenic silicone, shaped to each person’s ear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Prior to application of the electrodes, the skin was prepared through cleaning with medical wipes and abrasion with NuPrep gel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' A thin layer of Signa Gel conductive gel was applied to the surface of the fabric electrode in order to help establish good conduction at the skin-electrode interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' For the left and right ear single ear ECG measurements, a reference electrode was positioned on a second location which is suitable for monitoring through Hearable devices - the helix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' As with the previous electrode positions, the ipsi-lateral helix was prepared with medical wipes and NuPrep gel prior to electrode connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The GRASS Ag/AgCl electrodes were used to record the signal from the helix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' A layer of 10-20 conductive paste was applied to aid conduction at the skin-electrode interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' As a bias electrode, a second Ag/AgCl electrode was placed on the ipsi-lateral earlobe, with the same skin preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' For the cross-ear ECG measurement, the left concha electrode was referenced to the right helix electrode, with the left ear lobe electrode serving as the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' For the reference ECG measurement, for Subjects 1-3, a wrist ECG measurement was conducted, whereby the left wrist electrode was referenced to the right wrist electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' For Subjects 4 and 5, a reference ECG signal was obtained by referencing the left wrist electrode to the right helix electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' All reference channels were biased with the left ear lobe electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Vertical electro-oculogram (VEOG) channels were also recorded via one electrode below and another above each eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The previously described skin preparation and conductive paste was also applied for the VEOG measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Signal processing All signal processing was performed in MatLab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The processing steps for the recorded ECG data are described in Algorithm1, and consisted of the following procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The ECG data from both Experiment A and B were first bandpass-filtered between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='5 Hz and 95 Hz using a third-order Butterworth filter and notch-filtered with a second-order IIR-filter with a centre frequency of fc = 50 Hz and a band-width of w = 5 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' For data collected in Experiment B (during the driving task), an additional Algorithm 1: Signal processing steps 1 Record electric potential differences raw_ECG with one reference channel (wrist ECG, Lead I) and multiple head-ECG channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' 2 For data that was recorded as part of Experiment B, perform threshold- and blink-related artifact rejection on raw_ECG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' 3 Bandpass- and notch-filter the reference channel, respectively, using a third-order Butterworth filter with a lower cut-off frequency of fmin = 1 Hz and an upper cut-off frequency of fmax = 95 Hz, and a second-order IIR-filter with a centre frequency of fc = 50 Hz and a band-width of w = 5 Hz, to give filtered_reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' 4 Perform R-wave detection on filtered_reference according to [41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' 5 Bandpass-filter the signals in all channels in raw_ECG using a third-order Butterworth filter with a lower cut-off frequency of fmin = 1 Hz and an upper cut-off frequency of fmax = 30 Hz, to give filtered_ECG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' 6 Extract cardiac cycles from filtered_ECG around the identified R-waves (within a −200 ms to +400 ms window) using the R-wave timings obtained in Step 3 from the wrist ECG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' 7 Find the median the cardiac rhythms for different lengths of data - ranging from 2 − 540 cardiac cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' 8 Calculate four metrics for the quality assessment of the median cardiac cycles from individual channels: (i) correlation between cardiac rhythms in a given channel and Lead I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' (ii) ratios of the amplitude of P-,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Q-,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' S-,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' and T- waves relative to the R-wave in a given channel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' (iii) timing of P-,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Q-,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' S-,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' and T-waves relative to the timing of the R-wave in a given channel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' and (iv) normalised variance in the channels — the root-mean-square error (RMSE) of the differences between the individual cardiac rhythms and the grand-median cardiac rhythm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' divided by the standard deviation of the grand-median cardiac rhythm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' in the channel under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' cleaning procedure which removed blink artifacts and large amplitude deflections, such as those caused by motion and jaw- clenching, was applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Details of the artifact rejection are provided in the supplemental material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Averaging over multiple consecutive cardiac cycles was conducted in order to extract full cardiac rhythms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' In [11], a mutlimodal in-ear sensor was shown to provide a stand-alone solution for the reliable measurement and identification of cardiac cycles from ear ECG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' a collocated MEMS sensor placed beneath the surface of the fabric electrodes was used to detect pressure waves in the ear canal surface associated with the heart beating (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=', the ballistocardiogram).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' This served as a reference for the timing of the cardiac cycles in the ear ECG signal detected by the fabric electrodes and, in conjunction with a matched-filtering technique [9, 10], enabled reliable identification of cardiac cycle timing in the low SNR ear ECG signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Such techniques were not the subject of the present study, therefore timings of the R-peaks that were extracted from the reference channels were used here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' However, it is feasible that future developments of the single ear ECG presented in this study will incorporate the described multimodal sensing capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The use of the reference channel in this way is made possible by virtue of the ECG signals from all locations on the body sharing a common electrical source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The R-peak timings were identified using the Pan-Tompkins algorithm implementation in MatLab [42, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' From within the ear ECG signals, segments of 600 ms length containing the cardiac rhythms were extracted (200 ms prior to and 400 ms after the timing of the R-peak), enabling the identification of all the key components (P-, Q-, R-, S-, and T-waves) in the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Median cardiac rhythms were obtained for different numbers of cardiac cycles N, ranging from N = 2 to N = 540.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Since the success of the extraction of cardiac rhythms is dependant on the number of averages taken, and the number of cardiac cycles varies greatly during activities such as driving, different extraction procedures were categorised in terms of the number of cardiac cycles included in the segment, as opposed to the length of time in which the duration took place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' During normal heart function (roughly 60 bpm - 80 bpm), conducting averages over the values of N ranging from 2 to 540 produced results that provide an indication of the performance of the ear ECG ranging from recording duration of a few seconds up to roughly 9 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' For each value of N tested, the maximum number of median cardiac rhythms were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' For example, for a recording spanning M = 10 complete cardiac cycles, an N = 2 median cardiac rhythm could be extracted 8 times (M − N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' A grand-median cardiac rhythm was extracted from the reference-ECG signal of each subject to serve as a benchmark during analysis of ear ECG cardiac rhythm quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' For rigour, four performance metrics were calculated for each individual ear ECG cardiac rhythm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The four calculated metrics were as follows: i) The Pearson correlation coefficient between the cardiac rhythms from a given channel and a gran-median from the wrist ECG (Lead I);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' ii) Root mean square (RMS) of the ratios between the R-wave and remaining wave amplitudes in a given channel, bench- marked against equivalent ratios in Lead I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' iii) Root-mean-square error (RMSE) of the timings of the P-, Q-, S- and T-waves relative to the R-wave in a given channel, relative to timings in lead I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Details of the wave-timing calculation are provided in the supplemental material;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' iv) RMSE between the grand-median cardiac rhythm of a given channel and all individual cardiac rhythms recorded in that same channel, where the RMSE was normalised by dividing by the standard deviation of the grand-median cardiac rhythm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4 Surface potential [mV] (a) Time [s] Scalp Ear Neck 5µV ECG measurements Reference R-peak timing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' 2: Simulated and recorded cardiac electric potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' (a) Simulated electric potential on the head surface and upper torso at the time of the R-wave peak (in milliVolts) in the biophysics model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' (b) Recorded ECG traces from the scalp, ear, and neck ECG channels from a single subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The ECG potential at the instance of the R-peaks in the reference (Lead I) channel are indicated (red circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' RESULTS In order to provide rigorous theoretical support to the ear ECG measurements, simulations of ECG propagation throughout the body were conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' A single ECG cycle was simulated, ranging from the time t1 = −200 ms prior to the timing of the maximum of the R-wave, to t2 = 400 ms after (600 ms in total), at a temporal resolution of TR = 2 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' A snap-shot of the so-simulated cardiac cycle at the timing of the maximum of the R-wave is shown in Figure 6a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Notice the large potential difference between the left and right sides of the torso - positions where conventional Lead I electrodes are placed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Observe the reduction in potential difference across the surface of the head (where the considered ear ECG channels are based) relative to the torso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Scalp, ear, and neck ECG cardiac rhythms extracted from virtual sensor positions on a single side of the head (Figure 1c)) and are displayed in Figure 3a) (blue traces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Correspondence between simulation and measurement: Experiment A The narrow neck structure impedes ECG potential as it propagates from the heart to the head surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' It is therefore intuitive that the gradient in the ECG potential will be highest at the base of the neck, where large potential differences begin to occur, and lowest towards the top of the head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Measurements (from Subject 6) in Figure 2 demonstrate such a topography over the neck and head surface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' note that the P-,Q-,R-,S-, and T-waves waves in the Lead I cardiac rhythm are identifiable in the neck ECG channel, demonstrating a large potential difference at the position of the neck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' On the other hand, in the ear ECG trace, R-peaks are intermittently identifiable, and in the scalp trace, there is no evidence of ECG visible without further processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Averaging multiple cardiac cycles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=', as in Algorithm 1) enables extraction of the cardiac rhythms from the lower SNR ECG measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='3 Z x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='3 Z x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='3 Z x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4Time [s] Simulation Potential [µV] (a) Potential [µV] (b) Lead I Neck Ear Scalp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4 200 0 200 400 600 166 ms 34 ms 0 ms 32 ms 268 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4 10 0 10 20 30 166 ms 36 ms 2 ms 26 ms 268 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4 2 0 2 4 164 ms 38 ms 4 ms 24 ms 264 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4 2 0 2 4 158 ms 34 ms 4 ms 20 ms 260 ms 400 200 0 200 400 600 800 120 ms 32 ms 2 ms 32 ms 244 ms 5 0 5 10 15 116 ms 32 ms 2 ms 30 ms 242 ms 2 0 2 4 116 ms 32 ms 2 ms 32 ms 242 ms 2 0 2 4 116 ms 34 ms 2 ms 30 ms 242 ms Measurement Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' 3: Simulated and measured cardiac rhythms in scalp, ear, and neck ECG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' (a) Simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' (b) Measurements - cardiac rhythms based on averaging over a ten minute recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' ECG potentials as the timings of the P-, Q-, R-, S-, and T-waves (from left to right) are circled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The timing of each eave is indicated in ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The model predictions in Figure 3a) (blue traces) reveal the characteristic shape and timings of the ECG signal at each location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' If there were no other sources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=', brain activity and muscle activity) which contributed to the potential difference at these sites, these are the ECG signals that we would expect to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' As expected, the amplitude of the ECG on the head decreases the higher up the head the channels are placed, in both the measured and simulated channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Overall, in both measurements and simulations, the scalp, ear and neck signals closely match the Lead I signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' This can be attributed to the fact that the channels under consideration and Lead I form a similar projection plane for the heart vector (Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Therefore, in this study, the obtained scalp, ear, and neck rhythms were bench-marked against the Lead I equivalent wrist ECG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The measured traces in Figure 3b) were obtained by averaging over the entire set of cardiac cycles in the ten minute recording, according to Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' In this way, although high SNR and accurate measurements of cardiac rhythms were extracted, the performance of the channel is not well understood at shorter measurement lengths, when a lower number of cardiac cycles are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Ear ECG cardiac rhythms: Experiment A The scalp, ear, and neck traces in Figure 6 were based on averaging over 540 cardiac cycles (roughly ten minute recordings), and help establish the potential difference available to the scalp, ear, and neck ECG channels, due to their high SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' However, in practice, shorter recording lengths are desirable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' cardiac rhythms obtained through averaging over the first 240 recorded cardiac cycles (roughly 4 minutes of data) are shown in Figure 4 for 5 subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The cardiac rhythms are plotted in black for the reference-channel and blue for the scalp, ear, and neck ECG channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' While the neck ECG rhythms were consistent across subjects, more variation was observed for the ear and scalp channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' We have included examples of both good and bad ECG, from Subjects 1-3, and Subject 4-5, respectively, which reflect the level of variation observed across all subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Waveforms from the remaining 5 subjects are provided in the supplemental material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The scalp ECG results are considerably worse than the ear and neck ECG results, which primarily stems from i) poorer electrode skin contact through hair on the scalp, ii) the lower amplitude ECG signal at the location of the scalp relative to the ear and neck, and iii) the large amplitude background noise (EEG and temporal muscle-EMG) on the temporal scalp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Median cardiac cycles from the first N = 600 cycles are also plotted in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Minimal difference is observed for the high quality neck and ear ECG channels, however, little change is also observed for the lower quality scalp ECG for Subject 4 and 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' this indicates that a compact device attached on this position on the scalp might not be suitable for ECG recordings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' In instances where urgent care may be required following the diagnosis of an ECG abnormality (such as myocardial infarc- tion following ST elevation), variations in data requirements for the identification of the abnormality are crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The reliability of ECG extraction from each channel will be dependant on the number of cycles used to acquire the averaged ECG rhythm, since the averaging process improves the SNR for noise that is uncorrelated with the ECG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Therefore, performance metrics Lead I Neck Ear Scalp S1 S2 S3 S4 S5 Time [s] Potential [µV] (a) Potential [µV] (b) Potential [µV] (c) Potential [µV] (d) Potential [µV] (e) 200 0 200 400 124 ms 32 ms 2 ms 30 ms 224 ms 4 2 0 2 4 6 118 ms 42 ms 4 ms 24 ms 220 ms 1 0 1 2 120 ms 44 ms 0 ms 28 ms 192 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='5 1 114 ms 32 ms 2 ms 32 ms 216 ms 200 0 200 400 600 166 ms 34 ms 0 ms 32 ms 270 ms 10 0 10 20 168 ms 36 ms 2 ms 26 ms 268 ms 2 0 2 162 ms 40 ms 4 ms 28 ms 270 ms 1 0 1 2 198 ms 32 ms 4 ms 18 ms 226 ms 400 200 0 200 400 600 162 ms 30 ms 4 ms 34 ms 264 ms 5 0 5 164 ms 34 ms 2 ms 32 ms 254 ms 1 0 1 160 ms 30 ms 4 ms 32 ms 242 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='6 182 ms 32 ms 4 ms 34 ms 222 ms 100 0 100 200 178 ms 34 ms 2 ms 32 ms 222 ms 4 2 0 2 4 6 174 ms 36 ms 0 ms 28 ms 218 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='5 -188 ms 46 ms 2 ms 30 ms 186 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='5 178 ms 36 ms 4 ms 42 ms 164 ms 200 0 200 400 114 ms 30 ms 4 ms 32 ms 258 ms 10 0 10 20 114 ms 32 ms 2 ms 32 ms 262 ms 1 0 1 100 ms 32 ms 2 ms 32 ms 256 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4 -182 ms 8 ms 24 ms 42 ms 230 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' 4: Cardiac rhythms from wrist, scalp, ear, and neck ECG of 5 subjects ((a) - (d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Reference lead I cardiac cycles are displayed along the first column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' ECG acquired after N = 240 cycles (blue) and N = 540 cycles (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' ECG potentials at timings of P-, Q-, R-, S-, and T-waves in each channel are circled for the N = 240 cycle ECG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Timings of the waves are indicated in ms next to each wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' were calculated for averages formed from different numbers of cardiac cycles, ranging from N = 2 to N = 240 cardiac cycles (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The superiority of the neck ECG is evident, providing high fidelity Lead I ECG after averaging over a low number of cycles, while the scalp ECG performed the worst over the three evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Representative performance metrics after averaging over N = 240 cycles - a moderate data requirement - are provided in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Correspondence between simulation and measurement: Experiment B While the performance of the ear ECG at rest has been demonstrated through the analysis of data collected in Experiment A, the feasibility of recording theses signals in real-world environments must also be established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' To that end, measurements during a driving task were conducted on five subjects over the course of one hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Ear ECG data was collected from both the left and right ears (in a single ear, stand-alone fashion, and from between the ears, as was demonstrated in [11, 37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' In order to help provide a better understanding of the ECG potential available at single and cross ear locations, in Figure 2a), 102 0 10 20 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='8 1 1 2 3 4 102 Pearson corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' coeff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' [r] 102 1 10 1 10 Amplitude ratio Wave timing [ms] Cardiac cycles [N] (a) (b) (c) Scalp Ear Neck Performance metrics 1 10 r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='56 r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='86 r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='97 y=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='1 y=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='5 y=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='3 3 ms 10 ms 20 ms N=240 N=240 N=240 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' 5: Performance metrics for scalp, ear, and neck ECG after varying levels of averaging (N = cycles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' (a) correlation of the cardiac rhythms with the grand-median lead I cardiac rhythm, (b) RMS amplitude ratio between the R-peak and P-,Q-,S-, and T- peaks for a given channel, normalised by the values from lead I (c) RMSE of the timings of the P-,Q-,S-, and T- waves relative to the R-wave between a given channel and the lead I channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' A vertical line indicates the values at N = 240 cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Values for the scalp ECG and neck ECG at N = 240 cycles are displayed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Right ear [mV] (a) Concha Helix Left ear [mV] (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='35 Reference R-peak timing Left Right Cross-ear Ear-ECG measurements Time [s] (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='02mV Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' 6: Simulated and recorded cardiac electric potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Simulations of (a) left ear and (b) right ear surfaces on the model at the timing of the R-peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The outline of the mesh of the model is shown in black to improve the clarity of the shape of the ear surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' (c) Recorded ECG traces from left ear, right ear, and cross ear ECG channels from a single subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' ECG potentials in each channel at the timings of the R-peaks in the reference (lead I) channel are circled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' a magnified view of the simulated potential over the ears is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The range of potentials spanning the left and right ears (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=', the maximum available potential from any configuration of ear ECG electrodes) is considerably lower ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='03 mv) than that on the torso ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='8 mv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Moreover, the potential over the surface of a single ear (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=', the maximum available potential from any configuration of single ear ECG electrodes) is even further reduced to ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='001 mv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Measurements from Subject 4 exemplify the lower amplitude of single ear ECG relative to the cross ear channel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' note that the R-peaks are in the bandpass-filtered waveform of the cross ear ECG channel, however, for the single ear ECG, the SNR is too low enable such identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The reader should note that in the single ear data from Experiment A, R-peaks were identifiable in the bandpass-filtered signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The difference observed between these waveforms is likely down to the fact that the recordings during Experiment B were conducted in the presence of larger amplitude noise arising due to muscle activity, and after periods of motion of the user, during which the electrode-skin contact can be compromised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Figure 7 shows measurements and simulations revealing the characteristic shape and timing of the cardiac rhythms from the single ear and cross ear ECG channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' In both measurements and simulations, two salient features are apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The first is the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='28 Z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='3 y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='32-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='26 Z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='27 y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='28elevated amplitude in the cross ear signal relative to the single ear signals, explained clearly by the surface potentials in Figure 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Next, the good correspondence between the Lead I signal and the cross ear signal is evident (as demonstrated in [11, 37], as well as the correspondence between the Lead I signal and both single ear signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' However, there is a contradiction between the simulated and measured R-peak amplitudes in the left and right ears, whereby the amplitude of the right ear R-peak is observed to be marginally higher than that in the left ear signal, whereas the measurements across all subjects indicate that the left ear R-peak is the higher (mean = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4 µV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' This apparent discrepancy should be attributed to the high sensitivity of the model predictions when measuring potential differences across a small distance, whereby small changes in electrode positioning can lead to amplitude changes that are of the same magnitude as the potential difference being measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Based on the fact that the right ear is slightly further away from the heart than the left ear, one would expect the left ear signal to be slightly higher in amplitude than the right ear signal (as reflected in the measurements), however, further investigation would be needed to establish this difference, for example, through precisely positioned measurements on multiple subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Potential [µV] (a) Potential [µV] (b) Time [s] Simulation Lead I Cross ear Right ear Left ear 500 0 500 120 ms 30 ms 4 ms 34 ms 246 ms 10 0 10 116 ms 30 ms 4 ms 38 ms 246 ms 4 2 0 2 4 6 116 ms 30 ms 4 ms 34 ms 244 ms 4 2 0 2 4 6 114 ms 30 ms 4 ms 66 ms 248 ms Measurement 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4 0 500 1000 137 ms 37 ms 0 ms 30 ms 247 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4 10 0 10 20 137 ms 37 ms 0 ms 27 ms 247 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4 2 0 2 137 ms 37 ms 0 ms 30 ms 250 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4 2 0 2 137 ms 37 ms 0 ms 27 ms 240 ms Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' 7: Simulated and measured cardiac rhythms in wrist, left ear, right ear, and cross ear ECG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' (a) Simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' (b) Measure- ments - cardiac rhythms based on averaging over a sixty minute recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' ECG potentials as the timings of the P-, Q-, R-, S-, and T-waves (from left to right) are circled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The timing of each eave is indicated in ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Ear ECG cardiac rhythms: Experiment B Cardiac rhythms extracted from N = 240 cycles for all five subjects are shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The displayed cardiac rhythms were taken from the last 240 cycles recorded during the experiment - such that the impact of real-world recording scenarios (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=', the presence of muscle movement and physical movement of the user) would be present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Out of the ear ECG channels, the cross ear ECG was more robust, and faithfully retained the Lead I information in each example provided, demonstrating the suitability of this channel to real world recording scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' For example, the inter-subject variations in the shape of the Lead I signals are reflected well in the cross ear channel for all five subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' For the single ear measurements, the Lead I characteristics were still retained, however more distortion in the signal is evident (for Subjects 4 and 5 in particular).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Note that, for Subjects 4 and 5, the Lead I amplitude was relatively low compared with Subjects 1-3, indicating that the amplitude of the ECG was low across the whole body for these participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The implication of this result is that the single ear ECG quality might be highly dependant on the amplitude of electrical activity generated by the heart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' For regular ECG, and indeed, even cross-ear ECG, smaller amplitude heart potentials do not severely effect the ECG signal quality, which provides an explanation for the higher ECG quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Despite the higher distortion evident in the single ear traces in Figure 8, for the most part, the multiple waves were still readily discernible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Observe that, as previously discussed, the right ear amplitude was lower than the left ear amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' This is also reflected in more distortion visible in the right ear channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Over the course of long-term measurements in real-world scenarios, such as driving, regular movement of the user will induce motion and EMG artifacts in the signal, while also likely compromising the skin-electrode contacts of the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' In turn, this will all increase the number of cardiac cycles required in order to obtain faithful ECG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Therefore, we have provided an analysis of the performance of the channels with varying data lengths over the course of a one hour driving trial on five subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Lead I Cross ear Left ear Right ear 500 0 500 130 ms 30 ms 3 ms 27 ms 253 ms 10 0 10 133 ms 33 ms 3 ms 27 ms 260 ms 2 0 2 123 ms 33 ms 3 ms 30 ms 250 ms 1 0 1 2 157 ms 33 ms 3 ms 27 ms 253 ms 500 0 500 107 ms 27 ms 3 ms 27 ms 210 ms 10 0 10 110 ms 27 ms 3 ms 27 ms 210 ms 2 0 2 107 ms 27 ms 3 ms 27 ms 227 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='5 107 ms 27 ms 7 ms 30 ms 193 ms 0 500 1000 140 ms 33 ms 3 ms 27 ms 250 ms 0 10 20 137 ms 33 ms 0 ms 27 ms 243 ms 1 0 1 2 3 137 ms 33 ms 0 ms 27 ms 260 ms 1 0 1 2 137 ms 33 ms 0 ms 27 ms 223 ms 200 100 0 100 137 ms 27 ms 3 ms 30 ms 270 ms 10 5 0 5 123 ms 17 ms 13 ms 37 ms 260 ms 1 0 1 -150 ms 20 ms 10 ms 40 ms 297 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='5 -127 ms 17 ms 17 ms 43 ms 273 ms 200 0 200 123 ms 40 ms 7 ms 33 ms 247 ms 5 0 5 193 ms 57 ms 17 ms 10 ms 233 ms 1 0 1 200 ms 50 ms 13 ms 17 ms 223 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='5 -157 ms 53 ms 23 ms 7 ms 227 ms S1 S2 S3 S4 S5 Potential [µV] (a) Potential [µV] (b) Potential [µV] (c) Potential [µV] (d) Potential [µV] (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4 Time [s] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' 8: Cardiac rhythms from wrist, left ear, right ear, and cross ear ECG of 5 subjects ((a) - (d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Reference lead I cardiac cycles are displayed along the first column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' ECG acquired after N = 240 cycles (blue) and N = 600 cycles (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' ECG potentials at timings of P-, Q-, R-, S-, and T-waves in each channel are circled for the N = 240 cycle ECG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Timings of the waves are indicated in ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Performance metrics were calculated on all cardiac cycles recorded during the one hour trial from each subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Moreover, the performance was evaluated for different levels of averaging (expressed as the number cardiac cycles that were averaged over).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Figure 5a) displays the Pearson correlation between the ear ECG cardiac rhythms and Lead I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Correlation results in [11] (r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='96 for 4 minutes of cross-ear ECG) from a different cohort of subjects (N = 6) are in good agreement with the results in this study (r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='93 for N = 240 cycles - or roughly 4 minutes of data) from five subjects, demonstrating that the performance of the cross-ear ECG can be robust to real-world measurement noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' For single ear ECG, the performance during real world measurements was also stable, with the left single-ear channels performing almost identically in both experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' For example, the correlation at N = 240 cycles in experiment A and B, respectively, for the left ear single ear channel were 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='85 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='86, while the respective timing errors were 10 ms and 7 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Relative to the cross-ear ECG, observe the lower performance of the single ear channel, particularly at smaller values of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' However, at higher values of N, the performance of the single ear channels were similar to the cross-ear channel, indicating that the single ear ECG is also a viable option for cardiac monitoring in real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The observation of a slightly reduced amplitude of the right ear signal relative to that in the left ear is further supported by the performance metrics in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' However, it is also likely that poorer electrode skin contact would have contributed to the lower performance of the right ear channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Table 1 provides performance metrics for the channels under consideration from both Experiment A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The values represent the performance of each channel at a value of N = 240, which was laso used t obtain the average cardiac ryhtms displayed in Figures 4 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The performance metrics that are probided in Table 1 represent the mean across all cardiac rhtyhms from within each recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Performance metrics for the left ear ECG channel were evaluated for data collected in both Experiment A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Note that the the performance of the left ear ECG channel (with respect to the timing error and amplitude ratio metrics) was worse in Experiment A relative to Experiment B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' This result might seem counter-intuitive, since the data from Experiment A was collected during ideal conditions (with the subject at rest), whereas the data from Experiment B was collected in noise-prone conditions (driving in a simulator environment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' However, this discrepancy was possibly caused by the different recording duration over which the performance metrics were evaluated in the two Experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' For example, at N = 240 cycles, in recordings from experiment A (10 min duration), a total of 400 cardiac rhythms were evaluated, whereas in recordings from experiment B (60 min duration), roughly 3400 cardiac rhythms were evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' 102 0 10 20 30 102 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='5 1 102 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='5 Pearson corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' coeff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' [r] 1 10 1 10 1 10 Amplitude ratio Wave timing [ms] Cardiac cycles (N) (a) (b) (c) Performance metrics Left ear Right ear Cross-ear r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='76 r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='94 r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='85 y=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4 y=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 y=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='4 5 ms 7 ms 9 ms N=240 N=240 N=240 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' 9: Performance metrics for left ear, right ear, and cross ear ECG after varying levels of averaging (N = cycles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' (a) correlation of the cardiac rhythms with the grand-median lead I cardiac rhythm, (b) RMS amplitude ratio between the R-peak and P-,Q-,S-, and T- peaks for a given channel, normalised by the values from lead I (c) RMSE of the timings of the P-,Q-,S-, and T- waves relative to the R-wave between a given channel and the lead I channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' A vertical line indicates the values at N = 240 cycles, displayed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' i) Correlation ii) Amplitude difference ii) Time difference ECG Channel meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' (ms) sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' (ms) iii) var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Wrist - rest 1 1 1 1 0 0 1 Neck - rest 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='5 1 3 1 1 cross ear - driving 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='99 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='1 5 2 3 left ear - rest 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='99 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='1 10 2 6 left ear - driving 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='1 7 2 6 right ear - driving 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='2 9 2 9 Scalp - rest 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='97 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content='5 20 3 9 Table 1: Mean performance for cardiac rhythms (N = 240 cycles) for scalp, cross ear, left ear, right ear, and neck ECG channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' For the left ear channel, data (i) correlation of the cardiac rhythms with the grand-median lead I cardiac rhythm, (ii) RMS amplitude ratio between the R-peak and P-,Q-,S-, and T- peaks for a given channel, normalise dby the values from lead I iii) RMSE of the timings of the P-,Q-,S-, and T- waves relative to the R-wave between a given channel and the lead I channel, and (iv) normalised variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' The column heading ‘meas.’ denotes results for measured cardiac rhythms, whereas ’sim.’ denotes results for the simulated rhythms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Values for left ear, right ear, and cross ear ECG were calculated with data from Experiment B, while values for scalp, and neck ECG were calculated wit data from Experiment A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' CONCLUSION The ability to monitor ECG will be a key feature in future wearable health systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' A primary position to record physiological signals from is the ear as a result of its stable location relative to the vital organs during everyday activities and its ability to house commonplace accessories such as earbuds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' However, the ECG signal that is available over the surface of a single ear had not yet been established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' First, the difference in ECG potential on the ear and in surrounding regions of the neck and scalp was investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' These results can support both existing and prospective wearable ECG platforms that utilise scalp, ear, and neck locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Measurements of single ear ECG on ten subjects, during an ideal scenario of resting while sitting helped to demonstrate, for the first time, the characteristic timing and shape of the ECG signal available at the single ear location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Further measurements, including both single ear and cross ear ECG, on five subjects during a one hour driving task demonstrated real-world feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Both the cross ear and single ear ECG were demonstrated to be robust to real world environments over prolonged recording periods, providing valuable evidence for the use of such technology in society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Future work will consider the integration of additional cardiac monitoring sensors into the ear worn platform, such as the PPG and BCG, and consider various cardiac conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE0T4oBgHgl3EQfkgHa/content/2301.02475v1.pdf'} +page_content=' Acknowledgment This work was supported by the Racing Foundation grant 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Wagner,1 L. Crippa,2 A. Amaricci,3 P. Hansmann,4 M. Klett,5 E. J. K¨onig,5 +T. Sch¨afer,5 D. Di Sante,6 J. Cano,7 A. J. Millis,8 A. Georges,9 and G. Sangiovanni2 +1Institut f¨ur Theoretische Physik und Astrophysik, +Universit¨at W¨urzburg, 97074 W¨urzburg, Germany +2Institut f¨ur Theoretische Physik und Astrophysik and W¨urzburg-Dresden Cluster of Excellence ct.qmat, +Universit¨at W¨urzburg, 97074 W¨urzburg, Germany +3CNR-IOM DEMOCRITOS, Istituto Officina dei Materiali, +Consiglio Nazionale delle Ricerche, Via Bonomea 265, 34136 Trieste, Italy +4Department of Physics, Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg, 91058, Erlangen, Germany +5Max-Planck-Institut f¨ur Festk¨orperforschung, Heisenbergstr. 1, 70569 Stuttgart, Germany +6University of Bologna, Bologna, Italy and CCQ-Flatiron Institute, New York, NY, USA +7Department of Physics and Astronomy, Stony Brook University, Stony Brook, +New York 11974, USA and CCQ-Flatiron Institute, New York, NY, USA +8Columbia University, New York, NY, USA and CCQ-Flatiron Institute, New York, NY, USA +9Coll`ege de France, Paris, France and CCQ-Flatiron Institute, New York, NY, USA +The topological classification of electronic band +structures is based on symmetry properties of +Bloch eigenstates of single-particle Hamiltonians. +In parallel, topological field theory has opened +the doors to the formulation and characteriza- +tion of non-trivial phases of matter driven by +strong electron-electron interaction. Even though +important examples of topological Mott insula- +tors have been constructed, the relevance of the +underlying non-interacting band topology to the +physics of the Mott phase has remained unex- +plored. Here, we show that the momentum struc- +ture of the Green’s function zeros defining the +“Luttinger surface” provides a precise topolog- +ical characterization of the Mott phase surpris- +ingly related to the one of the single-particle elec- +tronic dispersion. +Considerations on the zeros +lead to the prediction of new phenomena: a topo- +logical Mott insulator with an inverted gap for +the bulk zeros must possess gapless zeros at the +boundary, which behave as a form of “topolog- +ical antimatter” annihilating conventional edge +states. +Placing band and Mott topological in- +sulators in contact produces distinctive observ- +able signatures at the interface, revealing the oth- +erwise spectroscopically elusive Green’s function +zeros. The theoretical description of topological order in +physical systems has progressed along the parallel routes +of band- and quantum field-theory [1–6]. +The former, +based on single-particle Hamiltonians, offers a clear ex- +planation of the origin of topological invariants but is +limited to the realm of weakly-interacting perturbation +theory. The latter, making use of the Green’s function +formalism, encompasses a wider range of phases, such as +topological Mott insulators [7–9], at the cost of a higher +theoretical complexity and a greater computational ef- +fort. This more sophisticated approach is, however, nec- +essary since single-particle wave functions are no longer +eigenvectors of the interacting many-electron Hamilto- +nian and a full-fledged group theory-based analysis of all +possible two-body terms is far too impractical. +Mott +insulators +(MIs) +are +characterized +by +an +interaction-driven gap opening occurring without explicit +breaking of an underlying symmetry or long-range or- +dering. Phenomena of this kind, which are intrisically +non-perturbative in the coupling constant [10], result in +gaps even when globally robust crossings of bands such +as Weyl cones or topological boundary states [11–15] are +present. In these cases [16, 17] as well as in MIs arising +from conventional topological insulators (TIs) with an in- +verted gap [18, 19], a key question is if and how the topo- +logical information encoded in the non-interacting elec- +tronic dispersion survives and reemerges after the Mott +transition. +Topological gapless excitations can occur in the con- +text of MI quantum spin liquids concomitant with non- +trivial entanglement of the fractionalized ground state +[20] allowing to circumvent the necessity of a Lut- +tinger’s theorem-imposed Fermi sea volume [21, 22]. +Strongly correlated counterparts to bulk semimetals dis- +close a relation to lattice symmetries in the case of +non-symmorphic space groups [23–25]. Topological Lut- +tinger invariants have been formulated specifically for +such systems [26], broadening the conceptual perimeter +of the Luttinger surface in a MI [27]. Further, a discrete +symmetry-breaking has been proposed to be associated +to a Mott-like transition [28] considering an interaction +term diagonal in momentum space [29]. +Recently, the +existence of massless quasiparticles approaching a pseu- +dogapped Luttinger surface has been demonstrated [30]. +Here, we search for symmetry-protected gapless modes +despite the presence of a hard Mott gap and for experi- +mentally observable fingerprints thereof. We present an- +alytic as well as numerical evidence that the zeros of the +Green’s functions have a dispersion in momentum space +that can be topologically classified exactly as the corre- +sponding non-interacting single-particle band structure. +This observation provides us with a clean signature of +the topological nature of Mott insulating states: first +of all, MIs arising from symmetry-protected semimetals +arXiv:2301.05588v1 [cond-mat.str-el] 13 Jan 2023 + +2 +FIG. 1: Bulk and edge zeros in strongly correlated topological phases. a-b, MIs originating at large values +of U from a semimetal with protected crossings and, c-d, from a conventional non-interacting TI. In both situations +the non-interacting dispersion represents the energy-momentum location of the Green’s function poles (light blue +surfaces). In the corresponding Mott phases, we find dispersive bulk zeros (red surfaces) inside the gap between the +lower and upper Hubbard bands (dark blue). In the case of the semimetallic bandstructure, b, the bulk zeros dis- +play a symmetry-protected crossing. For the topological insulating bandstructure, d, the bulk zeros form instead an +inverted gap, inherited from the non-interacting topological band structure. Therefore, gapless edge zeros appear, +as indicated by the red dashed lines. Note the opposite spin direction associated to the edge zeros of the TMI w.r.t. +to that of the edge states of the TI. +must have gapless bulk zeros and are thereby topologi- +cally distinct. Second, the zeros of a MI originating from +a gapped bandstructure are also gapped. When this bulk +gap is of inverted character, exotic gapless zeros localized +at the boundaries must exist. We give proof of their exis- +tence and propose how to detect them in an experiment, +relying on the intrinsically incoherent state that forms +when a pole and a boundary zero annihilate. +Momentum dispersion of Green’s function zeros – +A pole of the single-particle propagator close to the real- +frequency axis describes a conventional quasiparticle ex- +citation of the system. A zero eigenvalue of G(k, ω), with +k and ω indicating crystal momentum and complex fre- +quency respectively, corresponds instead to a divergence +of the self-energy Σ(k, ω) and causes the opening of a +Mott gap [31–34]. For an isolated atom, the pole of Σ +is momentum-independent. Moving away from this ex- +treme limit, information on the lattice dispersion enters +into the picture. Deep in the Mott phase, we show that +the self-energy acquires a remarkably simple form: +Σ(k, ω) = +U 2/4 +ω + � +H0(k) +. +(1) +� +H0(k) indicates the non-interacting tight-binding Hamil- +tonian with renormalized parameters. In the Supplemen- +tary Information we extensively describe how to obtain +these effective parameters in a controlled way, by relat- +ing them to the spatially non-local spin-density correla- +tors ⟨niσnjσ′⟩ − ⟨niσ⟩⟨njσ′⟩. These are only found to be +finite beyond local mean-field theories [35], as discussed +previously for single-orbital models [36], in the pseudogap +phase [37], to prove the breakdown of Luttinger’s theo- +rem in a MI [27] as well as in the context of magnetically +ordered phases [38] and doped spin liquids [39]. +In this expression for the self-energy, it is important to +note the + sign in front of � +H0(k), which is opposite to +that in the denominator of Green’s functions. Eq. 1 holds +for single- as well as multi-orbital Hamiltonians and cor- +rectly predicts that on-site and k-dependent terms are +differently affected by the renormalization (see Supple- +mentary Information). The same form of the self-energy +is obtained in “failed” (quantum disordered) supercon- +ductors or spin density-wave systems, in which fermions +couple to a strongly fluctuating boson which prevents +long-range ordering. In the latter case, for instance, the ++ sign is a consequence of H0(k + Q) = −H0(k) ap- +pearing in the denominator [38]. This observation sug- +gests an intimate connection between Eq. 1 and paradig- +matic quantum spin liquids, such as the resonating va- +lence bond theory [40] (a failed superconductor), and re- +cent theories proposing topological order for the pseu- +dogap phase [41] by means of failed antiferromagnetism. +In the present work we explore the surprising implica- +tions of Eq. 1 for the topology of strongly correlated +electronic systems. The link between Green’s function +zeros and interacting topology has been pioneered by Gu- +rarie [42] and their role has been considered for the topo- +logical classification of various systems [43–45] A sim- +ple connection between the topological properties of the +zeros and the microscopic non-interacting Hamiltonian +is however lacking. +In Fig. 1 we illustrate two proto- +typical cases predicted by Eq. 1: panels (a)-(b) show +a symmetry-protected semimetal with bulk bands that +meet at some momentum. If the bandwidth is finite, such +system in three dimensions turns into a MI at a critical +interaction strength [16]. As long as the interaction does +not break any symmetry that protects the cones of the +non-interacting dispersion, Eq. 1 dictates that the zeros +of G will display a crossing: Analogously to the non- +interacting case, a gap in the zeros can only be opened + +b +C TOPOLOGICAL INSULATOR +d + TOPOLOGICAL MOTT INSULATOR +a + DIRAC/WEYL SEMIMETAL WITH + MOTT INSULATOR WITH +PROTECTED CROSSING OF EIGENVALUES +PROTECTED CROSSING OF ZEROS +WITH GAPLESS EDGE MODES +WITH GAPLESS EDGE ZEROS +Hubbard bands +Hubbard bands + Hubbard bands +bulk zeros +bulk poles +bulk poles +bulk zeros + gapless +gapless +edge poles +edge zeros + Hubbard bands +Hubard bands +kx +Ky +Ky +kx + Hubbard bands3 +FIG. 2: Poles and zeros in an infinite chain. Spec- +tral representation of the determinant of GR(k, ω) on +the real axis highlighting poles (dark blue) and zeros +(dark red). In a, this is shown for a weakly interacting +SSH+U with hopping parameters v = 0.2 and w = 0.5 +in units of t, i.e. on the topologically non-trivial side. +The numerical solution is obtained with cluster dynam- +ical mean-field theory (CDMFT), whose details can be +found in Methods. In the Supplementary Information +we show also QMC solutions of longer SSH+U chains. +b, In the strongly interacting limit, in addition to the +Hubbard bands at about ±3t, dispersive zeros (in red) +are visible inside the spectral gap. The dashed black +lines show a fit of analytical formula of Eq. 1 to the po- +sition of the zeros of the CDMFT Green’s function. +when two “zero-nodes” meet in momentum space, due +to the renormalization of the parameters. This finding +offers a particularly transparent explanation of the Mott +transition in a Dirac or Weyl semimetal [16, 17]: even +if a gap between poles of G opens, the protected linear +crossing is in fact not lifted: it just occurs between spec- +troscopically invisible zeros. +The second implication regards a TI with an inverted +bulk gap and boundary Green’s function poles, as shown +in Fig. 1c-d, turning into a topological Mott insulator +(TMI) at large interaction strengths. The bulk zeros of +G responsible for the opening of the Mott gap again obey +Eq. 1: depending on the renormalization of � +H0(k), the +zeros can acquire an inverted gapped dispersion. +The +sketch also illustrates that the predicted gapless zeros are +spatially localized at the boundaries. +Their dispersion +inside the bulk gap follows the renormalized one of the +edge modes in the corresponding non-interacting TI with +opposite sign. +Infinite and finite SSH+U chains – Firstly we +test the validity of Eq. 1 with the example of a one- +dimensional Su-Schrieffer–Heeger (SSH) model [46] with +periodic boundary conditions. We supplement it by a lo- +cal Hubbard U in order to induce the Mott phase. For +an infinite SSH+U chain at small values of U, most of +the spectral weight A(k, ω) resembles the non-interacting +eigenvalues |v + e−ik w| (blue bands in Fig. 2a). At large +interactions instead, a gap of order U is sustained by low- +energy divergencies of Σ(k, ω) (red dispersive features in +Fig. 2b). This result, which is in agreement with previous +literature [43, 47], allows us to compare the momentum +dispersion of the zeros of G with those predicted by Eq. 1, +shown by the black dashed lines in Fig. 2b. The renor- +malized dispersion of the zeros perfectly describes our +numerical result (see Methods), confirming that Eq. 1 +captures the essence of the momentum-dependent self- +energy of a MI. Since our focus is on boundary zeros, +we need to validate Eq. 1 also in the case of an open +SSH+U chain. With a finite number of sites, the mo- +mentum dependence in the denominator is replaced by +the real-space matrix representation of � +H0(k). At U = 0 +and for nonzero winding number of the SSH Hamilto- +nian, G possesses zero-energy poles at the two ends of the +chain. This is signaled by the two blue states on the left- +and right-most sites of the chain in Fig. 3a. For large +U, we solve the finite chain with exact diagonalization +(ED) and obtain zeros at the ends of the chain instead, +shown in red in Fig. 3b. Interestingly, the topological na- +ture of the gap of the zeros, analogously to what happens +for the poles of G, implies the existence of two “in-gap” +zeros at the boundary of the system. +In the Supple- +mentary Information we compare these ED results with +those acquired using Eq. 1, demonstrating that the ana- +lytic formula is also well suited to describe the non-local +many-body features of a MI for finite size systems. To +address the case in which an edge pole and an edge zero +get spatially close to one another we look at the system +shown in Fig. 3c. The two non-trivial SSH chains (U=0 +on the right and finite U on the right) are connected by +a hopping in the center, that can be switched on and +off via a parameter dubbed f. Fig. 3c shows the fate of +pole and zero at the two ends which meet at the center +of the new chain. They hybridize giving rise to a finite +spectral weight which appears at the interface (sites 6 +and 7). This is an incoherent “bad metal”, i.e. neither a +peak nor a vanishing spectral weight, resulting from the +pole/zero annihilation. The solution obtained in Fig. 3 is +fully compatible with symmetry and topological require- +ments of having two gapless modes at the two ends of +the chain, due to the interchangeable role of poles and +zeros. From a spectroscopic point of view, instead, it is +highly unexpected as the new chain, seen as a whole, has +in fact only one “detectable” edge state, at the left end. +Its partner at the right end, is the dual zero which of +course would not be visible in a tunneling experiment. +Two-dimensional topological Mott insulator – In +two dimensions we find that Eq. 1 continues to give an +excellent description of the dispersion of the zeros of G in +MIs. A detailed analysis including a comparison with nu- +merical quantum cluster methods for different real-space +geometries, can be found in the Supplementary Informa- +tion. Analogously to the one-dimensional case, we define +a TMI through the bulk-boundary correspondence in an +extended sense: if the zeros of G have a topologically in- +verted gap in the bulk, then gapless zeros have to appear +at the edge. In contrast, conventional MIs never display + +Det G(k,w) +U = 0.1 +U=6 +b +a +104 +103 +4 +4 +102 +2 +2 +101 +100 +30 +30 +10-1 +10-2 +-2 +-2 +10-3 +-4 +-4 +10-4 +10-5 +0 +0 +H- +-TT +Tt +k +K4 +FIG. 3: Zero/pole annihilation in an SSH+U finite chain. a, Exact solution for a 12-site non-interacting +SSH chain in the topologically non-trivial phase (v = 0.1 and w = 1.0). Zero-dimensional states are visible at +the two ends of the chain with some finite propagation inside the chain. We visualize this by showing site-resolved +eigenvector components, weighted with the corresponding eigenvalues (see Methods). b, Exact solution for a 12-site +SSH+U chain in the topological phase with the same v and w but a large value of the on-site interaction U, i.e. in +the Mott phase. The ends of the chain now host two zeros inside the Mott gap (of order 4 in units of t). The topo- +logical invariant of these two chains is the same but the interacting chain has no edge pole, rather only zeros that +are nicely visualized via the weights. c, Interface (i.e. f = 1) between the two situations above; the position of +the interface is indicated with a dashed, vertical line. In order to align the chemical potential inside the global bulk +gap, we subtract the Hartree-shift on the interacting chain. +gapless zeros, regardless whether one looks at their inte- +rior or at the boundary. +Compared to zero-dimensional poles/zeros of the +SSH+U, the gapless edge zeros – living inside the gap +of the bulk zeros of a TMI – acquire a dispersion w.r.t. +the momentum parallel to the edge. This implies that +the pole/zero annihilation gets even more intriguing than +that shown in Fig. 3c. In the following we therefore an- +alyze a two-dimensional heterostructure between a con- +ventional quantum spin Hall (QSH) system and a TMI, +as illustrated in Fig. 4b. The two parts have a segment of +the edge in common and we ask what the consequences +are on the helical modes when they travel in this region. +We also consider two additional cases: a “benchmark”, in +which the two sides are completely disconnected (f=0) +and one where the TMI is replaced by a trivial Mott +(Fig. 4a). We initialize a wave packet at the very left +of the QSH side and let it evolve in time towards the +interface with the TMI. The time evolution is governed +by the full Green’s function of this hybrid 2D interacting +system, where the interaction is included using the ana- +lytic formula of Eq.1. As shown in Fig. 4b as well as in +the movie in the Supplementary Information, the wave +packet is well defined only up to the interface with the +TMI (marked in red in the inset to Fig. 4b). As soon +as it enters this region, the edge state becomes immedi- +ately incoherent and loses spectral weight. In the other +two cases (f=0 and trivial MI) the propagation proceeds +undisturbed, as standard topological arguments predict +(Fig. 4a and movie in the Supplementary Information). +We have thus described an experimental probe sen- +sitive to the otherwise invisible zeros. Based on the dy- +namics of the wave packet and its clearcut distinct coher- +ence, the propagation represents a perfect way to detect +the presence of the boundary zeros on the TMI side. We +have checked that when the wave packet starts to lose +weight, there is no component of the QSH edge state +that tries to circumvent the TMI part. This is strikingly +different from what would instead happen if we were to +replace a portion of the QSH with a trivial system. In +that case the edge state of the QSH would go around the +trivial region. Here instead, the QSH loses the helical +state even though nothing has happened to its topologi- +cal properties. +Operatively, one can first quantify the renormalization +of parameters of a single-particle tight-binding model for +a given material, as outlined in the Supplementary Infor- +mation. Then, Eq. 1 can be used to predict the nature +of the corresponding MI. Importantly, the renormaliza- +tion of the parameters in � +H0(k) can in principle be in + +a non-interacting topological SSH model +topological SSH model in Mott phase +b +weighted eigenvector components +3 +3 +2 +2 +10-2 +10-1 +100 +101 +102 +1 +1 +3 +0 +3 +0 +U +0 +-1 +V +.W +U +-2 +-2 +-3 +-3 +1 +4 +8 +12 +1 +4 +8 +12 +site +site +c +3 +2 +0 +-1 +-2 +-3 +1 +4 +8 +12 +site5 +FIG. 4: Evolution of wave packet along the edge. a, A wave packet is initialized from the leftmost part of +the edge of a QSH (green part, see inset) and let evolve towards the interface with a trivial Mott insulator (see in- +set). Upon entering the blue interface region, no appreciable change of the wave packet is observed, except for some +small loss of its relative weight due to reflections and scattering from the walls of the two-dimensional structure. b, +The left-moving wave packet evolving along the edge now enters the red region where a topological Mott insulator +is placed on the other side. The edge zeros of the TMI hybridize with the edge mode of the QSH and determine the +immediate collapse of the wave packet. This sudden loss of weight can be used as a detection protocol for the pres- +ence of boundary zeros on the Mott insulating side. In both cases the time evolution is calculated via the Green’s +function of a 40×40 slab where the interaction is taken into account via the analytic self-energy in Eq.1. +both directions: the non-trivial region on the Mott side +can be smaller or larger than that of H0(k), i.e. +one +can generate a TMI from a trivial band structure or get +vice-versa a trivial Mott from a QSH, depending on the +specific model and type of interaction. +Should a Mott material be non-trivial according to +Eq. 1, one can then exploit the annihilation phenomenon +discussed in the last part of this work to unambiguously +tell whether or not zeros exist at its boundary. These +results open interesting perspectives both in connection +to the theory of topological order and in the characteri- +zation of protected bulk and surface features in the realm +of non-Hermitian physics with strong correlation. +I. +METHODS +Cluster-DMFT. The numerical results for the bulk +systems presented in this work are obtained within the +framework of cluster-DMFT, an extension of Dynami- +cal Mean-Field Theory capable of grasping nonlocal cor- +relations. The Exact Diagonalization results have been +obtained using a cluster extension of the EDIPack code +[48], where the SSH model is mapped to a finite “cluster +impurity”, consisting of two or three interacting dimers, +coupled to a finite bath. This is structured as a number of +non-interacting clusters replicating the one-particle hop- +ping matrix of the impurity, each site of which is cou- +pled to the corresponding impurity one. The intra-replica +hopping amplitudes and bath-impurity couplings are ad- +justed self-consistently. Two such replicas have been used +for the 2-dimers case, and 1 replica for the 3-dimer case. +The BHZ model is solved through an asymmetric clus- +ter impurity consisting of two sites along the x direction, +coupled to two bath replicas. Benchmark tests with a +3x1-sites cluster plus 1 bath replica and a 2×2 cluster +plus 1 replica have confirmed analogous results, the lat- +ter restoring the symmetry of the model though at the +cost of a dramatically increased computational time. +Single-shot calculation for finite chains. The fi- +nite size SSH effects have been obtained through a single- +shot exact diagonalization of an impurity cluster of 6 +dimers decoupled from any bath. For the quantum Monte +Carlo calculations a single-shot solution of the impurity +problem, again decoupled from any bath, has been ac- +quired via the use of a continuous time quantum Monte +Carlo solver based on the interaction expansion (CT- +INT)[49]. For every QMC calculation a statistic of 50 +million Monte Carlo cycles is used. +Determination of the finite chain zeros. In or- +der to spatially resolve the zeros of real-space Green’s +function (cfr. Fig. 3) we look at weighted eigenvector +components, i.e. the quantity +wi = +� +j +��ψ(j) +i Ej +��� +(2) +where ψ(j) +i +is the i-th element of the j-th eigenvector and +Ej is the j-th eigenvalue. +Slab wavepacket evolution. For the 2D slab calcu- +lations the interaction is taken into account using the +analytic formula for the self-energy (Eq. 1). The time +evolution of a wave packet at location r and time t is +given by +ψ(r, t) = +� +dr′G(r, r′, t)f(r0 − r′) +(3) +where f(r0 − r′) is a gaussian centered about r0. + +time evolution along edge +time evolution along edge +0.05 +0.05 +trivial +topological +Mott insulator +Mott insulator +topological +topological +insulator +insulator +20 +40 +20 +40 +0 +position along edge +position alongedge +40 +406 +ACKNOWLEDGMENTS +Acknowledgments – N.W. is supported by the SFB 1170 +Tocotronics, funded by the Deutsche Forschungsgemein- +schaft (DFG, German Research Foundation) – Project- +ID 258499086. L.C. acknowledges financial support from +the DFG through the W¨urzburg-Dresden Cluster of Ex- +cellence on Complexity and Topology in Quantum Mat- +ter–ct.qmat (EXC 2147, project-id 390858490). G.S. ac- +knowledges support from the DFG through FOR 5249- +449872909 (Project P05). +We gratefully acknowledge +the Gauss Center for Supercomputing e.V. (www.gauss- +center.eu) for funding this project by providing comput- +ing time on the GCS Supercomputer SuperMUC at Leib- +niz Supercomputing Center (www.lrz.de). +Part of the +numerical calculations were carried out using the Julia +programming language [50]. J.C. acknowledges support +from the Air Force Office of Scientific Research under +Grant No. +FA9550-20-1-0260. +A.J.M. was supported +in part by Programmable Quantum Materials, an En- +ergy Frontier Research Center funded by the U.S. De- +partment of Energy (DOE), Office of Science, Basic En- +ergy Sciences (BES), under Award No. DE-SC0019443. +The research leading to these results has received funding +from the European Union’s Horizon 2020 research and in- +novation programme under the Marie Sk�lodowska-Curie +Grant Agreement No. 897276. The Flatiron Institute is +a division of the Simons Foundation. +[1] B. Bradlyn, +L. Elcoro, +J. Cano, +M. G. Vergniory, +Z. Wang, C. Felser, M. I. Aroyo, and B. A. Bernevig, +Topological quantum chemistry, Nature 547, 298 (2017). +[2] A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Lud- +wig, Classification of topological insulators and supercon- +ductors in three spatial dimensions, Physical Review B +78, 195125 (2008). +[3] X.-L. Qi and S.-C. 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Kane, Topological insulators with inver- +sion symmetry, Physical Review B 76, 045302 (2007). +[56] T. Yoshida, S. Fujimoto, and N. Kawakami, Correlation +effects on a topological insulator at finite temperatures, +Physical Review B 85, 125113 (2012). +[57] Y. Ando, Topological Insulator Materials, Journal of the +Physical Society of Japan 82, 102001 (2013). + +8 +SUPPLEMENTARY MATERIAL: MOTT INSULATORS WITH BOUNDARY ZEROS +This Supplementary Material is structured as follows: The first section (I) describes the derivation of Eq.(1) of +the main text, i.e. Σ(k, ω) = +U 2/4 +ω+ � +H0(k). The second section (II) gives further details about the numerical calculations +for the 2D TI slab calculations. Additional results and comparisons between the analytic self-energy formula and +numerical calculations are shown in the third section (III). +I. +HIGH-FREQUENCY EXPANSION +A. +Derivation of the expansion +In order to derive the high-frequency expansion we start by expanding the Green’s function. Following references +[51–53], the Matsubara Green’s function Gηρ(iωn, k) (where η and ρ describe the orbital and spin degrees of freedom, +ωn is the Matsubara frequency and k is crystal momentum) can be expressed using the Fourier transform of G(τ, k): +Gηρ(iωn, k) = +� β +0 +eiωnτGηρ(τ, k). +(4) +Here τ is the imaginary time and β is the inverse temperature. This expression can be rewritten as +Gηρ(iωn, k) = +1 +iωn +� β +0 +(∂τeiωnτ)Gηρ(τ, k). +(5) +Now, integration by parts yields +Gηρ(iωn, k) = +1 +iωn +� +−Gηρ(β−, k) − Gηρ(0+, k) − +� β +0 +eiωnτ(∂τGηρ(τ, k)) +� +. +(6) +This procedure can be repeated for the second term leading to +Gηρ(iωn, k) = − 1 +iωn +� +Gηρ(β−, k) + Gηρ(0+, k) +� ++ +1 +(iωn)2 (∂τGηρ(β, 0) + ∂τGηρ(0, k)) ++ +1 +(iωn)3 +� β +0 +(∂τeiωnτ)(∂2 +τGηρ(τ, k)). +(7) +Carrying out this procedure repeatedly we arrive at +Gηρ(iωn, k) = +1 +iωn +∞ +� +j=0 +(−1)j+1 +� +∂j +τGηρ(β−, k) + ∂j +τGηρ(0+, k) +� +(iωn)j += +1 +iωn +∞ +� +j=0 +(−1)j+1Cηρ +j +(iωn)j +. +(8) +The derivatives of Gηρ(τ, k) can be computed using +Gηρ(τ, k) = − +� +cη(k, τ)c† +ρ(k, 0) +� += −Tr(e−βHeHτcη(k, 0)e−Hτc† +ρ(k, 0)) +Tr(e−βH) +. +(9) +We get +Cηρ +0 += (Gηρ(0+, k) + Gηρ(β−, k)) = −1 +(10) +Cηρ +1 += (∂τG(0+, k) + ∂τG(β−, k)) = − +�� +[H, cη], c† +ρ +�� +(11) +and in general +Cηρ +i += − +�� +� +�[H, . . . [H, cη(k)]] +� +�� +� +i commutators +, c† +ρ(k) +� +� +� +� +. +(12) + +9 +The Green’s function is given by +G(iωn, k) = (iωn − ε(k) − Σ(iωn, k))−1 +(13) +where, depending on the degrees of freedom, all quantities can be matrices. Inverting the previous formula yields the +self-energy +Σ(iωn, k) = iωn − ε(k) − +1 +G(iωn, k) = iωn − ε(k) − +iωn +1 − �∞ +j=1(−1)j +Cj +(iωn)j +. +(14) +In the limit of iωn → ∞ the last term can be seen as the sum of a geometric series. Therefore: +Σ(iωn, k) = iωn − ε(k) − (iωn) +� +�1 + +∞ +� +n=1 +(−1)n +Cn +(iωn)n + +� ∞ +� +n=1 +(−1)n +Cn +(iωn)n +�2 ++ . . . +� +� += −ε(k) − (iωn) +� +� +∞ +� +n=1 +(−1)n +Cn +(iωn)n + +� ∞ +� +n=1 +(−1)n +Cn +(iωn)n +�2 ++ . . . +� +� . +(15) +Expanding in orders of 1 +ω we arrive at the following expression for the self-energy: +Σ(iω, k) = −ε(k) + µ + C1(k) − C2(k) + C2 +1(k) +iω ++ C3(k) + C1(k)C2(k) + C2(k)C1(k) + C3 +1(k) +iω2 ++ O +�� 1 +iω +�3� +. +(16) +Note that Ci are in general matrices. We can identify the poles of the self-energy by comparing with the expansion +A +iω − B = A +iω + AB +iω2 + O +�� 1 +iω +�3� +. +(17) +Close to the atomic limit, B is small. This will be justified below but is also apparent when comparing with the atomic +limit self-energy Σ = U 2 +4iω. This means that higher order corrections are not only suppressed by powers of +1 +iω but also +by increasing powers of B. For the Hubbard dimer discussed in section I D we have checked that the expansions of +Eq. 16 and Eq.17 coincide up to fourth order in +1 +iω. +B. +Application to the Hubbard-model +In order to arrive at Eq.(1) of the main text we have to compute the commutators in Eq. (12) for the Hubbard +model. +We consider a multi-orbital case with local Hubbard repulsion of strength U: +H = +� +kabσ +(εabσσ′ +k +− µδabδσσ′)c† +kaσckbσ′ + U +2 +� +iaσ +niaσnia−σ +(18) +where εabσσ′ +k +is the dispersion as a function of momentum k with orbital indices a and b as well as spin degrees of +freedom σ and σ′. The chemical potential is given by µ, i indicates a lattice site and n is the number operator. We +call the on-site, k-independent terms of the dispersion εabσσ′ +on-siteδab. +Up to second order in 1 +ω (ω being either iωn or ω + iδ), equation (16) leads to +Σabσσ′(k, ω) = U +2 δabδσσ′ + U 2 +4 +1 +ω δabδσσ′ + U 2 +4 +χabσσ′(k) +ω2 +(19) + +10 +for the half-filled case. χabσσ′(k) is given by +χabσσ′(k) = −εabσσ′ +k ++ 2 +N +� +q +εabσσ′ +q +δabδσσ′ + 2 +N +� +q +εabσσ′ +q +δabδσ,−σ′ +− 4 +N 2 +� +qlj +εabσσ′ +q +ei(k−q)(rl−rj) � +c† +la−σc† +jb−σ′cla−σcjb−σ′ +� +− 4 +N 2 +� +qlj +εabσσ′ +q +ei(k−q)(rl−rj) � +c† +la−σc† +jbσ′claσcjb−σ′ +� +(20) +where N is a normalization factor. Corrections to this expression depend either on the double occupations or on the +expectation value of hopping terms. As shown in section I C these are much smaller then the expectation values in +χabσσ′(k). +For on-site terms (with a = b) the expectation values in the last two terms vanish and we arrive at +χabσσ′ +on-site = −εabσσ′ +on-siteδab + 2 +N +� +q +εabσσ′ +q +δabδσσ′ + 2 +N +� +q +εabσσ′ +q +δabδσ,−σ′ +(21) +which under the assumption that all diagonal q-dependent terms in εabσσ′ +q +sum to zero reduces to +χabσσ′ +on-site = εabσσ′ +on-siteδab. +(22) +This is consistent with what one would expect for on-site terms. Those can be exactly described within the atomic +limit, leading only to constant shifts of poles and zeros respectively. Going beyond the atomic limit we therefore +would not expect any renormalization. +For the remaining terms we have to evaluate the expectation values instead: +χabσσ′ +rest (k) = −εabσσ′ +k +− 4 +N 2 +� +qlj +εabσσ′ +q +ei(k−q)(rl−rj) � +c† +la−σc† +jb−σ′cla−σcjb−σ′ +� +− 4 +N 2 +� +qlj +εabσσ′ +q +ei(k−q)(rl−rj) � +c† +la−σc† +jbσ′claσcjb−σ′ +� +. +(23) +The prefactor in front of the expectation values in Eq. (23) restricts j and l to sites connected by a hopping in the +non-interacting dispersion. This can be seen by defining α = l − j and explicitly writing the dispersion as a sum over +all hopping directions β: +4 +N 2 +� +q,l,j +εabσσ′ +q +ei(k−q)(rl−rj)f abσσ′ +lj += 4 +N +� +q,α,β +tabσσ′ +β +eiqdβei(k−q)dαf abσσ′ +α += 4 +� +β +tabσσ′ +β +eikdβf abσσ′ +β +(24) +where f stands for either one of the expectation values in Eq. (23) and dα indicates the distance between two sites +with indices differing by α. Here it is assumed that dβ ̸= 0 or a ̸= b, otherwise the corresponding expectation values +vanish (as already discussed for the on-site terms). +Considering +the +restrictions +for +l +and +j, +the +first +expectation +value +can +be +rewritten +as +� +c† +la−σc† +jb−σ′cla−σcjb−σ′ +� += − ⟨nla−σnjb−σ′⟩. +Now we define the correlation function ∆abσσ′ +ljσ +≡ ∆abσσ′ +l−j +which +fulfills +⟨nl,a,−σnj,b,−σ′⟩ = ⟨nl,a,−σ⟩ ⟨nj,b,−σ′⟩ − ∆abσσ′ +ljσ += 1 +4 − ∆abσσ′ +ljσ +. +(25) +As shown in section I C the second expectation value can be expressed using the same correlation function: +� +c† +la−σc† +jbσ′claσcjb−σ′ +� += 2∆abσσ′ +ljσ +. +(26) +Inserting the results into Eq. (23) we get +χabσσ′ +rest (k) = −12 +� +β +tabσσ′ +β +eikdβ∆abσσ′ +β +(27) + +11 +which is just the renormalized non-interacting dispersion. As will be shown in the section below for certain limits of +the parameters the correlation function can be determined analytically, yielding +∆abσσ′ +β += 1 +4 +(tabσσ′)2β +U +(28) +which leads to a renormalization factor of 3 (tabσσ′)2β +U +, consistent with the the results of Pairault et al. [36] for a +single-orbital cosine-dispersion. +The full self-energy is given by +Σabσσ′(ω, k) = U +2 δab + +� +U 2/4 +ω − χrest(k) − χon-site +� +abσσ′ = U +2 δab + +� +U 2/4 +ω + � +H0(k) +� +abσσ′ +(29) +where � +H0(k) has been identified with −χrest(k) − χon-site. Note that χrest(k) has the opposite sign in respect to the +non-interacting Hamiltonian terms while χon-site has the same sign. +C. +Correlation Values +As discussed above the result of the high-frequency expansion depends crucially on the correlation value ⟨niσnjσ⟩. +In order to justify the above result for the analytic expression for the self-energy, in this section we provide numerical +and analytic evidence that ⟨niσnjσ⟩ does indeed behave as assumed in the previous section. +Expansion about the atomic limit +In +order +to +estimate +∆lj +we +use +an +expansion +of +the +thermal +expectation +value +� +ˆO +� += +1 +Z +� +n ⟨n| exp(−βH + βµN) ˆO |n⟩. +The chemical potential term commutes with H and we can ignore it. +After +rewriting the exponential function as a series, +exp(−βH) = +� +x +(−βH)x +x! += +� +x +(−β)x +x! +(HU + Ht)x, +(30) +we restrict ourselves to the lowest order terms in t: +(HU + Ht)x = Hx +U + +� +a+b=x−1 +Ha +UHtHb +U + +� +a+b+c=x−2 +Ha +UHtHb +UHtHc +U + O(H3 +t ). +(31) +The U and t subscripts correspond to the interaction and hopping parts of the Hamiltonian respectively. +For +ˆO = nlσnjσ the first order in t vanishes and we get +exp(−βH) = +� +x +(−βU)x +x! +� +hx +U + Θ(x − 2) t2 +U 2 +� +a+b+c=x−2 +ha +Uhthb +Uhthc +U + O(t4) +� +(32) +where HU = UhU and Ht = tht and the second term contributes only for x ≥ 2. We can now make use of eq. (32) in +the expression for the expectation value, with particular attention posed to the effect of the series of operators defined +by hU, ht and ˆO on the number of doubly-occupied sites in the system. In this sense, it can be shown that at large βU +the contribution of each state to the sum is suppressed in a way proportional to the number of double occupations. +In first approximation, therefore, we are allowed to restrict our further analysis to states with no double occupations. +The final step is to determine the contributing states in the trace, by counting how many hopping processes defined +by Eq. (32) are allowed for a given electronic configuration. The result depends on the relation between i and j: +⟨nlσnjσ⟩ = 1 +4 +1 + t2 +U 2 βU( z +2N − δ) +1 + t2 +U 2 βU z +2N +(33) +where δ is non-zero only if l and j are neighbors (or, more generally, if there is a hopping term connecting l and +j). The factor z +2N and the correction δ comes from the number of states with non-zero contribution in the sum + +12 +over all possible states. Here z is the coordination number and N is the number of sites. We evaluate this expression +for small t keeping in mind that the number of sites N is large. Eq. (33) can then be seen as the expansion of +1 +4 +� +1 + t2 +U 2 βU( z +2 − δ +N +) +1 + t2 +U 2 βU z +2 +�N +≈ 1 +4 +� +1 − t2 +U 2 βU δ +N +�N +≈ 1 +4 − 1 +4 +t2 +U βδ. +(34) +Thus we can infer that ∆lj = 1 +4 +t2 +U βδ. This result is restricted to t2β +U +≪ 1 and βU ≫ 1 due to the assumptions +needed in the expansion. For the expectation value of the spin-hopping term +� +c† +la−σc† +jbσ′claσcjb−σ′ +� +a similar analysis +can be done. For the second-order term the two hopping terms coming from the expansion in Eq. (32) have to +compensate the spin-hopping. This is only possible if l and j are connected by a hopping in the non-interacting +dispersion, yielding +� +c† +la−σc† +jbσ′claσcjb−σ′ +� += 2 1 +4 +t2 +U βδ where the factor of 2 is due to the possibility to interchange +the two hopping terms. +Numerical results +In order to check the analytic results we used exact diagonalization (ED) calculations for a one dimensional chain +with periodic boundary conditions. As shown in Figure 5 the dependence of the correlation value on t, β and U +predicted by the analytic calculations of the previous section is very well reproduced (for small t, small β and large U +respectively). Further the double occupation and the expectation value of a hopping term are much smaller, justifying +to neglect corrections including these expressions in the derivation for the self-energy. +10-5 +10-4 +10-3 +10-2 +10-1 +100 +101 + 1 + 10 +t +0.25 - +double occupation + +β/(4U) t2 +10-4 +10-3 +10-2 +10-1 +100 +101 + 10 + 100 + 1000 +β +0.25 - +double occupation + +t2/(4U) β +10-6 +10-5 +10-4 +10-3 +10-2 +10-1 +100 +101 + 10 + 100 + 1000 +U +0.25 - +double occupation + +t2β/(4U) +FIG. 5: ED results for different expectection values as a function of hopping t, inverse temperature β and interac- +tion U. For the leftmost plot β = 10 and U = 100; for the center plot t = 1 and U = 100; and for the right hand +plot t = 1 and β = 100. + +13 +D. +Application to the Hubbard dimer +Applying the analytic self-energy formula to a finite system means replacing the k-dependence with the full matrix +structure of the Hamiltonian H0 in real space: +Σ(ω) = +U 2/4 +ω + αH0 +. +(35) +Here, the renormalization is taken into account by the prefactor α. +For the Hubbard dimer the full Hamiltonian H is given by +H = −t +� +σ +� +c† +1σc2σ + c† +2σc1σ +� ++ U +� +i=1,2 +ni↑ni↓ − U +2 +� +i=1,2;σ +niσ. +(36) +In the following the high-frequency approach is tested against the exact solution for the Hubbard dimer, both by +applying Eq.(16) and by explicitly solving the high-frequency expansion for the dimer, i.e. solving Eq.(12) with the +full dimer Hamiltonian. +Furthermore it is possible to describe the interface of two SSH models using a dimer where U is restricted to one site. +Exact Solution +At zero temperature the exact self-energy is given by [54] +Σ11 = U 2 +4 +� +ω +ω2 − 9t2 +� +Σ12 = U 2 +4 +� +3t +ω2 − 9t2 +� +, +(37) +Expanding in orders of 1 +ω leads to +Σ11 = U 2 +4ω + 9U 2t2 +4ω3 ++ O +� 1 +ω5 +� +Σ12 = 3U 2t +4ω2 + 27U 2t3 +4ω4 ++ O +� 1 +ω6 +� +. +(38) +Analytic Formula +With +H0 = +� +0 +−t +−t +0 +� +(39) +and α = 3 (due to ⟨niσnjσ⟩ = 0 leading to ∆ij = 1 +4) Eq.(35) yields the same self-energy as the exact solution. +Explicit High-Frequency Expansion +For the high-frequency expansion expectation values have to be evaluated (Eq. 12). This can also be done by +explicitly carrying out the calculations for the dimer Hamiltonian. Then it is enough to know that the ground state +has the form +|ψ0⟩ = γ |↑↓⟩ − γ |↓↑⟩ + β |0 ↑↓⟩ + β |↑↓ 0⟩ +(40) +with γ2 + β2 = 1 +2. Using this assumptions, we can confirm that the first four orders in 1 +ω are equivalent to Eq. (38). + +14 +Hubbard dimer as interface +By restricting the Hubbard interaction to one site and changing the coupling between the sites we can model the +interface between two SSH chains (one interacting, one non-interacting) in the atomic limit. For the analytic formula +the interaction is restricted to one site by setting all other elements of the self-energy to zero after the inversion: +Σ(ω) = +� +U 2 +4 +1 +ω− α2t2 +ω +0 +0 +0 +� +. +(41) +As shown in Figure 6 the agreement with ED results is perfect. This simple example captures the zero/pole annihilation +upon coupling the two atoms. Furthermore the difference between the ”absence of weight” and a zero becomes clear: +the weight of annihilated zeros and poles is orders of magnitude larger than the zeros but still orders of magnitude +smaller than the poles. +10-4 +10-2 +100 +102 +104 +106 +108 +-1.5 +-1 +-0.5 + 0 + 0.5 + 1 + 1.5 +weighted eigenvector components +ω +site 1; exact +site 2; exact +site 1; analytic formula +site 2; analytic formula +t=0.00 +10-4 +10-2 +100 +102 +104 +106 +108 +-1.5 +-1 +-0.5 + 0 + 0.5 + 1 + 1.5 +weighted eigenvector components +ω +site 1; exact +site 2; exact +site 1, analytic formula +site 2; analytic formula +t=0.30 +FIG. 6: Uncoupled Hubbard dimer (left) and coupled Hubbard dimer (right) with U only on site 1. + +15 +II. +NUMERICAL CALCULATIONS +A. +Slab calculations using the analytic formula +FIG. 7: Slab geometry consisting of an interface between two topological insulators, one with U = 0, the other +in the Mott phase. Along the gray edge of the interface there is no coupling between the two sides, while a finite +coupling exists along the black edge. The blue and red lines indicate the edge states and zeros respectively and are +only shown for one spin species, the other having all directions inverted. At the interface the edge channels have +the same direction which is a consequence of the plus sign in the denominator of Eq. (1) in the main text. +For the slab calculations we use a geometry as depicted in Fig. 7. The right hand side is a non-interacting topological +insulator, whereas the the left hand side is either a trivial or topological Mott insulator. We compute the Green’s +function using +G = (ω1 − H0 − Σ(ω))−1 +(42) +where the non-interacting Hamiltonian is given by a BHZ model[55–57]: +H0 = +� +iασ +Mτ z +ααc† +iασciασ − +� +iαβµσ +tαβ +µσc† +i+µασciβσ +(43) +with +tµσ = +� +t +σ i +2λeiσθµ +σ i +2λe−iσθµ +−t +� +(44) +where µ runs over the the neighbors (±x,±y), θµ is the angle between the x-axis and direction of the neighbor and +σ is the spin. The self-energy is computed using the analytic formula (Eq. (29)). In this case we do not have a +k-dependence. Instead the full matrix H0 (with orbital, spin and spatial degrees of freedom) is taken into account. +The renormalization of the non-local terms is set to one for simplicity. Due to the stability of topological edge states, +the exact parameters of the system aren’t important as long as there is no topological phase transition. Therefore we +can essentially choose the parameter almost arbitrarily as long as we stay in the topological phase because our aim +is to see the effect of the interplay of edge poles and zeros. The M-parameters are chosen with opposite signs for the +two slab sides in order to ensure that the zeros and poles can annihilate. The restriction of U to one side of the slab +is implemented by setting the components of the self-energy corresponding to a non-interacting site to zero after the +inversion. +At the interface the coupling is set to zero on the gray part and to one on the black part. The wave packet is +initialized (restricted to one spin species) using a gaussian on the uncoupled egde (gray): +ψ(r, t) = 1 +N +� +dr′G(r, r′, t)e−((r0,x−r′ +x)2/s2 +x+(r0,y−r′ +y)2/s2 +y)/2. +(45) +Here r and r′ are the slab coordinates and r0 is the point where the packet is initialized with variance sx and sy in +x- and y-direction respectively. N is a normalization factor. + +coupled +topological +topological +Mott insulator +insulator +y +X16 +III. +ADDITIONAL RESULTS +A. +Orbital and Spin Character of the Zeros +Analogously to the non-interacting band structure the zeros also retain an orbital/spin structure which is a conse- +quence of the analytic self-energy formula Eq. (29). This is illustrated in Fig. 8, showing the inverted (standard) gap +in the zeros for a topological insulator (band insulator). Analogously, the zeros also have a spin character as shown in +Fig. 9. As a consequence of the plus sign in the denominator of Eq. 29, their spin character is interchanged between +non-interacting and Mott system. +FIG. 8: Orbital character of the zeros obtained using the analytic formula Eq. (29) for a BHZ-model in the topo- +logical (left) and trivial (right) phase for ky = 0, t = 0.5 and λ = 0.3 (see also Eq. (46) in section III D). + +M = -0.5 +1.5 +1 +0.5 +orbital character +0.5 +3 +0 +0 +-0.5 +-0.5 +-1 +-1.5 +.1 +0 +1 +2 +3 +4 +5 +6 +kxM= 1 +3 +2 +0.5 +orbital character +1 +3 +0 +0 +-1 +-0.5 +-2 +-3 +0 +1 +2 +3 +4 +5 +6 +KX17 +FIG. 9: BHZ model with finite size in y-direction and periodic boundary conditions in x-directions. The first row +shows the poles and zeros for non-interacting and Mott Green’s function. The interaction is included by using the +analytic formula Eq.(29). The lower rows show the spin-momentum locking of the edge poles and zeros respectively +for the two edges of the model. The second column shows data where M has the opposite sign compared to the +non-interacting model in order to have the edge states/zeros at the same momenta. + +U=0 +Mott phase +det G +1×10-40 +1×10-30 +1×10-20 +1×10-10 +1 +1×1010 +3 +3 +2 +2 +1 +1 +3 +30 +-1 +-1 +-2 +-2 +-3 +-3 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +5 +6 +kx +kx +spin character of poles +spin character of zeros +edge 1 +edge 1 +0.4 +0.4 +0.3 +0.3 +0.2 +0.2 +0.1 +0.1 +3 +0 +3 +0 +-0.1 +-0.1 +-0.2 +-0.2 +-0.3 +-0.3 +-0.4 +-0.4 +2.6 +2.8 +3 +3.2 +3.4 +3.6 +2.6 +2.8 +3 +3.2 +3.4 +3.6 +kx +kx +edge 2 +edge 2 +0.4 +0.4 +0.3 +0.3 +0.2 +0.2 +0.1 +0.1 +3 +0 +3 +0 +-0.1 +-0.1 +-0.2 +-0.2 +-0.3 +-0.3 +-0.4 +-0.4 +1 +2.6 +2.8 +3 +3.2 +23.43.6 +2.6 +2.8 +3 +3.2 +3.4 +3.6 +kx +kx +down +dn18 +B. +QMC vs ED +As a check of the ED calculations we compare the results for a finite SSH chain with quantum Monte Carlo (QMC) +(Figure 10). In order to avoid an analytic continuation of the QMC data, we use the ED results on the imaginary +axis. As the figure is showing we find a very good agreement between the two methods and also with the results +obtained using the analytic formula. Especially these results are showing that the analytic approach is viable (at +least qualitatively) for a wide range of temperatures, starting from the zero temperature limit where ED is capable to +describe the system, to larger temperatures, accessible using QMC. With QMC we are also able to investigate much +larger chains. The results are shown in Fig.11, confirming that our results are stable also for large systems. + 0 + 2 + 4 + 6 + 8 + 10 + 12 + 14 + 2 + 4 + 6 + 8 + 10 + 12 +a +b +c +ED +QMC +analytic +ωn +site + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 + 1.2 + 1.4 + 0 + 2 + 4 + 6 + 8 + 10 + 12 + 14 + 2 + 4 + 6 + 8 + 10 + 12 +a +b +c +ED +QMC +analytic +ωn +site + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 + 1.2 + 1.4 + 0 + 2 + 4 + 6 + 8 + 10 + 12 + 14 + 2 + 4 + 6 + 8 + 10 + 12 +a +b +c +ED +QMC +analytic +ωn +site + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 + 1.2 + 1.4 +FIG. 10: Comparison between ED (a), QMC (b) and the analytic self-energy formula (c) for a non-trivial finite SSH +chain on the Matsubara axis. The interaction is U = 4 and the hopping parameters are v = 0.1 and w = 1.0. +The QMC results are for β = 10, the ED calculations are at zero temperature (where the spacing of the Matsubara +frequencies is set to be equivalent to β = 1000). For the analytic formula the spacing of the Matsubara frequencies +is set β = 100 and the renormaliazion of the parameters to 3. +FIG. 11: QMC results for a SSH chain with 48 sites for U = 4. The inverse temperature is given by β = 5 and +β = 10 respectively. The SSH model is in the topological phase with parameters v = 0.1 and w = 1.0. + +β= 5 +3 +14 +2.5 +12 +10 +2 +3 +8 +1.5 +9 +1 +4 +0.5 +2 +0 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +siteβ= 10 +3 +14 +2.5 +12 +10 +2 +8 +3 +1.5 +9 +1 +4 +0.5 +2 +0 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +site19 +C. +Comparison of analytic formula and numerical results for an interface between SSH chains +Figure 12 shows that we find a very good agreement between ED and the analytic formula for the self-energy +(Eq. 1 in the main text) for the coupled and uncoupled interface of two SSH chains (one in the Mott phase, the +other non-interacting), that have been discussed in the main text in Figure 3. The renormalization parameter for the +analytic formula has been chosen as α = 3. +FIG. 12: Comparison between ED (top) and the analytic formula (Eq.(29)) (down) for uncoupled SSH chains (left) +and coupled SSH chains (right). The right chain in each plot is for U = 0 and the left chain for U = 4. + +ED; uncoupled +3 +102 +2 +101 +1 +100 +3 +0 +-1 +10-1 +-2 +10-2 +-3 +1 +4 +8 +12 +siteED; coupled +3 +102 +2 +101 +1 +100 +3 +0 +-1 +10-1 +-2 +10-2 +-3 +1 +4 +8 +12 +siteanalytic formula; uncoupled +3 +102 +2 +101 +1 +3 +0 +100 +-1 +-2 +10-1 +-3 +1 +4 +8 +12 +siteanalytic formula; coupleo +3 +102 +2 +101 +1 +3 +0 +100 +-1 +-2 +10-1 +-3 +1 +4 +8 +12 +site20 +D. +Comparison of analytic formula and numerical results for a bulk BHZ model +We tested the analytic formula for a bulk BHZ model given by +H0(k) = (M − 2t(cos(kx) + cos(ky))τz + λ sin(kx)τx + λ sin(ky)τy +(46) +where τ describes the orbital degree of freedom. In comparing the analytical dispersion with the CDMFT data, +we have to be careful in including the effects the asymmetric cluster has on the results: computational memory +constraints render a complete CDMFT simulation for a 4-site cluster (8 energy levels per spin, to be supplemented +by at least as many bath levels) difficult to perform to convergence. We can however grasp the correctness of our +analytical formula by considering a 2 × 1-site cluster, which breaks the C2-symmetry of the square lattice. In this +case, the self-energy will have nonlocal components in the x-direction, but not in the y direction, where the problem +is effectively single-site. Hence, we expect the effective dispersion of the zeros to be comparable to an equation of +type (46), once the terms depending on ky have been completely neglected: +H2×1 +0 +(k) = (M − 2t cos(kx))τz + λ sin(kx)τx. +(47) +This will provide a gapped dispersion for M = 0, and a gap closing at M = 1 around the high-symmetry point Γ and +around M = −1 around high-symmetry point X. +We fit the zeros of the CDMFT results with Eq. (47) using t and λ as fitting parameters, taking the signs due to the +renormalization (Eq.(29)) into account. The crystal field-splitting M is not renormalized in the analytic self-energy +formula and is therefore kept constant during the fit. As shown in Fig. 13 the two approaches are in very good +agreement. The renormalization of the parameters t and λ is about 0.5 which explains the shifting of the topological +transition from M = 1 in the non-interacting case to M ≈ 0.5. Note that for CDMFT we use a Kanamori-type +interaction instead of the only-U case, which is used in the derivation of the analytic formula. In the only-U case the +parameter λ is strongly suppressed, which is consistent with the high-frequency expansion, since the corresponding +correlation values ⟨ninj⟩ remain uncorrelated. This suggests that the analytic formula can also be applied to more +complicated interactions as long as the correct behavior of the correlation values is taken into account. +Finally, we address the claim, mentioned in the main text, that in absence of perturbations that provoke the opening +of gapless Dirac or Weyl points in the noninteracting Hamiltonian the zeros of the Green’s function of the interacting +system in the Mott phase remain gapless. We can verify this claim by making again use of the 2D BHZ model of +eq. (46), which in the M = 0 case has a gapless dispersion at the X and Y high-symmetry points. As previously +mentioned, a comprehensive analysis of the topological phase diagram for the BHZ model using a 2×2 cluster through +CDMFT is hampered by the size of the impurity problem. Nevertheless, a converged result for selected M values is +still within reach. In Fig.14 we show the results after 10 DMFT steps for a 2×2 cluster for M = 0 as well as M = 0.2. +As for the 2 × 1 cluster in the first part of this section we fit the zeros with the non-interacting dispersion using t and +λ as fitting parameters. The resulting renormalization of the parameters is again given by a factor of about 0.5. + +21 +FIG. 13: Position of the zeros of G of a modified BHZ model (using a 2 × 1 cluster, see text) obtained using +CDMFT (U = 8) and fits with the analytic formula for the self-energy. The dashed lines show the non-interacting +band dispersion. +FIG. 14: Position of the zeros for a BHZ model obtained using CDMFT (U = 8) with a 2×2 cluster and fit with +the analytic formula for the self-energy. Dashed lines show the non-interacting dispersion. + +ED +fit with analytic formula +Ho(k) - +M=0.0 +M=0.3 +0.8 +G +0.6 +zeros +0.4 +0.2 +0 +position of +-0.2 +-0.4 +-0.6 +-0.8 +M=0.5 +M=0.7 +1.5 +G +position of zeros of +0.5 +0 +-0.5 +.1 +-1.5 +0 +2TT +0 +2π +TT +TT +XM=0.0 +4 +ED +G +3 +fit with analytic formula +of +Ho(k) +position of zeros ( +2 +1 +0 +-2 +X +M +YM=0.2 +4 +ED +G +3 +fit with analytic formula +Ho(k) +- +position of zeros c +2 +0 +-2 +X +M +Y22 +[1] B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang, C. Felser, M. I. Aroyo, and B. A. Bernevig, Topological quantum +chemistry, Nature 547, 298 (2017). +[2] A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Stony Brook,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' New York 11974,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' USA and CCQ-Flatiron Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' New York,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' NY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' USA 8Columbia University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' New York,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' NY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' USA and CCQ-Flatiron Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' New York,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' NY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' USA 9Coll`ege de France,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' France and CCQ-Flatiron Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' New York,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' NY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' USA The topological classification of electronic band structures is based on symmetry properties of Bloch eigenstates of single-particle Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' In parallel, topological field theory has opened the doors to the formulation and characteriza- tion of non-trivial phases of matter driven by strong electron-electron interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Even though important examples of topological Mott insula- tors have been constructed, the relevance of the underlying non-interacting band topology to the physics of the Mott phase has remained unex- plored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Here, we show that the momentum struc- ture of the Green’s function zeros defining the “Luttinger surface” provides a precise topolog- ical characterization of the Mott phase surpris- ingly related to the one of the single-particle elec- tronic dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Considerations on the zeros lead to the prediction of new phenomena: a topo- logical Mott insulator with an inverted gap for the bulk zeros must possess gapless zeros at the boundary, which behave as a form of “topolog- ical antimatter” annihilating conventional edge states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Placing band and Mott topological in- sulators in contact produces distinctive observ- able signatures at the interface, revealing the oth- erwise spectroscopically elusive Green’s function zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The theoretical description of topological order in physical systems has progressed along the parallel routes of band- and quantum field-theory [1–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The former, based on single-particle Hamiltonians, offers a clear ex- planation of the origin of topological invariants but is limited to the realm of weakly-interacting perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The latter, making use of the Green’s function formalism, encompasses a wider range of phases, such as topological Mott insulators [7–9], at the cost of a higher theoretical complexity and a greater computational ef- fort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' This more sophisticated approach is, however, nec- essary since single-particle wave functions are no longer eigenvectors of the interacting many-electron Hamilto- nian and a full-fledged group theory-based analysis of all possible two-body terms is far too impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Mott insulators (MIs) are characterized by an interaction-driven gap opening occurring without explicit breaking of an underlying symmetry or long-range or- dering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Phenomena of this kind, which are intrisically non-perturbative in the coupling constant [10], result in gaps even when globally robust crossings of bands such as Weyl cones or topological boundary states [11–15] are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' In these cases [16, 17] as well as in MIs arising from conventional topological insulators (TIs) with an in- verted gap [18, 19], a key question is if and how the topo- logical information encoded in the non-interacting elec- tronic dispersion survives and reemerges after the Mott transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Topological gapless excitations can occur in the con- text of MI quantum spin liquids concomitant with non- trivial entanglement of the fractionalized ground state [20] allowing to circumvent the necessity of a Lut- tinger’s theorem-imposed Fermi sea volume [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Strongly correlated counterparts to bulk semimetals dis- close a relation to lattice symmetries in the case of non-symmorphic space groups [23–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Topological Lut- tinger invariants have been formulated specifically for such systems [26], broadening the conceptual perimeter of the Luttinger surface in a MI [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Further, a discrete symmetry-breaking has been proposed to be associated to a Mott-like transition [28] considering an interaction term diagonal in momentum space [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Recently, the existence of massless quasiparticles approaching a pseu- dogapped Luttinger surface has been demonstrated [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Here, we search for symmetry-protected gapless modes despite the presence of a hard Mott gap and for experi- mentally observable fingerprints thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' We present an- alytic as well as numerical evidence that the zeros of the Green’s functions have a dispersion in momentum space that can be topologically classified exactly as the corre- sponding non-interacting single-particle band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' This observation provides us with a clean signature of the topological nature of Mott insulating states: first of all, MIs arising from symmetry-protected semimetals arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='05588v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='str-el] 13 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 1: Bulk and edge zeros in strongly correlated topological phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' a-b, MIs originating at large values of U from a semimetal with protected crossings and, c-d, from a conventional non-interacting TI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' In both situations the non-interacting dispersion represents the energy-momentum location of the Green’s function poles (light blue surfaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' In the corresponding Mott phases, we find dispersive bulk zeros (red surfaces) inside the gap between the lower and upper Hubbard bands (dark blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' In the case of the semimetallic bandstructure, b, the bulk zeros dis- play a symmetry-protected crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' For the topological insulating bandstructure, d, the bulk zeros form instead an inverted gap, inherited from the non-interacting topological band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Therefore, gapless edge zeros appear, as indicated by the red dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Note the opposite spin direction associated to the edge zeros of the TMI w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' to that of the edge states of the TI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' must have gapless bulk zeros and are thereby topologi- cally distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Second, the zeros of a MI originating from a gapped bandstructure are also gapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' When this bulk gap is of inverted character, exotic gapless zeros localized at the boundaries must exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' We give proof of their exis- tence and propose how to detect them in an experiment, relying on the intrinsically incoherent state that forms when a pole and a boundary zero annihilate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Momentum dispersion of Green’s function zeros – A pole of the single-particle propagator close to the real- frequency axis describes a conventional quasiparticle ex- citation of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' A zero eigenvalue of G(k, ω), with k and ω indicating crystal momentum and complex fre- quency respectively, corresponds instead to a divergence of the self-energy Σ(k, ω) and causes the opening of a Mott gap [31–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' For an isolated atom, the pole of Σ is momentum-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Moving away from this ex- treme limit, information on the lattice dispersion enters into the picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Deep in the Mott phase, we show that the self-energy acquires a remarkably simple form: Σ(k, ω) = U 2/4 ω + � H0(k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (1) � H0(k) indicates the non-interacting tight-binding Hamil- tonian with renormalized parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' In the Supplemen- tary Information we extensively describe how to obtain these effective parameters in a controlled way, by relat- ing them to the spatially non-local spin-density correla- tors ⟨niσnjσ′⟩ − ⟨niσ⟩⟨njσ′⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' These are only found to be finite beyond local mean-field theories [35], as discussed previously for single-orbital models [36], in the pseudogap phase [37], to prove the breakdown of Luttinger’s theo- rem in a MI [27] as well as in the context of magnetically ordered phases [38] and doped spin liquids [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' In this expression for the self-energy, it is important to note the + sign in front of � H0(k), which is opposite to that in the denominator of Green’s functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 1 holds for single- as well as multi-orbital Hamiltonians and cor- rectly predicts that on-site and k-dependent terms are differently affected by the renormalization (see Supple- mentary Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The same form of the self-energy is obtained in “failed” (quantum disordered) supercon- ductors or spin density-wave systems, in which fermions couple to a strongly fluctuating boson which prevents long-range ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' In the latter case, for instance, the + sign is a consequence of H0(k + Q) = −H0(k) ap- pearing in the denominator [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' This observation sug- gests an intimate connection between Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 1 and paradig- matic quantum spin liquids, such as the resonating va- lence bond theory [40] (a failed superconductor), and re- cent theories proposing topological order for the pseu- dogap phase [41] by means of failed antiferromagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' In the present work we explore the surprising implica- tions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 1 for the topology of strongly correlated electronic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The link between Green’s function zeros and interacting topology has been pioneered by Gu- rarie [42] and their role has been considered for the topo- logical classification of various systems [43–45] A sim- ple connection between the topological properties of the zeros and the microscopic non-interacting Hamiltonian is however lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 1 we illustrate two proto- typical cases predicted by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 1: panels (a)-(b) show a symmetry-protected semimetal with bulk bands that meet at some momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' If the bandwidth is finite, such system in three dimensions turns into a MI at a critical interaction strength [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' As long as the interaction does not break any symmetry that protects the cones of the non-interacting dispersion, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 1 dictates that the zeros of G will display a crossing: Analogously to the non- interacting case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' a gap in the zeros can only be opened b C TOPOLOGICAL INSULATOR d TOPOLOGICAL MOTT INSULATOR a DIRAC/WEYL SEMIMETAL WITH MOTT INSULATOR WITH PROTECTED CROSSING OF EIGENVALUES PROTECTED CROSSING OF ZEROS WITH GAPLESS EDGE MODES WITH GAPLESS EDGE ZEROS Hubbard bands Hubbard bands Hubbard bands bulk zeros bulk poles bulk poles bulk zeros gapless gapless edge poles edge zeros Hubbard bands Hubard bands kx Ky Ky kx Hubbard bands3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 2: Poles and zeros in an infinite chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Spec- tral representation of the determinant of GR(k, ω) on the real axis highlighting poles (dark blue) and zeros (dark red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' In a, this is shown for a weakly interacting SSH+U with hopping parameters v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='2 and w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5 in units of t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' on the topologically non-trivial side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The numerical solution is obtained with cluster dynam- ical mean-field theory (CDMFT), whose details can be found in Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' In the Supplementary Information we show also QMC solutions of longer SSH+U chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' b, In the strongly interacting limit, in addition to the Hubbard bands at about ±3t, dispersive zeros (in red) are visible inside the spectral gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The dashed black lines show a fit of analytical formula of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 1 to the po- sition of the zeros of the CDMFT Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' when two “zero-nodes” meet in momentum space, due to the renormalization of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' This finding offers a particularly transparent explanation of the Mott transition in a Dirac or Weyl semimetal [16, 17]: even if a gap between poles of G opens, the protected linear crossing is in fact not lifted: it just occurs between spec- troscopically invisible zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The second implication regards a TI with an inverted bulk gap and boundary Green’s function poles, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 1c-d, turning into a topological Mott insulator (TMI) at large interaction strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The bulk zeros of G responsible for the opening of the Mott gap again obey Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 1: depending on the renormalization of � H0(k), the zeros can acquire an inverted gapped dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The sketch also illustrates that the predicted gapless zeros are spatially localized at the boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Their dispersion inside the bulk gap follows the renormalized one of the edge modes in the corresponding non-interacting TI with opposite sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Infinite and finite SSH+U chains – Firstly we test the validity of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 1 with the example of a one- dimensional Su-Schrieffer–Heeger (SSH) model [46] with periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' We supplement it by a lo- cal Hubbard U in order to induce the Mott phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' For an infinite SSH+U chain at small values of U, most of the spectral weight A(k, ω) resembles the non-interacting eigenvalues |v + e−ik w| (blue bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' At large interactions instead, a gap of order U is sustained by low- energy divergencies of Σ(k, ω) (red dispersive features in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' This result, which is in agreement with previous literature [43, 47], allows us to compare the momentum dispersion of the zeros of G with those predicted by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 1, shown by the black dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The renor- malized dispersion of the zeros perfectly describes our numerical result (see Methods), confirming that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 1 captures the essence of the momentum-dependent self- energy of a MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Since our focus is on boundary zeros, we need to validate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 1 also in the case of an open SSH+U chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' With a finite number of sites, the mo- mentum dependence in the denominator is replaced by the real-space matrix representation of � H0(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' At U = 0 and for nonzero winding number of the SSH Hamilto- nian, G possesses zero-energy poles at the two ends of the chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' This is signaled by the two blue states on the left- and right-most sites of the chain in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' For large U, we solve the finite chain with exact diagonalization (ED) and obtain zeros at the ends of the chain instead, shown in red in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Interestingly, the topological na- ture of the gap of the zeros, analogously to what happens for the poles of G, implies the existence of two “in-gap” zeros at the boundary of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' In the Supple- mentary Information we compare these ED results with those acquired using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 1, demonstrating that the ana- lytic formula is also well suited to describe the non-local many-body features of a MI for finite size systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' To address the case in which an edge pole and an edge zero get spatially close to one another we look at the system shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The two non-trivial SSH chains (U=0 on the right and finite U on the right) are connected by a hopping in the center, that can be switched on and off via a parameter dubbed f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 3c shows the fate of pole and zero at the two ends which meet at the center of the new chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' They hybridize giving rise to a finite spectral weight which appears at the interface (sites 6 and 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' This is an incoherent “bad metal”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' neither a peak nor a vanishing spectral weight, resulting from the pole/zero annihilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The solution obtained in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 3 is fully compatible with symmetry and topological require- ments of having two gapless modes at the two ends of the chain, due to the interchangeable role of poles and zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' From a spectroscopic point of view, instead, it is highly unexpected as the new chain, seen as a whole, has in fact only one “detectable” edge state, at the left end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Its partner at the right end, is the dual zero which of course would not be visible in a tunneling experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Two-dimensional topological Mott insulator – In two dimensions we find that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 1 continues to give an excellent description of the dispersion of the zeros of G in MIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' A detailed analysis including a comparison with nu- merical quantum cluster methods for different real-space geometries, can be found in the Supplementary Informa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Analogously to the one-dimensional case, we define a TMI through the bulk-boundary correspondence in an extended sense: if the zeros of G have a topologically in- verted gap in the bulk, then gapless zeros have to appear at the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' In contrast, conventional MIs never display Det G(k,w) U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='1 U=6 b a 104 103 4 4 102 2 2 101 100 30 30 10-1 10-2 2 2 10-3 4 4 10-4 10-5 0 0 H- TT Tt k K4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 3: Zero/pole annihilation in an SSH+U finite chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' a, Exact solution for a 12-site non-interacting SSH chain in the topologically non-trivial phase (v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='1 and w = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Zero-dimensional states are visible at the two ends of the chain with some finite propagation inside the chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' We visualize this by showing site-resolved eigenvector components, weighted with the corresponding eigenvalues (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' b, Exact solution for a 12-site SSH+U chain in the topological phase with the same v and w but a large value of the on-site interaction U, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' in the Mott phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The ends of the chain now host two zeros inside the Mott gap (of order 4 in units of t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The topo- logical invariant of these two chains is the same but the interacting chain has no edge pole, rather only zeros that are nicely visualized via the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' c, Interface (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' f = 1) between the two situations above;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' the position of the interface is indicated with a dashed, vertical line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' In order to align the chemical potential inside the global bulk gap, we subtract the Hartree-shift on the interacting chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' gapless zeros, regardless whether one looks at their inte- rior or at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Compared to zero-dimensional poles/zeros of the SSH+U, the gapless edge zeros – living inside the gap of the bulk zeros of a TMI – acquire a dispersion w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' the momentum parallel to the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' This implies that the pole/zero annihilation gets even more intriguing than that shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' In the following we therefore an- alyze a two-dimensional heterostructure between a con- ventional quantum spin Hall (QSH) system and a TMI, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The two parts have a segment of the edge in common and we ask what the consequences are on the helical modes when they travel in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' We also consider two additional cases: a “benchmark”, in which the two sides are completely disconnected (f=0) and one where the TMI is replaced by a trivial Mott (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' We initialize a wave packet at the very left of the QSH side and let it evolve in time towards the interface with the TMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The time evolution is governed by the full Green’s function of this hybrid 2D interacting system, where the interaction is included using the ana- lytic formula of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 4b as well as in the movie in the Supplementary Information, the wave packet is well defined only up to the interface with the TMI (marked in red in the inset to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' As soon as it enters this region, the edge state becomes immedi- ately incoherent and loses spectral weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' In the other two cases (f=0 and trivial MI) the propagation proceeds undisturbed, as standard topological arguments predict (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 4a and movie in the Supplementary Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' We have thus described an experimental probe sen- sitive to the otherwise invisible zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Based on the dy- namics of the wave packet and its clearcut distinct coher- ence, the propagation represents a perfect way to detect the presence of the boundary zeros on the TMI side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' We have checked that when the wave packet starts to lose weight, there is no component of the QSH edge state that tries to circumvent the TMI part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' This is strikingly different from what would instead happen if we were to replace a portion of the QSH with a trivial system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' In that case the edge state of the QSH would go around the trivial region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Here instead, the QSH loses the helical state even though nothing has happened to its topologi- cal properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Operatively, one can first quantify the renormalization of parameters of a single-particle tight-binding model for a given material, as outlined in the Supplementary Infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Then, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 1 can be used to predict the nature of the corresponding MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Importantly, the renormaliza- tion of the parameters in � H0(k) can in principle be in a non-interacting topological SSH model topological SSH model in Mott phase b weighted eigenvector components 3 3 2 2 10-2 10-1 100 101 102 1 1 3 0 3 0 U 0 1 V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='W U 2 2 3 3 1 4 8 12 1 4 8 12 site site c 3 2 0 1 2 3 1 4 8 12 site5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 4: Evolution of wave packet along the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' a, A wave packet is initialized from the leftmost part of the edge of a QSH (green part, see inset) and let evolve towards the interface with a trivial Mott insulator (see in- set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Upon entering the blue interface region, no appreciable change of the wave packet is observed, except for some small loss of its relative weight due to reflections and scattering from the walls of the two-dimensional structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' b, The left-moving wave packet evolving along the edge now enters the red region where a topological Mott insulator is placed on the other side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The edge zeros of the TMI hybridize with the edge mode of the QSH and determine the immediate collapse of the wave packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' This sudden loss of weight can be used as a detection protocol for the pres- ence of boundary zeros on the Mott insulating side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' In both cases the time evolution is calculated via the Green’s function of a 40×40 slab where the interaction is taken into account via the analytic self-energy in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' both directions: the non-trivial region on the Mott side can be smaller or larger than that of H0(k), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' one can generate a TMI from a trivial band structure or get vice-versa a trivial Mott from a QSH, depending on the specific model and type of interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Should a Mott material be non-trivial according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 1, one can then exploit the annihilation phenomenon discussed in the last part of this work to unambiguously tell whether or not zeros exist at its boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' These results open interesting perspectives both in connection to the theory of topological order and in the characteri- zation of protected bulk and surface features in the realm of non-Hermitian physics with strong correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' METHODS Cluster-DMFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The numerical results for the bulk systems presented in this work are obtained within the framework of cluster-DMFT, an extension of Dynami- cal Mean-Field Theory capable of grasping nonlocal cor- relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The Exact Diagonalization results have been obtained using a cluster extension of the EDIPack code [48], where the SSH model is mapped to a finite “cluster impurity”, consisting of two or three interacting dimers, coupled to a finite bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' This is structured as a number of non-interacting clusters replicating the one-particle hop- ping matrix of the impurity, each site of which is cou- pled to the corresponding impurity one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The intra-replica hopping amplitudes and bath-impurity couplings are ad- justed self-consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Two such replicas have been used for the 2-dimers case, and 1 replica for the 3-dimer case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The BHZ model is solved through an asymmetric clus- ter impurity consisting of two sites along the x direction, coupled to two bath replicas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Benchmark tests with a 3x1-sites cluster plus 1 bath replica and a 2×2 cluster plus 1 replica have confirmed analogous results, the lat- ter restoring the symmetry of the model though at the cost of a dramatically increased computational time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Single-shot calculation for finite chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The fi- nite size SSH effects have been obtained through a single- shot exact diagonalization of an impurity cluster of 6 dimers decoupled from any bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' For the quantum Monte Carlo calculations a single-shot solution of the impurity problem, again decoupled from any bath, has been ac- quired via the use of a continuous time quantum Monte Carlo solver based on the interaction expansion (CT- INT)[49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' For every QMC calculation a statistic of 50 million Monte Carlo cycles is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Determination of the finite chain zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' In or- der to spatially resolve the zeros of real-space Green’s function (cfr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 3) we look at weighted eigenvector components, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' the quantity wi = � j ��ψ(j) i Ej ��� (2) where ψ(j) i is the i-th element of the j-th eigenvector and Ej is the j-th eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Slab wavepacket evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' For the 2D slab calcu- lations the interaction is taken into account using the analytic formula for the self-energy (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The time evolution of a wave packet at location r and time t is given by ψ(r, t) = � dr′G(r, r′, t)f(r0 − r′) (3) where f(r0 − r′) is a gaussian centered about r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' time evolution along edge time evolution along edge 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='05 trivial topological Mott insulator Mott insulator topological topological insulator insulator 20 40 20 40 0 position along edge position alongedge 40 406 ACKNOWLEDGMENTS Acknowledgments – N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' is supported by the SFB 1170 Tocotronics, funded by the Deutsche Forschungsgemein- schaft (DFG, German Research Foundation) – Project- ID 258499086.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' acknowledges financial support from the DFG through the W¨urzburg-Dresden Cluster of Ex- cellence on Complexity and Topology in Quantum Mat- ter–ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='qmat (EXC 2147, project-id 390858490).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' ac- knowledges support from the DFG through FOR 5249- 449872909 (Project P05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' We gratefully acknowledge the Gauss Center for Supercomputing e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='gauss- center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='eu) for funding this project by providing comput- ing time on the GCS Supercomputer SuperMUC at Leib- niz Supercomputing Center (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='lrz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='de).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Part of the numerical calculations were carried out using the Julia programming language [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' acknowledges support from the Air Force Office of Scientific Research under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' FA9550-20-1-0260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' was supported in part by Programmable Quantum Materials, an En- ergy Frontier Research Center funded by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' De- partment of Energy (DOE), Office of Science, Basic En- ergy Sciences (BES), under Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' DE-SC0019443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The research leading to these results has received funding from the European Union’s Horizon 2020 research and in- novation programme under the Marie Sk�lodowska-Curie Grant Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 897276.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The Flatiron Institute is a division of the Simons Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 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S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Fujimoto, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Kawakami, Correlation effects on a topological insulator at finite temperatures, Physical Review B 85, 125113 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' [57] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Ando, Topological Insulator Materials, Journal of the Physical Society of Japan 82, 102001 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 8 SUPPLEMENTARY MATERIAL: MOTT INSULATORS WITH BOUNDARY ZEROS This Supplementary Material is structured as follows: The first section (I) describes the derivation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (1) of the main text, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Σ(k, ω) = U 2/4 ω+ � H0(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The second section (II) gives further details about the numerical calculations for the 2D TI slab calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Additional results and comparisons between the analytic self-energy formula and numerical calculations are shown in the third section (III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' HIGH-FREQUENCY EXPANSION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Derivation of the expansion In order to derive the high-frequency expansion we start by expanding the Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Following references [51–53], the Matsubara Green’s function Gηρ(iωn, k) (where η and ρ describe the orbital and spin degrees of freedom, ωn is the Matsubara frequency and k is crystal momentum) can be expressed using the Fourier transform of G(τ, k): Gηρ(iωn, k) = � β 0 eiωnτGηρ(τ, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (4) Here τ is the imaginary time and β is the inverse temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' This expression can be rewritten as Gηρ(iωn, k) = 1 iωn � β 0 (∂τeiωnτ)Gηρ(τ, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (5) Now, integration by parts yields Gηρ(iωn, k) = 1 iωn � −Gηρ(β−, k) − Gηρ(0+, k) − � β 0 eiωnτ(∂τGηρ(τ, k)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (6) This procedure can be repeated for the second term leading to Gηρ(iωn, k) = − 1 iωn � Gηρ(β−, k) + Gηρ(0+, k) � + 1 (iωn)2 (∂τGηρ(β, 0) + ∂τGηρ(0, k)) + 1 (iωn)3 � β 0 (∂τeiωnτ)(∂2 τGηρ(τ, k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (7) Carrying out this procedure repeatedly we arrive at Gηρ(iωn, k) = 1 iωn ∞ � j=0 (−1)j+1 � ∂j τGηρ(β−, k) + ∂j τGηρ(0+, k) � (iωn)j = 1 iωn ∞ � j=0 (−1)j+1Cηρ j (iωn)j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (8) The derivatives of Gηρ(τ, k) can be computed using Gηρ(τ, k) = − � cη(k, τ)c† ρ(k, 0) � = −Tr(e−βHeHτcη(k, 0)e−Hτc† ρ(k, 0)) Tr(e−βH) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (9) We get Cηρ 0 = (Gηρ(0+, k) + Gηρ(β−, k)) = −1 (10) Cηρ 1 = (∂τG(0+, k) + ∂τG(β−, k)) = − �� [H, cη], c† ρ �� (11) and in general Cηρ i = − �� � �[H, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' [H, cη(k)]] � �� � i commutators , c† ρ(k) � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (12) 9 The Green’s function is given by G(iωn, k) = (iωn − ε(k) − Σ(iωn, k))−1 (13) where, depending on the degrees of freedom, all quantities can be matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Inverting the previous formula yields the self-energy Σ(iωn, k) = iωn − ε(k) − 1 G(iωn, k) = iωn − ε(k) − iωn 1 − �∞ j=1(−1)j Cj (iωn)j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (14) In the limit of iωn → ∞ the last term can be seen as the sum of a geometric series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Therefore: Σ(iωn, k) = iωn − ε(k) − (iωn) � �1 + ∞ � n=1 (−1)n Cn (iωn)n + � ∞ � n=1 (−1)n Cn (iωn)n �2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' � � = −ε(k) − (iωn) � � ∞ � n=1 (−1)n Cn (iωn)n + � ∞ � n=1 (−1)n Cn (iωn)n �2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (15) Expanding in orders of 1 ω we arrive at the following expression for the self-energy: Σ(iω, k) = −ε(k) + µ + C1(k) − C2(k) + C2 1(k) iω + C3(k) + C1(k)C2(k) + C2(k)C1(k) + C3 1(k) iω2 + O �� 1 iω �3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (16) Note that Ci are in general matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' We can identify the poles of the self-energy by comparing with the expansion A iω − B = A iω + AB iω2 + O �� 1 iω �3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (17) Close to the atomic limit, B is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' This will be justified below but is also apparent when comparing with the atomic limit self-energy Σ = U 2 4iω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' This means that higher order corrections are not only suppressed by powers of 1 iω but also by increasing powers of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' For the Hubbard dimer discussed in section I D we have checked that the expansions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 16 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='17 coincide up to fourth order in 1 iω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Application to the Hubbard-model In order to arrive at Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (1) of the main text we have to compute the commutators in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (12) for the Hubbard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' We consider a multi-orbital case with local Hubbard repulsion of strength U: H = � kabσ (εabσσ′ k − µδabδσσ′)c† kaσckbσ′ + U 2 � iaσ niaσnia−σ (18) where εabσσ′ k is the dispersion as a function of momentum k with orbital indices a and b as well as spin degrees of freedom σ and σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The chemical potential is given by µ, i indicates a lattice site and n is the number operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' We call the on-site, k-independent terms of the dispersion εabσσ′ on-siteδab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Up to second order in 1 ω (ω being either iωn or ω + iδ), equation (16) leads to Σabσσ′(k, ω) = U 2 δabδσσ′ + U 2 4 1 ω δabδσσ′ + U 2 4 χabσσ′(k) ω2 (19) 10 for the half-filled case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' χabσσ′(k) is given by χabσσ′(k) = −εabσσ′ k + 2 N � q εabσσ′ q δabδσσ′ + 2 N � q εabσσ′ q δabδσ,−σ′ − 4 N 2 � qlj εabσσ′ q ei(k−q)(rl−rj) � c† la−σc† jb−σ′cla−σcjb−σ′ � − 4 N 2 � qlj εabσσ′ q ei(k−q)(rl−rj) � c† la−σc† jbσ′claσcjb−σ′ � (20) where N is a normalization factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Corrections to this expression depend either on the double occupations or on the expectation value of hopping terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' As shown in section I C these are much smaller then the expectation values in χabσσ′(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' For on-site terms (with a = b) the expectation values in the last two terms vanish and we arrive at χabσσ′ on-site = −εabσσ′ on-siteδab + 2 N � q εabσσ′ q δabδσσ′ + 2 N � q εabσσ′ q δabδσ,−σ′ (21) which under the assumption that all diagonal q-dependent terms in εabσσ′ q sum to zero reduces to χabσσ′ on-site = εabσσ′ on-siteδab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (22) This is consistent with what one would expect for on-site terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Those can be exactly described within the atomic limit, leading only to constant shifts of poles and zeros respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Going beyond the atomic limit we therefore would not expect any renormalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' For the remaining terms we have to evaluate the expectation values instead: χabσσ′ rest (k) = −εabσσ′ k − 4 N 2 � qlj εabσσ′ q ei(k−q)(rl−rj) � c† la−σc† jb−σ′cla−σcjb−σ′ � − 4 N 2 � qlj εabσσ′ q ei(k−q)(rl−rj) � c† la−σc† jbσ′claσcjb−σ′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (23) The prefactor in front of the expectation values in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (23) restricts j and l to sites connected by a hopping in the non-interacting dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' This can be seen by defining α = l − j and explicitly writing the dispersion as a sum over all hopping directions β: 4 N 2 � q,l,j εabσσ′ q ei(k−q)(rl−rj)f abσσ′ lj = 4 N � q,α,β tabσσ′ β eiqdβei(k−q)dαf abσσ′ α = 4 � β tabσσ′ β eikdβf abσσ′ β (24) where f stands for either one of the expectation values in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (23) and dα indicates the distance between two sites with indices differing by α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Here it is assumed that dβ ̸= 0 or a ̸= b, otherwise the corresponding expectation values vanish (as already discussed for the on-site terms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Considering the restrictions for l and j, the first expectation value can be rewritten as � c† la−σc† jb−σ′cla−σcjb−σ′ � = − ⟨nla−σnjb−σ′⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Now we define the correlation function ∆abσσ′ ljσ ≡ ∆abσσ′ l−j which fulfills ⟨nl,a,−σnj,b,−σ′⟩ = ⟨nl,a,−σ⟩ ⟨nj,b,−σ′⟩ − ∆abσσ′ ljσ = 1 4 − ∆abσσ′ ljσ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (25) As shown in section I C the second expectation value can be expressed using the same correlation function: � c† la−σc† jbσ′claσcjb−σ′ � = 2∆abσσ′ ljσ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (26) Inserting the results into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (23) we get χabσσ′ rest (k) = −12 � β tabσσ′ β eikdβ∆abσσ′ β (27) 11 which is just the renormalized non-interacting dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' As will be shown in the section below for certain limits of the parameters the correlation function can be determined analytically, yielding ∆abσσ′ β = 1 4 (tabσσ′)2β U (28) which leads to a renormalization factor of 3 (tabσσ′)2β U , consistent with the the results of Pairault et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' [36] for a single-orbital cosine-dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The full self-energy is given by Σabσσ′(ω, k) = U 2 δab + � U 2/4 ω − χrest(k) − χon-site � abσσ′ = U 2 δab + � U 2/4 ω + � H0(k) � abσσ′ (29) where � H0(k) has been identified with −χrest(k) − χon-site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Note that χrest(k) has the opposite sign in respect to the non-interacting Hamiltonian terms while χon-site has the same sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Correlation Values As discussed above the result of the high-frequency expansion depends crucially on the correlation value ⟨niσnjσ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' In order to justify the above result for the analytic expression for the self-energy, in this section we provide numerical and analytic evidence that ⟨niσnjσ⟩ does indeed behave as assumed in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Expansion about the atomic limit In order to estimate ∆lj we use an expansion of the thermal expectation value � ˆO � = 1 Z � n ⟨n| exp(−βH + βµN) ˆO |n⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The chemical potential term commutes with H and we can ignore it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' After rewriting the exponential function as a series, exp(−βH) = � x (−βH)x x!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' = � x (−β)x x!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (HU + Ht)x, (30) we restrict ourselves to the lowest order terms in t: (HU + Ht)x = Hx U + � a+b=x−1 Ha UHtHb U + � a+b+c=x−2 Ha UHtHb UHtHc U + O(H3 t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (31) The U and t subscripts correspond to the interaction and hopping parts of the Hamiltonian respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' For ˆO = nlσnjσ the first order in t vanishes and we get exp(−βH) = � x (−βU)x x!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' � hx U + Θ(x − 2) t2 U 2 � a+b+c=x−2 ha Uhthb Uhthc U + O(t4) � (32) where HU = UhU and Ht = tht and the second term contributes only for x ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' We can now make use of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (32) in the expression for the expectation value, with particular attention posed to the effect of the series of operators defined by hU, ht and ˆO on the number of doubly-occupied sites in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' In this sense, it can be shown that at large βU the contribution of each state to the sum is suppressed in a way proportional to the number of double occupations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' In first approximation, therefore, we are allowed to restrict our further analysis to states with no double occupations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The final step is to determine the contributing states in the trace, by counting how many hopping processes defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (32) are allowed for a given electronic configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The result depends on the relation between i and j: ⟨nlσnjσ⟩ = 1 4 1 + t2 U 2 βU( z 2N − δ) 1 + t2 U 2 βU z 2N (33) where δ is non-zero only if l and j are neighbors (or, more generally, if there is a hopping term connecting l and j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The factor z 2N and the correction δ comes from the number of states with non-zero contribution in the sum 12 over all possible states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Here z is the coordination number and N is the number of sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' We evaluate this expression for small t keeping in mind that the number of sites N is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (33) can then be seen as the expansion of 1 4 � 1 + t2 U 2 βU( z 2 − δ N ) 1 + t2 U 2 βU z 2 �N ≈ 1 4 � 1 − t2 U 2 βU δ N �N ≈ 1 4 − 1 4 t2 U βδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (34) Thus we can infer that ∆lj = 1 4 t2 U βδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' This result is restricted to t2β U ≪ 1 and βU ≫ 1 due to the assumptions needed in the expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' For the expectation value of the spin-hopping term � c† la−σc† jbσ′claσcjb−σ′ � a similar analysis can be done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' For the second-order term the two hopping terms coming from the expansion in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (32) have to compensate the spin-hopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' This is only possible if l and j are connected by a hopping in the non-interacting dispersion, yielding � c† la−σc† jbσ′claσcjb−σ′ � = 2 1 4 t2 U βδ where the factor of 2 is due to the possibility to interchange the two hopping terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Numerical results In order to check the analytic results we used exact diagonalization (ED) calculations for a one dimensional chain with periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' As shown in Figure 5 the dependence of the correlation value on t, β and U predicted by the analytic calculations of the previous section is very well reproduced (for small t, small β and large U respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Further the double occupation and the expectation value of a hopping term are much smaller, justifying to neglect corrections including these expressions in the derivation for the self-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 10-5 10-4 10-3 10-2 10-1 100 101 1 10 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='25 - double occupation β/(4U) t2 10-4 10-3 10-2 10-1 100 101 10 100 1000 β 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='25 - double occupation t2/(4U) β 10-6 10-5 10-4 10-3 10-2 10-1 100 101 10 100 1000 U 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='25 - double occupation t2β/(4U) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 5: ED results for different expectection values as a function of hopping t, inverse temperature β and interac- tion U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' For the leftmost plot β = 10 and U = 100;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' for the center plot t = 1 and U = 100;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' and for the right hand plot t = 1 and β = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 13 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Application to the Hubbard dimer Applying the analytic self-energy formula to a finite system means replacing the k-dependence with the full matrix structure of the Hamiltonian H0 in real space: Σ(ω) = U 2/4 ω + αH0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (35) Here, the renormalization is taken into account by the prefactor α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' For the Hubbard dimer the full Hamiltonian H is given by H = −t � σ � c† 1σc2σ + c† 2σc1σ � + U � i=1,2 ni↑ni↓ − U 2 � i=1,2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='σ niσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (36) In the following the high-frequency approach is tested against the exact solution for the Hubbard dimer, both by applying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (16) and by explicitly solving the high-frequency expansion for the dimer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (12) with the full dimer Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Furthermore it is possible to describe the interface of two SSH models using a dimer where U is restricted to one site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Exact Solution At zero temperature the exact self-energy is given by [54] Σ11 = U 2 4 � ω ω2 − 9t2 � Σ12 = U 2 4 � 3t ω2 − 9t2 � , (37) Expanding in orders of 1 ω leads to Σ11 = U 2 4ω + 9U 2t2 4ω3 + O � 1 ω5 � Σ12 = 3U 2t 4ω2 + 27U 2t3 4ω4 + O � 1 ω6 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (38) Analytic Formula With H0 = � 0 −t −t 0 � (39) and α = 3 (due to ⟨niσnjσ⟩ = 0 leading to ∆ij = 1 4) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (35) yields the same self-energy as the exact solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Explicit High-Frequency Expansion For the high-frequency expansion expectation values have to be evaluated (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' This can also be done by explicitly carrying out the calculations for the dimer Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Then it is enough to know that the ground state has the form |ψ0⟩ = γ |↑↓⟩ − γ |↓↑⟩ + β |0 ↑↓⟩ + β |↑↓ 0⟩ (40) with γ2 + β2 = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Using this assumptions, we can confirm that the first four orders in 1 ω are equivalent to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 14 Hubbard dimer as interface By restricting the Hubbard interaction to one site and changing the coupling between the sites we can model the interface between two SSH chains (one interacting, one non-interacting) in the atomic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' For the analytic formula the interaction is restricted to one site by setting all other elements of the self-energy to zero after the inversion: Σ(ω) = � U 2 4 1 ω− α2t2 ω 0 0 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (41) As shown in Figure 6 the agreement with ED results is perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' This simple example captures the zero/pole annihilation upon coupling the two atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Furthermore the difference between the ”absence of weight” and a zero becomes clear: the weight of annihilated zeros and poles is orders of magnitude larger than the zeros but still orders of magnitude smaller than the poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 10-4 10-2 100 102 104 106 108 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5 weighted eigenvector components ω site 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' exact site 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' exact site 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' analytic formula site 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' analytic formula t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='00 10-4 10-2 100 102 104 106 108 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5 weighted eigenvector components ω site 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' exact site 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' exact site 1, analytic formula site 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' analytic formula t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='30 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 6: Uncoupled Hubbard dimer (left) and coupled Hubbard dimer (right) with U only on site 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 15 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' NUMERICAL CALCULATIONS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Slab calculations using the analytic formula FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 7: Slab geometry consisting of an interface between two topological insulators, one with U = 0, the other in the Mott phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Along the gray edge of the interface there is no coupling between the two sides, while a finite coupling exists along the black edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The blue and red lines indicate the edge states and zeros respectively and are only shown for one spin species, the other having all directions inverted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' At the interface the edge channels have the same direction which is a consequence of the plus sign in the denominator of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (1) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' For the slab calculations we use a geometry as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The right hand side is a non-interacting topological insulator, whereas the the left hand side is either a trivial or topological Mott insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' We compute the Green’s function using G = (ω1 − H0 − Σ(ω))−1 (42) where the non-interacting Hamiltonian is given by a BHZ model[55–57]: H0 = � iασ Mτ z ααc† iασciασ − � iαβµσ tαβ µσc† i+µασciβσ (43) with tµσ = � t σ i 2λeiσθµ σ i 2λe−iσθµ −t � (44) where µ runs over the the neighbors (±x,±y), θµ is the angle between the x-axis and direction of the neighbor and σ is the spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The self-energy is computed using the analytic formula (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (29)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' In this case we do not have a k-dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Instead the full matrix H0 (with orbital, spin and spatial degrees of freedom) is taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The renormalization of the non-local terms is set to one for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Due to the stability of topological edge states, the exact parameters of the system aren’t important as long as there is no topological phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Therefore we can essentially choose the parameter almost arbitrarily as long as we stay in the topological phase because our aim is to see the effect of the interplay of edge poles and zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The M-parameters are chosen with opposite signs for the two slab sides in order to ensure that the zeros and poles can annihilate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The restriction of U to one side of the slab is implemented by setting the components of the self-energy corresponding to a non-interacting site to zero after the inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' At the interface the coupling is set to zero on the gray part and to one on the black part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The wave packet is initialized (restricted to one spin species) using a gaussian on the uncoupled egde (gray): ψ(r, t) = 1 N � dr′G(r, r′, t)e−((r0,x−r′ x)2/s2 x+(r0,y−r′ y)2/s2 y)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (45) Here r and r′ are the slab coordinates and r0 is the point where the packet is initialized with variance sx and sy in x- and y-direction respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' N is a normalization factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' coupled topological topological Mott insulator insulator y X16 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' ADDITIONAL RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Orbital and Spin Character of the Zeros Analogously to the non-interacting band structure the zeros also retain an orbital/spin structure which is a conse- quence of the analytic self-energy formula Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' This is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 8, showing the inverted (standard) gap in the zeros for a topological insulator (band insulator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Analogously, the zeros also have a spin character as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' As a consequence of the plus sign in the denominator of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 29, their spin character is interchanged between non-interacting and Mott system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 8: Orbital character of the zeros obtained using the analytic formula Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (29) for a BHZ-model in the topo- logical (left) and trivial (right) phase for ky = 0, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5 and λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='3 (see also Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (46) in section III D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' M = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5 orbital character 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5 3 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='1 0 1 2 3 4 5 6 kxM= 1 3 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5 orbital character 1 3 0 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5 2 3 0 1 2 3 4 5 6 KX17 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 9: BHZ model with finite size in y-direction and periodic boundary conditions in x-directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The first row shows the poles and zeros for non-interacting and Mott Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The interaction is included by using the analytic formula Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='(29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The lower rows show the spin-momentum locking of the edge poles and zeros respectively for the two edges of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The second column shows data where M has the opposite sign compared to the non-interacting model in order to have the edge states/zeros at the same momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' U=0 Mott phase det G 1×10-40 1×10-30 1×10-20 1×10-10 1 1×1010 3 3 2 2 1 1 3 30 1 1 2 2 3 3 0 1 2 3 4 5 6 0 1 2 3 4 5 6 kx kx spin character of poles spin character of zeros edge 1 edge 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='1 3 0 3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='2 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='8 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='6 kx kx edge 2 edge 2 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='4 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='8 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='2 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='8 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='6 kx kx down dn18 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' QMC vs ED As a check of the ED calculations we compare the results for a finite SSH chain with quantum Monte Carlo (QMC) (Figure 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' In order to avoid an analytic continuation of the QMC data, we use the ED results on the imaginary axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' As the figure is showing we find a very good agreement between the two methods and also with the results obtained using the analytic formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Especially these results are showing that the analytic approach is viable (at least qualitatively) for a wide range of temperatures, starting from the zero temperature limit where ED is capable to describe the system, to larger temperatures, accessible using QMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' With QMC we are also able to investigate much larger chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='11, confirming that our results are stable also for large systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 0 2 4 6 8 10 12 14 2 4 6 8 10 12 a b c ED QMC analytic ωn site 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='4 0 2 4 6 8 10 12 14 2 4 6 8 10 12 a b c ED QMC analytic ωn site 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='4 0 2 4 6 8 10 12 14 2 4 6 8 10 12 a b c ED QMC analytic ωn site 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 10: Comparison between ED (a), QMC (b) and the analytic self-energy formula (c) for a non-trivial finite SSH chain on the Matsubara axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The interaction is U = 4 and the hopping parameters are v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='1 and w = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The QMC results are for β = 10, the ED calculations are at zero temperature (where the spacing of the Matsubara frequencies is set to be equivalent to β = 1000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' For the analytic formula the spacing of the Matsubara frequencies is set β = 100 and the renormaliazion of the parameters to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 11: QMC results for a SSH chain with 48 sites for U = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The inverse temperature is given by β = 5 and β = 10 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The SSH model is in the topological phase with parameters v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='1 and w = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' β= 5 3 14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5 12 10 2 3 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5 9 1 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5 2 0 0 5 10 15 20 25 30 35 40 45 siteβ= 10 3 14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5 12 10 2 8 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5 9 1 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5 2 0 0 5 10 15 20 25 30 35 40 45 site19 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Comparison of analytic formula and numerical results for an interface between SSH chains Figure 12 shows that we find a very good agreement between ED and the analytic formula for the self-energy (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 1 in the main text) for the coupled and uncoupled interface of two SSH chains (one in the Mott phase, the other non-interacting), that have been discussed in the main text in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The renormalization parameter for the analytic formula has been chosen as α = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 12: Comparison between ED (top) and the analytic formula (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (29)) (down) for uncoupled SSH chains (left) and coupled SSH chains (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The right chain in each plot is for U = 0 and the left chain for U = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' ED;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' uncoupled 3 102 2 101 1 100 3 0 1 10-1 2 10-2 3 1 4 8 12 siteED;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' coupled 3 102 2 101 1 100 3 0 1 10-1 2 10-2 3 1 4 8 12 siteanalytic formula;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' uncoupled 3 102 2 101 1 3 0 100 1 2 10-1 3 1 4 8 12 siteanalytic formula;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' coupleo 3 102 2 101 1 3 0 100 1 2 10-1 3 1 4 8 12 site20 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Comparison of analytic formula and numerical results for a bulk BHZ model We tested the analytic formula for a bulk BHZ model given by H0(k) = (M − 2t(cos(kx) + cos(ky))τz + λ sin(kx)τx + λ sin(ky)τy (46) where τ describes the orbital degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' In comparing the analytical dispersion with the CDMFT data, we have to be careful in including the effects the asymmetric cluster has on the results: computational memory constraints render a complete CDMFT simulation for a 4-site cluster (8 energy levels per spin, to be supplemented by at least as many bath levels) difficult to perform to convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' We can however grasp the correctness of our analytical formula by considering a 2 × 1-site cluster, which breaks the C2-symmetry of the square lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' In this case, the self-energy will have nonlocal components in the x-direction, but not in the y direction, where the problem is effectively single-site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Hence, we expect the effective dispersion of the zeros to be comparable to an equation of type (46), once the terms depending on ky have been completely neglected: H2×1 0 (k) = (M − 2t cos(kx))τz + λ sin(kx)τx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (47) This will provide a gapped dispersion for M = 0, and a gap closing at M = 1 around the high-symmetry point Γ and around M = −1 around high-symmetry point X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' We fit the zeros of the CDMFT results with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (47) using t and λ as fitting parameters, taking the signs due to the renormalization (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (29)) into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The crystal field-splitting M is not renormalized in the analytic self-energy formula and is therefore kept constant during the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 13 the two approaches are in very good agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The renormalization of the parameters t and λ is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5 which explains the shifting of the topological transition from M = 1 in the non-interacting case to M ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Note that for CDMFT we use a Kanamori-type interaction instead of the only-U case, which is used in the derivation of the analytic formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' In the only-U case the parameter λ is strongly suppressed, which is consistent with the high-frequency expansion, since the corresponding correlation values ⟨ninj⟩ remain uncorrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' This suggests that the analytic formula can also be applied to more complicated interactions as long as the correct behavior of the correlation values is taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Finally, we address the claim, mentioned in the main text, that in absence of perturbations that provoke the opening of gapless Dirac or Weyl points in the noninteracting Hamiltonian the zeros of the Green’s function of the interacting system in the Mott phase remain gapless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' We can verify this claim by making again use of the 2D BHZ model of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' (46), which in the M = 0 case has a gapless dispersion at the X and Y high-symmetry points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' As previously mentioned, a comprehensive analysis of the topological phase diagram for the BHZ model using a 2×2 cluster through CDMFT is hampered by the size of the impurity problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Nevertheless, a converged result for selected M values is still within reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='14 we show the results after 10 DMFT steps for a 2×2 cluster for M = 0 as well as M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' As for the 2 × 1 cluster in the first part of this section we fit the zeros with the non-interacting dispersion using t and λ as fitting parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The resulting renormalization of the parameters is again given by a factor of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 21 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 13: Position of the zeros of G of a modified BHZ model (using a 2 × 1 cluster, see text) obtained using CDMFT (U = 8) and fits with the analytic formula for the self-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' The dashed lines show the non-interacting band dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' 14: Position of the zeros for a BHZ model obtained using CDMFT (U = 8) with a 2×2 cluster and fit with the analytic formula for the self-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Dashed lines show the non-interacting dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' ED fit with analytic formula Ho(k) - M=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='0 M=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='8 G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='6 zeros 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='2 0 position of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='8 M=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5 M=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5 G position of zeros of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='5 0 2TT 0 2π TT TT XM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content='0 4 ED G 3 fit 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Spectral and Optical Properties of Correlated Materials (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' [55] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Fu and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Kane, Topological insulators with inversion symmetry, Physical Review B 76, 045302 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' [56] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Yoshida, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Fujimoto, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Kawakami, Correlation effects on a topological insulator at finite temperatures, Physical Review B 85, 125113 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' [57] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} +page_content=' Ando, Topological Insulator Materials, Journal of the Physical Society of Japan 82, 102001 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfag8H/content/2301.05588v1.pdf'} diff --git a/UdE3T4oBgHgl3EQfEgk6/vector_store/index.pkl b/UdE3T4oBgHgl3EQfEgk6/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..5aa141f5e2ba3a00c0eb26463c05a62ba50113fc --- /dev/null +++ b/UdE3T4oBgHgl3EQfEgk6/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:315d7f51f8feb2d77a9be8b769d9653e859546a67ef8a2becdd1c334cef5c249 +size 273113 diff --git a/VNE5T4oBgHgl3EQfbw9b/content/tmp_files/2301.05598v1.pdf.txt b/VNE5T4oBgHgl3EQfbw9b/content/tmp_files/2301.05598v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..26def391d9c3e7c50a55fddad7311c84f659a7bc --- /dev/null +++ b/VNE5T4oBgHgl3EQfbw9b/content/tmp_files/2301.05598v1.pdf.txt @@ -0,0 +1,159 @@ +Nonprofit Adopt a Star: Lessons from 15 years of Crowdfunding +Travis S. Metcalfe∗ (White Dwarf Research Corporation) +Summary: In the past 15 years, the number of known planets outside of our solar +system has grown from about 200 to more than 5000. +During that time, we have +conducted one of the longest crowdfunding campaigns in history, a nonprofit adopt a +star program that supports astronomy research. The program includes the targets of +NASA space telescopes that are searching for planets around other stars, and it uses the +proceeds to help determine the properties of those stars and their planetary systems. +I summarize how this innovative program has evolved over the years and engaged the +public worldwide to support an international team of astronomers. +Background +In January 2008, we started a crowdfunding program known as the Pale Blue Dot Project to +support an international team of astronomers who were preparing for the launch of the Kepler space +telescope the following year. The program offered an “adopt a star” service featuring the targets +of the Kepler mission, with all of the proceeds supporting the work of the team to characterize the +stars and their planetary systems [1]. Donors received a personalized “Certificate of Adoption” by +email, and their selected target was updated in our online database—ensuring that each star could +only be adopted once. The database showed an image of the star in Google Sky, along with the +constellation name and coordinates, a link to a star chart, and a link to additional information +from the SIMBAD astronomical database [2]. We have previously described our experiences during +the first seven years of the program [3], so here we focus on developments after 2014. +Evolution of the Program +To align the name of our program with the service that it provided, in 2014 we rebranded as Adopt +a Star (adoptastar.org). This decision was motivated in part by a new generation of supporters who +were not familiar with astronomer Carl Sagan, who coined the phrase “pale blue dot” to describe +the Earth as viewed from a distance. The original name alluded to our search for pale blue dots +around other stars, but the reference was obscure to many people. The “adopt a star” concept +had previously been introduced [4], but it had not been actively publicized. We began promoting +our adopt a star program in 2008 using keyword-based advertising sponsored by an in-kind grant +from a leading search engine company. The phrase has now been embraced by deceptive “name a +star” companies, to the extent that we recently rebranded our program Nonprofit Adopt a Star to +distinguish it from the for-profit competitors. +Our original adopt a star program included only the targets of the Kepler space telescope. +The Kepler mission came to an abrupt end in 2013 after the spacecraft lost the second of four +reaction wheels that provided pointing stability. However, some clever engineers realized that the +two remaining wheels could still allow the telescope to observe stars in the ecliptic plane for up to +a few months at a time [5]. After demonstrating that it could work, the repurposed Kepler mission +(known as K2) began operating in this mode in 2014, and went on to observe a series of 19 fields +near the ecliptic plane over the next four years. Several hundred thousand targets were selected +from a newly-created Ecliptic Plane Input Catalog [6], including many stars that are visible to the +unaided eye. Our adopt a star program initially offered all targets for $10, but we began providing +∗+1 720-310-5180; travis@wdrc.org +1 +arXiv:2301.05598v1 [physics.soc-ph] 13 Jan 2023 + +Figure 1: Weekly visits during the 15 years of our crowdfunding campaign. Although visits stayed +relatively constant after 2014, program revenue quadrupled as a higher fraction of visitors adopted +stars and made larger donations for value-added targets. +value-added targets for larger donations in 2012: $15 for double stars, $25 for stars with suspected +planets, and $50 for confirmed planetary systems. The K2 mission gave us the first opportunity +to add bright stars (visible without a telescope) for a $100 donation. The launch of the Transiting +Exoplanet Survey Satellite (TESS) in 2018 extended this opportunity to bright stars in almost +every constellation in the sky [7]. +Although only 10 percent of our donors chose to adopt a bright star, the option eventually +yielded half of the total revenue from the program. While we had attracted about $100,000 in the +first seven years of the Pale Blue Dot Project, we raised an additional $450,000 over the next eight +years under the Adopt a Star brand. As illustrated in Figure 1, weekly visits to our website stayed +relatively constant during this time, so the increase in revenue probably resulted from converting +a higher fraction of our visitors into donors, and from convincing supporters to donate more for +value-added targets. We believe that improvements to our website inspired more visitors to adopt +a star, primarily a streamlining of the donation process (allowing the donor to select the brightest +available star in any category by default) and enhancements of the product that we provided (better +certificate design and additional features in the Google Sky interface through our database). We +recently launched a refreshed website (adoptastar.org) that amplifies these advantages, and couples +the website and database into a more unified user experience. +New Successes & Challenges +We have previously described [3] how there is an interesting story behind almost every peak in +Figure 1. +In addition to the recurring features that correspond to popular gift-giving holidays +(Christmas and Valentine’s Day), there are episodic spikes that coincide with traditional media +coverage or social media exposure. A few of these successes triggered some of our early challenges, +including institutional pressure from NASA, threats of legal action by the estate of Carl Sagan, and +an insult to Russian president Vladimir Putin by Ukrainian astronomers [8]. More recent spikes +have been largely positive, tapping into worldwide enthusiasm for our program. A large spike in +August 2018 corresponds to a surge in star adoptions for Qixi festival (Chinese Valentine’s Day). +2 + +2500 +Weekly Visits +1500 +500 +2008 +2011 +2014 +2017 +2020 +2023Figure 2: Custom star adoption certificate designed for the Gucci Cosmogonie fashion show in +May 2022. A bright star in a zodiac constellation was adopted for each of several hundred guests, +yielding the largest single donation in the history of our crowdfunding campaign. +A peak in mid-2020 resulted from a birthday star adoption for Filipino social media influencer +Bretman “Da Baddest” Rock, who featured the gift on Instagram. A surge of visitors in early +2021 arrived on the fifteenth anniversary of Twitter, when a marketing company released a website +featuring the top tweets of 140 people in the tech Twitterverse, with a blue star adoption in the +Pleiades cluster for each person (including Elon Musk). +The greatest success of our Nonprofit Adopt a Star program came in the spring of 2022, when +we were contacted by the press office of Gucci in Italy. They wanted to discuss a collaboration +opportunity, so they set up a video call for later in the day. Ahead of the call they asked us to sign +a non-disclosure agreement, citing the discussion of highly confidential information. It turned out +that Gucci was organizing a space-themed fashion show at a castle in Italy [9], and they wanted to +adopt a bright star in a zodiac constellation for each of their several hundred guests. We worked +with the team to coordinate the star adoptions, ensuring that none of the details appeared in our +database until the day of the fashion show. Gucci designed the custom star adoption certificate +shown in Figure 2, which was included in a gift bag for each of their guests. The resulting payment +was the largest single donation in the history of our crowdfunding program, representing nearly +half of the support that we had received in the previous year. +All proceeds from our Nonprofit Adopt a Star program support the research efforts of the +Kepler/TESS Asteroseismic Science Consortium (tasoc.dk), a large collaboration led by scientists +in Denmark. The leadership of this organization has generally tolerated our crowdfunding cam- +paign, and one of the greatest challenges has been to get the broader membership to invest in the +3 + +GUCCI +COSMOGONIE +STAR ADOPTION CERTIFICATE +We hereby declare that a star with the identification code +has been officially registered in the name ofGUCCICOSMOGONIE +Mo-nday, May 18, 2022 at Bpm CEST +Castel del Mnnte +Andria,ltaly +#GucciCosmogonie +The proceeds from the star adoption will support the scientifie research efforts of an international team of astronom +from the White Dwarf Research Corporation. To locate your star, visit https:/nnprofit.adoptastar.orgconcept. The collaboration as a whole has gladly accepted our co-sponsorship of annual science +conferences, which has given many students and early career researchers the opportunity to attend +and present their work without paying a registration fee. Some individual members in developing +countries have also accepted support for the publication charges on their scientific papers, helping +them publish in top-tier journals. But efforts to engage the collaboration in direct promotion of +the campaign for their own benefit have largely failed. This may reflect a difference in the cul- +ture of philanthropy in European countries, where investment in scientific research is seen as the +responsibility of governments rather than individuals, and where there are relatively few incentives +for charitable giving. +Future Outlook +Considering the history of our well-established brand, there is enormous potential for current and +future planet-search missions to benefit from our crowdfunding program. The TESS mission is +approved for continued operations through September 2025, and NASA has already invited the +mission to propose an additional extension through 2028. The European PLATO mission is cur- +rently scheduled for launch in late 2026, and the baseline observing plan includes two years of +monitoring for two different fields that each cover an area 20 times larger than the original Kepler +field (platomission.com). This mission will certainly rely on international teams to help analyze +the observations, but there is presently no NASA program to support the participation of U.S. +scientists. With cooperation from the PLATO mission, citizens worldwide will be able to support +the next wave of discoveries through Nonprofit Adopt a Star. +Acknowledgments: +White Dwarf Research Corporation receives in-kind support from Google, +Stripe, and Dreamhost. We would like to thank Sjors Provoost and Robert Piller for contribu- +tions that substantially improved our crowdfunding program. +References +[1] Metcalfe, T. S. 2009, Bulletin of the American Astronomical Society, 41, 410 (ADS) +[2] Wenger, M. 2000, Astronomy & Astrophysics Supplement, 143, 9 (ADS) +[3] Metcalfe, T. S. 2015, Journal of Astronomy & Earth Sciences Education, 2, 109 (ADS) +[4] The Inhabited Sky service (my.sky-map.org/adopt services.jsp) +[5] Howell, S. B. et al. 2014, Publications of the Astronomical Society of the Pacific, 126, 398 (ADS) +[6] Huber, D. et al. 2016, The Astrophysical Journal Supplement Series, 224, 2 (ADS) +[7] Stassun, K. G. et al. 2018, The Astronomical Journal, 156, 102 (ADS) +[8] Feeney, N. 2014, “A Star With a Not-So-Nice Nickname for Putin Won’t Have to Change” +(time.com/2964352/vladimir-putin-huilo-star) +[9] The Gucci Cosmogonie Fashion Show (youtu.be/BUej93sSeng) +4 + diff --git a/VNE5T4oBgHgl3EQfbw9b/content/tmp_files/load_file.txt b/VNE5T4oBgHgl3EQfbw9b/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2be7395d8deca3eddae0fdd8e5436162f4be0c62 --- /dev/null +++ b/VNE5T4oBgHgl3EQfbw9b/content/tmp_files/load_file.txt @@ -0,0 +1,98 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf,len=97 +page_content='Nonprofit Adopt a Star: Lessons from 15 years of Crowdfunding Travis S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' Metcalfe∗ (White Dwarf Research Corporation) Summary: In the past 15 years, the number of known planets outside of our solar system has grown from about 200 to more than 5000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' During that time, we have conducted one of the longest crowdfunding campaigns in history, a nonprofit adopt a star program that supports astronomy research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' The program includes the targets of NASA space telescopes that are searching for planets around other stars, and it uses the proceeds to help determine the properties of those stars and their planetary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' I summarize how this innovative program has evolved over the years and engaged the public worldwide to support an international team of astronomers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' Background In January 2008, we started a crowdfunding program known as the Pale Blue Dot Project to support an international team of astronomers who were preparing for the launch of the Kepler space telescope the following year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' The program offered an “adopt a star” service featuring the targets of the Kepler mission, with all of the proceeds supporting the work of the team to characterize the stars and their planetary systems [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' Donors received a personalized “Certificate of Adoption” by email, and their selected target was updated in our online database—ensuring that each star could only be adopted once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' The database showed an image of the star in Google Sky, along with the constellation name and coordinates, a link to a star chart, and a link to additional information from the SIMBAD astronomical database [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' We have previously described our experiences during the first seven years of the program [3], so here we focus on developments after 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' Evolution of the Program To align the name of our program with the service that it provided, in 2014 we rebranded as Adopt a Star (adoptastar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' This decision was motivated in part by a new generation of supporters who were not familiar with astronomer Carl Sagan, who coined the phrase “pale blue dot” to describe the Earth as viewed from a distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' The original name alluded to our search for pale blue dots around other stars, but the reference was obscure to many people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' The “adopt a star” concept had previously been introduced [4], but it had not been actively publicized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' We began promoting our adopt a star program in 2008 using keyword-based advertising sponsored by an in-kind grant from a leading search engine company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' The phrase has now been embraced by deceptive “name a star” companies, to the extent that we recently rebranded our program Nonprofit Adopt a Star to distinguish it from the for-profit competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' Our original adopt a star program included only the targets of the Kepler space telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' The Kepler mission came to an abrupt end in 2013 after the spacecraft lost the second of four reaction wheels that provided pointing stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' However, some clever engineers realized that the two remaining wheels could still allow the telescope to observe stars in the ecliptic plane for up to a few months at a time [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' After demonstrating that it could work, the repurposed Kepler mission (known as K2) began operating in this mode in 2014, and went on to observe a series of 19 fields near the ecliptic plane over the next four years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' Several hundred thousand targets were selected from a newly-created Ecliptic Plane Input Catalog [6], including many stars that are visible to the unaided eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' Our adopt a star program initially offered all targets for $10, but we began providing ∗+1 720-310-5180;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' travis@wdrc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content='org 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content='05598v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content='soc-ph] 13 Jan 2023 Figure 1: Weekly visits during the 15 years of our crowdfunding campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' Although visits stayed relatively constant after 2014, program revenue quadrupled as a higher fraction of visitors adopted stars and made larger donations for value-added targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' value-added targets for larger donations in 2012: $15 for double stars, $25 for stars with suspected planets, and $50 for confirmed planetary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' The K2 mission gave us the first opportunity to add bright stars (visible without a telescope) for a $100 donation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' The launch of the Transiting Exoplanet Survey Satellite (TESS) in 2018 extended this opportunity to bright stars in almost every constellation in the sky [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' Although only 10 percent of our donors chose to adopt a bright star, the option eventually yielded half of the total revenue from the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' While we had attracted about $100,000 in the first seven years of the Pale Blue Dot Project, we raised an additional $450,000 over the next eight years under the Adopt a Star brand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' As illustrated in Figure 1, weekly visits to our website stayed relatively constant during this time, so the increase in revenue probably resulted from converting a higher fraction of our visitors into donors, and from convincing supporters to donate more for value-added targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' We believe that improvements to our website inspired more visitors to adopt a star, primarily a streamlining of the donation process (allowing the donor to select the brightest available star in any category by default) and enhancements of the product that we provided (better certificate design and additional features in the Google Sky interface through our database).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' We recently launched a refreshed website (adoptastar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content='org) that amplifies these advantages, and couples the website and database into a more unified user experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' New Successes & Challenges We have previously described [3] how there is an interesting story behind almost every peak in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' In addition to the recurring features that correspond to popular gift-giving holidays (Christmas and Valentine’s Day), there are episodic spikes that coincide with traditional media coverage or social media exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' A few of these successes triggered some of our early challenges, including institutional pressure from NASA, threats of legal action by the estate of Carl Sagan, and an insult to Russian president Vladimir Putin by Ukrainian astronomers [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' More recent spikes have been largely positive, tapping into worldwide enthusiasm for our program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' A large spike in August 2018 corresponds to a surge in star adoptions for Qixi festival (Chinese Valentine’s Day).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' 2 2500 Weekly Visits 1500 500 2008 2011 2014 2017 2020 2023Figure 2: Custom star adoption certificate designed for the Gucci Cosmogonie fashion show in May 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' A bright star in a zodiac constellation was adopted for each of several hundred guests, yielding the largest single donation in the history of our crowdfunding campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' A peak in mid-2020 resulted from a birthday star adoption for Filipino social media influencer Bretman “Da Baddest” Rock, who featured the gift on Instagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' A surge of visitors in early 2021 arrived on the fifteenth anniversary of Twitter, when a marketing company released a website featuring the top tweets of 140 people in the tech Twitterverse, with a blue star adoption in the Pleiades cluster for each person (including Elon Musk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' The greatest success of our Nonprofit Adopt a Star program came in the spring of 2022, when we were contacted by the press office of Gucci in Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' They wanted to discuss a collaboration opportunity, so they set up a video call for later in the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' Ahead of the call they asked us to sign a non-disclosure agreement, citing the discussion of highly confidential information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' It turned out that Gucci was organizing a space-themed fashion show at a castle in Italy [9], and they wanted to adopt a bright star in a zodiac constellation for each of their several hundred guests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' We worked with the team to coordinate the star adoptions, ensuring that none of the details appeared in our database until the day of the fashion show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' Gucci designed the custom star adoption certificate shown in Figure 2, which was included in a gift bag for each of their guests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' The resulting payment was the largest single donation in the history of our crowdfunding program, representing nearly half of the support that we had received in the previous year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' All proceeds from our Nonprofit Adopt a Star program support the research efforts of the Kepler/TESS Asteroseismic Science Consortium (tasoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content='dk), a large collaboration led by scientists in Denmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' The leadership of this organization has generally tolerated our crowdfunding cam- paign,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' and one of the greatest challenges has been to get the broader membership to invest in the 3 GUCCI COSMOGONIE STAR ADOPTION CERTIFICATE We hereby declare that a star with the identification code has been officially registered in the name ofGUCCICOSMOGONIE Mo-nday,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' May 18,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' 2022 at Bpm CEST Castel del Mnnte Andria,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content='ltaly #GucciCosmogonie The proceeds from the star adoption will support the scientifie research efforts of an international team of astronom from the White Dwarf Research Corporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' To locate your star, visit https:/nnprofit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content='adoptastar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content='orgconcept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' The collaboration as a whole has gladly accepted our co-sponsorship of annual science conferences, which has given many students and early career researchers the opportunity to attend and present their work without paying a registration fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' Some individual members in developing countries have also accepted support for the publication charges on their scientific papers, helping them publish in top-tier journals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' But efforts to engage the collaboration in direct promotion of the campaign for their own benefit have largely failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' This may reflect a difference in the cul- ture of philanthropy in European countries, where investment in scientific research is seen as the responsibility of governments rather than individuals, and where there are relatively few incentives for charitable giving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' Future Outlook Considering the history of our well-established brand, there is enormous potential for current and future planet-search missions to benefit from our crowdfunding program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' The TESS mission is approved for continued operations through September 2025, and NASA has already invited the mission to propose an additional extension through 2028.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' The European PLATO mission is cur- rently scheduled for launch in late 2026, and the baseline observing plan includes two years of monitoring for two different fields that each cover an area 20 times larger than the original Kepler field (platomission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' This mission will certainly rely on international teams to help analyze the observations, but there is presently no NASA program to support the participation of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' scientists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' With cooperation from the PLATO mission, citizens worldwide will be able to support the next wave of discoveries through Nonprofit Adopt a Star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' Acknowledgments: White Dwarf Research Corporation receives in-kind support from Google, Stripe, and Dreamhost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' We would like to thank Sjors Provoost and Robert Piller for contribu- tions that substantially improved our crowdfunding program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' References [1] Metcalfe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' 2009, Bulletin of the American Astronomical Society, 41, 410 (ADS) [2] Wenger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' 2000, Astronomy & Astrophysics Supplement, 143, 9 (ADS) [3] Metcalfe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' 2015, Journal of Astronomy & Earth Sciences Education, 2, 109 (ADS) [4] The Inhabited Sky service (my.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content='sky-map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content='org/adopt services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content='jsp) [5] Howell, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' 2014, Publications of the Astronomical Society of the Pacific, 126, 398 (ADS) [6] Huber, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' 2016, The Astrophysical Journal Supplement Series, 224, 2 (ADS) [7] Stassun, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' 2018, The Astronomical Journal, 156, 102 (ADS) [8] Feeney, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content=' 2014, “A Star With a Not-So-Nice Nickname for Putin Won’t Have to Change” (time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content='com/2964352/vladimir-putin-huilo-star) [9] The Gucci Cosmogonie Fashion Show (youtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} +page_content='be/BUej93sSeng) 4' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE5T4oBgHgl3EQfbw9b/content/2301.05598v1.pdf'} diff --git a/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf b/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..7eddb4e90a39bd011e6ccffdfbd20699ea8425d9 --- /dev/null +++ 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sha256:b462eb167fe2b41211bfa93c897cc1872d2bef40cc48ac267944e8ea84d3a453 +size 192267 diff --git a/YtFAT4oBgHgl3EQf3h4G/content/tmp_files/2301.08720v1.pdf.txt b/YtFAT4oBgHgl3EQf3h4G/content/tmp_files/2301.08720v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..437fe17faf83dfeef5dae98643dc8776cc4a16ea --- /dev/null +++ b/YtFAT4oBgHgl3EQf3h4G/content/tmp_files/2301.08720v1.pdf.txt @@ -0,0 +1,5226 @@ +arXiv:2301.08720v1 [math.RT] 20 Jan 2023 +OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX +SYSTEMS +DANIEL ALPAY AND ILWOO CHO +Abstract. In this paper, we consider natural Hilbert-space representations +�� +C2, πt +�� +t∈R of the hypercomplex system {Ht}t∈R, and study the realizations +πt (h) of hypercomplex numbers h ∈ Ht, as (2 × 2)-matrices acting on C2, +for an arbitrarily fixed scale t ∈ R. Algebraic, operator-theoretic, spectral- +analytic, and free-probabilistic properties of them are considered. +1. Introduction +In this paper, we study representations of the hypercomplex numbers (a, b) of +complex numbers a and b, constructing a ring, +Ht = +� +C2, +, ·t +� +, +scaled by a real number t ∈ R, where (+) is the usual vector addition on the 2- +dimensional vector space C2, and (·t) is the t-scaled vector-multiplication on C2, +defined by +(a1, b1) ·t (a2, b2) = +� +a1a2 + tb1b2, a1b2 + b1a2 +� +, +where z are the conjugates of z in C. +Motivated by the canonical Hilbert-space representation +� +C2, π +� +of the quater- +nions H, introduced in [2], [3] and [19], we consider the canonical representation, +Πt = +� +C2, πt +� +, +of the ring Ht, and understand each element h = (a, b) of Ht as its realization, +πt (h) +denote += +[h]t +def += +� a +tb +b +a +� +in M2 (C) , +where M2 (C) = B +� +C2� +is the matricial algebra (or, the operator algebra acting on +C2) of all (2 × 2)-matrices over C (respectively, all bounded linear transformations, +or simply operators on C2), for each t ∈ R. Under our setting, one can check that +the ring H−1 is nothing but the noncommutative field H of all quaternions (e.g., +[2], [3] and [19]), and the ring H1 is the ring of all bicomplex numbers (e.g., [1]). +The spectral-analytic, operator-theoretic (or, matrix-theoretic), and free-probabilistic +properties of Ht are considered and characterized under the canonical representa- +tion Πt. In particular, certain decompositional properties on Ht are studied alge- +braically, and spectral-theoretically. And then, it is considered how those properties +affect the spectral-analytic, operator-theoretic, and free-probabilistic properties of +hypercomplex numbers of Ht, for t ∈ R. +2000 Mathematics Subject Classification. 20G20, 46S10, 47S10. +Key words and phrases. Scaled Hypercomplex Ring, Scaled Hypercomplex Monoids, Repre- +sentations, Scaled-Spectral Forms, Scaled-Spectralization, Spectral Theory, Free Probability. +1 + +2 +DANIEL ALPAY AND ILWOO CHO +1.1. Motivation. The quaternions H is an interesting object not only in pure +mathematics (e.g., [5], [10], [11], [12], [13] [14], [17], [19], [23]), but also in applied +mathematics (e.g., [4], [7], [15], [16], [20] and [21]). Independently, spectral analysis +on H is considered in [2] and [3], under representation, “over C,” different from the +usual quaternion-eigenvalue problems of quaternion-matrices studied in [13], [15] +and 16[]. +Motivated by the generalized setting of the quaternions so-called the split-quaternions +of [1], and by the main results of [2] and [3], we study a new type of hypercom- +plex numbers induced by the pairs of C2. Especially, we construct a system of +the scaled hypercomplex rings {Ht}t∈R, and study how the hypercomplex num- +bers act as (2 × 2)-matrices over C for given scales t ∈ R, under our canonical +Hilbert-space representations +� +Πt = +� +C2, πt +�� +t∈R. We are interested in algebraic, +operator-theoretic, spectral-theoretic, free-probabilistic properties of Ht under Πt, +for t ∈ R. Are they similar to those of the quaternions H−1 = H, shown in [2] and +[3]? The answers are determined differently case-by-case, up to scales (See below). +1.2. Overview. In Section 2, we define our main objects, the scaled hypercomplex +rings {Ht}t∈R, and their canonical Hilbert-space representations {Πt}t∈R. We un- +derstand each hypercomplex number of Ht as an operator, a (2 × 2)-matrix over C. +We concentrate on studying the invertibility on Ht, for an arbitrarily fixed scale t. +It is shown that if t < 0, then Ht forms a noncommutative field like the quaternions +H = H−1, however, if t ≥ 0, then it becomes a ring with unity, which is not a +noncommutative field. +In Section 3, the spectral theory on (the realizations of) Ht is studied over +C. After finding the spectra of hypercomplex numbers, we define so-called the t- +spectral forms whose main diagonal entries are from the spectra, and off-diagonal +entries are 0’s. As we have seen in [2] and [3], such spectral forms are similar to the +realizations of quaternions of H−1. However, if a scale t ∈ R\{−1} is arbitrary, then +such a similarity does not hold in general. We focus on studying such a similarity +in detail. +In Section 4, we briefly discuss about how the usual adjoint on M2 (C) acts +on the sub-structure Ht +2 of M2 (C), consisting of all realizations of Ht, for a scale +t ∈ R. Different from the quaternionic case of [2] and [3], in general, the adjoints +(conjugate-transposes) of many matrices of Ht +2 are not contained in Ht +2, especially, if +t ̸= −1. It shows that a bigger, operator-algebraically-better ∗-algebraic structure +generated by Ht +2 is needed in M2 (C), to consider operator-theoretic, and free- +probabilistic properties on Ht +2. +In the final Section 5, on the C∗-algebraic structure of Section 4, we study +operator-theoretic, and free-probabilistic properties up to the usual trace, and the +normalized trace. +2. The Scaled Hypercomplex Systems {Ht}t∈R +In this section, we define a ring Ht of hypercomplex numbers, and establish the +corresponding canonical Hilbert-space representations Πt, for an arbitrary fixed +scale t ∈ R. Throughout this section, we let +C2 = {(a, b) : a, b ∈ C} + +OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS +3 +be the Cartesian product of two copies of the complex field C. One may understand +C2 as the usual 2-dimensional Hilbert space equipped with its canonical orthonor- +mal basis, {(1, 0) , (0, 1)} . +2.1. A t-Scaled Hypercomplex Ring Ht. In this section, we fix an arbitrary +real number t in the real field R. On the vector space C2 (over C), define the +t-scaled vector-multiplication (·t) by +(a1, b1) ·t (a2, b2) +def += +� +a1a2 + tb1b2, a1b2 + b1a2 +� +, +(2.1.1) +for (al, bl) ∈ C2, for all l = 1, 2, where z are the conjugates of z in C. It is not +difficult to check that such an operation (·t) is closed on C2. Moreover, it satisfies +that +((a1, b1) ·t (a2, b2)) ·t (a3, b3) += +� +a1a2 + tb1b2, a1b2 + b1a2 +� +·t (a3, b3) += +� +a1a2a3 + t +� +b1b2a3 + a1b2b3 + b1a2b3 +� +, +a1a2b3 + a1b2a3 + b1a2a3 + tb1b2b3 +� +, +and +(a1, b1) ·t ((a2, b2) ·t (a3, b3)) += (a1, b1) ·t +� +a2a3 + tb2b3, a2b3 + b2a3 +� += +� +a1 +� +a2a3 + tb2b3 +� ++ tb1 +� +a2b3 + b2a3 +� +, +a1 (a2b3 + b2a3) + b1 +� +a2a3 + tb2b3 +�� +, +implying the equality, +(2.1.2) +((a1, b1) ·t (a2, b2)) ·t (a3, b3) = (a1, b1) ·t ((a2, b2) ·t (a2, b3)) , +in C2, for (al, bl) ∈ C2, for all l = 1, 2, 3. +Furthermore, if ϑ = (1, 0) ∈ C2, then +ϑ ·t (a, b) = (a, b) = (a, b) ·t ϑ +(2.1.3) +by (2.1.1), for all (a, b) ∈ C2. +By (2.1.2) and (2.1.3), if +C2× = C2 \ {(0, 0)} , +then the pair +� +C2×, ·t +� +forms a monoid (i.e., semigroup with its identity (1, 0)). +Lemma 1. Let C2× = C2 \ {(0, 0)}, and (·t) be the closed operation (2.1.1) on C2. +Then the algebraic structure +� +C2×, ·t +� +forms a monoid with its identity (1, 0). +Proof. The proof is done by (2.1.2) and (2.1.3). +Therefore, one can obtain the following ring structure. +Proposition 2. The algebraic triple +� +C2, +, ·t +� +forms a unital ring with its unity +(or, the multiplication-identity) (1, 0), where (+) is the usual vector addition on +C2, and (·t) is the vector multiplication (2.1.1). + +4 +DANIEL ALPAY AND ILWOO CHO +Proof. Clearly, the algebraic pair +� +C2, + +� +is an abelian group under the usual addi- +tion (+) with its (+)-identity (0, 0). While, by Lemma 1, the pair +� +C2×, ·t +� +forms +a monoid (and hence, a semigroup). Observe now that +(a1, b1) ·t ((a2, b2) + (a3, b3)) = (a1, b1) ·t (a2 + a3, b2 + b3) += +� +a1 (a2 + a3) + tb1 +� +b2 + b3 +� +, a1 (b2 + b3) + b1 (a2 + a3) +� += +� +a1a2 + a1a3 + tb1b2 + tb1b3, a1b2 + a1b3 + b1a2 + b1a3 +� += +� +a1a2 + tb1b2, a1b2 + b1a2 +� ++ +� +a1a3 + tb1b3, a1b3 + b1a3 +� += (a1, b1) ·t (a2, b2) + (a1, b1) ·t (a3, b3), +and, similarly, +(2.1.4) +((a1, b1) + (a2, b2)) ·t (a3, b3) = (a1, b1) ·t (a3, b3) + (a2, b2) ·t (a3, b3) , +in C2. So, the operations (+) and (·t) are left-and-right distributive by (2.1.4). +Therefore, the algebraic triple +� +C2, +, ·t +� +forms a unital ring with its unity +(1, 0). +The above proposition characterizes the algebraic structure of +� +C2, +, ·t +� +as a +well-defined unital ring for a fixed t ∈ R. +Remark here that, since a scale t is +arbitrary in R, in fact, we obtain the unital rings {Ht}t∈R. +Definition 3. For a fixed t ∈ R, the ring +� +C2, +, ·t +� +is called the hypercomplex +ring with its scale t (in short, the t-scaled hypercomplex ring). By Ht, we denote +the t-scaled hypercomplex ring. +2.2. The Canonical Representation Πt = +� +C2, πt +� +of Ht. In this section, we +fix t ∈ R, and the corresponding t-scaled hypercomplex ring, +Ht = +� +C2, +, ·t +� +, +where (·t) is the vector-multiplication (2.1.1). We consider a natural finite-dimensional- +Hilbert-space representation Πt of Ht, and understand each hypercomplex number +h ∈ Ht as an operator acting on a Hilbert space determined by Πt. In particu- +lar, as in the quaternionic case of [2], [3] and [19], a 2-dimensional-Hilbert-space +representation of the hypercomplex ring Ht is established naturally. +Define now a morphism, +πt : Ht → B +� +C2� += M2 (C) , +by +(2.2.1) +πt ((a, b)) = +� a +tb +b +a +� +, ∀ (a, b) ∈ Ht, +where B (H) is the operator algebra consisting of all bounded (or, continuous linear) +operators on a Hilbert space H, and Mk (C) is the matricial algebra of all (k × k)- +matrices over C, isomorphic to B +� +Ck� +, for all k ∈ N (e.g., [8] and [9]). +By definition, the function πt of (2.2.1) is an injective map from Ht into M2 (C). +Indeed, if +(a1, b1) ̸= (a2, b2) in Ht, +then +(2.2.2) +πt ((a1, b1)) = +� a1 +tb1 +b1 +a1 +� +̸= +� a2 +tb2 +b2 +a2 +� += πt ((a2, b2)) , + +OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS +5 +in M2 (C). Furthermore, it satisfies that +πt ((a1, b1) + (a2, b2)) = + + +a1 + a2 +t (b1 + b2) +b1 + b2 +a1 + a2 + + += +� a1 +tb1 +b1 +b2 +� ++ +� a2 +tb2 +b2 +a2 +� += πt ((a1, b1)) + πt ((a2, b2)) . +(2.2.3) +Also, one has +πt ((a1, b1) ·t (a2, b2)) = πt +�� +a1a2 + tb1b2, a1b2 + b1a2 +�� +by (2.1.1) += + + +a1a2 + tb1b2 +t (a1b2 + b1a2) +a1b2 + b1a2 +a1a2 + tb1b2 + + += +� a1 +tb1 +b1 +a1 +� � a2 +tb2 +b2 +a2 +� += πt ((a1, b1)) πt ((a2, b2)) , +(2.2.4) +where the multiplication (·) in the far-right-hand side of (2.2.4) is the usual matricial +multiplication on M2 (C). +Since our t-scaled hypercomplex ring Ht = +� +C2, +, ·t +� +is identified with the 2- +dimensional space C2 (set-theoretically), one may / can understand this ring Ht as +a topological ring equipped with the usual topology for C2, for any t ∈ R. From +below, we regard the ring Ht as a topological unital ring under the usual topology +for C2. +Lemma 4. The pair +� +C2, πt +� +is an injective Hilbert-space representation of the +t-scaled hypercomplex ring Ht, where πt is an action (2.2.1). +Proof. The morphism πt : Ht → M2 (C) of (2.2.1) is a well-defined injective function +by (2.2.2). Moreover, this map πt satisfies the relations (2.2.3) and (2.2.4), and +hence, it is a(n algebraic) ring-action of Ht, acting on the 2-dimensional vector space +C2. So, the pair +� +C2, πt +� +forms an algebraic representation of Ht. By regarding +Ht and M2 (C) as topological spaces equipped with their usual topologies, then it +is not difficult to check that the ring-action πt is continuous from Ht (which is +homeomorphic to C2 as a topological space) into M2 (C) (which is ∗-isomorphic +to the C∗-algebra B +� +C2� +). Thus, the algebraic representation +� +C2, πt +� +forms a +Hilbert-space representation of Ht acting on C2 via πt. +The above lemma shows that the t-scaled hypercomplex ring Ht is realized in +the matricial algebra M2 (C) as +πt (Ht) = +�� a +tb +b +a +� +∈ M2 (C) : (a, b) ∈ Ht +� +, +as an embedded topological ring in M2 (C). +Definition 5. The realization πt (Ht) of the t-scaled hypercomplex ring Ht is called +the t-scaled (hypercomplex-)realization of Ht (in M2 (C)), for a scale t ∈ R. And +we denote πt (Ht) by Ht +2. i.e., +Ht +2 +denote += +πt (Ht) = +�� a +tb +b +a +� +: (a, b) ∈ Ht +� +. + +6 +DANIEL ALPAY AND ILWOO CHO +Also, by [ξ]t, we denote πt (ξ) ∈ Ht +2, for all ξ ∈ Ht. +By the above lemma and definition, we obtain the following result. +Theorem 6. For t ∈ R, the corresponding t-scaled hypercomplex ring Ht is topological- +ring-isomorphic to the t-scaled realization Ht +2 in M2 (C). i.e., +Ht +T.R += Ht +2 +in +M2 (C), +(2.2.5) +where “ +T.R += ” means “being topological-ring-isomorphic to.” +Proof. The relation (2.2.5) is proven by Lemma 4 and the injectivity (2.2.2) of πt. +By the above theorem, one can realize that Ht and Ht +2 as an identical topological +ring, for a fixed t ∈ R. Recall that the relation (2.2.5) is independently shown in +[2] and [3], only for the quaternionic case where t = −1. +2.3. Scaled Hypercomplex Monoids. Throughout this section, we fix a scale +t ∈ R, and the corresponding t-scaled hypercomplex ring, +Ht = +� +C2, +, ·t +� +, +which is isomorphic to the t-scaled realization, +Ht +2 = +�� a +tb +b +a +� +∈ M2 (C) : (a, b) ∈ Ht +� +, +in M2 (C). Let +H× +t +denote += +Ht \ {(0, 0)} , +set-theoretically, where (0, 0) ∈ Ht is the (+)-identity of the abelian group +� +C2, + +� +. +Thus, by Proposition 2, this set forms a well-defined semigroup, +H× +t +denote += +� +H× +t , ·t +� +, +equipped with its (·t)-identity (1, 0), and hence, the pair H× +t is the maximal monoid +embedded in Ht +2 up to the operation (·t). +Definition 7. The maximal monoid H× +t += +� +H× +t , ·t +� +, embedded in the t-scaled +hypercomplex ring Ht, is called the t-scaled hypercomplex monoid. +By (2.2.5), it is trivial that: +Corollary 8. The t-scaled hypercomplex monoid H× +t is monoid-isomorphic to the +monoid Ht× +2 +denote += +� +Ht× +2 , · +� +, equipped with its identity, +I2 = +� 1 +0 +0 +1 +� += +� 1 +t · 0 +0 +1 +� += [(1, 0)]t , +the (2 × 2)-identity matrix of M2 (C), where (·) is the usual matricial multiplication +inherited from that on M2 (C). i.e., +H× +t = +� +H× +t , ·t +� Monoid += +� +Ht× +2 , · +� += Ht× +2 , +(2.3.1) +where “ +Monoid += +” means “being monoid-isomorphic.” +Proof. The isomorphic relation (2.3.1) is proven by the proof of Proposition 2, and +that of Theorem 6. + +OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS +7 +2.4. Invertibility on Ht. In this section, by identifying our t-scaled hypercomplex +ring Ht as its isomorphic realization Ht +2, we consider invertibility of elements of Ht, +for an arbitrarily fixed t ∈ R. +Observe first that, for any (a, b) ∈ Ht realized to be [(a, b)]t ∈ Ht +2, one can get +that +det ([(a, b)]t) = det +� a +tb +b +a +� += |a|2 − t |b|2 , +i.e., +(2.4.1) +det ([(a, b)]t) = |a|2 − t |b|2 , +where det : M2 (C) → C is the determinant, and |.| is the modulus on C. +Theorem 9. Let (a, b) ∈ Ht, realized to be [(a, b)]t ∈ Ht +2. +(2.4.2) det ([(a, b)]t) = |a|2 − t |b|2. +(2.4.3) If either |a|2 > t |b|2, or |a|2 < t |b|2, then [(a, b)]t is invertible “in M2 (C),” +with its inverse matrix, +[(a, b)]−1 +t += +1 +|a|2 − t |b|2 +� +a +t (−b) +(−b) +a +� +. +(2.4.4) If |a|2 − t |b|2 ̸= 0, then (a, b) ∈ Ht is invertible in the sense that there exists +a unique (c, d) ∈ Ht, such that +(a, b) ·t (c, d) = (1, 0) = (c, d) ·t (a, b) . +In particular, one has that +(c, d) = +� +a +|a|2 − t |b|2 , +−b +|a|2 − t |b|2 +� +∈ C2 +(2.4.5) Assume that (a, b) is invertible in Ht in the sense of (2.4.4). +Then the +inverse is also contained “in Ht.” +Proof. The statement (2.4.2) is shown by (2.4.1). +Note-and-recall that a matrix A ∈ Mn (C) is invertible in Mn (C), if and only if +det (A) ̸= 0, for all n ∈ N. Therefore, +det ([(a, b)]t) ̸= 0 ⇐⇒ [(a, b)]t is invertible in M2 (C) . +So, by (2.4.2), +|a|2 − t |b|2 ̸= 0, ⇐⇒ [(a, b)]t is invertible in M2 (C) . +Moreover, |a|2 − t |b|2 ̸= 0, if and only if +[(a, b)]−1 +t += +� a +tb +b +a +�−1 += +1 +|a|2 − t |b|2 +� +a +−tb +−b +a +� +, +in M2 (C). Therefore, the statement (2.4.3) holds true in M2 (C). +By (2.4.3), one has det ([(a, b)]t) ̸= 0, if and only if +[(a, b)]−1 +t += + + + + +a +|a|2−t|b|2 +t +� +−b +|a|2−t|b|2 +� +� +−b +|a|2−t|b|2 +� +a +|a|2−t|b|2 + + + + ∈ M2 (C) , + +8 +DANIEL ALPAY AND ILWOO CHO +and it is actually contained ”in Ht +2,” satisfying +π−1 +t + + + + +a +|a|2−t|b|2 +t +� +−b +|a|2−t|b|2 +� +� +−b +|a|2−t|b|2 +� +a +|a|2−t|b|2 + + + + = +� +a +|a|2 − t |b|2 , +−b +|a|2 − t |b|2 +� +, +in Ht, by the injectivity of πt. It shows that [(a, b)]−1 +t +exists in M2 (C), if and only if +it is contained “in Ht +2.” i.e., if [(a, b)]t is invertible, then its inverse is also contained +in Ht +2, too, and vice versa. So, the statements (2.2.4) and (2.2.5) hold. +The above theorem not only characterizes the invertibility of the monoidal ele- +ments of the t-scaled hypercomplex monoid H× +t , but also confirms that the inverses +(if exist) are contained in the monoid H× +t . i.e., +(a, b)−1 exists, ⇐⇒ (a, b)−1 = +� +a +|a|2 − t |b|2 , +−b +|a|2 − t |b|2 +� +, +”in H× +t ,” equivalently, +� +(a, b)−1� +t = [(a, b)]−1 +t +in H× +2 . +Corollary 10. Let (a, b) ∈ H× +t . Then it is invertible, if and only if +� +(a, b)−1� +t = +�� +a +|a|2−t|b|2 , +−b +|a|2−t|b|2 +�� +t = [(a, b)]−1 +t +, +(2.4.6) +in H× +2 , where [(a, b)]−1 +t +means the matricial inverse in M2 (C). +Proof. The proof of (2.4.6) is immediately done by (2.4.3), (2.4.4) and (2.4.5). +The above corollary can be re-stated by that: if ξ ∈ H× +t is invertible, then +πt +� +ξ−1� += (πt (ξ))−1 in Ht× +2 . +Now consider the cases where +|a|2 − t |b|2 = 0 ⇐⇒ |a|2 = t |b|2 , +(2.4.7) +in R. As we have seen above, the condition (2.4.7) holds for (a, b) ∈ Ht, if and only +if (a, b) is not invertible in Ht (and hence, its realization [(a, b)]t is not invertible in +M2 (C), and hence, in Ht +2). Clearly, we are not interested in the (+)-identity (0, 0) of +Ht automatically satisfying the condition (2.4.7). So, without loss of generality, we +focus on elements (a, b) of the t-scaled hypercomplex monoid H× +t (or, its realizations +[(a, b)]t of Ht× +2 ), satisfying the condition (2.4.7). +Recall that an algebraic triple, (X, +, ·), is a noncommutative field, if (i) (X, +) +is an abelian group, (ii) (X×, ·) forms a non-abelian group, and (iii) the operations +(+) and (·) are left-and-right distributive. For instance, the quaternions H = H−1 +is a noncommutative field (e.g., [2] and [3]). +Theorem 11. Suppose the fixed scale t ∈ R is negative, i.e., t < 0 in R. Then +“all” elements (a, b) of the t-scaled hypercomplex monoid H× +t are invertible in Ht, +with their inverses, +� +a +|a|2 − t |b|2 , +−b +|a|2 − t |b|2 +� +∈ H× +t . + +OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS +9 +i.e., +(2.4.8) +t < 0 in R =⇒ Ht is a noncommutative field. +Proof. Suppose the scale t ∈ R is negative. Then, for any (a, b) ∈ H× +t , +|a|2 ̸= t |b|2 ⇐⇒ |a|2 − t |b|2 > 0, +since (a, b) ̸= (0, 0). i.e., if t < 0, then every element (a, b) ∈ H× +t does “not” satisfy +the condition (2.4.7). It implies that if t < 0, then every element (a, b) ∈ H× +t is +invertible in H× +t , by (2.4.4) and (2.4.5); and the inverse is determined to be (2.4.6) +in H× +t . Thus, the pair H× +t = +� +H× +t , ·t +� +forms a group which is not abelian by (2.1.1) +and (2.2.4). +Therefore, if t < 0 in R, then the t-scaled hypercomplex ring Ht becomes a +noncommutative field, proving the statement (2.4.8). +The above theorem characterizes that the algebraic structure of scaled hyper- +complex rings {Ht}t<0 as noncommutative fields. +Theorem 12. Suppose t = 0 in R. Then an element (a, b) of the 0-scaled hyper- +complex monoid H× +0 is invertible in H0, with their inverses, +� +a +|a|2 , −b +|a|2 +� +∈ H× +0 , +if and only if a ̸= 0 in C, if and only if only the elements of the subset, +� +(a, b) ∈ H× +0 : a ̸= 0 +� +of H× +0 +(2.4.9) +are invertible in H× +0 , if and only if (0, b) ∈ H× +0 are not invertible in H× +0 , for all +b ∈ C. +Proof. Assume that we have the zero scale, i.e., t = 0 in R. Then, by (2.4.7), +|a|2 = 0 · |b|2 ⇐⇒ |a|2 = 0 ⇐⇒ a = 0 in C, +if and only if (0, b) ∈ H× +0 are not invertible in H× +0 , for all b ∈ C, if and only if all +elements (a, b), contained in the subset (2.4.9), are invertible in H× +0 . +Observe that (a, b) is contained in the subset (2.4.9) of H× +0 , if and only if +[(a, b)]0 +�� +a +|a|2 , +−b +|a|2 +�� +0 = +� a +0 +b +a +� + + + +a +|a|2 +0 +−b +|a|2 +a +|a|2 + + + += +� 1 +0 +0 +1 +� += + + + +a +|a|2 +0 +−b +|a|2 +a +|a|2 + + + +� a +0 +b +a +� += +�� +a +|a|2 , +−b +|a|2 +�� +0 [(a, b)]0 , +in H× +0 . Therefore, if exists, (a, b)−1 = +� +a +|a|2 , +−b +|a|2 +� +in H× +0 . +The above theorem shows that if we have the zero-scale in R, then our 0-scaled +hypercomplex ring H0 cannot be a noncommutative field. It directly illustrates +that the algebra on the quaternions H = H−1, and the algebra on the scaled- +hypercomplex rings {Ht}t∈R\{−1} can be different in general, especially, when t ≥ 0. + +10 +DANIEL ALPAY AND ILWOO CHO +Theorem 13. Suppose the scale t ∈ R is positive, i.e., t > 0 in R. +Then an +element (a, b) ∈ H× +t is invertible in H× +t with its inverse, +� +a +|a|2 − t |b|2 , +−b +|a|2 − t |b|2 +� +∈ H× +t , +if and only if |a|2 ̸= t |b|2 in R+ +0 = {r ∈ R : r ≥ 0}, if and only if (a, b) is contained +in the subset, +� +(a, b) : |a|2 ̸= t |b|2 in R+ +0 +� +, +(2.4.10) +of H× +t . As application, if t > 0 in R, then the all elements of +{(a, 0) ∈ Ht : a ∈ C×} ∪ {(0, b) ∈ Ht : b ∈ C×} , +(2.4.11) +are invertible in Ht, where C× = C \ {0}. +Proof. Assume that t > 0 in R, and H× +t , the corresponding t-scaled hypercomplex +monoid. Then (a, b) ∈ H× +t is invertible in H× +t , if and only if the condition (2.4.7) +does not hold, if and only if +|a|2 ̸= t |b|2 ⇐⇒ either |a|2 > t |b|2 , or |a|2 < t |b|2 , +in R+ +0 , since t > 0. Therefore, if t > 0 in R, then an element (a, b) is invertible in +H× +t , if and only if +either |a|2 > t |b|2 , or |a|2 < t |b|2 in R+ +0 , +if and only if (a, b) is contained in the subset (2.4.10) in H× +t . +In particular, for t > 0 in R, (i) if (a, 0) ∈ H× +t with a ∈ C×, then |a|2 > 0; and +(ii) if (0, b) ∈ H× +t with b ∈ C×, then 0 < t |b|2. Therefore, the subset (2.4.11) is +properly contained in the subset (2.4.10) in H× +t , whenever t > 0. So, all elements, +formed by (a, 0) ,or by (0, b) with a, b ∈ C×, are invertible in H× +t . +The above theorem characterizes the invertibility on the t-scaled hypercomplex +monoid H× +t , where the scale t is positive in R. +Theorems 11, 12 and 13 refine +Theorem 8, case-by-case. We again summarize the main results. +Corollary 14. Let H× +t +be the t-scaled hypercomplex monoid. If t < 0, then all +nonzero elements of H× +t are invertible; and if t = 0, then +� +(a, b) ∈ H× +0 : a ̸= 0 +� +is the invertible proper subset of H× +0 ; and if t > 0, then +� +(a, b) : |a|2 ̸= t |b|2 in R+ +0 +� +is the invertible proper subset of H× +t , where “invertible subset of H× +t ” means “a +subset of H× +t containing of all invertible elements.” +Proof. This corollary is nothing but a summary of Theorems 11, 12 and 13. + +OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS +11 +2.5. Decompositions of the Nonnegatively-Scaled Hypercomplex Rings. +In this section, we consider a certain decomposition of the t-scaled hypercomplex +ring Ht, for an arbitrary fixed “positive” scale t > 0 in R. +Recall that, as we +have seen in Section 2.4, the negatively-scaled hypercomplex rings {Hs}s<0 are +noncommutative fields by (2.4.8), equivalently, the negatively-scaled hypercomplex +monoids {H× +s }s<0 are non-abelian groups. However, if t ≥ 0, then Ht cannot be a +noncommutative field in general, by (2.4.9) and (2.4.10). We here concentrate on +such cases. +Let t ≥ 0 and Ht, the corresponding t-scaled hypercomplex ring. Partition Ht +by +Ht = Hinv +t +⊔ Hsing +t +with +(2.5.1) +Hinv +t += +� +(a, b) : |a|2 ̸= t |b|2� +, +and +Hsing +t += +� +(a, b) : |a|2 = t |b|2� +, +where ⊔ is the disjoint union. By (2.4.9) and (2.4.10), (a, b) ∈ Hinv +t +, if and only if +it is invertible, equivalently, (a, b) ∈ Hsing +t +, if and only if it is not invertible, in Ht. +Recall-and-note that the determinant is a multiplicative map on Mn (C), for all +n ∈ N, in the sense that: +det (AB) = det (A) det (B) , ∀A, B ∈ Mn (C) . +(2.5.2) +Thus, by (2.5.2), one has +ξ, η ∈ Hinv +t +⇒ det ([ξ ·t η]t) = det ([ξ]t [η]t) ̸= 0. +(2.5.3) +Lemma 15. Let t ≥ 0 in R. Then the subset Hinv +t +denote += +� +Hinv +t +, ·t +� +of the t-scaled +hypercomplex monoid H× +t forms a non-abelian group. i.e., Hinv +t +is not only a sub- +monoid, but also an embedded group in H× +t . +Proof. By (2.5.2), if ξ, η ∈ Hinv +t +, then ξ ·t η ∈ Hinv +t +, too. i.e., the operation (·t) is +closed, and associative on Hinv +t +. Also, the (·t)-identity (1, 0) is contained in Hinv +t +by (2.5.1). Therefore, the sub-structure +� +Hinv +t +, ·t +� +forms a sub-monoid of H× +t . But, +by (2.4.8) and (2.5.3), each element ξ ∈ Hinv +t +has its (·t)-inverse ξ−1 contained in +Hinv +t +. It shows that Hinv +t +forms a non-abelian group in the monoid H× +t . +By the partition (2.5.1) and the multiplicativity (2.5.3), one can obtain the +following equivalent result of the above theorem. +Lemma 16. Let t ≥ 0 in R. Then the pair +H×sing +t +denote += +� +Hsing +t +∩ H× +t , ·t +� += +� +Hsing +t +\ {(0, 0)} , ·t +� +forms a semigroup without identity in the t-scaled hypercomplex monoid H× +t . +Proof. By (2.5.2) and (2.5.3), the operation (·t) is closed and associative on the set, +H×sing +t +def += H× +t ∩ Hsing +t += Hsing +t +\ {(0, 0)} . +However, the (·t)-identity (1, 0) is not contained in H×sing +t +, since I2 = [(1, 0)]t is +in Hinv +t +. So, in the monoid H× +t , the sub-structure +� +H×sing +t +, ·t +� +forms a semigroup +(without identity). + +12 +DANIEL ALPAY AND ILWOO CHO +The above lemma definitely includes the fact that: +� +Hsing +t +, ·t +� +is just a semigroup +(without identity), which is not a sub-monoid of H× +t (and hence, not a group). +The above two algebraic characterizations show that the set-theoretical decom- +position (2.5.1) induces an algebraic decomposition of the t-scaled hypercomplex +monoid H× +t , +H× +t = +� +Hinv +t +, ·t +� +⊔ +� +H×sing +t +, ·t +� +, +where +(2.5.4) +Hinv +t += +� +(a, b) ∈ H× +t : |a|2 ̸= t |b|2� +, +and +H×sing +t += +� +(a, b) ∈ H× +t : |a|2 = t |b|2� +, +whenever t ≥ 0 in R. +Theorem 17. For t ≥ 0 in R, the t-scaled hypercomplex monoid H× +t is algebraically +decomposed to be +H× +t = Hinv +t +⊔ H×sing +t +, +where Hinv +t +is the group, and H×sing +t +is the semigroup without identity in (2.5.4). +Proof. The algebraic decomposition, +H× +t = Hinv +t +⊔ H×sing +t +, +of the t-scaled hypercomplex monoid H× +t is obtained by the set-theoretic decompo- +sition (2.5.1) of H× +t , the above two lemmas, and (2.5.4). +By the above theorem, one can have the following concepts whenever a given +scale t is nonnegative in R. +Definition 18. Let t ≥ 0 in R, and H× +t , the t-scaled hypercomplex monoid. The +algebraic block, +Hinv +t += +�� +(a, b) ∈ H× +t : |a|2 ̸= t |b|2� +, ·t +� +, +is called the group-part of H× +t (or, of Ht), and the other algebraic block, +H×sing +t += +�� +(a, b) ∈ H× +t : |a|2 = t |b|2� +, ·t +� +, +is called the semigroup-part of H× +t (or, of Ht). +By the above definition, Theorem 17 can be re-stated that: if a scale t is non- +negative in R, then the t-scaled hypercomplex monoid H× +t is decomposed to be the +group-part Hinv +t +and the semigroup-part H×sing +t +. +One may / can say that if t < 0 in R, then the semigroup-part H×sing +t +is empty +in H× +t . Indeed, for any scale t ∈ R, the t-scaled hypercomplex monoid Ht is decom- +posed to be (2.5.4). As we have seen in this section, if t ≥ 0, then the semigroup-part +H×sing +t +is nonempty, meanwhile, as we considered in Section 2.4, if t < 0, then the +semigroup-part H×sing +t +is empty, equivalently, the t-scaled hypercomplex monoid +H× +t is identified with its group-part Hinv +t +, i.e., H× +t = Hinv +t +in Ht, whenever t < 0. + +OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS +13 +Corollary 19. For every t ∈ R, the t-scaled hypercomplex monoid H× +t is partitioned +by +H× +t = Hinv +t +⊔ H×sing +t +, +where the group-part Hinv +t +and the semigroup-part H×sing +t +are in the sense of (2.5.4). +In particular, if t < 0, then +H×sing +t += Ø ⇐⇒ H× +t = Hinv +t +; +meanwhile, if t ≥ 0, then H×sing +t +is a non-empty proper subset of H× +t . +Proof. It is shown conceptually by the discussion of the very above paragraph. +Also, see Theorems 11 and 17. +3. Spectral Analysis on {Ht}t∈R Under +�� +C2, πt +�� +t∈R +Throughout this section, we fix an arbitrary scale t ∈ R, and the corresponding +t-scaled hypercomplex ring, +Ht = +� +C2, +, ·t +� +, +containing its hypercomplex monoid H× +t = +� +H× +t , ·t +� +. In Section 2, we showed that +for a scale t ∈ R, the monoid H× +t is partitioned by +H× +t = Hinv +t +⊔ H×sing +t +, +where Hinv +t +is the group-part, and H×sing +t +is the semigroup-part of Ht. In particular, +if t < 0, then the semigroup-part H×sing +t +is empty in H× +t , equivalently, H× +t = Hinv +t +in Ht, meanwhile, if t ≥ 0, then H×sing +t +is a non-empty proper subset of H× +t . +Motivated by such an analysis of invertibility on Ht, we here consider spectral +analysis on Ht. +3.1. Hypercomplex-Spectral Forms on Ht. For t ∈ R, let Ht be the t-scaled +hypercomplex ring realized to be +Ht +2 = πt (Ht) = +�� a +tb +b +a +� +∈ M2 (C) : (a, b) ∈ Ht +� +, +in M2 (C) under the Hilbert-space representation Πt = +� +C2, πt +� +of Ht. +Let (a, b) ∈ Ht be an arbitrary element with +πt (a, b) = [(a, b)]t = +� a +tb +b +a +� +∈ Ht +2. +Then, in a variable z on C, +det ([(a, b)]t − z [(1, 0)]t) = det + + +a − z +tb +b +a − z + + += (a − z) (a − z) − t |b|2 += |a|2 − az − az + z2 − t |b|2 += z2 − (a + a) z + +� +|a|2 − t |b|2� += z2 − 2Re (a) z + det ([(a, b)]t), +(3.1.1) +where Re (a) is the real part of a in C, and +det ([(a, b)]t) = |a|2 − t |b|2 , + +14 +DANIEL ALPAY AND ILWOO CHO +by (2.4.2). Thus, the equation, +det ([(a, b)]t − z [(1, 0)]t) = 0, +in a variable z on C, has its solutions, +z = +2Re (a) ± +� +4Re (a)2 − 4det ([(a, b)]t) +2 +, +⇐⇒ +(3.1.2) +z = Re (a) ± +� +Re (a)2 − det ([(a, b)]t). +Recall that a matrix A ∈ Mn (C), for any n ∈ N, has its spectrum, +spec (A) = {λ ∈ C : det (A − λIn) = 0} , +equivalently, +(3.1.3) +spec (A) = {λ ∈ C : ∃η ∈ Cn, s.t., Aη = λη} , +if and only if +spec (A) = {λ ∈ C : A − λIn is not invertible in Mn (C)} , +as a nonempty discrete (compact) subset of C, where In is the identity matrix of +Mn (C) (e.g., [8]). More generally, if T ∈ B (H) is an operator on a Hilbert space +H, then the spectrum σ (T ) of T is defined to be a nonempty compact subset, +σ (T ) = {z ∈ C : T − zIH is not invertible on H} , +where IH is the identity operator of B (H). Remark that if H is infinite-dimensional, +then σ (T ) is not a discrete subset of C as in (3.1.3), in general (e.g., [9]). +Theorem 20. Let (a, b) ∈ Ht realized to be [(a, b)]t ∈ Ht +2. Then +spec ([(a, b)]t) = +� +Re (a) ± +� +Re (a)2 − det ([(a, b)]t) +� +, +in C. More precisely, if +a = x + yi, b = u + vi ∈ C, +with x, y, u, v ∈ R and i = √−1 in C, then +spec ([(a, b)]t) = +� +x ± i +� +y2 − tu2 − tv2 +� +in C. +(3.1.4) +Proof. The realization [(a, b)]t = +� a +tb +b +a +� +∈ Ht +2 of a hypercomplex number +(a, b) ∈ Ht has its spectrum, +spec ([(a, b)]t) = +� +Re (a) ± +� +Re (a)2 − +� +|a|2 − t |b|2�� +, +in C, by (3.1.2) and (3.1.3). If +a = x + yi, and b = u + vi in C, +with x, y, u, v ∈ R and i = √−1 in C, then +Re (a) = x, +and +|a|2 − t |b|2 = +� +x2 + y2� +− t +� +u2 + v2� +, + +OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS +15 +in R, and hence, +spec ([(a, b)]t) = +� +x ± +� +−y2 + tu2 + tv2 +� +, +if and only if +spec ([(a, b)]t) = +� +x ± i +� +y2 − tu2 − tv2 +� +, +in C. Therefore, the set-equality (3.1.4) holds. +From below, for our purposes, we let +a = x + yi and b = u + vi in C, +with +(3.1.5) +x, y, u, v ∈ R, and i = +√ +−1. +The above theorem can be refined by the following result. +Corollary 21. Let (a, b) ∈ Ht, realized to be [(a, b)]t ∈ Ht +2, satisfy (3.1.5). +(3.1.6) If Im (a)2 = t |b|2 in R, where Im (a) is the imaginary part of a in C, then +spec ([(a, b)]t) = {x} = {Re (a)} in R. +(3.1.7) If Im (a)2 < t |b|2 in R, then +spec ([(a, b)]t) = +� +x ± +� +tu2 + tv2 − y2 +� +in R. +(3.1.8) If Im (a)2 > t |b|2 in R, then +spec ([(a, b)]t) = +� +x ± i +� +y2 − tu2 − tv2 +� +in C \ R. +Proof. For (a, b) ∈ Ht, satisfying (3.1.5), one has +spec ([(a, b)]t) = +� +x ± i +� +y2 − tu2 − tv2 +� +, +by (3.1.4). So, one can verify that: (i) if y2 − tu2 − tv2 = 0, equivalently, if +Im (a)2 = t |b|2 in R, +then spec ([(a, b)]t) = +� +x ± i +√ +0 +� += {x} in R; (ii) if y2 − tu2 − tv2 < 0, equivalently, +if +Im (a)2 < t |b|2 in R, +then +x ± i +� +y2 − tu2 − tv2 = x ± i +� +− |y2 − tu2 − tv2|, +implying that +x ± i +� +y2 − tu2 − tv2 = x ± i2� +tu2 + tv2 − y2, +and hence, +spec ([(a, b)]t) = +� +x ∓ +� +tu2 + tv2 − y2 +� +in R; +and, finally, (iii) if y2 − tu2 − tv2 > 0, equivalently, if +Im (a)2 > t |b|2 in R, +then +spec ([(a, b)]t) = +� +x ± i +� +y2 − tu2 − tv2 +� +, +contained in C \ R. + +16 +DANIEL ALPAY AND ILWOO CHO +Therefore, the refined statements (3.1.6), (3.1.7) and (3.1.8) of the spectrum +(3.1.4) of [(a, b)]t hold true. +By the above corollary, one immediately obtains the following result. +Corollary 22. Suppose (a, b) ∈ Ht. If Im (a)2 ≤ t |b|2, then +spec ([(a, b)]t) ⊂ R; +meanwhile, if Im (b)2 > t |b|2, then +spec ([(a, b)]t) ⊂ (C \ R) , in C. +Proof. It is shown by (3.1.6), (3.1.7) and (3.1.8). +Also, we have the following result. +Theorem 23. Assume that the fixed scale t ∈ R is negative, i.e., t < 0 in R. If +(a, b) ∈ Ht, with b ̸= 0 in C, +then +spec ([(a, b)]t) ⊂ (C \ R) in C. +(3.1.9) +Meanwhile, if b = 0 in C for (a, b) ∈ Ht, then +a ∈ R =⇒ spec ([(a, 0)]t) = {a} in R, +and +(3.1.10) +a ∈ C \ R =⇒ spec ([(a, 0)]t) = {a, a} in C \ R. +Proof. Assume that the scale t is given to be negative in R. Then, for any (a, b) ∈ +Ht, one immediately obtains that +Im (a)2 ≥ t |b|2 , +because the left-hand side, Im (a)2, is nonnegative, but the right-hand side, t |b|2 +is either negative or zero in R by the negativity of t. +Suppose b ̸= 0 in C, equivalently, |b|2 > 0, implying t |b|2 < 0 in R. Then +Im (a)2 > t |b|2 in R. +Thus, by (3.1.8), the spectra, spec ([(a, b)]t), of the realizations [(a, b)]t of (a, b) ∈ +Ht, with b ̸= 0, is contained in C \ R. It proves the relation (3.1.9). +Meanwhile, if a = Re (a), and b = 0 in C, then +0 = Im (a)2 ≤ 0 = t · 0 in R, +implying that +spec ([(a, 0)]t) ⊂ R in C, +by (3.1.6). However, if Im (a) ̸= 0, and b = 0, then +Im (a)2 > 0 = t · 0 in R, +and hence, +spec ([(a, 0)]t) ⊂ (C \ R) in C. +So, the relation (3.1.10) is proven. +The above theorem specifies Theorem 19 for the case where t < 0 in R, by (3.1.9) +and (3.1.10). + +OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS +17 +Theorem 24. Assume that t = 0 in R. If (a, b) ∈ H0 with Im (a) ̸= 0 in C, then +spec ([(a, b)]t) ⊂ (C \ R) in C. +(3.1.11) +Meanwhile, if Im (a) = 0, then +spec ([(a, b)]t) ⊂ R in C. +(3.1.12) +Proof. Suppose the fixed scale t is zero in R. Then, for any hypercomplex number +(a, b) ∈ H0, one has +[(a, b)]0 = +� a +0 +b +a +� +∈ H0 +2, +and hence, +Im (a)2 ≥ 0 = 0 · |b|2 in R. +In particular, if Im (a) ̸= 0 in C, then the above inequality becomes +Im (a)2 > 0 in R, +implying that +spec ([(a, b)]t) ⊂ (C \ R) in C, +by (3.1.8). i.e., for all (a, b) ∈ H0, with a ∈ C with Im (a) ̸= 0, and b ∈ C arbitrary, +the spectra of the realizations of such (a, b) are contained in C \ R. It shows the +relation (3.1.11) holds. +Meanwhile, if Im (a) = 0 in C, then one has +Im (a)2 = 0 ≥ 0 = 0 · |b|2 in R. +So, by (3.1.6), we have +spec ([(a, b)]t) ⊂ R in C. +Therefore, the relation (3.1.12) holds true, too. +The above theorem specifies Theorem 19 for the case where a scale t is zero in +R, by (3.1.11) and (3.1.12). +Theorem 25. Assume that the fixed scale t is positive in R. Then the t-scaled +hypercomplex ring Ht is decomposed to be +Ht = H+ +t ⊔ H−0 +t , +with +(3.1.13) +H+ +t = +� +(a, b) ∈ Ht : Im (a)2 > t |b|2� +, +and +H−0 +t += +� +(a, b) ∈ Ht : Im (a)2 ≤ t |b|2� +, +where ⊔ is the disjoint union. Moreover, if (a, b) ∈ H+ +t , then +spec ([(a, b)]t) ⊂ (C \ R) ; +(3.1.14) +meanwhile, if (a, b) ∈ H−0 +t , then +spec ([(a, b)]t) ⊂ R in C. +(3.1.15) + +18 +DANIEL ALPAY AND ILWOO CHO +Proof. Suppose that t > 0 in R. Then one can decompose the t-scaled hypercomplex +ring Ht by +Ht = H+ +t ⊔ H−0 +t , +with +(3.1.16) +H+ +t = +� +(a, b) ∈ Ht : Im (a)2 > t |b|2� +, +and +H−0 +t += +� +(a, b) ∈ Ht : Im (a)2 ≤ t |b|2� +, +set-theoretically. Thus, the partition (3.1.13) holds by (3.1.16). +By Theorem 19 and Corollary 20, if (a, b) ∈ H+ +t , then +spec ([(a, b)]t) ⊂ (C \ R) , +meanwhile, if (a, b) ∈ H−0 +t , then +spec ([(a, b)]t) ⊂ R, in C. +So, the relations (3.1.14) and (3.1.15) are proven. +The above theorem specifies Theorem 19 for the cases where a fixed scale t is +positive in R, by (3.1.14) and (3.1.15), up to the decomposition (3.1.13). +In fact, one can realize that, for “all” t ∈ R, the corresponding t-scaled hyper- +complex ring Ht is partitioned to be +Ht = H+ +t ⊔ H−0 +t , +where H+ +t and H−0 +t +are in the sense of (3.1.13). Especially, Theorems 22, 23 and +24 characterize the above decomposition case-by-case, based on Theorem 19 and +Corollary 20. So, we obtain the following universal spectral properties on Ht. +Corollary 26. Let t ∈ R be an arbitrarily fixed scale for Ht. Then +Ht = H+ +t ⊔ H−0 +t , set-theoretically, +where +� +H+ +t , H−0 +t +� +is a partition in the sense of (3.1.13) for t. Moreover, if (a, b) ∈ +H+ +t , then +spec ([(a, b)]t) ⊂ (C \ R) , +meanwhile, if (a, b) ∈ H−0 +t , then +spec ([(a, b)]t) ⊂ R in C. +Especially, if t < 0, then H−0 +t += {(0, 0)}, equivalently, H× +t = H+ +t . +Proof. This corollary is nothing but a summary of Theorems 22, 23 and 24. +It is not hard to check the converses of the statements of Corollary 25 hold true, +too. +Theorem 27. Let Ht = H+ +t ⊔H−0 +t +be the fixed t-scaled hypercomplex ring for t ∈ R. +(3.1.17) (a, b) ∈ H+ +t , if and only if spec ([(a, b)]t) ⊂ (C \ R). +(3.1.18) (a, b) ∈ H−0 +t , if and only if spec ([(a, b)]t) ⊂ R. + +OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS +19 +Proof. First, assume that (a, b) ∈ H+ +t in Ht. Then, by Corollary 25, +spec ([a, b]t) ⊂ (C \ R) . +Now, suppose that +spec ([a, b]t) ⊂ R in C, +and assume that (a, b) ∈ H+ +t . Then, (a, b) is contained in H−0 +t , equivalently, it +cannot be an element of H+ +t , by (3.1.6), (3.1.7), (3.1.10), (3.1.12) and (3.1.15). It +contradicts our assumption. Therefore, +(a, b) ∈ H+ +t ⇐⇒ spec ([(a, b)]t) ⊂ (C \ R) . +Thus, the statement (3.1.17) holds. +By the decomposition (3.1.13), the statement (3.1.18) holds true, by (3.1.17). +By the above theorem, we obtain the following result. +Corollary 28. Let Ht be the t-scaled hypercomplex ring for an arbitrary t ∈ R, +and suppose it is decomposed to be +Ht = H+ +t ⊔ H−0 +t , +as in (3.1.13). Assume that a given element (a, b) satisfies the condition (3.1.5). +Then +(3.1.19) (a, b) ∈ H+ +t , if and only if +spec ([(a, b)]t) = +� +x ± i +� +y2 − tu2 − tv2 +� +⊂ (C \ R) . +(3.1.20) (a, b) ∈ H−0 +t , if and only if either +spec ([(a, b)]t) = + + + + + +{x} +if Im (a)2 = t |b|2 +� +x ± +� +tu2 + tv2 − y2 +� +if Im (a)2 < t |b|2 , +in R. +Proof. The statement (3.1.19) holds by (3.1.9) and (3.1.17). Meanwhile, the state- +ment (3.1.20) holds by (3.1.10) and (3.1.18). +Recall that a Hilbert-space operator T ∈ B (H) is self-adjoint, if T ∗ = T in +B (H), where T ∗ is the adjoint of T (See Section 5 below). It is well-known that T +is self-adjoint, if and only if its spectrum is contained in R in C. So, one obtains +the following result. +Proposition 29. A hypercomplex number (a, b) ∈ H−0 +t +in Ht, if and only if the +realization [(a, b)]t ∈ Ht +2 is self-adjoint “in M2 (C).” +Proof. (⇒) Suppose (a, b) ∈ H−0 +t +in Ht. Then spec ([(a, b)]t) ⊂ R in C, implying +that [(a, b)]t is self-adjoint in M2 (C). +(⇐) Suppose [(a, b)]t ∈ Ht +2 is self-adjoint in M2 (C), and assume that (a, b) /∈ H−0 +t , +equivalently, (a, b) ∈ H+ +t in Ht. Then, +spec ([(a, b)]t) ⊂ (C \ R) in C, +and hence, [(a, b)]t is not self-adjoint in M2 (C). It contradicts our assumption that +it is self-adjoint. + +20 +DANIEL ALPAY AND ILWOO CHO +Equivalent to the above proposition, one can conclude that (a, b) ∈ H+ +t in Ht, +if and only if [(a, b)]t is not be self-adjoint in M2 (C). The self-adjointness of re- +alizations of hypercomplex numbers would be considered more in detail in Section +5. +3.2. The Scaled-Spectralizations {σt}t∈R. In this section, we fix an arbitrary +scale t ∈ R, and the corresponding hypercomplex ring Ht, containing the t-scaled +hypercomplex monoid H× +t += (Ht \ {(0, 0)} , ·t). +Recall that H× +t +is algebraically +decomposed to be +H× +t = Hinv +t +⊔ H×sing +t +, +with +(3.2.1) +Hinv +t += +� +(a, b) : |a|2 ̸= t |b|2� +, the group-part, +and +H×sing +t += +� +(a, b) : |a|2 = t |b|2� +, the semigroup-part, +as in (2.5.4). Therefore, the t-scaled hypercomplex ring is set-theoretically decom- +posed to be +Ht = Hinv +t +⊔ {(0, 0)} ⊔ H×sing +t += Hinv +t +⊔ Hsing +t +, +(3.2.2) +by (3.2.1), where +Hsing +t +denote += +{(0, 0)} ⊔ H×sing +t +in (3.2.2). +Also, the ring Ht is spectrally decomposed to be +Ht = H+ +t ⊔ H−0 +t , +with +(3.2.3) +H+ +t = +� +(a, b) : Im (a)2 > t |b|2� +, +and +H−0 +t += +� +(a, b) : Im (a)2 ≤ t |b|2� +, +satisfying that: (a, b) ∈ H+ +t +if and only if spec ([(a, b)]t) ⊂ (C \ R); meanwhile, +(a, b) ∈ H−0 +t +if and only if spec ([(a, b)]t) ⊂ R, by (3.1.19) and (3.1.20). +Corollary 30. Let Ht be the t-scaled hypercomplex ring for t ∈ R. Then it is +decomposed to be +Ht = +� +Hinv +t +∩ H+ +t +� +⊔ +� +Hinv +t +∩ H−0 +t +� +� +Hsing +t +∩ H+ +t +� +⊔ +� +Hsing +t +∩ H−0 +t +� +, +(3.2.4) +set-theoretically. +Proof. It is proven by (3.2.2) and (3.2.3). +Observe now that if (a, 0) ∈ Ht, then +[(a, 0)]t = +� a +0 +0 +a +� +in Ht +2, +satisfying +(3.2.5) +spec ([(a, 0)]t) = {a, a} in C. + +OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS +21 +Indeed, by (3.1.4), if (a, 0) ∈ Ht satisfying a = x + yi ∈ C with x, y ∈ R, then +spec ([(a, b)]t) = +� +x ± i +� +y2 +� += {x ± |y| i} = {x ± yi} , +implying (3.2.5), where |y| is the absolute value of y in R. +Motivated by (3.2.3), (3.2.4) and (3.2.5), we define a certain C-valued function +σt from Ht. Define a function, +σt : Ht → C, +by +(3.2.6) +σt ((a, b)) +def += + + + +a = x + yi +if b = 0 in C +x + i +� +y2 − tu2 − tv2 +if b ̸= 0 in C, +for all (a, b) ∈ Ht satisfying the condition (3.1.5): +a = x + yi and b = u + vi in C, +with x, y, u, v ∈ R and i = √−1. +Remark that such a morphism σt is indeed a well-defined function assigning all +hypercomplex numbers of Ht to complex numbers of C. Moreover, by the very +definition (3.2.6), it is surjective. But it is definitely not injective. For instance, +even though +ξ = (1 + 3i, −1 + i) and η = (1 − 3i, 1 − i) +are distinct in Ht, one has +σt (ξ) = 1 + i +√ +9 − 2t = σt (η) , +by (3.2.6). +Definition 31. The surjection σt : Ht → C of (3.2.6) is called the t(-scaled)- +spectralization on Ht. The images {σt (ξ)}ξ∈Ht are said to be t(-scaled)-spectral +values. +From below, we also understand each t-spectral value σt (ξ) ∈ C of a +hypercomplex number ξ ∈ Ht as a hypercomplex number (σt (ξ) , 0) in Ht. i.e., +such an assigned hypercomplex number (σt (ξ) , 0) from the t-spectral value σt (ξ) +of ξ is also called the t-spectral value of ξ. +By definition, all t-spectral values are not only C-quantities for many hypercom- +plex numbers of Ht whose realizations of Ht +2 share the same eigenvalues, but also +hypercomplex numbers of Ht, whose first coordinates are the value and the second +coordinates are 0. +Definition 32. Let ξ ∈ Ht be a hypercomplex number inducing its t-spectral value +w +denote += +σt (ξ) ∈ C, also understood to be η = (w, 0) ∈ Ht. The corresponding +realization, +[η]t = +� +w +t · 0 +0 +w +� += + + +σt (ξ) +0 +0 +σt (ξ) + + ∈ Ht +2 +is called the t(-scaled)-spectral form of ξ. By Σt (ξ), we denote the t-spectral form +of ξ ∈ Ht. + +22 +DANIEL ALPAY AND ILWOO CHO +Note that the conjugate-notation in Definition 30 is symbolic in the sense that: +if t > 0, and +σt (ξ) = 1 + i +√ +1 − 5t = 1 − +√ +5t − 1, +(and hence, σt (ξ) ∈ R), then the symbol, +σt (ξ) +means += +1 − i +√ +1 − 5t = 1 + +√ +5t − 1, +in R. i.e., the conjugate-notation in Definition 30 has a symbolic meaning containing +not only the usual conjugate on C, but also the above computational meaning on +R. +Remark-and-Assumption 3.2.1. (From below, RA 3.2.1) The conjugate-notation +in Definition 30 is symbolic case-by-case. If the t-spectral value σt (ξ) is in C, then +σt (ξ) means the usual conjugate. Meanwhile, if t-spectral value +σt (ξ) = x + +� +tu2 + tv2 − y2, +with +tu2 + tv2 − y2 ≥ 0, in R, +then +σt (ξ) = x − +� +tu2 + tv2 − y2 in R, +where ξ ∈ Ht satisfies the condition (3.1.5). + +For instance, if ξ1 = (−2 − i, 0) ∈ Ht, then the t-spectral value is +σt (ξ1) = −2 − i in C, +inducing the t-spectral form, +Σt (ξ1) = + + +−2 − i +0 +0 +−2 + i + + in Ht +2; +meanwhile, if ξ2 = (−2 − i, 1 + 3i) ∈ Ht, then the t-spectral value is +w +denote += +σt (ξ2) = −2 + i +√ +1 − 10t, +inducing the t-spectral form, +Σt (ξ2) = +� w +0 +0 +w +� += + + +−2 + i√1 − 10t +0 +0 +−2 − i√1 − 10t + + , +where w is symbolic in the sense of RA 3.2.1; if t ≤ 0, then +Σt (ξ2) = + + +−2 + i√1 − 10t +0 +0 +−2 − i√1 − 10t + + , +meanwhile, if t > 0, then +Σt (ξ2) = + + +−2 + √10t − 1 +0 +0 +−2 − √10t − 1 + + , +in Ht +2. + +OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS +23 +Definition 33. Two hypercomplex numbers ξ, η ∈ Ht are said to be t(-scaled)- +spectral-related, if +σt (ξ) = σt (η) in C, +equivalently, +Σt (ξ) = Σt (η) in Ht +2. +On the t-hypercomplex ring Ht, the t-spectral relation of Definition 31 is an +equivalent relation. Indeed, +σt (ξ) = σt (ξ) , ∀ξ ∈ Ht; +and if ξ and η are t-spectral related in Ht, then +σt (ξ) = σt (η) ⇐⇒ σt (η) = σt (ξ) , +and hence, η and ξ are t-spectral related in Ht; and if ξ1 and ξ2 are t-spectral +related, and if ξ2 and ξ3 are t-spectral related, then +σt (ξ1) = σt (ξ2) = σt (ξ3) in C, +and hence, ξ1 and ξ3 are t-spectral related. +Proposition 34. The t-spectral relation on Ht is an equivalence relation. +Proof. The t-spectral relation is reflexive, symmetric and transitive on Ht, by the +discussion of the very above paragraph. +Since the t-spectral relation is an equivalence relation, each element ξ of Ht has +its equivalence class, +�ξ +def += {η ∈ Ht : η is t-related to ξ} , +and hence, the corresponding quotient set, +� +Ht +def += +� +�ξ : ξ ∈ Ht +� +, +(3.2.7) +is well-defined to be the set of all equivalence classes. +Theorem 35. Let � +Ht be the quotient set (3.2.7) induced by the t-spectral relation +on Ht. Then +� +Ht and C are equipotent. +(3.2.8) +Proof. It is not difficult to check that, for any z ∈ C, there exist ξ ∈ Ht, such that +z = σt (ξ) by the surjectivity of the t-spectralization σt. It implies that there exists +(z, 0) ∈ Ht, such that +� +(z, 0) = �ξ in � +Ht, whenever z = σt (ξ) . +Thus, set-theoretically, we have +� +Ht = +� +� +(z, 0) : z ∈ C +� equip += +C, +where “ +equip += ” means “being equipotent (or, bijective) to.” Therefore, the relation +(3.2.8) holds. +The above equipotence (3.2.8) of the quotient set � +Ht of (3.2.7) with the complex +numbers C shows that the set C classifies Ht, for “every” t ∈ R, up to the t-spectral +relation. + +24 +DANIEL ALPAY AND ILWOO CHO +3.3. Similarity on M2 (C) and The t-Scaled-Spectral Relation on Ht. In +Section 3.2, we defined the t-spectralization σt on the t-scaled hypercomplex ring +Ht, for a fixed scale t∈ R, and it induces the t-spectral forms {Σt (ξ)}ξ∈Ht in +Ht +2 as complex diagonal matrices whose main diagonals are the eigenvalues of the +realizations {[ξ]t}ξ∈Ht, under the symbolic understanding RA 3.2.1. Moreover, σt +lets the set C classify Ht by (3.2.8) under the t-spectral relation. +Independently, we showed in [2] and [3] that: on the quaternions H = H−1, +the (−1)-spectral relation, called the quaternion-spectral relation in [2] and [3], is +equivalent to the similarity “on H−1 +2 ,” as equivalence relations. Here, the similarity +“on H−1 +2 ” means that: the realizations [q1]−1 and [q2]−1 of two quaternions q1, q2 ∈ +H−1 are similar “in H−1 +2 ,” if there exists invertible element U “in H−1 +2 ,” such that +[q2]−1 = U −1 [q1]−1 U in H−1 +2 . +Here, we consider such property for an arbitrary scale t ∈ R. +Recall that, we +showed in [2] and [3] that: the (−1)-spectral form Σ−1 (η) and the realization [η]−1 +are similar “in H−1 +2 ,” for “all” quaternions which are the (−1)-scaled hypercomplex +numbers η ∈ H−1 = H. Are the t-spectral relation on Ht and the similarity on Ht +2 +same as equivalence relations? In conclusion, the answer is negative in general. +Two matrices A and B of Mn (C), for any n ∈ N, are said to be similar, if there +exists an invertible matrix U ∈ Mn (C), such that +B = U −1AU in Mn (C) . +Remember that if two matrices A and B are similar, then (i) they share the same +eigenvalues, (ii) they have the same traces, and (iii) their determinants are same +(e.g., [8] and [9]). We here focus on the fact (iii): the similarity of matrices implies +their identical determinants, equivalently, if +det (A) ̸= det (B) , +then A and B are not similar in Mn (C). +Definition 36. Let A, B ∈ Ht +2 be realizations of certain hypercomplex numbers +of Ht, for t ∈ R. They are said to be similar “in Ht +2,” if there exists an invertible +U ∈ Ht +2, such that +B = U −1AU in Ht +2. +By abusing notation, we say that two hypercomplex numbers ξ and η are similar +in Ht, if their realizations [ξ]t and [η]t are similar in Ht +2. +Let (a, b) ∈ Ht be a hypercomplex number satisfying the condition (3.1.5) and +(a, b) ̸= (0, 0). Then it has +[(a, b)]t = +� a +tb +b +a +� +∈ Ht +2, +σt ((a, b)) = x + i +� +y2 − tu2 − tv2 let += w ∈ C, +and +(3.3.1) +Σt ((a, b)) = +� +w +0 +0 +w +� +∈ Ht +2, +where w is symbolic under RA 3.2.1. Observe that +det ([(a, b)]t) = |a|2 − t |b|2 = +� +x2 + y2� +− t +� +u2 + v2� +, + +OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS +25 +and +(3.3.2) +det (Σt ((a, b))) = |w|2 = x2 + +��y2 − tu2 − tv2�� , +by (3.3.1). +These computations in (3.3.2) show that, in general, [(a, b)]t and +Σt ((a, b)) are “not” similar “as matrices of M2 (C),” and hence, not similar in +Ht +2. Indeed, for instance, if +t > 0, and |a|2 < t |b|2 , +then det ([(a, b)]t) < 0, but det (Σt ((a, b))) > 0 in R, by (3.3.2), implying that +det ([(a, b)]t) ̸= det (Σt ((a, b))) in general, +showing that [(a, b)]t and Σt ((a, b)) are not similar in M2 (C), and hence, they are +not similar in Ht +2, in general. +Proposition 37. Let (a, b) ∈ Ht be “nonzero” hypercomplex number satisfying +|a|2 < t |b|2 in R. Then the realization [(a, b)]t and the t-spectral form Σt ((a, b)) +are not similar “in Ht +2.” +Proof. Suppose (a, b) ∈ Ht satisfies (a, b) ̸= (0, 0) and |a|2 < t |b|2, for t > 0. And +assume that [(a, b)]t and Σt ((a, b)) are similar in Ht +2. Since they are assumed to be +similar, their determinants are identically same. However, +det ([(a, b)]t) < 0 and det (Σt ((a, b))) > 0, +by (3.3.2). It contradicts our assumption that they are similar in Ht +2. +The above proposition confirms that the realizations and the corresponding t- +spectral forms of a t-scaled hypercomplex number are not similar in Ht +2, in general. +Consider that, in the quaternions H = H−1, since the scale is t = −1 < 0 in R, +det +� +[ξ]−1 +� += det (Σ−1 (ξ)) ≥ 0, ∀ξ ∈ H−1, +and it is proven that [ξ]−1 and Σ−1 (ξ) are indeed similar in H−1 +2 , for “all” ξ ∈ H−1 +in [2] and [3], which motivates a question: if a scale t < 0 in R, then +det ([η]t) = det (Σt (η)) ≥ 0, ∀η ∈ Ht, +by (3.3.2); so, are the realizations [η]t and the corresponding t-spectral forms Σt (η) +similar in Ht +2 as in the case of t = −1? +First of all, we need to recall that if t < 0, then the t-scaled hypercomplex ring +Ht forms a noncommutative field, since the t-scaled hypercomplex monoid H× +t is a +non-abelian group, by (2.4.8). It allows us to use similar techniques of [2] and [3]. +Assumption. In the rest part of this section, a given scale t ∈ R is automatically +assumed to be negative in R. + +Assume that (a, 0) ∈ Ht, where t < 0. Then +[(a, 0)]t = +� a +0 +0 +a +� += Σt ((a, 0)) , +in Ht +2, since σt ((a, 0)) = a in C. So, clearly, [(a, 0)]t and Σt ((a, 0)) are similar +in Ht +2, because they are equal in Ht +2. Indeed, there exist diagonal matrices with +nonzero real entries, +X = [(x, 0)]t ∈ Ht +2, with x = x + 0i ∈ C, x ̸= 0, + +26 +DANIEL ALPAY AND ILWOO CHO +such that +[(a, 0)]t = X−1 (Σt (a, 0)) X in Ht +2. +Thus, we are interested in the cases where (a, b) ∈ Ht with b ∈ C× = C \ {0}. +Lemma 38. Let t < 0 in R, and (a, 0) ∈ Ht, a hypercomplex number. +Then +the realization [(a, 0)]t and the t-spectral form Σt ((a, 0)) are identically same in +Ht +2, and hence, they are similar in Ht +2. (Remark that, in fact, the scale t is not +necessarily negative in R here.) +Proof. It is proven by the discussion of the very above paragraph. Indeed, one has +[(a, 0)]t = Σt ((a, 0)) in Ht +2, +since σt ((a, 0)) = a in C. +Let h = (a, b) ∈ Ht with b ∈ C×, satisfying the condition (3.1.5), where t < 0, +having its realization, +[h]t = +� a +tb +b +a +� += + + +x + yi +t (u + vi) +u − vi +x − yi + + , +and its t-spectral form, +Σt (h) = + + +x + i +� +y2 − tu2 − tv2 +0 +0 +x − i +� +y2 − tu2 − tv2 + + let += +� w +0 +0 +w +� +, +in Ht +2. Since t < 0 and b ̸= 0 (by assumption), the t-spectral value w = σt (h) is a +C-quantity with its conjugate w. Define now a matrix, +Qh +def += + + + +1 +t +� +w−a +tb +� +w−a +tb +1 + + + in M2 (C) . +Remark that, by the assumption that t < 0 and b ̸= 0, this matrix is well-defined. +Furthermore, one can immediately recognize that Qh ∈ Ht +2. i.e., +Qh = +�� +1, +� w−a +tb +��� +t ∈ Ht +2. +(3.3.3) +One can find that the element Qh ∈ Ht +2 of (3.3.3) is indeed invertible by our +negative-scale assumption, since +det (Qh) = 1 − t +���� +w − a +tb +���� +2 +≥ 1, since t < 0, +implying that +det (Qh) ̸= 0 ⇐⇒ Qh is invertible in Ht +2. +Observe now that +QhΣt (h) = + + + +w +t +� +w2−aw +tb +� +w2−aw +tb +w + + + + +OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS +27 +and +(3.3.4) +[h]t Qh = + + + +w +t +� +a +� +w−a +tb +� ++ b +� +a +� w−a +tb +� ++ b +w + + + , +in Ht +2. Now, let’s compare the (1, 2)-entries of resulted matrices in (3.3.4). The +(1, 2)-entry of the element QhΣt (h) is +t +� +w2−aw +tb +� += w(w−a) +b += +� +x+i√ +y2−tu2−tv2 +�� +i√ +y2−tu2−tv2−yi +� +u+vi += ix +√ +R−xyi−R+y +√ +R +u+vi +, +where +(3.3.5) +R +denote += +y2 − tu2 − tv2 in R, +and the (1, 2)-entry of the matrix [h]t Qh is +t +� +a +� +w−a +tb +� ++ b +� += t +� +a +� w−a +tb +� ++ b +� += t +� +aw−|a|2+t|b|2 +tb +� += aw−|a|2+t|b|2 +b += +(x−yi) +� +x+i√ +y2−tu2−tv2 +� +−(x2+y2)−t(u2+v2) +u+vi += x2+ix +√ +R−xyi+y +√ +R−x2−y2−tu2−tv2 +u+vi += x2+ix +√ +R−xyi+y +√ +R−x2−R +u+vi += ix +√ +R−xyi−R+y +√ +R +u+vi +, +(3.3.6) +where the R-quantity R is in the sense of (3.3.5). As one can see in (3.3.5) and +(3.3.6), the (1, 2)-entries of [h]t Qh and QhΣt (h) are identically same. i.e., +QhΣt (h) = [h]t Qh in Ht +2, +(3.3.7) +where the matrix Qh ∈ Ht +2 is in the sense of (3.3.3). +Lemma 39. Let t < 0 in R, and let h = (a, b) ∈ Ht with b ∈ C×. Then the +realization [h]t and the t-spectral form Σt (h) are similar in Ht +2. +In particular, +there exists +qh = +� +1, t +�w − a +tb +�� +∈ Ht, +having its realization, +Qh = [qh]t = + + + +1 +t +� +w−a +tb +� +w−a +tb +1 + + + ∈ Ht +2, +such that +(3.3.8) +Σt (h) = Q−1 +h [h]t Qh in Ht +2. + +28 +DANIEL ALPAY AND ILWOO CHO +Proof. Under the hypothesis, one obtains that +QhΣt (h) = [h]t Qb in Ht +2, +by (3.3.7). By the invertibility of Qh, we have +Σt (h) = Q−1 +h [h]t Qh in Ht +2, +implying the relation (3.3.8). +The above lemma shows that if a scale t is negative in R, then the realization [h]t +and the t-spectral form Σt (h) are similar in Ht +2, whenever h = (a, b) ∈ Ht satisfies +b ̸= 0 in C. +Theorem 40. If t < 0 in R, then every hypercomplex number h ∈ Ht is similar to +its t-spectral value (σt (h) , 0) ∈ Ht, in the sense that: +[h]t and Σt (h) are similar in Ht +2. +(3.3.9) +Proof. Let h = (a, b) ∈ Ht, for t < 0. If b = 0 in C, then [(a, 0)]t and Σt ((a, 0)) are +similar in Ht +2, by Lemma 38. Indeed, if b = 0, then these matrices are identically +same in Ht +2. Meanwhile, if b ̸= 0 in C, then [h]t and Σt (h) are similar in Ht +2 by +Lemma 39. In particular, if b ̸= 0, then there exists +qh = +� +1, w − a +tb +� +∈ Ht, +such that +Σt (h) = [qh]−1 +t +[h]t [qh]t , +in Ht +2, by (3.3.8). +Therefore, if t < 0, then [h]t and Σt (h) are similar in Ht +2, +equivalently, two hypercomplex numbers h and (σt (h) , 0) are similar in Ht, for all +h ∈ Ht. +The above theorem guarantees that the negative-scale condition on hypercom- +plex numbers implies the similarity of the realizations and the scaled-spectral forms +of them, just like the quaternionic case (whose scale is −1), shown in [2] and [3]. +Theorem 41. If t < 0 in R, then the t-spectral relation on Ht and the similarity +on Ht are same as equivalence relations on Ht. i.e., +t < 0 =⇒ t-spectral relation +equi += similarity on Ht, +(3.3.10) +where “ +equi += ” means “being equivalent to, as equivalence relations.” +Proof. Suppose a negative scale t < 0 is fixed, and let Ht be the corresponding +t-scaled hypercomplex ring. Assume that two hypercomplex numbers h1 and h2 +are t-spectral related. Then their t-spectral values are identical in C, i.e., +σt (h1) = σt (h2) +let += w in C. +Thus the realizations [h1]t and [h2]t are similar to +Σt (h1) = +� w +0 +0 +w +� += Σt (h2) +let += W, +in Ht +2, by (3.3.9). i.e., there exist q1, q2 ∈ Ht such that +[q1]−1 +t +[h1]t [q1]t = W = [q2]−1 +t +[h2]t [q2]t , + +OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS +29 +in Ht +2. So, one obtains that +[h1]t = +� +[q1]t [q2]−1 +t +� +[h2]t +� +[q2]t [q1]−1 +t +� +, +⇐⇒ +[h1]t = +� +[q2]t [q1]−1 +t +�−1 +[h2]t +� +[q2]t [q1]−1 +t +� +, +in Ht +2, implying that [h1]t and [h2]t are similar in Ht +2. +Thus, if h1 and h2 are +t-spectral related, then they are similar in Ht. +Conversely, suppose T1 +denote += +[h1]t and T2 +denote += +[h2]t are similar in Ht +2. Then +there exists U ∈ Ht +2, such that +T1 = U −1T2U in Ht +2. +Since the realizations Tl and the corresponding t-spectral forms Sl +denote += +Σt (hl) are +similar by (3.3.9), say, +Tl = V −1 +l +SlVl in Ht +2, for some Vl ∈ Ht +2, +for all l = 1, 2. Thus, +T1 = U −1T2U = U −1 � +V −1 +2 +S2V2 +� +U, +⇐⇒ +V1S1V −1 +1 += T1 = (V2U)−1 S2 (V2U) , +⇐⇒ +S1 = V −1 +1 +(V2U)−1 S2 (V2U) V1, +⇐⇒ +S1 = (V2UV1)−1 S2 (V2UV1) , +and hence, two matrices S1 and S2 are similar in Ht +2. It means that S1 and S2 +share the same eigenvalues. So, it ie either +S1 = +� +w +0 +0 +w +� += S2, +for some w ∈ C, or +S1 = +� w +0 +0 +w +� +, and S2 = +� w +0 +0 +w +� +, +in Ht +2. However, by the assumption that t < 0, we have +S1 = S2 in Ht +2, +by (3.1.8). It shows that, if the realizations T1 and T2 are similar, then the t-spectral +forms S1 and S2 are identically same in Ht +2, implying that +σt (h1) = σt (h2) in C, +thus h1 and h2 are t-spectral related in Ht. +Therefore, the equivalence (3.3.10) between the t-spectral relation and the simi- +larity on Ht holds, whenever t < 0 in R. + +30 +DANIEL ALPAY AND ILWOO CHO +The above theorem generalizes the equivalence between the quaternion-spectral +relation, which is the (−1)-spectral relation, and the similarity on the quaternions +H−1 = H (e.g., [2] and [3]). +Discussion. How about the cases where given scale t are nonnegative in R, i.e., +t ≥ 0? One may need to consider the decomposition (3.2.4), +Ht = +� +Hinv +t +∩ H+ +t +� +⊔ +� +Hinv +t +∩ H−0 +t +� +� +Hsing +t +∩ H+ +t +� +⊔ +� +Hsing +t +∩ H−0 +t +� +, +of Ht, for t ≥ 0, where +Hinv +t += +� +(a, b) : |a|2 ̸= t |b|2� +, +Hsing +t += +� +(a, b) : |a|2 = t |b|2� +, +H+ +t = +� +(a, b) : Im (a)2 > t |b|2� +, +and +H−0 +t += +� +(a, b) : Im (a)2 ≤ t |b|2� +, +block-by-block. In particular, if +h ∈ Hinv +t +∩ H+ +t , +then it “seems” that the realization [h]t and the t-spectral form Σt (h) are similar +in Ht +2. The proof “may” be similar to the above proofs for negative scales. We +leave this problem for a future project. + +3.4. The t-Spectral Mapping Theorem. In this section, we let a scale t be +arbitrary in R, and let Ht be the t-scaled hypercomplex ring. Let h = (a, b) ∈ Ht +satisfy the condition (3.1.5), and suppose it has its t-spectral value, +σt (h) = x + i +� +y2 − tu2 − tv2 let += w, +and hence, its t-spectral form +Σt (h) = +� +w +0 +0 +w +� +in Ht +2, +under NA 3.2.1. +Now recall that if n ∈ N, and A ∈ Mn (C), and if +f ∈ C[z] +def += + + +g : +g = +m +� +k=0 +zkzk, with +z1, ..., zm ∈ C, for m ∈ N + + + , +then +(3.4.1) +spec (f (A)) = {f (w) : w ∈ spec (A)} , +in C, where C[z] is the polynomial ring in a variable z over C, consisting of all +polynomials in z whose coefficients are from C, and +f (A) = +N +� +k=0 +skAk, with A0 = In, + +OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS +31 +whenever +f (z) = +N +� +k=0 +skzk ∈ C [z] , with s1, ..., sN ∈ C, +where In is the identity matrix of Mn (C), by the spectral mapping theorem (e.g., +[8] and [9]). By (3.4.1), if R[x] is the polynomial ring in a variable x over the real +field R, then +spec (g (A)) = {g (w) : w ∈ spec (A)} in C, +(3.4.2) +for all g ∈ R[x], because R[z] is a subring of C[z] if we identify x to z. +It is shown in [2] and [3] that, for f ∈ C[z], +spec +� +f +� +[ξ]−1 +�� += +� +f (σ−1 (ξ)) , f +� +σ−1 (ξ) +�� +in C, by (3.4.1), but +f +� +σ−1 (ξ) +� +̸= f (σ−1 (ξ)), in general, +and hence, even though the spectral mapping theorem (3.4.1) holds “on M2 (C), +for [ξ]−1 ∈ H−1 +2 ,” it does not hold “on H−1 +2 ,” in general. It demonstrates that, in +a similar manner, the spectral mapping theorem (3.4.1) holds “on M2 (C) ,” but it +does not hold “on the t-scaled realization Ht +2 of Ht,” for t ∈ R, because the spectra +of hypercomplex numbers satisfy +spec ([η]t) = {w, w} , with w = σt (η) , +by (3.1.4), for all η ∈ Ht under RA 3.2.1, just like the quaternionic case of [2] and +[3]. +Observation. For an arbitrary scale t ∈ R, the spectral mapping theorem (3.4.1) +does not hold “on Ht +2.” + +However, in [2] and [3], it is proven that, for all g ∈ R[x], one has +spec +� +g +� +[ξ]−1 +�� += +� +g (σt (ξ)) , g (σt (ξ)) +� +, +in C, by (3.4.2), since +g ∈ R[x] =⇒ g (w) = g (w), ∀w ∈ C. +It means that the “restricted” spectral mapping theorem of (3.4.2) holds “on the +realization H−1 +2 +of the quaternions H−1.” Similarly, we obtain the following result. +Theorem 42. Let ξ ∈ Ht, realized to be [ξ]t ∈ Ht +2. Then, for any g ∈ R[x], +spec (g ([ξ]t)) = +� +g (σt (ξ)) , g (σt (ξ)) +� +, +i.e., +(3.4.3) +spec (g ([ξ]t)) = {g (w) : w ∈ spec ([ξ]t)} in C, ∀t ∈ R. +Proof. By (3.1.4) and (3.2.6), if ξ ∈ Ht, then +spec ([ξ]t) = {w, w} , with w = σt (ξ) , +in C (under the symbolic understanding of RA 3.2.1). For any g = +N +� +k=1 +skxk ∈ R[x], +with s1, ..., sN ∈ R, and N ∈ N, one has that + +32 +DANIEL ALPAY AND ILWOO CHO +g (w) = +N +� +k−1 +skwk = +N +� +k=1 +skwk = +N +� +k=1 +skwk = g (w), +(3.4.4) +in C. It implies that +spec (g ([ξ]t)) = {g (w) , g (w)} = +� +g (w) , g (w) +� +, +in C, by (3.4.2) and (3.4.4). Therefore, the relation (3.4.3) holds true. +One may call the relation (3.4.3), the hypercomplex-spectral mapping theorem, +since it holds for all scales t ∈ R. +4. The Usual Adjoint on Ht +2 in M2 (C) +In this section, we consider how the usual adjoint on M2 (C) = B +� +C2� +acts on +the t-scaled realization Ht +2 of the t-scaled hypercomplex numbers. Throughout this +section, we fix an arbitrary scale t ∈ R, and the corresponding t-scaled hypercom- +plex ring Ht realized to be Ht +2 in M2 (C) under the representation Πt = +� +C2, πt +� +. +Recall that every Hilbert-space operator T acting on a Hilbert space H has its +unique adjoint T ∗ on H. Especially, if T ∈ Mn (C) = B (Cn), for n ∈ N, is a +matrix which is an operator on Cn, then its adjoint T ∗ is determined to be the +conjugate-transpose of T in Mn (C). For instance, +T = +� +a11 +a12 +a21 +a22 +� +∈ M2 (C) ⇐⇒ T ∗ = +� +a11 +a21 +a12 +a22 +� +∈ M2 (C) . +It says that, if we understand our t-scaled realization Ht +2 as a sub-structure of +M2 (C), then each hypercomplex number (a, b) ∈ Ht assigns a unique adjoint +[(a, b)]∗ +t of the realization [(a, b)]t “in M2 (C).” +Let (a, b) ∈ Ht realized to be +[(a, b)]t = +� a +tb +b +a +� +∈ Ht +2. +Then, as a matrix of M2 (C), this realization has its adjoint, +[(a, b)]∗ +t = +� a +b +tb +a +� +in M2 (C) . +It shows that the usual adjoint (conjugate-transpose) of [(a, b)]t is not contained +“in Ht +2,” in general. In particular, if +t2 ̸= 1 ⇐⇒ either t ̸= 1 or t ̸= −1, in R, +then +[(a, b)]t /∈ Ht +2 in general. +Theorem 43. The scale t ∈ R satisfies that t2 = 1 in R, if and only if the adjoint +of every realization of a hypercomplex number Ht is contained in Ht +2. i.e., +either t = 1, or t = −1⇐⇒ [ξ]∗ +t ∈ Ht +2, ∀ξ ∈ Ht. +(4.1) +Proof. For an arbitrary scale t ∈ R, if (a, b) ∈ Ht, then +[(a, b)]∗ +t = +� a +b +tb +a +� +in M2 (C) . + +OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS +33 +(⇒) Assume that either t = 1, or t = −1, equivalently, suppose t2 = 1 in R. Then +[(a, b)]∗ +t = +� a +b +tb +a +� += +� +a +t +� b +t +� +t2� b +t +� +a +� += +� +a +t +� b +t +� +� b +t +� +a +� +, +contained in Ht +2. So, if either t = 1, or t = −1, then [(a, b)]∗ +t ∈ Ht +2, for all (a, b) ∈ Ht. +Moreover, in such a case, +[(a, b)]∗ +t = +�� +a, b +t +�� +t in Ht +2. +(4.2) +(⇐) Assume now that t2 ̸= 1 in R. Then the adjoint [(a, b)]∗ +t of [(a, b)]t is identical +to the matrix, +[(a, b)]∗ +t = +� a +b +tb +a +� +in M2 (C) , +which “can” be +� +a +t +� b +t +� +t2 � +b +t +� +a +� +in Ht +2. +However, by the assumption that t2 ̸= 1, the adjoint [(a, b)]∗ +t is not contained in Ht +2, +in general. In particular, if b ̸= 0 in C, then the adjoint [(a, b)]∗ +t /∈ Ht +2 in M2 (C), +i.e., +t2 ̸= 1 and b ̸= 0 in C =⇒ [(a, b)]∗ +t ∈ (M2 (C) \ Ht +2) . +(4.3) +Therefore, the characterization (4.1) holds by (4.2) and (4.3). +Note that, if t = −1, then H−1 is the quaternions; and if t = 1, then H1 is +the bicomplex numbers. +The above theorem shows that, only when the scaled +hypercomplex ring Ht is either the quaternions H−1, or the bicomplex numbers H1, +the usual adjoint (∗) is closed on Ht +2, as a well-defined unary operation, by (4.1). +5. Free Probability on Ht +In this section, we establish a universal free-probabilistic model on our t-scaled +hypercomplex ring Ht, for “every” scale t ∈ R. First, recall that, on M2 (C), we +have the usual trace tr, defined by +tr +�� a11 +a12 +a21 +a22 +�� += a11 + a22, +for all +� +a11 +a12 +a21 +a22 +� +∈ M2 (C); and the normalized trace τ, +τ = 1 +2tr on M2 (C) . +i.e., we have two typical free-probabilistic models, +(M2 (C) , tr) and (M2 (C) , τ) . + +34 +DANIEL ALPAY AND ILWOO CHO +5.1. Free Probability. For more about free probability theory, see e.g., [19] and +[22]. +Let A be an noncommutative algebra over C, and ϕ : A → C, a linear +functional on A. Then the pair (A, ϕ) is called a (noncommutative) free probability +space. By definition, free probability spaces are the noncommutative version of +classic measure spaces (X, µ) consisting of a set X and a measure µ on the σ- +algebra of X. +As in measure theory, the (noncommutative) free probability on +(A, ϕ) is dictated by the linear functional ϕ. Meanwhile, if (A, ϕ) is unital in the +sense that (i) the unity 1A of A exists, and (ii) ϕ (1A) = 1, then it is called a unital +free probability space. These unital free probability spaces are the noncommutative +analogue of classical probability spaces (Y, ρ) where the given measures ρ are the +probability measures satisfying ρ (Y ) = 1. +If A is a topological algebra, and if ϕ is bounded (and hence, continuous under +linearity), then the corresponding free probability space (A, ϕ) is said to be a topo- +logical free probability space. Similarly, if A is a topological ∗-algebra equipped +with the adjoint (∗), then the topological free probability space (A, ϕ) is said to +be a topological (free) ∗-probability space. More in detail, if A is a C∗-algebra, or +a von Neumann algebra, or a Banach ∗-algebra, we call (A, ϕ), a C∗-probability +space, respectively, a W ∗-probability space, respectively, a Banach ∗-probability +space, etc.. For our main purposes, we focus on C∗-probability spaces from below. +If (A, ϕ) is a C∗-probability space, and a ∈ A, then the algebra-element a is +said to be a free random variable of (A, ϕ). For any arbitrarily fixed free random +variables a1, ..., as ∈ (A, ϕ) for s ∈ N, one can get the corresponding free distribution +of a1, ..., as, characterized by the joint free moments, +ϕ +� n +� +l=1 +ari +il +� += ϕ +� +ar1 +i1 ar2 +i2 ...arn +in +� +, +for all (i1, ..., in) ∈ {1, ..., s}n and (r1, ..., rn) ∈ {1, ∗}n, for all n ∈ N, where a∗ +l +are the adjoints of al, for all l = 1, ..., s. +For instance, if a ∈ (A, ϕ) is a free +random variable, then the free distribution of a is fully characterized by the joint +free moments of {a, a∗}, +ϕ +� n +� +l=1 +arl +� += ϕ (ar1ar2...arn) , +for all (r1, ..., rn) ∈ {1, ∗}n, for all n ∈ N (e.g., [19] and [22]). So, if a free random +variable a ∈ (A, ϕ) is self-adjoint in the sense that: a∗ = a in A, then the free +distribution of a is determined by the free-moment sequence, +(ϕ (an))∞ +n=1 = +� +ϕ (a) , ϕ +� +a2� +, ϕ +� +a3� +, ... +� +(e.g., [19] and [22]). +5.2. Free-Probabilistic Models Induced by Ht. By identifying the t-scaled +hypercomplex ring Ht and its realization Ht +2 as the same ring, we identify the t- +scaled hypercomplex monoid H× +t and its realization Ht× +2 +as the same monoid. As +a subset in M2 (C), we define a subset, +Ht× +2 (∗) +def += +� +[ξ]∗ +t ∈ M2 (C) : ξ ∈ H× +t +� +, + +OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS +35 +i.e., +(5.2.1) +Ht× +2 (∗) = +�� a +b +tb +a +� +∈ M2 (C) : (a, b) ∈ H× +t +� +, +by the subset of all adjoints of realizations in H×t +2 . Indeed, +[(a, b)]∗ +t = +� a +tb +b +a +�∗ += +� a +b +tb +a +� +in M2 (C) . +As we have seen in Section 4, the adjoint is not closed on Ht +2 in general, and hence, +Ht× +2 (∗) ̸= Ht× +2 +in M2 (C) , +in general. In particular, the scale t satisfies t2 ̸= 1 in R, if and only if the above +non-equality holds in M2 (C), by (4.1). Now, let +Ht× +2 (1, ∗) +denote += +Ht× +2 +∪ Ht× +2 (∗), +i.e., +(5.2.2) +Ht× +2 (1, ∗) = +�� a +tb +b +a +� +, +� a +b +tb +a +� +: (a, b) ∈ H× +t +� +, +in M2 (C), set-theoretically. By (4.1), (5.2.1) and (5.2.2), +Ht× +2 (1, ∗) ⫌ Ht× +2 +in M2 (C) , in general. +Define now the C∗-algebra Ht +2 by the C∗-subalgebra of M2 (C) generated by the +set Ht× +2 (1, ∗) of (5.2.2). i.e., +Ht +2 +denote += +C∗ � +Ht× +2 +� def += C +� +Ht× +2 (1, ∗) +� +, +(5.2.3) +in M2 (C), where C∗ (Z) means the C∗-subalgebra of B +� +C2� +generated by the +subset Z and their adjoints, and C[X] is the (pure-algebraic) algebra (over C) +generated by a subset X of M2 (C), and Y means the operator-norm-topology +closure of a subset Y of the operator algebra M2 (C) = B +� +C2� +, which is a C∗- +algebra over C. +Definition 44. The C∗-algebra Ht +2 of (5.2.3), generated by the t-scaled hyper- +complex monoid H× +t +monoid += +Ht× +2 , is called the t-scaled(-hypercomplex)-monoidal +C∗-algebra of H× +t (or, of Ht). +Clearly, by the definition (5.2.3), the t-scaled-monoidal C∗-algebra Ht +2 is well- +determined in M2 (C). So, the usual trace tr and the normalized trace τ on M2 (C) +are well-defined on Ht +2. i.e., we have two trivial free-probabilistic models of Ht +2, +� +Ht +2, tr +� +and +� +Ht +2, τ +� +, +as C∗-probability spaces (e.g., see Section 5.1). Note that such free-probabilistic +structures are independent from the choice of the scales t ∈ R. +Observe that, if +� al +bl +tbl +al +� +∈ Ht× +2 (∗) in Ht +2, for l = 1, 2, then +� a1 +b1 +tb1 +a1 +� � a2 +b2 +tb2 +a2 +� += + + +a1a2 + tb1b2 +a1b2 + b1a2 +t +� +b1a2 + a1b2 +� +tb1b2 + a1a2 + + , + +36 +DANIEL ALPAY AND ILWOO CHO +identifying to be +(5.2.4) + + +a1a2 + tb1b2 +b1a2 + a1b2 +t +� +b1a2 + a1b2 +� +a1a2 + tb1b2 + + in Ht +2. +Therefore, +� a1 +b1 +tb1 +a1 +� � a2 +b2 +tb2 +a2 +� +∈ Ht× +2 (∗), too. +i.e., the matricial multiplication is closed on the set Ht× +2 (∗) of (5.2.2), by (5.2.4). +In fact, under the closed-ness (5.2.4), the algebraic pair, +Ht× +2 (∗) +denote += +� +Ht× +2 (∗), · +� +, +forms a monoid with its identity I2. +So, the generating set Ht× +2 (1, ∗) of the t- +scaled-monoidal C∗-algebra Ht +2 is the set-theoretical union of two monoids Ht× +2 +and Ht× +2 (∗), under the matricial multiplication. Note, however, that the matricial +multiplication is not closed on the generating set Ht× +2 (1, ∗) of (5.2.2). Indeed, if +� a1 +tb1 +b1 +a1 +� +∈ Ht× +2 , +� a2 +b2 +tb2 +a2 +� +∈ Ht× +2 (∗) +in Ht +2, then +� a1 +tb1 +b1 +a1 +� � a2 +b2 +tb2 +a2 +� += + + +a1a2 + t2b1b2 +a1b2 + ta2b1 +a2b1 + ta1b2 +b1b2 + a1a2 + + , +and +(5.2.5) +� a2 +b2 +tb2 +a2 +� � a1 +tb1 +b1 +a1 +� += + + +a1a2 + b1b2 +tb1a2 + a1b2 +ta1b2 + b1a2 +t2b1b2 + a1a2 + + , +in Ht +2. However, the resulted products of (5.2.5), contained in Ht +2, are not contained +in Ht× +2 (1, ∗), in general. +Observation. By (5.2.4) and (5.2.5), one can realize that: (i) if A, B ∈ Ht× +2 , then +AB ∈ Ht× +2 , (ii) if C, D ∈ Ht× +2 (∗), then CD ∈ Ht× +2 (∗), and (iii) if T, S ∈ Ht× +2 (1, ∗), +then T S /∈ Ht× +2 (1, ∗), in general, as elements of the t-scaled-monoidal C∗-algebra +Ht +2. Even though the non-closed rule (iii) is satisfied “on Ht +2 (1, ∗),” at least, we +have a multiplication rule (5.2.5) “in the C ∗-algebra Ht +2.” + +Assume that [(a, b)]t ∈ Ht× +2 +in Ht +2. Then +tr ([(a, b)]t) = a + a = 2Re (a) , +and +(5.2.6) +τ ([(a, b)]t) = 1 +2tr ([(a, b)]t) = Re (a) , +where Re (a) is the real part of a in C. Similarly, if [(a, b)]∗ +t ∈ Ht× +2 (∗) in Ht +2, then +we have +tr +� +[(a, b)]∗ +t +� += tr +� a +b +tb +a +� += a + a = 2Re (a) , + +OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS +37 +and +(5.2.7) +τ +� +[(a, b)]∗ +t +� += 1 +2 (2Re (a)) = Re (a) . +Remark that, since tr and τ are well-defined linear functional on the C∗-algebra +Ht +2, they satisfy +tr (T ∗) = tr (T ), and τ (T ∗) = τ (T ), +for all T ∈ Ht +2. So, the relation (5.2.7) is well-verified, too. +Also, if [(a1, b1)]t , [(a2, b2)]∗ +t ∈ Ht× +2 (1, ∗) in Ht +2, then +tr +� +[(a1, b1)]t [(a2, b2)]∗ +t +� += tr + + + + +a1a2 + t2b1b2 +a1b2 + ta2b1 +a2b1 + ta1b2 +b1b2 + a1a2 + + + + +by (5.2.5) += a1a2 + t2b1b2 + b1b2 + a1a2 += 2Re (a1a2) + t2b1b2 + b1b2, +and similarly, +(5.2.8) +tr +� +[(a1, b1)]∗ +t [(a2, b2)]t +� += 2Re (a1a2) + t2b1b2 + b1b2, +and hence, +τ +� +[(a1, b1)]t [(a2, b2)]∗ +t +� += Re (a1a2) + t2b1b2 + b1b2 +2 +, +and +(5.2.9) +τ +� +[(a1, b1)]∗ +t [(a2, b2)]t +� += Re (a1a2) + t2b1b2 + b1b2 +2 +, +by (5.2.8). +Proposition 45. Let (a, b) , (al, bl) ∈ Ht, for l = 1, 2, and let A = [(a, b)]t and +Al = [(al, bl)]t be the corresponding realizations of Ht +2, regarded as elements of the +t-scaled-monoidal C∗-algebra Ht +2. Then +τ (A) = 1 +2tr (A) = Re (a) = 1 +2tr (A∗) = τ (A∗) , +and +(5.2.10) +τ (A1A∗ +2) = 1 +2tr (A1A∗ +2) = Re (a1a2) + t2b1b2 + b1b2 +2 +, +and +τ (A∗ +1A2) = 1 +2tr (A∗ +1A2) = Re (a1a2) + t2b1b2 + b1b2 +2 +. +Proof. The joint free moments in (5.2.10) are proven by (5.2.6), (5.2.7), (5.2.8) and +(5.2.9). +The above computations in (5.2.10) provide a general way to compute free- +distributional data, in particular, the joint free moments of matrices in the t-scaled- +monoidal C∗-algebra Ht +2, up to the trace tr, and up to the normalized trace τ. And, +they demonstrate that computing such free-distributional data is not easy. So, we +will restrict our interests to a certain specific case. + +38 +DANIEL ALPAY AND ILWOO CHO +5.3. Free Probability on (Ht +2, tr). In this section, we fix a scale t ∈ R, and the +corresponding t-scaled-monoidal C∗-algebra Ht +2 generated by the t-scaled hyper- +complex monoid H× +t . Let (Ht +2, tr) be the C∗-probability space with respect to the +usual trace tr on Ht +2. +Recall that if a scale t is negative, then the realization [ξ]t and the t-spectral form +Σt (ξ) are similar “in Ht +2” by (3.3.9), for all ξ ∈ Ht. It implies that the similarity +“on Ht +2” is equivalent to the t-spectral relation on Ht by (3.3.10). Also, recall that +if two matrices A and B are similar in Mn (C), for any n ∈ N, +tr (A) = tr (B) . +So, if the realization [ξ]t and the t-spectral form Σt (ξ) are similar in Ht +2, then +the free-moment computations would be much simpler than the computations of +(5.2.10). Note again that if (a, b) ∈ Ht satisfies the condition (3.1.5), then +tr ([(a, b)]t) = 2Re (a) = 2x = +� +x + i +√ +R +� ++ +� +x − i +√ +R +� += tr (Σt (a, b)) , +where +(5.3.1) +R = y2 − tu2 − tv2 in R, +under RA 3.2.1. Even though the identical results hold in (5.3.1) (without simi- +larity), if [(a, b)]t and Σt (a, b) are not similar in Ht +2, then +tr ([(a, b)]n +t ) ̸= tr ((Σt (a, b))n) , +for some n ∈ N, by (5.2.5). It implies that some (joint) free-moments of [(a, b)]t +and those of Σt (a, b) are not identical, and hence, the free distributions of them +are distinct. +Lemma 46. Suppose the realization [(a, b)]t and the t-spectral form Σt (a, b) are +similar in Ht +2 for (a, b) ∈ Ht. Then +tr ([(a, b)]n +t ) = 2Re (σt (a, b)n) = tr +�� +[(a, b)]∗ +t +�n� +(5.3.2) +for all n ∈ N, where σt (a, b) is the t-spectral value of (a, b). +Proof. Suppose (a, b) ∈ Ht satisfies the condition (3.1.5). Then +[(a, b)]t = +� a +tb +b +a +� +and Σt ((a, b)) = +� σt (a, b) +0 +0 +σt (a, b) +� +, +in Ht +2, where +σt (a, b) = x + i +� +y2 − tu2 − tv2, +under RA 3.2.1. Assume that [(a, b)]t and Σt ((a, b)) are similar in Ht +2. Then the +matrices [(a, b)]n +t and Σt ((a, b))n are similar in Ht +2, for all n ∈ N. Indeed, if two +elements A and B are similar in Ht +2, satisfying B = U −1AU in Ht +2, for an invertible +element U ∈ Ht +2, then +Bn = +� +U −1AU +�n = U −1AnU in Ht +2, +implying the similarity of An and Bn, for n ∈ N. Thus, +tr ([(a, b)]n +t ) = tr (Σt ((a, b))n) , +and +tr (Σt ((a, b))n) = tr +�� σt (a, b)n +0 +0 +σt (a, b)n +�� +, + +OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS +39 +implying that +tr ([(a, b)]n +t ) = tr (Σt ((a, b))n) = 2Re (σt (a, b)n) , +for all n ∈ N. Therefore, the first equality in (5.3.2) holds. +Since tr is a well-defined linear functional on the C∗-algebra Ht +2, one has +tr (A∗) = tr (A), for all A ∈ Ht +2. +Since +tr +�� +[(a, b)]∗ +t +�n� += tr +� +([(a, b)]n +t )∗� += tr ([(a, b)]n +t ), +one has +tr +�� +[(a, b)]∗ +t +�n� += 2Re (σt (a, b)n) = 2Re (σt (a, b)n) , +for all n ∈ N. So, the second equality in (5.3.2) holds, too. +Note that the formula (5.3.2) holds true under the similarity assumption of the +realization and the t-spectral form. +Remark that every complex number w ∈ C is polar-decomposed to be +w = |w| wo with wo ∈ T, +uniquely, where T = {z ∈ C : |z| = 1} is the unit circle in C. So, all our t-spectral +values σt (ξ) are polar-decomposed to be +σt (ξ) = |σt (ξ)| σt (ξ)o with σt (ξ)o ∈ T, +for all ξ ∈ Ht. In such a sense, we have that +tr ([ξ]n +t ) = 2 |σt (ξ)|n Re (σt (ξ)n +o) , +for all n ∈ N, by (5.3.2). +Corollary 47. Suppose the realization [ξ]t and the t-spectral form Σt (ξ) are similar +in Ht +2 for ξ ∈ Ht. Then +tr ([ξ]n +t ) = 2 |σt (ξ)|n Re (σt (ξ)n +o ) = tr +�� +[ξ]∗ +t +�n� +, +(5.3.3) +for all n ∈ N, where σt (ξ) = |σt (ξ)| σt (ξ)o is the polar decomposition of σt (ξ), +with σt (ξ)o ∈ T. +Proof. The free-distributional data (5.3.3) is immediately obtained by (5.3.2) under +the polar decomposition of the t-spectral value σt (ξ) in C. +Assume again that a hypercomplex number (a, b) ∈ Ht satisfies our similarity +assumption, i.e., T +denote += +[(a, b)]t and S +denote += +Σt ((a, b)) are similar in Ht +2. Then, +for any +(r1, ..., rn) ∈ {1, ∗}n , for n ∈ N, +the matrix +n� +l=1 +T rl is similar to +n� +l=1 +Srl in Ht +2 (and hence, in Ht +2). +Theorem 48. Let (a, b) ∈ Ht satisfy the similarity assumption that: T +denote += +[(a, b)]t and S +denote += +Σt ((a, b)) are similar in Ht +2. If +σt (a, b) = rwo, polar decomposition, +with +(5.3.4) +r = |σt (a, b)| and wo ∈ T, +then + +40 +DANIEL ALPAY AND ILWOO CHO +tr +� n� +l=1 +T rl +� += 2rnRe + +w +n +� +l=1 +el +o + + , +(5.3.5) +for all (r1, ..., rn) ∈ {1, ∗}n, for all n ∈ N, where +el = +� +1 +if rl = 1 +−1 +if rl = ∗, +for all l = 1, ..., n. +Proof. Since the realization T and the t-spectral form S are assumed to be similar +in Ht +2, their adjoints T ∗ and S∗ are similar in Ht× +2 (∗) ∪ {[(0, 0)]t}; and hence, the +matrix +n� +l=1 +T rl and +n� +l=1 +Srl are similar “in Ht +2.” Consider that +S = + + +σt (a, b) +0 +0 +σt (a, b) + + = +� rwo +0 +0 +rwo +� += r +� wo +0 +0 +w−1 +o +� +, +under hypotheses, because z = 1 +z = z−1 in T, whenever z ∈ T in C. It shows that +Sj = rj +� +wj +o +0 +0 +w−j +o +� +, for all j ∈ N ∪ {0} , +and +S∗ = r +� wo +0 +0 +wo +� += r +� w−1 +o +0 +0 +wo +� +, +satisfying that +(S∗)j = +� +Sj�∗ , for all j ∈ N. +It implies that, for any (r1, ..., rn) ∈ {1, ∗}n, for n ∈ N, there exists (e1, ..., en) ∈ +{±1}n, such that +el = +� +1 +if rl = 1 +−1 +if rl = ∗, +for all l = 1, ..., n, and +(5.3.6) +n +� +l=1 +Srl = rn + + + + + + +w +n +� +l=1 +el +o +0 +0 +w +− +� n +� +l=1 +el +� +o + + + + + + +, +in Ht +2. Thus, under our similarity assumption, +tr +� n +� +l=1 +T rl +� += tr +� n +� +l=1 +Srl +� += rn + +w +n +� +l=1 +el +o ++ w +− +� n +� +l=1 +el +� +o + + , +implying that +tr +� n +� +l=1 +T rl +� += rn + +2Re + +w +n +� +l=1 +el +o + + + + , +for all (r1, ..., rn) ∈ {1, ∗}n, for all n ∈ N, where (e1, ..., en) ∈ {±1}n satisfies (5.3.6). +Therefore, under our similarity assumption and the polar decomposition (5.3.4), +the free-distributional data (5.3.5) holds. + +OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS +41 +By the above theorem, one immediately obtain the following result. +Corollary 49. Let (a, b) ∈ Ht satisfy the similarity assumption that: T +denote += +[(a, b)]t and S +denote += +Σt ((a, b)) are similar in Ht +2. If +σt (a, b) = rwo, polar decomposition, +with +(5.3.7) +r = |σt (a, b)| and wo ∈ T, +then +τ +� n� +l=1 +T rl +� += rnRe + +w +n +� +l=1 +el +o + + , +(5.3.8) +for all (r1, ..., rn) ∈ {1, ∗}n, for all n ∈ N, where +el = +� +1 +if rl = 1 +−1 +if rl = ∗, +for all l = 1, ..., n. +Proof. By (5.3.5), the free-distributional data (5.3.8) holds up to the normalized +trace τ = 1 +2tr on Ht +2, under (5.3.7). +Under our similarity assumption and the condition (5.3.7), the free-distributional +data (5.3.8) fully characterizes the free distribution of [(a, b)]t ∈ Ht +2 in the C∗- +probability space (Ht +2, τ). +Corollary 50. Suppose a given scale t is negative in R. Let (a, b) ∈ Ht, and let +T +denote += +[(a, b)]t and S +denote += +Σt ((a, b)) in Ht +2. If +σt (a, b) = rwo, polar decomposition, +with +(5.3.9) +r = |σt (a, b)| and wo ∈ T, +then +tr +� n� +l=1 +T rl +� += 2rnRe + +w +n +� +l=1 +el +o + + = 2τ +� n� +l=1 +T rl +� +, +(5.3.10) +for all (r1, ..., rn) ∈ {1, ∗}n, for all n ∈ N, where +el = +� +1 +if rl = 1 +−1 +if rl = ∗, +for all l = 1, ..., n. +Proof. In Theorem 48 and Corollary 49, we showed that if T and S are similar in +Ht +2, then the free-distributional data (5.3.10) holds under the condition (5.3.9), by +(5.3.5) and (5.3.8), respectively. So, it suffices to show that the realization T and +the t-spectral form S are similar in Ht +2. However, since t < 0 in R, the matrices T +and S are similar in Ht +2 by (3.3.9). + +42 +DANIEL ALPAY AND ILWOO CHO +The above corollary shows that, if a given scale t is negative in R, then the +free-distributional data (5.3.10) fully characterizes the free distributions of the re- +alizations [ξ]t in the t-scaled-monoidal C∗-algebra Ht +2 up to the usual trace tr, and +the normalized trace τ, for “all” ξ ∈ Ht. In other words, it illustrates that, if t < 0 +in R, then the free-distributional data on the C∗-probability spaces, +� +Ht +2, tr +� +and +� +Ht +2, τ +� +, +are fully characterized by the spectra of hypercomplex numbers of Ht, by (5.3.9) +and (5.3.10). +But, if t ≥ 0, and hence, there are some hypercomplex numbers η of Ht whose +realization and spectral form are not similar in Ht +2, then computing joint free mo- +ments of [η]t in Ht +2 would not be easy e.g., see (5.2.10). +5.4. More Free-Distributional Data on (Ht +2, τ) for t < 0. In this section, a +fixed scale t is automatically assumed to be negative, i.e., t < 0 in R. At this +moment, we emphasize that most main results of this section would hold even +though t is not negative in R. +However, we assume a given scale t is negative +for convenience (e.g., see (5.3.10)). Let Ht +2 be the t-scaled-monoidal C∗-algebra +inducing a C∗-probability space (Ht +2, τ), where τ is the normalized trace on Ht +2. +Since t is assumed to be negative in R, the realizations T = [η]t and the t-spectral +forms S = Σt (η) are similar in Ht +2 by (3.3.9), and hence, +τ +� n +� +l=1 +T rl +� += rnRe + +w +n +� +l=1 +el +o + + = τ +� n +� +l=1 +Srl +� +, +by (5.3.5), where +(5.4.1) +σt (η) = rwo ∈ C, polar decomposition, +with r = |σt (η)| and wo ∈ T, for all (r1, ..., rn) ∈ {1, ∗}n, where (e1, ..., en) ∈ {±1}n +satisfies (5.3.6), for all n ∈ N, for “all” η ∈ Ht. And the free-distributional data +(5.4.1) fully characterizes the free distribution of [η]t ∈ (Ht +2, τ), for all η ∈ Ht. +In this section, we refine (5.4.1) case-by-case, up to operator-theoretic properties +of elements of (Ht +2, τ). +Definition 51. Let A be a unital C∗-algebra with its unity 1A, and let T ∈ A, +and T ∗ ∈ A, the adjoint of T . +(1) T is said to be self-adjoint, if T ∗ = T in A. +(2) T is a projection, if T ∗ = T = T 2 in A. +(3) T is normal, if T ∗T = T T ∗ in A. +(4) T is a unitary, if T ∗T = 1A = T T ∗ in A. +Let (a, b) ∈ Ht, satisfying the condition (3.1.5), and T +denote += +[(a, b)]t ∈ Ht +2, as an +element of (Ht +2, τ). Then its adjoint, +T ∗ = +� a +b +tb +a +� +∈ Ht +2(∗), +is well-defined in (Ht +2, τ), and the corresponding t-spectral form, +S +denote += +Σt ((a, b)) = +� w +0 +0 +w +� +∈ Ht +2, + +OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS +43 +is contained in (Ht +2, τ), where w is determined by RA 3.2.1, and +w = σt (a, b) = x + i +� +y2 − tu2 − tv2 +is the t-spectral value, uniqely polar-decomposed to be +w = rwo with r = |σt (a, b)| and wo ∈ T. +Assumption and Notation 5.4.1. (from below AN 5.4.1) From now on, if we say +that “a given hypercomplex number (a, b) ∈ Ht satisfies AN 5.4.1,” then it means +it has its realization denoted by T , its t-spectral form denoted by S, determined by +the t-spectral value denoted by w, which is polar-decomposed to be w = rwo, as +indicated in the very above paragraph. + +Let (a, b) ∈ Ht satisfy AN 5.4.1. Then the self-adjointness of the realization +T ∈ Ht +2 in Ht +2 says that +T ∗ = T ⇐⇒ +� a +b +tb +a +� += +� a +tb +b +a +� +, +if and only if +a = a and tb = b in C, +if and only if +(5.4.2) +a ∈ R and b = 0. +Especially, the equality b = 0 in (5.4.2) is obtained by our negative-scale assump- +tion: t < 0 in R. +Proposition 52. Let (a, b) ∈ Ht satisfy AN 5.4.1. Then the realization T ∈ Ht +2 +is self-adjoint in Ht +2, if and only if +a ∈ R and b = 0 ⇐⇒ (a, b) = (Re (a) , 0) in Ht. +(5.4.3) +Proof. The self-adjointness (5.4.3) is shown by (5.4.2). +The self-adjointness (5.4.3) illustrates that the self-adjoint generating elements +T ∈ Ht +2 of (Ht +2, τ) have their forms, +T = +� x +0 +0 +x +� +∈ Ht +2 (1, ∗) with x ∈ R. +Remark and Observation. The above self-adjointness characterization (5.4.3) is +obtained for the case where t < 0 in R. How about the other cases? Generally, one +has T is self-adjont in Ht +2, if and only if +a = a and tb = b, +like (5.4.2). Thus one can verify that: (i) if t = 0, then T is self-adjoint, if and only +if a ∈ R and b = 0, just like (5.4.3); (ii) if t > 0 and t ̸= 1, then T is self-adjoint, if +and only if a ∈ R and b = 0, just like (5.4.3); meanwhile, (iii) if t = 1 (equivalently, +if (a, b) is a bicomplex number of H1), then T is self-adjoint in H1 +2, if and only if +a ∈ R, if and only if (a, b) = (Re (a) , b) in H1. In summary, +T is self-adjoint in Ht +2 ⇐⇒ (a, b) = (Re (a) , 0) in Ht, +like (5.4.3), whenever t ∈ R \ {1}, meanwhile, +T is self-adjoint in H1 +2 ⇐⇒ (a, b) = (Re (a) , b) ∈ H1. + +44 +DANIEL ALPAY AND ILWOO CHO + +Now, let (a, b) ∈ Ht, under AN 5.4.1 and our negative-scale assumption, satisfy +the self-adjointness (5.4.3), i.e., it is actually (a, 0) with a ∈ R. Then +T = +� +a +0 +0 +a +� += S in Ht +2 (1, ∗) , +as an element of Ht +2. +Theorem 53. Let (a, b) ∈ Ht satisfy AN 5.4.1, and assume that the realization T +is self-adjoint in (Ht +2, τ). Then +τ +� n� +l=1 +T rl +� += τ (T n) = an in R, +(5.4.4) +for all (r1, ..., rn) ∈ {1, ∗}n, for all n ∈ N. +Proof. By the self-adjointness (5.4.3) of the realization T of (a, b) ∈ Ht, one has +(a, b) = (a, 0) in Ht, with a ∈ R, and +T = S = +� a +0 +0 +a +� += S∗ = T ∗ in Ht +2. +So, +τ +� n +� +l=1 +T rl +� += τ (T n) = τ (Sn) = τ +�� +an +0 +0 +an +�� +, +for all (r1, ..., rn) ∈ {1, ∗}n, for all n ∈ N. Therefore, the free-distributional data +(5.4.4) holds true. +Observation. Similar to the above theorem, one can verify that: if t ∈ R\{1}, then +the free-distributional data (5.4.4) holds for self-adjoint realizations T ∈ (Ht +2, τ) of +(a, 0) ∈ Ht with a ∈ R. + +By (5.4.3), the realization T of a hypercomplex number (a, b) ∈ Ht, satisfying +AN 5.4.1, is self-adjoint, if and only if (a, b) = (a, 0) with a ∈ R. And, by definition, +such a self-adjoint matrix T can be a projection, if and only if it is idempotent in +the sense that +T 2 = T in Ht +2. +Observe that a self-adjoint realization T satisfies the above idempotence, if and +only if +T 2 = +� a2 +0 +0 +a2 +� += +� a +0 +0 +a +� += T, +if and only if +(5.4.5) +a2 = a ⇐⇒ a = 0, or a = 1, in R. +Proposition 54. Let (a, b) ∈ Ht satisfy AN 5.4.1. Then the realization T is a +projection, if and only if +either T = I2, or T = O2 in Ht +2, +(5.4.6) +where I2 = [(1, 0)]t is the identity matrix, and O2 = [(0, 0)]t is the zero matrix of +Ht +2. + +OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS +45 +Proof. The operator-equality (5.4.6) holds in Ht +2 (and hence, in Ht +2) by (5.4.5). +Observation. Like the above proposition, one can conclude that: if t ∈ R \ {1}, +then the realization T is a projection in Ht +2, if and only if it is either the identity +matrix I2, or the zero matrix O2 of Ht +2. How about the case where t = 1? As we +discussed above, T ∈ H1 +2 is self-adjoint, if and only if (a, b) = (Re (a) , b) in H1, if +and only if +T = +� x +b +b +x +� +∈ H1 +2, and S = + + +x + i +√ +−u2 − v2 +0 +0 +x − i +√ +−u2 − v2 + + , +implying that +S = + + +x − |b| +0 +0 +x + |b| + + in H1 +2, +under AN 5.4.1. Such a self-adjoint T is a projection, if and only if T 2 = T in H1 +2, +if and only if +x2 + |b|2 = x and 2xb = b. +Thus if b = 0, then x ∈ {0, 1}, meanwhile, if b ̸= 0, then +x2 + |b|2 = x and x = 1 +2, +⇐⇒ +x = 1 +2 and 1 +4 + |b|2 = 1 +2, +⇐⇒ +x = 1 +2 +and +|b|2 = 1 +4, +if and only if +(a, b) = +�1 +2, b +� +with |b|2 = 1 +4. +It implies that T is a projection in H1 +2, if and only if +(a, b) = (0, 0) , or (a, b) = (1, 0) , +or +(a, b) = +�1 +2, b +� +with |b|2 = 1 +4, +in H1. + +The above proposition says that, under our negative-scale assumption, the only +projections of Ht +2 induced by hypercomplex numbers of Ht are the identity ele- +ment I2 = [(1, 0)]t, and the zero element O2 = [(0, 0)]t in Ht +2. For any unital +C∗-probability spaces (A, ϕ), the unity 1A has its free distributions characterized +by its free-moment sequence, +(ϕ (1n +A) = ϕ (1A))∞ +n=1 = (1, 1, 1, 1, 1, ...); +and the free distribution of the zero element 0A is nothing but the zero-free distri- +bution, characterized by the free-moment sequence, +(ϕ (0n +A) = ϕ (0A))∞ +n=1 = (0, 0, 0, 0, ...). + +46 +DANIEL ALPAY AND ILWOO CHO +Theorem 55. Let (a, b) ∈ Ht, satisfying AN 5.4.1, have its realization T ∈ Ht +2, +which is a “non-zero” projection in Ht +2. Then +τ (T n) = 1, ∀n ∈ N. +(In fact, this result holds true for all t ∈ R \ {1}.) +Proof. Under hypothesis, the realization T ∈ Ht +2 is a projection in Ht +2, if and only +if (a, b) = (1, 0), or (0, 0) in Ht, by (5.4.6). Since T ∈ Ht +2 is assumed to a non-zero +projection in Ht +2, we have +(a, b) = (1, 0) in Ht, ⇐⇒ T = I2 = S in Ht +2. +Therefore, +τ (T n) = τ (In +2 ) = 1, ∀n ∈ N. +(Note that it holds true for all t ∈ R \ {1}.) +Let (a, b) ∈ Ht satisfy AN 5.4.1, and let T ∈ Ht +2 be the realization in Ht +2. +Observe that +T ∗T = +� a +b +tb +a +� � a +tb +b +a +� += + + +|a|2 + |b|2 +(t + 1) ab +(t + 1) ab +t2 |b|2 + |a|2 + + , +and +(5.4.7) +T T ∗ = +� a +tb +b +a +� � a +b +tb +a +� += + + +|a|2 + t2 |b|2 +(t + 1) ab +(t + 1) ab +|b|2 + |a|2 + + , +in Ht +2. So, the realization T of (a, b) is normal in Ht +2, if and only if +|a|2 + t2 |b|2 = |a|2 + |b|2 and (t + 1) ab = (t + 1) ab, +(5.4.8) +in C, by (5.4.7). +Proposition 56. Let (a, b) ∈ Ht satisfy AN 5.4.1. Then the realization T ∈ Ht +2 +is normal in Ht +2, if and only if +t2 |b|2 = |b|2 and (t + 1) ab = (t + 1) ab, +(5.4.9) +in C. In particular, if t = −1 (equivalently, if (a, b) ∈ H−1 is a quaternion), then +T is normal in H−1 +2 ; if t = 1, (equivalently, if (a, b) ∈ H1 is a bicomplex number), +then T is normal in H1 +2, if and only if +either (a, b) = (Re (a) , b) or (a, b) = (a, 0) in H1; +(5.4.10) +meanwhile, if t ∈ R \ {±1}, then T is normal in Ht +2, if and only if +b = 0 in C ⇐⇒ (a, b) = (a, 0) ∈ Ht. +(5.4.11) +Proof. By (5.4.8), the normality characterization (5.4.9) holds. +By (5.4.9), if t = −1 in R, and hence, if (a, b) ∈ H−1 is a quaternion, then the +condition (5.4.9) is identified with +|b|2 = |b|2 , and 0 = 0, +which are the identities on C. These identities demonstrate that the realization of +every quaternion is automatically normal in H−1 +2 . + +OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS +47 +Suppose t = 1 in R. Then the condition (5.4.9) is equivalent to +|b|2 = |b|2 and 2ab = 2ab, +if and only if either +a = a in C ⇐⇒ (a, b) = (Re (a) , b) ∈ H1 (if b ̸= 0), +or +(a, b) = (a, 0) ∈ H1 +(if b = 0). +Thus, if t = 1, then T is normal, if and only if the condition (5.4.10) holds. +Assume now that both t ̸= 1 and t ̸= −1, i.e., suppose t2 ̸= 1 in R. So, the first +condition of (5.4.9) is identified with +t2 |b|2 = |b|2 ⇐⇒ b = 0 in C. +So, the second condition of (5.4.9) automatically holds, since +(t + 1) a · 0 = (t + 1) a · 0 ⇐⇒ 0 = 0. +Therefore, the realization T ∈ Ht +2 of (a, b) ∈ Ht is normal in Ht +2, if and only if +(a, b) = (a, 0) in Ht, whenever t ∈ R \ {±1}. i.e., the normality (5.4.11) holds. +The above proposition illustrates that: (i) the realizations of “all” quaternions +are normal in H−1 +2 , (ii) the realizations of bicomplex numbers are normal in H1 +2, if +and only if either (a, b) = (Re (a) , b), or (a, b) = (a, 0) in H1, by (5.4.10), and (iii) +the only realizations [(a, 0)]t are normal in Ht +2, whenever t ∈ R \ {±1}, by (5.4.11). +Theorem 57. Let (a, b) ∈ Ht satisfy AN 5.4.1. +(5.4.12) Suppose t = −1. Then T is normal in H−1 +2 , and its free distribution is +characterized by the formula (5.3.10). +(5.4.13) Let t ∈ R \ {±1}. If T is “non-zero” normal in Ht +2, then +τ +� n +� +l=1 +T rl +� += RnRe + +W +n +� +l=1 +el +o + + , +with +(5.4.14) +R = |a| and Wo = a +|a| ∈ T, +where +el = +� +1 +if rl = 1 +−1 +if rl = ∗, +for l = 1, ..., n, for all (r1, ..., rn) ∈ {1, ∗}n, for all n ∈ N. +Proof. The statement (5.4.12) holds by (5.3.10). +Of course, if t < 0, and if T ∈ Ht +2, then the free-distributional data (5.4.14) holds +by (5.3.10), because T and the t-spectral form S are similar in Ht +2 as elements of +(Ht +2, τ). However, in the statement (5.4.13), the normality works for all the scales +t ∈ R \ {±1}. Assume that the realization T is a “non-zero,” “normal” element of +Ht +2. Then +(a, b) = (a, 0) ∈ Ht, with a ̸= 0, +by (5.4.11). Therefore, +T = +� a +0 +0 +a +� += S, + +48 +DANIEL ALPAY AND ILWOO CHO +because σt (a, 0) = a in C. i.e., the realization T and the t-spectral form S are +identical in Ht +2, implying the similarity of them. So, under AN 5.4.1, +a = w +denote += +σt (a, 0) , +polar-decomposed to be +w = a = |a| +� a +|a| +� +∈ C, +i.e., r = |a| and wo = +a +|a| under AN 5.4.1. +Therefore, similar to (5.3.10), the +free-distributional data (5.4.14) holds. +Note that, in the proof of the statement (5.4.13), we did not use our negative- +scale assumption for the cases where t < 0, but t ̸= −1. Indeed, even though t ≥ 0, +but t ̸= 1, the normality (5.4.11) shows that the realization T is a diagonal matrix +not affected by the scale t. So, whatever scales t are given in R \ {±1}, the free- +distributional data (5.4.14) holds in (Ht +2, τ), under normality. Then, how about the +case where t = 1? Recall that if t = 1, then the realization T of (a, b) ∈ H1 is +normal in H1 +2, if and only if either +(a, b) = (Re (a) , b) , if b ̸= 0, +or +(a, b) = (a, 0) , if b = 0, +in H1, by (5.4.10). So, if (a, b) = (a, 0) in H1, the joint free moments of T are deter- +mined similarly by the formula (5.4.14), by the identity (and hence, the similarity) +of T and S (under AN 5.4.1). However, if (a, b) = (Re (a) , b) with b ̸= 0, then we +need a better tool than (5.2.10) to compute the corresponding free-distributional +data, because we cannot use our similarity technique (of Theorem 48) here. +By the definition of the unitarity, if an element U of a C∗-algebra A is a unitary, +then it is automatically normal. +i.e., the unitarity implies the normality. +Let +(a, b) ∈ Ht satisfy AN 5.4.1 with its realization T ∈ Ht +2 in (Ht +2, τ), and suppose it +is a unitary in Ht +2. By the assumption that T is a unitary in Ht +2, it is normal. +Assume first that t = −1 in R, and hence, (a, b) ∈ H−1 is a quaternion. Then +the realization T is automatically normal in Ht +2 by (5.4.12). Indeed, in this case, +T = +� a +−b +b +a +� +with T ∗ = +� +a +b +−b +a +� += [(a, −b)]−1 , +in H−1 +2 , as elements of H−1 +2 . So, the normality is guaranteed; +T ∗T = + + +|a|2 + |b|2 +0 +0 +|a|2 + |b|2 + + = T T ∗, +in H−1 +2 , as elements of H−1 +2 . It shows that T is a unitary in H−1 +2 , if and only if +|a|2 + |b|2 = 1. +(5.4.15) +Meanwhile, if t ∈ R \ {±1} in R, then T is normal, if and only if (a, b) = (a, 0) +in Ht by (5.4.11), if and only if +T = +� a +0 +0 +a +� +∈ Ht +2, + +OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS +49 +which is identical (and hence, similar) to the t-spectral form S of (a, 0) in Ht +2. This +normal element T is a unitary in Ht +2, if and only if +T ∗T = I2 = T T ∗ ⇐⇒ +� |a|2 +0 +0 +|a|2 +� += +� 1 +0 +0 +1 +� +, +if and only if +(5.4.16) +|a|2 = 1 +in +C. +Proposition 58. Let (a, b) ∈ Ht satisfy AN 5.4.1. +(5.4.17) Let t = −1. Then T is a unitary in Ht +2, if and only if |a|2 + |b|2 = 1. +(5.4.18) Let t ∈ R \ {±1}. Then T is a unitary in Ht +2, if and only if |a|2 = 1 and +b = 0. +Proof. The statements (5.4.17) and (5.4.18) hold by (5.4.15) and (5.4.16), respec- +tively, because a unitary realization T of (a, b) automatically satisfies the normality +(5.4.9). +Observation. Now, assume that t = 1, and let (a, b) ∈ H1 be a bicomplex number +satisfying AN 5.4.1. By (5.4.10), the realization T ∈ H1 +2 is normal in H1 +2, if and +only if either +(a, b) = (Re (a) , b) , or (a, b) = (a, 0) , +in H1. So, if (a, b) = (a, 0) in H1, then one obtains the unitarity that: T is a unitary +in H1 +2, if and only if |a|2 = 1, just like (5.4.18). However, if +(a, b) = (Re (a) , b) = (x, b) in H1, +with b ̸= 0 in C, then T is a unitary in H1 +2, if and only if +� +x +b +b +x +� � x +b +b +x +� += + + +x2 + b2 +2xRe (b) +2xRe (b) +x2 + b2 + + = I2, +and +� x +b +b +x +� � +x +b +b +x +� += + + +x2 + b2 +2xRe (b) +2xRe (b) +x2 + b2 + + = I2, +in H1 +2, if and only if +x2 + b2 = x2 + b2 = 1 and 2xRe (b) = 0, +if and only if +b2 = b2 = 1 − x2 and 2xRe (b) = 0, +if and only if +b2 = 1 − x2 ∈ R and x = 0, +because b is assumed not to be zero in C, if and only if +x = 0 and b = ±1 +in +R, +if and only if +T = +� 0 +1 +1 +0 +� +, or T = +� +0 +−1 +−1 +0 +� +in H1 +2, +if and only if +(a, b) = (0, 1) , or (a, b) = (0, −1) in H1. + +50 +DANIEL ALPAY AND ILWOO CHO +i.e., if (a, b) = (Re (a) , b) in H1, then T is a unitary in H1 +2, if and only if +(a, b) = (0, 1) , or (a, b) = (0, −1), +in H1. In summary, the realization T ∈ H1 +2 of a bicomplex number (a, b) ∈ H1 is a +unitary in Ht +2, if and only if either +(a, b) = (a, 0) with |a|2 = 1, +or +(a, b) = (0, 1) , or (a, b) = (0, −1), +in H1. + +By the unitarity (5.4.17) and (5.4.18), one has the following result. +Theorem 59. Let (a, b) ∈ Ht satisfy AN 5.4.1. +(5.4.19) Suppose t = −1. +If T is a unitary in Ht +2, then its free distribution is +characterized by the formula (5.3.10) with r = 1. +(5.4.20) Let t ∈ R \ {±1}. If T is a unitary in Ht +2, then +τ +� n +� +l=1 +T rl +� += Re +� +a +n +� +l=1 +el +� +, with a ∈ T in C, +where +(5.4.21) +el = +� +1 +if rl = 1 +−1 +if rl = ∗, +for l = 1, ..., n, for all (r1, ..., rn) ∈ {1, ∗}n, for all n ∈ N. +Proof. The statement (5.4.19) holds by (5.3.11). In particular, by the unitarity +characterization (5.4.17), the free-distributional data in (5.3.11) must have r = 1, +since +|σt (a, b)| = |w| +denote += +r = 1, +under the similarity of T and S, by (5.4.17). +By (5.4.13), if t ̸= ±1, then the free-distributional data (5.4.21) holds by (5.4.14). +Indeed, under the unitarity of T, the formula (5.4.14) satisfies +R = |a| = 1 and Wo = a ∈ T. +Therefore, the joint free moments (5.4.21) holds. +The above theorem characterizes the free distributions of unitary elements of +(Ht +2, τ) induced by Ht, where t ∈ R \ {1}. +Suppose t = 1, and (a, b) ∈ H1 satisfies AN 5.4.1. In the above Observation, +we showed that the realization T ∈ H1 +2 of (a, b) is a unitary, if and only if either +(a, b) = (a, 0) with a ∈ T, +or +(a, b) = (0, 1) , or (a, b) = (0, −1), +in H1, equivalently, either +T = +� +a +0 +0 +a +� +with a ∈ T, +or +T = +� 0 +1 +1 +0 +� +, or T = +� +0 +−1 +−1 +0 +� +, + +OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS +51 +in H1 +2 (as an element of H1 +2). Thus, if (a, b) = (a, 0) ∈ H1 with |a|2 = 1, then the +free distribution of T is similarly characterized by the formula (5.4.21). Meanwhile, +if T = [(0, 1)]1, then +T ∗ = T ∈ H1 +2 ⊂ H1 +2 (1, ∗) in H1 +2, +and +T 2 = +� +0 +1 +1 +0 +� � +0 +1 +1 +0 +� += +� +1 +0 +0 +1 +� += I2, +in H1 +2, satisfying that +(5.4.22) +(T n)∞ +n=1 = (T, I2, T, I2, T, I2, ...) ; +and, if T = [(0, −1)]1, then +T ∗ = T ∈ H1 +2 ⊂ H1 +2 (1, ∗) in H1 +2, +and +T 2 = +� +0 +−1 +−1 +0 +� � +0 +−1 +−1 +0 +� += +� 1 +0 +0 +1 +� += I2, +in H1 +2, satisfying that +(5.4.23) +(T n)∞ +n=1 = (T, I2, T, I2T, I2, ...) . +Therefore, one obtains the following result in addition with Theorem 59. +Theorem 60. Let (a, b) ∈ H1 be a bicomplex number satisfying AN 5.4.1. Then +the realization T is a unitary in +� +H1 +2, τ +� +, if and only if either +(a, b) = (a, 0) , with |a|2 = 1, +or +(5.4.24) +(a, b) = (0, 1) , or (a, b) = (0, −1) in H1. +(5.4.25) If (a, b) = (a, 0), with |a|2 = 1, in H1, then the free distribution of T is +characterized by the formula (5.4.21). +(5.4.26) If either (a, b) = (0, 1), or (a, b) = (0, −1) in H1, then the free distribution +of the unitary realization T is fully characterized by the free-moment sequence, +(τ (T n))∞ +n=1 = (0, 1, 0, 1, 0, 1, 0, 1, ...). +(5.4.27) +Proof. By the very above Observation after Proposition 58, it is shown that the +realization T ∈ H1 +2 of a bicomplex number (a, b) ∈ H1 is a unitary in H1 +2, if and +only if the condition (5.4.24) holds true. +The statement (5.4.25) is shown similarly by the proof of the statement (5.4.20). +So, the free-distributional data (5.4.21) holds. +Now, if either T = [(0, 1)]1, or T = [(0, −1)]1 in H1 +2, it is not only a unitary, but +also a self-adjoint element of +� +H1 +2, τ +� +, and hence, the free distribution of T is fully +characterized by the free-moment sequence (τ (T n))∞ +n=1. However, by (5.4.22) and +(5.4.23), one immediately obtain the free-moment sequence (5.4.27). Therefore, the +statement (5.4.26) holds. +The above theorem fully characterizes the free distributions of the unitaries of +� +H1 +2, τ +� +induced by bicomplex numbers of H1. + +52 +DANIEL ALPAY AND ILWOO CHO +References +[1] D. Alpay, M. E. Luna-Elizarraras, M. Shapiro, and D. 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Forum, vol. 7, no. 17, (2012) 831 - 838. +[15] S. D. Leo, G. Scolarici, and L. Solombrino, Quaternionic Eigenvalue Problem, J. Math. Phy., +43, (2002) 5815 - 5829. +[16] L. Rodman, Topics in Quaternion Linear Algebra, ISBN:978-0-691-16185-3, (2014) Published +by Prinston Univ. Press, NJ. +[17] B. A. Rozenfeld, The History of non-Eucledean Geometry: Evolution of the Concept of a +Geometric Spaces, ISBN: 978-038-796458-4, (1988) Published by Springer. +[18] R. Speicher, Combinatorial Theory of the Free Product with Amalgamation and Operator- +Valued Free Probability Theory, Amer. Math. Soc., Memoire, ISBN:978-0-8218-0693-7, (1998) +Published by Amer. Math. Soc.. +[19] A. Sudbery, Quaternionic Analysis, DOI: 10.1017/s0305004100053638, Math. Proc. Cam- +bridge Philosop. Soc., (1998) +[20] L. Taosheng, Eigenvalues and Eigenvectors of Quaternion Matrices, J. Central China Normal +Univ., 29, no. 4, (1995) 407 - 411. +[21] J. A. Vince, Geometric Algebra for Computer Graphics, ISBN: 978-1-84628-996-5, (2008) +Published by Springer. +[22] D.V. Voiculescu, K. J. Dykema, and A. Nica, Free Random Variables, ISBN:978-0-8218-1140- +5, (1992) Published by Amer. Math. Soc.. +[23] J. Voight, Quaternion Algebra, Available on http://math.dartmouth.edu/˜jvoight/quat-book.pdf, +(2019) Dept. of Math., Dartmouth Univ.. +Chapman Univ., Dept. +of Math., 1 University Dr., Orange, CA, 92866, U. S. A. / +St. Ambrose Univ., Dept. of Math. and Stat., 421 Ambrose Hall, 518 W. Locust St., +Davenport, Iowa, 52803, U. S. A. +Email address: alpay@chapman.edu / choilwoo@sau.edu + diff --git a/YtFAT4oBgHgl3EQf3h4G/content/tmp_files/load_file.txt b/YtFAT4oBgHgl3EQf3h4G/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..246410b2bf5341fc5a9ae07b22faab81b662a98f --- /dev/null +++ b/YtFAT4oBgHgl3EQf3h4G/content/tmp_files/load_file.txt @@ -0,0 +1,2236 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf,len=2235 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='08720v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='RT] 20 Jan 2023 OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS DANIEL ALPAY AND ILWOO CHO Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In this paper, we consider natural Hilbert-space representations �� C2, πt �� t∈R of the hypercomplex system {Ht}t∈R, and study the realizations πt (h) of hypercomplex numbers h ∈ Ht, as (2 × 2)-matrices acting on C2, for an arbitrarily fixed scale t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Algebraic, operator-theoretic, spectral- analytic, and free-probabilistic properties of them are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Introduction In this paper, we study representations of the hypercomplex numbers (a, b) of complex numbers a and b, constructing a ring, Ht = � C2, +, ·t � , scaled by a real number t ∈ R, where (+) is the usual vector addition on the 2- dimensional vector space C2, and (·t) is the t-scaled vector-multiplication on C2, defined by (a1, b1) ·t (a2, b2) = � a1a2 + tb1b2, a1b2 + b1a2 � , where z are the conjugates of z in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Motivated by the canonical Hilbert-space representation � C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' π � of the quater- nions H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' introduced in [2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' [3] and [19],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' we consider the canonical representation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Πt = � C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' πt � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' of the ring Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' and understand each element h = (a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' b) of Ht as its realization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' πt (h) denote = [h]t def = � a tb b a � in M2 (C) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' where M2 (C) = B � C2� is the matricial algebra (or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' the operator algebra acting on C2) of all (2 × 2)-matrices over C (respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' all bounded linear transformations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' or simply operators on C2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' for each t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Under our setting, one can check that the ring H−1 is nothing but the noncommutative field H of all quaternions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', [2], [3] and [19]), and the ring H1 is the ring of all bicomplex numbers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The spectral-analytic, operator-theoretic (or, matrix-theoretic), and free-probabilistic properties of Ht are considered and characterized under the canonical representa- tion Πt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In particular, certain decompositional properties on Ht are studied alge- braically, and spectral-theoretically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' And then, it is considered how those properties affect the spectral-analytic, operator-theoretic, and free-probabilistic properties of hypercomplex numbers of Ht, for t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 2000 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 20G20, 46S10, 47S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Scaled Hypercomplex Ring, Scaled Hypercomplex Monoids, Repre- sentations, Scaled-Spectral Forms, Scaled-Spectralization, Spectral Theory, Free Probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 1 2 DANIEL ALPAY AND ILWOO CHO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The quaternions H is an interesting object not only in pure mathematics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', [5], [10], [11], [12], [13] [14], [17], [19], [23]), but also in applied mathematics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', [4], [7], [15], [16], [20] and [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Independently, spectral analysis on H is considered in [2] and [3], under representation, “over C,” different from the usual quaternion-eigenvalue problems of quaternion-matrices studied in [13], [15] and 16[].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Motivated by the generalized setting of the quaternions so-called the split-quaternions of [1], and by the main results of [2] and [3], we study a new type of hypercom- plex numbers induced by the pairs of C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Especially, we construct a system of the scaled hypercomplex rings {Ht}t∈R, and study how the hypercomplex num- bers act as (2 × 2)-matrices over C for given scales t ∈ R, under our canonical Hilbert-space representations � Πt = � C2, πt �� t∈R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' We are interested in algebraic, operator-theoretic, spectral-theoretic, free-probabilistic properties of Ht under Πt, for t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Are they similar to those of the quaternions H−1 = H, shown in [2] and [3]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The answers are determined differently case-by-case, up to scales (See below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In Section 2, we define our main objects, the scaled hypercomplex rings {Ht}t∈R, and their canonical Hilbert-space representations {Πt}t∈R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' We un- derstand each hypercomplex number of Ht as an operator, a (2 × 2)-matrix over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' We concentrate on studying the invertibility on Ht, for an arbitrarily fixed scale t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It is shown that if t < 0, then Ht forms a noncommutative field like the quaternions H = H−1, however, if t ≥ 0, then it becomes a ring with unity, which is not a noncommutative field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In Section 3, the spectral theory on (the realizations of) Ht is studied over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' After finding the spectra of hypercomplex numbers, we define so-called the t- spectral forms whose main diagonal entries are from the spectra, and off-diagonal entries are 0’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' As we have seen in [2] and [3], such spectral forms are similar to the realizations of quaternions of H−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' However, if a scale t ∈ R\\{−1} is arbitrary, then such a similarity does not hold in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' We focus on studying such a similarity in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In Section 4, we briefly discuss about how the usual adjoint on M2 (C) acts on the sub-structure Ht 2 of M2 (C), consisting of all realizations of Ht, for a scale t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Different from the quaternionic case of [2] and [3], in general, the adjoints (conjugate-transposes) of many matrices of Ht 2 are not contained in Ht 2, especially, if t ̸= −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It shows that a bigger, operator-algebraically-better ∗-algebraic structure generated by Ht 2 is needed in M2 (C), to consider operator-theoretic, and free- probabilistic properties on Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In the final Section 5, on the C∗-algebraic structure of Section 4, we study operator-theoretic, and free-probabilistic properties up to the usual trace, and the normalized trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The Scaled Hypercomplex Systems {Ht}t∈R In this section, we define a ring Ht of hypercomplex numbers, and establish the corresponding canonical Hilbert-space representations Πt, for an arbitrary fixed scale t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Throughout this section, we let C2 = {(a, b) : a, b ∈ C} OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS 3 be the Cartesian product of two copies of the complex field C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' One may understand C2 as the usual 2-dimensional Hilbert space equipped with its canonical orthonor- mal basis, {(1, 0) , (0, 1)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' A t-Scaled Hypercomplex Ring Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In this section, we fix an arbitrary real number t in the real field R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' On the vector space C2 (over C), define the t-scaled vector-multiplication (·t) by (a1, b1) ·t (a2, b2) def = � a1a2 + tb1b2, a1b2 + b1a2 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1) for (al, bl) ∈ C2, for all l = 1, 2, where z are the conjugates of z in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It is not difficult to check that such an operation (·t) is closed on C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Moreover, it satisfies that ((a1, b1) ·t (a2, b2)) ·t (a3, b3) = � a1a2 + tb1b2, a1b2 + b1a2 � t (a3, b3) = � a1a2a3 + t � b1b2a3 + a1b2b3 + b1a2b3 � , a1a2b3 + a1b2a3 + b1a2a3 + tb1b2b3 � , and (a1, b1) ·t ((a2, b2) ·t (a3, b3)) = (a1, b1) ·t � a2a3 + tb2b3, a2b3 + b2a3 � = � a1 � a2a3 + tb2b3 � + tb1 � a2b3 + b2a3 � , a1 (a2b3 + b2a3) + b1 � a2a3 + tb2b3 �� , implying the equality, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2) ((a1, b1) ·t (a2, b2)) ·t (a3, b3) = (a1, b1) ·t ((a2, b2) ·t (a2, b3)) , in C2, for (al, bl) ∈ C2, for all l = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Furthermore, if ϑ = (1, 0) ∈ C2, then ϑ ·t (a, b) = (a, b) = (a, b) ·t ϑ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3) by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1), for all (a, b) ∈ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3), if C2× = C2 \\ {(0, 0)} , then the pair � C2×, ·t � forms a monoid (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', semigroup with its identity (1, 0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let C2× = C2 \\ {(0, 0)}, and (·t) be the closed operation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1) on C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then the algebraic structure � C2×, ·t � forms a monoid with its identity (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The proof is done by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Therefore, one can obtain the following ring structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The algebraic triple � C2, +, ·t � forms a unital ring with its unity (or, the multiplication-identity) (1, 0), where (+) is the usual vector addition on C2, and (·t) is the vector multiplication (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 4 DANIEL ALPAY AND ILWOO CHO Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Clearly, the algebraic pair � C2, + � is an abelian group under the usual addi- tion (+) with its (+)-identity (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' While, by Lemma 1, the pair � C2×, ·t � forms a monoid (and hence, a semigroup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Observe now that (a1, b1) ·t ((a2, b2) + (a3, b3)) = (a1, b1) ·t (a2 + a3, b2 + b3) = � a1 (a2 + a3) + tb1 � b2 + b3 � , a1 (b2 + b3) + b1 (a2 + a3) � = � a1a2 + a1a3 + tb1b2 + tb1b3, a1b2 + a1b3 + b1a2 + b1a3 � = � a1a2 + tb1b2, a1b2 + b1a2 � + � a1a3 + tb1b3, a1b3 + b1a3 � = (a1, b1) ·t (a2, b2) + (a1, b1) ·t (a3, b3), and, similarly, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4) ((a1, b1) + (a2, b2)) ·t (a3, b3) = (a1, b1) ·t (a3, b3) + (a2, b2) ·t (a3, b3) , in C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, the operations (+) and (·t) are left-and-right distributive by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Therefore, the algebraic triple � C2, +, ·t � forms a unital ring with its unity (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The above proposition characterizes the algebraic structure of � C2, +, ·t � as a well-defined unital ring for a fixed t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Remark here that, since a scale t is arbitrary in R, in fact, we obtain the unital rings {Ht}t∈R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' For a fixed t ∈ R, the ring � C2, +, ·t � is called the hypercomplex ring with its scale t (in short, the t-scaled hypercomplex ring).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By Ht, we denote the t-scaled hypercomplex ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The Canonical Representation Πt = � C2, πt � of Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In this section, we fix t ∈ R, and the corresponding t-scaled hypercomplex ring, Ht = � C2, +, ·t � , where (·t) is the vector-multiplication (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' We consider a natural finite-dimensional- Hilbert-space representation Πt of Ht, and understand each hypercomplex number h ∈ Ht as an operator acting on a Hilbert space determined by Πt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In particu- lar, as in the quaternionic case of [2], [3] and [19], a 2-dimensional-Hilbert-space representation of the hypercomplex ring Ht is established naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Define now a morphism, πt : Ht → B � C2� = M2 (C) , by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1) πt ((a, b)) = � a tb b a � , ∀ (a, b) ∈ Ht, where B (H) is the operator algebra consisting of all bounded (or, continuous linear) operators on a Hilbert space H, and Mk (C) is the matricial algebra of all (k × k)- matrices over C, isomorphic to B � Ck� , for all k ∈ N (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', [8] and [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By definition, the function πt of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1) is an injective map from Ht into M2 (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Indeed, if (a1, b1) ̸= (a2, b2) in Ht, then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2) πt ((a1, b1)) = � a1 tb1 b1 a1 � ̸= � a2 tb2 b2 a2 � = πt ((a2, b2)) , OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS 5 in M2 (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Furthermore, it satisfies that πt ((a1, b1) + (a2, b2)) = \uf8eb \uf8ed a1 + a2 t (b1 + b2) b1 + b2 a1 + a2 \uf8f6 \uf8f8 = � a1 tb1 b1 b2 � + � a2 tb2 b2 a2 � = πt ((a1, b1)) + πt ((a2, b2)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3) Also, one has πt ((a1, b1) ·t (a2, b2)) = πt �� a1a2 + tb1b2, a1b2 + b1a2 �� by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1) = \uf8eb \uf8ed a1a2 + tb1b2 t (a1b2 + b1a2) a1b2 + b1a2 a1a2 + tb1b2 \uf8f6 \uf8f8 = � a1 tb1 b1 a1 � � a2 tb2 b2 a2 � = πt ((a1, b1)) πt ((a2, b2)) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4) where the multiplication (·) in the far-right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4) is the usual matricial multiplication on M2 (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Since our t-scaled hypercomplex ring Ht = � C2, +, ·t � is identified with the 2- dimensional space C2 (set-theoretically), one may / can understand this ring Ht as a topological ring equipped with the usual topology for C2, for any t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' From below, we regard the ring Ht as a topological unital ring under the usual topology for C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The pair � C2, πt � is an injective Hilbert-space representation of the t-scaled hypercomplex ring Ht, where πt is an action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The morphism πt : Ht → M2 (C) of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1) is a well-defined injective function by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Moreover, this map πt satisfies the relations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4), and hence, it is a(n algebraic) ring-action of Ht, acting on the 2-dimensional vector space C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, the pair � C2, πt � forms an algebraic representation of Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By regarding Ht and M2 (C) as topological spaces equipped with their usual topologies, then it is not difficult to check that the ring-action πt is continuous from Ht (which is homeomorphic to C2 as a topological space) into M2 (C) (which is ∗-isomorphic to the C∗-algebra B � C2� ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Thus, the algebraic representation � C2, πt � forms a Hilbert-space representation of Ht acting on C2 via πt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The above lemma shows that the t-scaled hypercomplex ring Ht is realized in the matricial algebra M2 (C) as πt (Ht) = �� a tb b a � ∈ M2 (C) : (a, b) ∈ Ht � , as an embedded topological ring in M2 (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The realization πt (Ht) of the t-scaled hypercomplex ring Ht is called the t-scaled (hypercomplex-)realization of Ht (in M2 (C)), for a scale t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' And we denote πt (Ht) by Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', Ht 2 denote = πt (Ht) = �� a tb b a � : (a, b) ∈ Ht � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 6 DANIEL ALPAY AND ILWOO CHO Also, by [ξ]t, we denote πt (ξ) ∈ Ht 2, for all ξ ∈ Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By the above lemma and definition, we obtain the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' For t ∈ R, the corresponding t-scaled hypercomplex ring Ht is topological- ring-isomorphic to the t-scaled realization Ht 2 in M2 (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', Ht T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='R = Ht 2 in M2 (C), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5) where “ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='R = ” means “being topological-ring-isomorphic to.” Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The relation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5) is proven by Lemma 4 and the injectivity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2) of πt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By the above theorem, one can realize that Ht and Ht 2 as an identical topological ring, for a fixed t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Recall that the relation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5) is independently shown in [2] and [3], only for the quaternionic case where t = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Scaled Hypercomplex Monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Throughout this section, we fix a scale t ∈ R, and the corresponding t-scaled hypercomplex ring, Ht = � C2, +, ·t � , which is isomorphic to the t-scaled realization, Ht 2 = �� a tb b a � ∈ M2 (C) : (a, b) ∈ Ht � , in M2 (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let H× t denote = Ht \\ {(0, 0)} , set-theoretically, where (0, 0) ∈ Ht is the (+)-identity of the abelian group � C2, + � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Thus, by Proposition 2, this set forms a well-defined semigroup, H× t denote = � H× t , ·t � , equipped with its (·t)-identity (1, 0), and hence, the pair H× t is the maximal monoid embedded in Ht 2 up to the operation (·t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The maximal monoid H× t = � H× t , ·t � , embedded in the t-scaled hypercomplex ring Ht, is called the t-scaled hypercomplex monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5), it is trivial that: Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The t-scaled hypercomplex monoid H× t is monoid-isomorphic to the monoid Ht× 2 denote = � Ht× 2 , · � , equipped with its identity, I2 = � 1 0 0 1 � = � 1 t · 0 0 1 � = [(1, 0)]t , the (2 × 2)-identity matrix of M2 (C), where (·) is the usual matricial multiplication inherited from that on M2 (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', H× t = � H× t , ·t � Monoid = � Ht× 2 , · � = Ht× 2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1) where “ Monoid = ” means “being monoid-isomorphic.” Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The isomorphic relation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1) is proven by the proof of Proposition 2, and that of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Invertibility on Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In this section, by identifying our t-scaled hypercomplex ring Ht as its isomorphic realization Ht 2, we consider invertibility of elements of Ht, for an arbitrarily fixed t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Observe first that, for any (a, b) ∈ Ht realized to be [(a, b)]t ∈ Ht 2, one can get that det ([(a, b)]t) = det � a tb b a � = |a|2 − t |b|2 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1) det ([(a, b)]t) = |a|2 − t |b|2 , where det : M2 (C) → C is the determinant, and |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='| is the modulus on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let (a, b) ∈ Ht, realized to be [(a, b)]t ∈ Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2) det ([(a, b)]t) = |a|2 − t |b|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3) If either |a|2 > t |b|2, or |a|2 < t |b|2, then [(a, b)]t is invertible “in M2 (C),” with its inverse matrix, [(a, b)]−1 t = 1 |a|2 − t |b|2 � a t (−b) (−b) a � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4) If |a|2 − t |b|2 ̸= 0, then (a, b) ∈ Ht is invertible in the sense that there exists a unique (c, d) ∈ Ht, such that (a, b) ·t (c, d) = (1, 0) = (c, d) ·t (a, b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In particular, one has that (c, d) = � a |a|2 − t |b|2 , −b |a|2 − t |b|2 � ∈ C2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5) Assume that (a, b) is invertible in Ht in the sense of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then the inverse is also contained “in Ht.” Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The statement (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2) is shown by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Note-and-recall that a matrix A ∈ Mn (C) is invertible in Mn (C), if and only if det (A) ̸= 0, for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Therefore, det ([(a, b)]t) ̸= 0 ⇐⇒ [(a, b)]t is invertible in M2 (C) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2), |a|2 − t |b|2 ̸= 0, ⇐⇒ [(a, b)]t is invertible in M2 (C) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Moreover, |a|2 − t |b|2 ̸= 0, if and only if [(a, b)]−1 t = � a tb b a �−1 = 1 |a|2 − t |b|2 � a −tb −b a � , in M2 (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Therefore, the statement (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3) holds true in M2 (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3), one has det ([(a, b)]t) ̸= 0, if and only if [(a, b)]−1 t = \uf8eb \uf8ec \uf8ec \uf8ed a |a|2−t|b|2 t � −b |a|2−t|b|2 � � −b |a|2−t|b|2 � a |a|2−t|b|2 \uf8f6 \uf8f7 \uf8f7 \uf8f8 ∈ M2 (C) , 8 DANIEL ALPAY AND ILWOO CHO and it is actually contained ”in Ht 2,” satisfying π−1 t \uf8eb \uf8ec \uf8ec \uf8ed a |a|2−t|b|2 t � −b |a|2−t|b|2 � � −b |a|2−t|b|2 � a |a|2−t|b|2 \uf8f6 \uf8f7 \uf8f7 \uf8f8 = � a |a|2 − t |b|2 , −b |a|2 − t |b|2 � , in Ht, by the injectivity of πt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It shows that [(a, b)]−1 t exists in M2 (C), if and only if it is contained “in Ht 2.” i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', if [(a, b)]t is invertible, then its inverse is also contained in Ht 2, too, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, the statements (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The above theorem not only characterizes the invertibility of the monoidal ele- ments of the t-scaled hypercomplex monoid H× t , but also confirms that the inverses (if exist) are contained in the monoid H× t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', (a, b)−1 exists, ⇐⇒ (a, b)−1 = � a |a|2 − t |b|2 , −b |a|2 − t |b|2 � , ”in H× t ,” equivalently, � (a, b)−1� t = [(a, b)]−1 t in H× 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let (a, b) ∈ H× t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then it is invertible, if and only if � (a, b)−1� t = �� a |a|2−t|b|2 , −b |a|2−t|b|2 �� t = [(a, b)]−1 t , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='6) in H× 2 , where [(a, b)]−1 t means the matricial inverse in M2 (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='6) is immediately done by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The above corollary can be re-stated by that: if ξ ∈ H× t is invertible, then πt � ξ−1� = (πt (ξ))−1 in Ht× 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Now consider the cases where |a|2 − t |b|2 = 0 ⇐⇒ |a|2 = t |b|2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='7) in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' As we have seen above, the condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='7) holds for (a, b) ∈ Ht, if and only if (a, b) is not invertible in Ht (and hence, its realization [(a, b)]t is not invertible in M2 (C), and hence, in Ht 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Clearly, we are not interested in the (+)-identity (0, 0) of Ht automatically satisfying the condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, without loss of generality, we focus on elements (a, b) of the t-scaled hypercomplex monoid H× t (or, its realizations [(a, b)]t of Ht× 2 ), satisfying the condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Recall that an algebraic triple, (X, +, ·), is a noncommutative field, if (i) (X, +) is an abelian group, (ii) (X×, ·) forms a non-abelian group, and (iii) the operations (+) and (·) are left-and-right distributive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' For instance, the quaternions H = H−1 is a noncommutative field (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', [2] and [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Suppose the fixed scale t ∈ R is negative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', t < 0 in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then “all” elements (a, b) of the t-scaled hypercomplex monoid H× t are invertible in Ht, with their inverses, � a |a|2 − t |b|2 , −b |a|2 − t |b|2 � ∈ H× t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS 9 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='8) t < 0 in R =⇒ Ht is a noncommutative field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Suppose the scale t ∈ R is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then, for any (a, b) ∈ H× t , |a|2 ̸= t |b|2 ⇐⇒ |a|2 − t |b|2 > 0, since (a, b) ̸= (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', if t < 0, then every element (a, b) ∈ H× t does “not” satisfy the condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It implies that if t < 0, then every element (a, b) ∈ H× t is invertible in H× t , by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' and the inverse is determined to be (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='6) in H× t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Thus, the pair H× t = � H× t , ·t � forms a group which is not abelian by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Therefore, if t < 0 in R, then the t-scaled hypercomplex ring Ht becomes a noncommutative field, proving the statement (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The above theorem characterizes that the algebraic structure of scaled hyper- complex rings {Ht}t<0 as noncommutative fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Suppose t = 0 in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then an element (a, b) of the 0-scaled hyper- complex monoid H× 0 is invertible in H0, with their inverses, � a |a|2 , −b |a|2 � ∈ H× 0 , if and only if a ̸= 0 in C, if and only if only the elements of the subset, � (a, b) ∈ H× 0 : a ̸= 0 � of H× 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='9) are invertible in H× 0 , if and only if (0, b) ∈ H× 0 are not invertible in H× 0 , for all b ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Assume that we have the zero scale, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', t = 0 in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='7), |a|2 = 0 · |b|2 ⇐⇒ |a|2 = 0 ⇐⇒ a = 0 in C, if and only if (0, b) ∈ H× 0 are not invertible in H× 0 , for all b ∈ C, if and only if all elements (a, b), contained in the subset (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='9), are invertible in H× 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Observe that (a, b) is contained in the subset (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='9) of H× 0 , if and only if [(a, b)]0 �� a |a|2 , −b |a|2 �� 0 = � a 0 b a � \uf8eb \uf8ec \uf8ed a |a|2 0 −b |a|2 a |a|2 \uf8f6 \uf8f7 \uf8f8 = � 1 0 0 1 � = \uf8eb \uf8ec \uf8ed a |a|2 0 −b |a|2 a |a|2 \uf8f6 \uf8f7 \uf8f8 � a 0 b a � = �� a |a|2 , −b |a|2 �� 0 [(a, b)]0 , in H× 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Therefore, if exists, (a, b)−1 = � a |a|2 , −b |a|2 � in H× 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The above theorem shows that if we have the zero-scale in R, then our 0-scaled hypercomplex ring H0 cannot be a noncommutative field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It directly illustrates that the algebra on the quaternions H = H−1, and the algebra on the scaled- hypercomplex rings {Ht}t∈R\\{−1} can be different in general, especially, when t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 10 DANIEL ALPAY AND ILWOO CHO Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Suppose the scale t ∈ R is positive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', t > 0 in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then an element (a, b) ∈ H× t is invertible in H× t with its inverse, � a |a|2 − t |b|2 , −b |a|2 − t |b|2 � ∈ H× t , if and only if |a|2 ̸= t |b|2 in R+ 0 = {r ∈ R : r ≥ 0}, if and only if (a, b) is contained in the subset, � (a, b) : |a|2 ̸= t |b|2 in R+ 0 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10) of H× t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' As application, if t > 0 in R, then the all elements of {(a, 0) ∈ Ht : a ∈ C×} ∪ {(0, b) ∈ Ht : b ∈ C×} , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='11) are invertible in Ht, where C× = C \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Assume that t > 0 in R, and H× t , the corresponding t-scaled hypercomplex monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then (a, b) ∈ H× t is invertible in H× t , if and only if the condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='7) does not hold, if and only if |a|2 ̸= t |b|2 ⇐⇒ either |a|2 > t |b|2 , or |a|2 < t |b|2 , in R+ 0 , since t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Therefore, if t > 0 in R, then an element (a, b) is invertible in H× t , if and only if either |a|2 > t |b|2 , or |a|2 < t |b|2 in R+ 0 , if and only if (a, b) is contained in the subset (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10) in H× t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In particular, for t > 0 in R, (i) if (a, 0) ∈ H× t with a ∈ C×, then |a|2 > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' and (ii) if (0, b) ∈ H× t with b ∈ C×, then 0 < t |b|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Therefore, the subset (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='11) is properly contained in the subset (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10) in H× t , whenever t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, all elements, formed by (a, 0) ,or by (0, b) with a, b ∈ C×, are invertible in H× t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The above theorem characterizes the invertibility on the t-scaled hypercomplex monoid H× t , where the scale t is positive in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Theorems 11, 12 and 13 refine Theorem 8, case-by-case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' We again summarize the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Corollary 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let H× t be the t-scaled hypercomplex monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' If t < 0, then all nonzero elements of H× t are invertible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' and if t = 0, then � (a, b) ∈ H× 0 : a ̸= 0 � is the invertible proper subset of H× 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' and if t > 0, then � (a, b) : |a|2 ̸= t |b|2 in R+ 0 � is the invertible proper subset of H× t , where “invertible subset of H× t ” means “a subset of H× t containing of all invertible elements.” Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' This corollary is nothing but a summary of Theorems 11, 12 and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Decompositions of the Nonnegatively-Scaled Hypercomplex Rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In this section, we consider a certain decomposition of the t-scaled hypercomplex ring Ht, for an arbitrary fixed “positive” scale t > 0 in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Recall that, as we have seen in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4, the negatively-scaled hypercomplex rings {Hs}s<0 are noncommutative fields by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='8), equivalently, the negatively-scaled hypercomplex monoids {H× s }s<0 are non-abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' However, if t ≥ 0, then Ht cannot be a noncommutative field in general, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' We here concentrate on such cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let t ≥ 0 and Ht, the corresponding t-scaled hypercomplex ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Partition Ht by Ht = Hinv t ⊔ Hsing t with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1) Hinv t = � (a, b) : |a|2 ̸= t |b|2� , and Hsing t = � (a, b) : |a|2 = t |b|2� , where ⊔ is the disjoint union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10), (a, b) ∈ Hinv t , if and only if it is invertible, equivalently, (a, b) ∈ Hsing t , if and only if it is not invertible, in Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Recall-and-note that the determinant is a multiplicative map on Mn (C), for all n ∈ N, in the sense that: det (AB) = det (A) det (B) , ∀A, B ∈ Mn (C) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2) Thus, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2), one has ξ, η ∈ Hinv t ⇒ det ([ξ ·t η]t) = det ([ξ]t [η]t) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3) Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let t ≥ 0 in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then the subset Hinv t denote = � Hinv t , ·t � of the t-scaled hypercomplex monoid H× t forms a non-abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', Hinv t is not only a sub- monoid, but also an embedded group in H× t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2), if ξ, η ∈ Hinv t , then ξ ·t η ∈ Hinv t , too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', the operation (·t) is closed, and associative on Hinv t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Also, the (·t)-identity (1, 0) is contained in Hinv t by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Therefore, the sub-structure � Hinv t , ·t � forms a sub-monoid of H× t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' But, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='8) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3), each element ξ ∈ Hinv t has its (·t)-inverse ξ−1 contained in Hinv t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It shows that Hinv t forms a non-abelian group in the monoid H× t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By the partition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1) and the multiplicativity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3), one can obtain the following equivalent result of the above theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let t ≥ 0 in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then the pair H×sing t denote = � Hsing t ∩ H× t , ·t � = � Hsing t \\ {(0, 0)} , ·t � forms a semigroup without identity in the t-scaled hypercomplex monoid H× t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3), the operation (·t) is closed and associative on the set, H×sing t def = H× t ∩ Hsing t = Hsing t \\ {(0, 0)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' However, the (·t)-identity (1, 0) is not contained in H×sing t , since I2 = [(1, 0)]t is in Hinv t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, in the monoid H× t , the sub-structure � H×sing t , ·t � forms a semigroup (without identity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 12 DANIEL ALPAY AND ILWOO CHO The above lemma definitely includes the fact that: � Hsing t , ·t � is just a semigroup (without identity), which is not a sub-monoid of H× t (and hence, not a group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The above two algebraic characterizations show that the set-theoretical decom- position (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1) induces an algebraic decomposition of the t-scaled hypercomplex monoid H× t , H× t = � Hinv t , ·t � ⊔ � H×sing t , ·t � , where (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4) Hinv t = � (a, b) ∈ H× t : |a|2 ̸= t |b|2� , and H×sing t = � (a, b) ∈ H× t : |a|2 = t |b|2� , whenever t ≥ 0 in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Theorem 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' For t ≥ 0 in R, the t-scaled hypercomplex monoid H× t is algebraically decomposed to be H× t = Hinv t ⊔ H×sing t , where Hinv t is the group, and H×sing t is the semigroup without identity in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The algebraic decomposition, H× t = Hinv t ⊔ H×sing t , of the t-scaled hypercomplex monoid H× t is obtained by the set-theoretic decompo- sition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1) of H× t , the above two lemmas, and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By the above theorem, one can have the following concepts whenever a given scale t is nonnegative in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Definition 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let t ≥ 0 in R, and H× t , the t-scaled hypercomplex monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The algebraic block, Hinv t = �� (a, b) ∈ H× t : |a|2 ̸= t |b|2� , ·t � , is called the group-part of H× t (or, of Ht), and the other algebraic block, H×sing t = �� (a, b) ∈ H× t : |a|2 = t |b|2� , ·t � , is called the semigroup-part of H× t (or, of Ht).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By the above definition, Theorem 17 can be re-stated that: if a scale t is non- negative in R, then the t-scaled hypercomplex monoid H× t is decomposed to be the group-part Hinv t and the semigroup-part H×sing t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' One may / can say that if t < 0 in R, then the semigroup-part H×sing t is empty in H× t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Indeed, for any scale t ∈ R, the t-scaled hypercomplex monoid Ht is decom- posed to be (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' As we have seen in this section, if t ≥ 0, then the semigroup-part H×sing t is nonempty, meanwhile, as we considered in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4, if t < 0, then the semigroup-part H×sing t is empty, equivalently, the t-scaled hypercomplex monoid H× t is identified with its group-part Hinv t , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', H× t = Hinv t in Ht, whenever t < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS 13 Corollary 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' For every t ∈ R, the t-scaled hypercomplex monoid H× t is partitioned by H× t = Hinv t ⊔ H×sing t , where the group-part Hinv t and the semigroup-part H×sing t are in the sense of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In particular, if t < 0, then H×sing t = Ø ⇐⇒ H× t = Hinv t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' meanwhile, if t ≥ 0, then H×sing t is a non-empty proper subset of H× t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It is shown conceptually by the discussion of the very above paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Also, see Theorems 11 and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Spectral Analysis on {Ht}t∈R Under �� C2, πt �� t∈R Throughout this section, we fix an arbitrary scale t ∈ R, and the corresponding t-scaled hypercomplex ring, Ht = � C2, +, ·t � , containing its hypercomplex monoid H× t = � H× t , ·t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In Section 2, we showed that for a scale t ∈ R, the monoid H× t is partitioned by H× t = Hinv t ⊔ H×sing t , where Hinv t is the group-part, and H×sing t is the semigroup-part of Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In particular, if t < 0, then the semigroup-part H×sing t is empty in H× t , equivalently, H× t = Hinv t in Ht, meanwhile, if t ≥ 0, then H×sing t is a non-empty proper subset of H× t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Motivated by such an analysis of invertibility on Ht, we here consider spectral analysis on Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Hypercomplex-Spectral Forms on Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' For t ∈ R, let Ht be the t-scaled hypercomplex ring realized to be Ht 2 = πt (Ht) = �� a tb b a � ∈ M2 (C) : (a, b) ∈ Ht � , in M2 (C) under the Hilbert-space representation Πt = � C2, πt � of Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let (a, b) ∈ Ht be an arbitrary element with πt (a, b) = [(a, b)]t = � a tb b a � ∈ Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then, in a variable z on C, det ([(a, b)]t − z [(1, 0)]t) = det \uf8eb \uf8ed a − z tb b a − z \uf8f6 \uf8f8 = (a − z) (a − z) − t |b|2 = |a|2 − az − az + z2 − t |b|2 = z2 − (a + a) z + � |a|2 − t |b|2� = z2 − 2Re (a) z + det ([(a, b)]t), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1) where Re (a) is the real part of a in C, and det ([(a, b)]t) = |a|2 − t |b|2 , 14 DANIEL ALPAY AND ILWOO CHO by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Thus, the equation, det ([(a, b)]t − z [(1, 0)]t) = 0, in a variable z on C, has its solutions, z = 2Re (a) ± � 4Re (a)2 − 4det ([(a, b)]t) 2 , ⇐⇒ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2) z = Re (a) ± � Re (a)2 − det ([(a, b)]t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Recall that a matrix A ∈ Mn (C), for any n ∈ N, has its spectrum, spec (A) = {λ ∈ C : det (A − λIn) = 0} , equivalently, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3) spec (A) = {λ ∈ C : ∃η ∈ Cn, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', Aη = λη} , if and only if spec (A) = {λ ∈ C : A − λIn is not invertible in Mn (C)} , as a nonempty discrete (compact) subset of C, where In is the identity matrix of Mn (C) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' More generally, if T ∈ B (H) is an operator on a Hilbert space H, then the spectrum σ (T ) of T is defined to be a nonempty compact subset, σ (T ) = {z ∈ C : T − zIH is not invertible on H} , where IH is the identity operator of B (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Remark that if H is infinite-dimensional, then σ (T ) is not a discrete subset of C as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3), in general (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Theorem 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let (a, b) ∈ Ht realized to be [(a, b)]t ∈ Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then spec ([(a, b)]t) = � Re (a) ± � Re (a)2 − det ([(a, b)]t) � , in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' More precisely, if a = x + yi, b = u + vi ∈ C, with x, y, u, v ∈ R and i = √−1 in C, then spec ([(a, b)]t) = � x ± i � y2 − tu2 − tv2 � in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The realization [(a, b)]t = � a tb b a � ∈ Ht 2 of a hypercomplex number (a, b) ∈ Ht has its spectrum, spec ([(a, b)]t) = � Re (a) ± � Re (a)2 − � |a|2 − t |b|2�� , in C, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' If a = x + yi, and b = u + vi in C, with x, y, u, v ∈ R and i = √−1 in C, then Re (a) = x, and |a|2 − t |b|2 = � x2 + y2� − t � u2 + v2� , OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS 15 in R, and hence, spec ([(a, b)]t) = � x ± � −y2 + tu2 + tv2 � , if and only if spec ([(a, b)]t) = � x ± i � y2 − tu2 − tv2 � , in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Therefore, the set-equality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' From below, for our purposes, we let a = x + yi and b = u + vi in C, with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5) x, y, u, v ∈ R, and i = √ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The above theorem can be refined by the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Corollary 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let (a, b) ∈ Ht, realized to be [(a, b)]t ∈ Ht 2, satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='6) If Im (a)2 = t |b|2 in R, where Im (a) is the imaginary part of a in C, then spec ([(a, b)]t) = {x} = {Re (a)} in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='7) If Im (a)2 < t |b|2 in R, then spec ([(a, b)]t) = � x ± � tu2 + tv2 − y2 � in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='8) If Im (a)2 > t |b|2 in R, then spec ([(a, b)]t) = � x ± i � y2 − tu2 − tv2 � in C \\ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' For (a, b) ∈ Ht, satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5), one has spec ([(a, b)]t) = � x ± i � y2 − tu2 − tv2 � , by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, one can verify that: (i) if y2 − tu2 − tv2 = 0, equivalently, if Im (a)2 = t |b|2 in R, then spec ([(a, b)]t) = � x ± i √ 0 � = {x} in R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (ii) if y2 − tu2 − tv2 < 0, equivalently, if Im (a)2 < t |b|2 in R, then x ± i � y2 − tu2 − tv2 = x ± i � − |y2 − tu2 − tv2|, implying that x ± i � y2 − tu2 − tv2 = x ± i2� tu2 + tv2 − y2, and hence, spec ([(a, b)]t) = � x ∓ � tu2 + tv2 − y2 � in R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' and, finally, (iii) if y2 − tu2 − tv2 > 0, equivalently, if Im (a)2 > t |b|2 in R, then spec ([(a, b)]t) = � x ± i � y2 − tu2 − tv2 � , contained in C \\ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 16 DANIEL ALPAY AND ILWOO CHO Therefore, the refined statements (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='6), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='8) of the spectrum (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4) of [(a, b)]t hold true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By the above corollary, one immediately obtains the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Corollary 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Suppose (a, b) ∈ Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' If Im (a)2 ≤ t |b|2, then spec ([(a, b)]t) ⊂ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' meanwhile, if Im (b)2 > t |b|2, then spec ([(a, b)]t) ⊂ (C \\ R) , in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It is shown by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='6), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Also, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Theorem 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Assume that the fixed scale t ∈ R is negative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', t < 0 in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' If (a, b) ∈ Ht, with b ̸= 0 in C, then spec ([(a, b)]t) ⊂ (C \\ R) in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='9) Meanwhile, if b = 0 in C for (a, b) ∈ Ht, then a ∈ R =⇒ spec ([(a, 0)]t) = {a} in R, and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10) a ∈ C \\ R =⇒ spec ([(a, 0)]t) = {a, a} in C \\ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Assume that the scale t is given to be negative in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then, for any (a, b) ∈ Ht, one immediately obtains that Im (a)2 ≥ t |b|2 , because the left-hand side, Im (a)2, is nonnegative, but the right-hand side, t |b|2 is either negative or zero in R by the negativity of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Suppose b ̸= 0 in C, equivalently, |b|2 > 0, implying t |b|2 < 0 in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then Im (a)2 > t |b|2 in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Thus, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='8), the spectra, spec ([(a, b)]t), of the realizations [(a, b)]t of (a, b) ∈ Ht, with b ̸= 0, is contained in C \\ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It proves the relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Meanwhile, if a = Re (a), and b = 0 in C, then 0 = Im (a)2 ≤ 0 = t · 0 in R, implying that spec ([(a, 0)]t) ⊂ R in C, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' However, if Im (a) ̸= 0, and b = 0, then Im (a)2 > 0 = t · 0 in R, and hence, spec ([(a, 0)]t) ⊂ (C \\ R) in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, the relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10) is proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The above theorem specifies Theorem 19 for the case where t < 0 in R, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='9) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS 17 Theorem 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Assume that t = 0 in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' If (a, b) ∈ H0 with Im (a) ̸= 0 in C, then spec ([(a, b)]t) ⊂ (C \\ R) in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='11) Meanwhile, if Im (a) = 0, then spec ([(a, b)]t) ⊂ R in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='12) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Suppose the fixed scale t is zero in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then, for any hypercomplex number (a, b) ∈ H0, one has [(a, b)]0 = � a 0 b a � ∈ H0 2, and hence, Im (a)2 ≥ 0 = 0 · |b|2 in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In particular, if Im (a) ̸= 0 in C, then the above inequality becomes Im (a)2 > 0 in R, implying that spec ([(a, b)]t) ⊂ (C \\ R) in C, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', for all (a, b) ∈ H0, with a ∈ C with Im (a) ̸= 0, and b ∈ C arbitrary, the spectra of the realizations of such (a, b) are contained in C \\ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It shows the relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='11) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Meanwhile, if Im (a) = 0 in C, then one has Im (a)2 = 0 ≥ 0 = 0 · |b|2 in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='6), we have spec ([(a, b)]t) ⊂ R in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Therefore, the relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='12) holds true, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The above theorem specifies Theorem 19 for the case where a scale t is zero in R, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='11) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Theorem 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Assume that the fixed scale t is positive in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then the t-scaled hypercomplex ring Ht is decomposed to be Ht = H+ t ⊔ H−0 t , with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='13) H+ t = � (a, b) ∈ Ht : Im (a)2 > t |b|2� , and H−0 t = � (a, b) ∈ Ht : Im (a)2 ≤ t |b|2� , where ⊔ is the disjoint union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Moreover, if (a, b) ∈ H+ t , then spec ([(a, b)]t) ⊂ (C \\ R) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='14) meanwhile, if (a, b) ∈ H−0 t , then spec ([(a, b)]t) ⊂ R in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='15) 18 DANIEL ALPAY AND ILWOO CHO Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Suppose that t > 0 in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then one can decompose the t-scaled hypercomplex ring Ht by Ht = H+ t ⊔ H−0 t , with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='16) H+ t = � (a, b) ∈ Ht : Im (a)2 > t |b|2� , and H−0 t = � (a, b) ∈ Ht : Im (a)2 ≤ t |b|2� , set-theoretically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Thus, the partition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='13) holds by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By Theorem 19 and Corollary 20, if (a, b) ∈ H+ t , then spec ([(a, b)]t) ⊂ (C \\ R) , meanwhile, if (a, b) ∈ H−0 t , then spec ([(a, b)]t) ⊂ R, in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, the relations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='14) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='15) are proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The above theorem specifies Theorem 19 for the cases where a fixed scale t is positive in R, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='14) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='15), up to the decomposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In fact, one can realize that, for “all” t ∈ R, the corresponding t-scaled hyper- complex ring Ht is partitioned to be Ht = H+ t ⊔ H−0 t , where H+ t and H−0 t are in the sense of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Especially, Theorems 22, 23 and 24 characterize the above decomposition case-by-case, based on Theorem 19 and Corollary 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, we obtain the following universal spectral properties on Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Corollary 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let t ∈ R be an arbitrarily fixed scale for Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then Ht = H+ t ⊔ H−0 t , set-theoretically, where � H+ t , H−0 t � is a partition in the sense of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='13) for t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Moreover, if (a, b) ∈ H+ t , then spec ([(a, b)]t) ⊂ (C \\ R) , meanwhile, if (a, b) ∈ H−0 t , then spec ([(a, b)]t) ⊂ R in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Especially, if t < 0, then H−0 t = {(0, 0)}, equivalently, H× t = H+ t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' This corollary is nothing but a summary of Theorems 22, 23 and 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It is not hard to check the converses of the statements of Corollary 25 hold true, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Theorem 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let Ht = H+ t ⊔H−0 t be the fixed t-scaled hypercomplex ring for t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='17) (a, b) ∈ H+ t , if and only if spec ([(a, b)]t) ⊂ (C \\ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='18) (a, b) ∈ H−0 t , if and only if spec ([(a, b)]t) ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS 19 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' First, assume that (a, b) ∈ H+ t in Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then, by Corollary 25, spec ([a, b]t) ⊂ (C \\ R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Now, suppose that spec ([a, b]t) ⊂ R in C, and assume that (a, b) ∈ H+ t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then, (a, b) is contained in H−0 t , equivalently, it cannot be an element of H+ t , by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='6), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='7), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='12) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It contradicts our assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Therefore, (a, b) ∈ H+ t ⇐⇒ spec ([(a, b)]t) ⊂ (C \\ R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Thus, the statement (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='17) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By the decomposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='13), the statement (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='18) holds true, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By the above theorem, we obtain the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Corollary 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let Ht be the t-scaled hypercomplex ring for an arbitrary t ∈ R, and suppose it is decomposed to be Ht = H+ t ⊔ H−0 t , as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Assume that a given element (a, b) satisfies the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='19) (a, b) ∈ H+ t , if and only if spec ([(a, b)]t) = � x ± i � y2 − tu2 − tv2 � ⊂ (C \\ R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='20) (a, b) ∈ H−0 t , if and only if either spec ([(a, b)]t) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 {x} if Im (a)2 = t |b|2 � x ± � tu2 + tv2 − y2 � if Im (a)2 < t |b|2 , in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The statement (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='19) holds by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='9) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Meanwhile, the state- ment (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='20) holds by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Recall that a Hilbert-space operator T ∈ B (H) is self-adjoint, if T ∗ = T in B (H), where T ∗ is the adjoint of T (See Section 5 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It is well-known that T is self-adjoint, if and only if its spectrum is contained in R in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, one obtains the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proposition 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' A hypercomplex number (a, b) ∈ H−0 t in Ht, if and only if the realization [(a, b)]t ∈ Ht 2 is self-adjoint “in M2 (C).” Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (⇒) Suppose (a, b) ∈ H−0 t in Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then spec ([(a, b)]t) ⊂ R in C, implying that [(a, b)]t is self-adjoint in M2 (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (⇐) Suppose [(a, b)]t ∈ Ht 2 is self-adjoint in M2 (C), and assume that (a, b) /∈ H−0 t , equivalently, (a, b) ∈ H+ t in Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then, spec ([(a, b)]t) ⊂ (C \\ R) in C, and hence, [(a, b)]t is not self-adjoint in M2 (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It contradicts our assumption that it is self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 20 DANIEL ALPAY AND ILWOO CHO Equivalent to the above proposition, one can conclude that (a, b) ∈ H+ t in Ht, if and only if [(a, b)]t is not be self-adjoint in M2 (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The self-adjointness of re- alizations of hypercomplex numbers would be considered more in detail in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The Scaled-Spectralizations {σt}t∈R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In this section, we fix an arbitrary scale t ∈ R, and the corresponding hypercomplex ring Ht, containing the t-scaled hypercomplex monoid H× t = (Ht \\ {(0, 0)} , ·t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Recall that H× t is algebraically decomposed to be H× t = Hinv t ⊔ H×sing t , with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1) Hinv t = � (a, b) : |a|2 ̸= t |b|2� , the group-part, and H×sing t = � (a, b) : |a|2 = t |b|2� , the semigroup-part, as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Therefore, the t-scaled hypercomplex ring is set-theoretically decom- posed to be Ht = Hinv t ⊔ {(0, 0)} ⊔ H×sing t = Hinv t ⊔ Hsing t , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2) by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1), where Hsing t denote = {(0, 0)} ⊔ H×sing t in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Also, the ring Ht is spectrally decomposed to be Ht = H+ t ⊔ H−0 t , with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3) H+ t = � (a, b) : Im (a)2 > t |b|2� , and H−0 t = � (a, b) : Im (a)2 ≤ t |b|2� , satisfying that: (a, b) ∈ H+ t if and only if spec ([(a, b)]t) ⊂ (C \\ R);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' meanwhile, (a, b) ∈ H−0 t if and only if spec ([(a, b)]t) ⊂ R, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='19) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Corollary 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let Ht be the t-scaled hypercomplex ring for t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then it is decomposed to be Ht = � Hinv t ∩ H+ t � ⊔ � Hinv t ∩ H−0 t � � Hsing t ∩ H+ t � ⊔ � Hsing t ∩ H−0 t � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4) set-theoretically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It is proven by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Observe now that if (a, 0) ∈ Ht, then [(a, 0)]t = � a 0 0 a � in Ht 2, satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5) spec ([(a, 0)]t) = {a, a} in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS 21 Indeed, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4), if (a, 0) ∈ Ht satisfying a = x + yi ∈ C with x, y ∈ R, then spec ([(a, b)]t) = � x ± i � y2 � = {x ± |y| i} = {x ± yi} , implying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5), where |y| is the absolute value of y in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Motivated by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5), we define a certain C-valued function σt from Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Define a function, σt : Ht → C, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='6) σt ((a, b)) def = \uf8f1 \uf8f2 \uf8f3 a = x + yi if b = 0 in C x + i � y2 − tu2 − tv2 if b ̸= 0 in C, for all (a, b) ∈ Ht satisfying the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5): a = x + yi and b = u + vi in C, with x, y, u, v ∈ R and i = √−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Remark that such a morphism σt is indeed a well-defined function assigning all hypercomplex numbers of Ht to complex numbers of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Moreover, by the very definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='6), it is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' But it is definitely not injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' For instance, even though ξ = (1 + 3i, −1 + i) and η = (1 − 3i, 1 − i) are distinct in Ht, one has σt (ξ) = 1 + i √ 9 − 2t = σt (η) , by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Definition 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The surjection σt : Ht → C of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='6) is called the t(-scaled)- spectralization on Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The images {σt (ξ)}ξ∈Ht are said to be t(-scaled)-spectral values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' From below, we also understand each t-spectral value σt (ξ) ∈ C of a hypercomplex number ξ ∈ Ht as a hypercomplex number (σt (ξ) , 0) in Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', such an assigned hypercomplex number (σt (ξ) , 0) from the t-spectral value σt (ξ) of ξ is also called the t-spectral value of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By definition, all t-spectral values are not only C-quantities for many hypercom- plex numbers of Ht whose realizations of Ht 2 share the same eigenvalues, but also hypercomplex numbers of Ht, whose first coordinates are the value and the second coordinates are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Definition 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let ξ ∈ Ht be a hypercomplex number inducing its t-spectral value w denote = σt (ξ) ∈ C, also understood to be η = (w, 0) ∈ Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The corresponding realization, [η]t = � w t · 0 0 w � = \uf8eb \uf8ed σt (ξ) 0 0 σt (ξ) \uf8f6 \uf8f8 ∈ Ht 2 is called the t(-scaled)-spectral form of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By Σt (ξ), we denote the t-spectral form of ξ ∈ Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 22 DANIEL ALPAY AND ILWOO CHO Note that the conjugate-notation in Definition 30 is symbolic in the sense that: if t > 0, and σt (ξ) = 1 + i √ 1 − 5t = 1 − √ 5t − 1, (and hence, σt (ξ) ∈ R), then the symbol, σt (ξ) means = 1 − i √ 1 − 5t = 1 + √ 5t − 1, in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', the conjugate-notation in Definition 30 has a symbolic meaning containing not only the usual conjugate on C, but also the above computational meaning on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Remark-and-Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (From below, RA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1) The conjugate-notation in Definition 30 is symbolic case-by-case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' If the t-spectral value σt (ξ) is in C, then σt (ξ) means the usual conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Meanwhile, if t-spectral value σt (ξ) = x + � tu2 + tv2 − y2, with tu2 + tv2 − y2 ≥ 0, in R, then σt (ξ) = x − � tu2 + tv2 − y2 in R, where ξ ∈ Ht satisfies the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' \x08 For instance, if ξ1 = (−2 − i, 0) ∈ Ht, then the t-spectral value is σt (ξ1) = −2 − i in C, inducing the t-spectral form, Σt (ξ1) = \uf8eb \uf8ed −2 − i 0 0 −2 + i \uf8f6 \uf8f8 in Ht 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' meanwhile, if ξ2 = (−2 − i, 1 + 3i) ∈ Ht, then the t-spectral value is w denote = σt (ξ2) = −2 + i √ 1 − 10t, inducing the t-spectral form, Σt (ξ2) = � w 0 0 w � = \uf8eb \uf8ed −2 + i√1 − 10t 0 0 −2 − i√1 − 10t \uf8f6 \uf8f8 , where w is symbolic in the sense of RA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' if t ≤ 0, then Σt (ξ2) = \uf8eb \uf8ed −2 + i√1 − 10t 0 0 −2 − i√1 − 10t \uf8f6 \uf8f8 , meanwhile, if t > 0, then Σt (ξ2) = \uf8eb \uf8ed −2 + √10t − 1 0 0 −2 − √10t − 1 \uf8f6 \uf8f8 , in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS 23 Definition 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Two hypercomplex numbers ξ, η ∈ Ht are said to be t(-scaled)- spectral-related, if σt (ξ) = σt (η) in C, equivalently, Σt (ξ) = Σt (η) in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' On the t-hypercomplex ring Ht, the t-spectral relation of Definition 31 is an equivalent relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Indeed, σt (ξ) = σt (ξ) , ∀ξ ∈ Ht;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' and if ξ and η are t-spectral related in Ht, then σt (ξ) = σt (η) ⇐⇒ σt (η) = σt (ξ) , and hence, η and ξ are t-spectral related in Ht;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' and if ξ1 and ξ2 are t-spectral related, and if ξ2 and ξ3 are t-spectral related, then σt (ξ1) = σt (ξ2) = σt (ξ3) in C, and hence, ξ1 and ξ3 are t-spectral related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proposition 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The t-spectral relation on Ht is an equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The t-spectral relation is reflexive, symmetric and transitive on Ht, by the discussion of the very above paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Since the t-spectral relation is an equivalence relation, each element ξ of Ht has its equivalence class, �ξ def = {η ∈ Ht : η is t-related to ξ} , and hence, the corresponding quotient set, � Ht def = � �ξ : ξ ∈ Ht � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='7) is well-defined to be the set of all equivalence classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Theorem 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let � Ht be the quotient set (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='7) induced by the t-spectral relation on Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then � Ht and C are equipotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='8) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It is not difficult to check that, for any z ∈ C, there exist ξ ∈ Ht, such that z = σt (ξ) by the surjectivity of the t-spectralization σt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It implies that there exists (z, 0) ∈ Ht, such that � (z, 0) = �ξ in � Ht, whenever z = σt (ξ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Thus, set-theoretically, we have � Ht = � � (z, 0) : z ∈ C � equip = C, where “ equip = ” means “being equipotent (or, bijective) to.” Therefore, the relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='8) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The above equipotence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='8) of the quotient set � Ht of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='7) with the complex numbers C shows that the set C classifies Ht, for “every” t ∈ R, up to the t-spectral relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 24 DANIEL ALPAY AND ILWOO CHO 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Similarity on M2 (C) and The t-Scaled-Spectral Relation on Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2, we defined the t-spectralization σt on the t-scaled hypercomplex ring Ht, for a fixed scale t∈ R, and it induces the t-spectral forms {Σt (ξ)}ξ∈Ht in Ht 2 as complex diagonal matrices whose main diagonals are the eigenvalues of the realizations {[ξ]t}ξ∈Ht, under the symbolic understanding RA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Moreover, σt lets the set C classify Ht by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='8) under the t-spectral relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Independently, we showed in [2] and [3] that: on the quaternions H = H−1, the (−1)-spectral relation, called the quaternion-spectral relation in [2] and [3], is equivalent to the similarity “on H−1 2 ,” as equivalence relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Here, the similarity “on H−1 2 ” means that: the realizations [q1]−1 and [q2]−1 of two quaternions q1, q2 ∈ H−1 are similar “in H−1 2 ,” if there exists invertible element U “in H−1 2 ,” such that [q2]−1 = U −1 [q1]−1 U in H−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Here, we consider such property for an arbitrary scale t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Recall that, we showed in [2] and [3] that: the (−1)-spectral form Σ−1 (η) and the realization [η]−1 are similar “in H−1 2 ,” for “all” quaternions which are the (−1)-scaled hypercomplex numbers η ∈ H−1 = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Are the t-spectral relation on Ht and the similarity on Ht 2 same as equivalence relations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In conclusion, the answer is negative in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Two matrices A and B of Mn (C), for any n ∈ N, are said to be similar, if there exists an invertible matrix U ∈ Mn (C), such that B = U −1AU in Mn (C) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Remember that if two matrices A and B are similar, then (i) they share the same eigenvalues, (ii) they have the same traces, and (iii) their determinants are same (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', [8] and [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' We here focus on the fact (iii): the similarity of matrices implies their identical determinants, equivalently, if det (A) ̸= det (B) , then A and B are not similar in Mn (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Definition 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let A, B ∈ Ht 2 be realizations of certain hypercomplex numbers of Ht, for t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' They are said to be similar “in Ht 2,” if there exists an invertible U ∈ Ht 2, such that B = U −1AU in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By abusing notation, we say that two hypercomplex numbers ξ and η are similar in Ht, if their realizations [ξ]t and [η]t are similar in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let (a, b) ∈ Ht be a hypercomplex number satisfying the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5) and (a, b) ̸= (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then it has [(a, b)]t = � a tb b a � ∈ Ht 2, σt ((a, b)) = x + i � y2 − tu2 − tv2 let = w ∈ C, and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1) Σt ((a, b)) = � w 0 0 w � ∈ Ht 2, where w is symbolic under RA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Observe that det ([(a, b)]t) = |a|2 − t |b|2 = � x2 + y2� − t � u2 + v2� , OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS 25 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2) det (Σt ((a, b))) = |w|2 = x2 + ��y2 − tu2 − tv2�� , by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' These computations in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2) show that, in general, [(a, b)]t and Σt ((a, b)) are “not” similar “as matrices of M2 (C),” and hence, not similar in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Indeed, for instance, if t > 0, and |a|2 < t |b|2 , then det ([(a, b)]t) < 0, but det (Σt ((a, b))) > 0 in R, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2), implying that det ([(a, b)]t) ̸= det (Σt ((a, b))) in general, showing that [(a, b)]t and Σt ((a, b)) are not similar in M2 (C), and hence, they are not similar in Ht 2, in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proposition 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let (a, b) ∈ Ht be “nonzero” hypercomplex number satisfying |a|2 < t |b|2 in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then the realization [(a, b)]t and the t-spectral form Σt ((a, b)) are not similar “in Ht 2.” Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Suppose (a, b) ∈ Ht satisfies (a, b) ̸= (0, 0) and |a|2 < t |b|2, for t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' And assume that [(a, b)]t and Σt ((a, b)) are similar in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Since they are assumed to be similar, their determinants are identically same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' However, det ([(a, b)]t) < 0 and det (Σt ((a, b))) > 0, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It contradicts our assumption that they are similar in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The above proposition confirms that the realizations and the corresponding t- spectral forms of a t-scaled hypercomplex number are not similar in Ht 2, in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Consider that, in the quaternions H = H−1, since the scale is t = −1 < 0 in R, det � [ξ]−1 � = det (Σ−1 (ξ)) ≥ 0, ∀ξ ∈ H−1, and it is proven that [ξ]−1 and Σ−1 (ξ) are indeed similar in H−1 2 , for “all” ξ ∈ H−1 in [2] and [3], which motivates a question: if a scale t < 0 in R, then det ([η]t) = det (Σt (η)) ≥ 0, ∀η ∈ Ht, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' so, are the realizations [η]t and the corresponding t-spectral forms Σt (η) similar in Ht 2 as in the case of t = −1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' First of all, we need to recall that if t < 0, then the t-scaled hypercomplex ring Ht forms a noncommutative field, since the t-scaled hypercomplex monoid H× t is a non-abelian group, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It allows us to use similar techniques of [2] and [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In the rest part of this section, a given scale t ∈ R is automatically assumed to be negative in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' \x08 Assume that (a, 0) ∈ Ht, where t < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then [(a, 0)]t = � a 0 0 a � = Σt ((a, 0)) , in Ht 2, since σt ((a, 0)) = a in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, clearly, [(a, 0)]t and Σt ((a, 0)) are similar in Ht 2, because they are equal in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Indeed, there exist diagonal matrices with nonzero real entries, X = [(x, 0)]t ∈ Ht 2, with x = x + 0i ∈ C, x ̸= 0, 26 DANIEL ALPAY AND ILWOO CHO such that [(a, 0)]t = X−1 (Σt (a, 0)) X in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Thus, we are interested in the cases where (a, b) ∈ Ht with b ∈ C× = C \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Lemma 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let t < 0 in R, and (a, 0) ∈ Ht, a hypercomplex number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then the realization [(a, 0)]t and the t-spectral form Σt ((a, 0)) are identically same in Ht 2, and hence, they are similar in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (Remark that, in fact, the scale t is not necessarily negative in R here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It is proven by the discussion of the very above paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Indeed, one has [(a, 0)]t = Σt ((a, 0)) in Ht 2, since σt ((a, 0)) = a in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let h = (a, b) ∈ Ht with b ∈ C×, satisfying the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5), where t < 0, having its realization, [h]t = � a tb b a � = \uf8eb \uf8ed x + yi t (u + vi) u − vi x − yi \uf8f6 \uf8f8 , and its t-spectral form, Σt (h) = \uf8eb \uf8ed x + i � y2 − tu2 − tv2 0 0 x − i � y2 − tu2 − tv2 \uf8f6 \uf8f8 let = � w 0 0 w � , in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Since t < 0 and b ̸= 0 (by assumption), the t-spectral value w = σt (h) is a C-quantity with its conjugate w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Define now a matrix, Qh def = \uf8eb \uf8ec \uf8ed 1 t � w−a tb � w−a tb 1 \uf8f6 \uf8f7 \uf8f8 in M2 (C) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Remark that, by the assumption that t < 0 and b ̸= 0, this matrix is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Furthermore, one can immediately recognize that Qh ∈ Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', Qh = �� 1, � w−a tb ��� t ∈ Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3) One can find that the element Qh ∈ Ht 2 of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3) is indeed invertible by our negative-scale assumption, since det (Qh) = 1 − t ���� w − a tb ���� 2 ≥ 1, since t < 0, implying that det (Qh) ̸= 0 ⇐⇒ Qh is invertible in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Observe now that QhΣt (h) = \uf8eb \uf8ec \uf8ed w t � w2−aw tb � w2−aw tb w \uf8f6 \uf8f7 \uf8f8 OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS 27 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4) [h]t Qh = \uf8eb \uf8ec \uf8ed w t � a � w−a tb � + b � a � w−a tb � + b w \uf8f6 \uf8f7 \uf8f8 , in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Now, let’s compare the (1, 2)-entries of resulted matrices in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The (1, 2)-entry of the element QhΣt (h) is t � w2−aw tb � = w(w−a) b = � x+i√ y2−tu2−tv2 �� i√ y2−tu2−tv2−yi � u+vi = ix √ R−xyi−R+y √ R u+vi , where (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5) R denote = y2 − tu2 − tv2 in R, and the (1, 2)-entry of the matrix [h]t Qh is t � a � w−a tb � + b � = t � a � w−a tb � + b � = t � aw−|a|2+t|b|2 tb � = aw−|a|2+t|b|2 b = (x−yi) � x+i√ y2−tu2−tv2 � −(x2+y2)−t(u2+v2) u+vi = x2+ix √ R−xyi+y √ R−x2−y2−tu2−tv2 u+vi = x2+ix √ R−xyi+y √ R−x2−R u+vi = ix √ R−xyi−R+y √ R u+vi , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='6) where the R-quantity R is in the sense of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' As one can see in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='6), the (1, 2)-entries of [h]t Qh and QhΣt (h) are identically same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', QhΣt (h) = [h]t Qh in Ht 2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='7) where the matrix Qh ∈ Ht 2 is in the sense of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Lemma 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let t < 0 in R, and let h = (a, b) ∈ Ht with b ∈ C×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then the realization [h]t and the t-spectral form Σt (h) are similar in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In particular, there exists qh = � 1, t �w − a tb �� ∈ Ht, having its realization, Qh = [qh]t = \uf8eb \uf8ec \uf8ed 1 t � w−a tb � w−a tb 1 \uf8f6 \uf8f7 \uf8f8 ∈ Ht 2, such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='8) Σt (h) = Q−1 h [h]t Qh in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 28 DANIEL ALPAY AND ILWOO CHO Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Under the hypothesis, one obtains that QhΣt (h) = [h]t Qb in Ht 2, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By the invertibility of Qh, we have Σt (h) = Q−1 h [h]t Qh in Ht 2, implying the relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The above lemma shows that if a scale t is negative in R, then the realization [h]t and the t-spectral form Σt (h) are similar in Ht 2, whenever h = (a, b) ∈ Ht satisfies b ̸= 0 in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Theorem 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' If t < 0 in R, then every hypercomplex number h ∈ Ht is similar to its t-spectral value (σt (h) , 0) ∈ Ht, in the sense that: [h]t and Σt (h) are similar in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='9) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let h = (a, b) ∈ Ht, for t < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' If b = 0 in C, then [(a, 0)]t and Σt ((a, 0)) are similar in Ht 2, by Lemma 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Indeed, if b = 0, then these matrices are identically same in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Meanwhile, if b ̸= 0 in C, then [h]t and Σt (h) are similar in Ht 2 by Lemma 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In particular, if b ̸= 0, then there exists qh = � 1, w − a tb � ∈ Ht, such that Σt (h) = [qh]−1 t [h]t [qh]t , in Ht 2, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Therefore, if t < 0, then [h]t and Σt (h) are similar in Ht 2, equivalently, two hypercomplex numbers h and (σt (h) , 0) are similar in Ht, for all h ∈ Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The above theorem guarantees that the negative-scale condition on hypercom- plex numbers implies the similarity of the realizations and the scaled-spectral forms of them, just like the quaternionic case (whose scale is −1), shown in [2] and [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Theorem 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' If t < 0 in R, then the t-spectral relation on Ht and the similarity on Ht are same as equivalence relations on Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', t < 0 =⇒ t-spectral relation equi = similarity on Ht, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10) where “ equi = ” means “being equivalent to, as equivalence relations.” Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Suppose a negative scale t < 0 is fixed, and let Ht be the corresponding t-scaled hypercomplex ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Assume that two hypercomplex numbers h1 and h2 are t-spectral related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then their t-spectral values are identical in C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', σt (h1) = σt (h2) let = w in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Thus the realizations [h1]t and [h2]t are similar to Σt (h1) = � w 0 0 w � = Σt (h2) let = W, in Ht 2, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', there exist q1, q2 ∈ Ht such that [q1]−1 t [h1]t [q1]t = W = [q2]−1 t [h2]t [q2]t , OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS 29 in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, one obtains that [h1]t = � [q1]t [q2]−1 t � [h2]t � [q2]t [q1]−1 t � , ⇐⇒ [h1]t = � [q2]t [q1]−1 t �−1 [h2]t � [q2]t [q1]−1 t � , in Ht 2, implying that [h1]t and [h2]t are similar in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Thus, if h1 and h2 are t-spectral related, then they are similar in Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Conversely, suppose T1 denote = [h1]t and T2 denote = [h2]t are similar in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then there exists U ∈ Ht 2, such that T1 = U −1T2U in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Since the realizations Tl and the corresponding t-spectral forms Sl denote = Σt (hl) are similar by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='9), say, Tl = V −1 l SlVl in Ht 2, for some Vl ∈ Ht 2, for all l = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Thus, T1 = U −1T2U = U −1 � V −1 2 S2V2 � U, ⇐⇒ V1S1V −1 1 = T1 = (V2U)−1 S2 (V2U) , ⇐⇒ S1 = V −1 1 (V2U)−1 S2 (V2U) V1, ⇐⇒ S1 = (V2UV1)−1 S2 (V2UV1) , and hence, two matrices S1 and S2 are similar in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It means that S1 and S2 share the same eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, it ie either S1 = � w 0 0 w � = S2, for some w ∈ C, or S1 = � w 0 0 w � , and S2 = � w 0 0 w � , in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' However, by the assumption that t < 0, we have S1 = S2 in Ht 2, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It shows that, if the realizations T1 and T2 are similar, then the t-spectral forms S1 and S2 are identically same in Ht 2, implying that σt (h1) = σt (h2) in C, thus h1 and h2 are t-spectral related in Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Therefore, the equivalence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10) between the t-spectral relation and the simi- larity on Ht holds, whenever t < 0 in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 30 DANIEL ALPAY AND ILWOO CHO The above theorem generalizes the equivalence between the quaternion-spectral relation, which is the (−1)-spectral relation, and the similarity on the quaternions H−1 = H (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', [2] and [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' How about the cases where given scale t are nonnegative in R, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', t ≥ 0?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' One may need to consider the decomposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4), Ht = � Hinv t ∩ H+ t � ⊔ � Hinv t ∩ H−0 t � � Hsing t ∩ H+ t � ⊔ � Hsing t ∩ H−0 t � , of Ht, for t ≥ 0, where Hinv t = � (a, b) : |a|2 ̸= t |b|2� , Hsing t = � (a, b) : |a|2 = t |b|2� , H+ t = � (a, b) : Im (a)2 > t |b|2� , and H−0 t = � (a, b) : Im (a)2 ≤ t |b|2� , block-by-block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In particular, if h ∈ Hinv t ∩ H+ t , then it “seems” that the realization [h]t and the t-spectral form Σt (h) are similar in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The proof “may” be similar to the above proofs for negative scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' We leave this problem for a future project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' \x08 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The t-Spectral Mapping Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In this section, we let a scale t be arbitrary in R, and let Ht be the t-scaled hypercomplex ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let h = (a, b) ∈ Ht satisfy the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5), and suppose it has its t-spectral value, σt (h) = x + i � y2 − tu2 − tv2 let = w, and hence, its t-spectral form Σt (h) = � w 0 0 w � in Ht 2, under NA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Now recall that if n ∈ N, and A ∈ Mn (C), and if f ∈ C[z] def = \uf8f1 \uf8f2 \uf8f3g : g = m � k=0 zkzk, with z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', zm ∈ C, for m ∈ N \uf8fc \uf8fd \uf8fe , then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1) spec (f (A)) = {f (w) : w ∈ spec (A)} , in C, where C[z] is the polynomial ring in a variable z over C, consisting of all polynomials in z whose coefficients are from C, and f (A) = N � k=0 skAk, with A0 = In, OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS 31 whenever f (z) = N � k=0 skzk ∈ C [z] , with s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', sN ∈ C, where In is the identity matrix of Mn (C), by the spectral mapping theorem (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', [8] and [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1), if R[x] is the polynomial ring in a variable x over the real field R, then spec (g (A)) = {g (w) : w ∈ spec (A)} in C, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2) for all g ∈ R[x], because R[z] is a subring of C[z] if we identify x to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It is shown in [2] and [3] that, for f ∈ C[z], spec � f � [ξ]−1 �� = � f (σ−1 (ξ)) , f � σ−1 (ξ) �� in C, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1), but f � σ−1 (ξ) � ̸= f (σ−1 (ξ)), in general, and hence, even though the spectral mapping theorem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1) holds “on M2 (C), for [ξ]−1 ∈ H−1 2 ,” it does not hold “on H−1 2 ,” in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It demonstrates that, in a similar manner, the spectral mapping theorem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1) holds “on M2 (C) ,” but it does not hold “on the t-scaled realization Ht 2 of Ht,” for t ∈ R, because the spectra of hypercomplex numbers satisfy spec ([η]t) = {w, w} , with w = σt (η) , by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4), for all η ∈ Ht under RA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1, just like the quaternionic case of [2] and [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' For an arbitrary scale t ∈ R, the spectral mapping theorem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1) does not hold “on Ht 2.” \x08 However, in [2] and [3], it is proven that, for all g ∈ R[x], one has spec � g � [ξ]−1 �� = � g (σt (ξ)) , g (σt (ξ)) � , in C, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2), since g ∈ R[x] =⇒ g (w) = g (w), ∀w ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It means that the “restricted” spectral mapping theorem of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2) holds “on the realization H−1 2 of the quaternions H−1.” Similarly, we obtain the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Theorem 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let ξ ∈ Ht, realized to be [ξ]t ∈ Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then, for any g ∈ R[x], spec (g ([ξ]t)) = � g (σt (ξ)) , g (σt (ξ)) � , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3) spec (g ([ξ]t)) = {g (w) : w ∈ spec ([ξ]t)} in C, ∀t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='6), if ξ ∈ Ht, then spec ([ξ]t) = {w, w} , with w = σt (ξ) , in C (under the symbolic understanding of RA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' For any g = N � k=1 skxk ∈ R[x], with s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', sN ∈ R, and N ∈ N, one has that 32 DANIEL ALPAY AND ILWOO CHO g (w) = N � k−1 skwk = N � k=1 skwk = N � k=1 skwk = g (w), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4) in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It implies that spec (g ([ξ]t)) = {g (w) , g (w)} = � g (w) , g (w) � , in C, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Therefore, the relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' One may call the relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3), the hypercomplex-spectral mapping theorem, since it holds for all scales t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The Usual Adjoint on Ht 2 in M2 (C) In this section, we consider how the usual adjoint on M2 (C) = B � C2� acts on the t-scaled realization Ht 2 of the t-scaled hypercomplex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Throughout this section, we fix an arbitrary scale t ∈ R, and the corresponding t-scaled hypercom- plex ring Ht realized to be Ht 2 in M2 (C) under the representation Πt = � C2, πt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Recall that every Hilbert-space operator T acting on a Hilbert space H has its unique adjoint T ∗ on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Especially, if T ∈ Mn (C) = B (Cn), for n ∈ N, is a matrix which is an operator on Cn, then its adjoint T ∗ is determined to be the conjugate-transpose of T in Mn (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' For instance, T = � a11 a12 a21 a22 � ∈ M2 (C) ⇐⇒ T ∗ = � a11 a21 a12 a22 � ∈ M2 (C) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It says that, if we understand our t-scaled realization Ht 2 as a sub-structure of M2 (C), then each hypercomplex number (a, b) ∈ Ht assigns a unique adjoint [(a, b)]∗ t of the realization [(a, b)]t “in M2 (C).” Let (a, b) ∈ Ht realized to be [(a, b)]t = � a tb b a � ∈ Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then, as a matrix of M2 (C), this realization has its adjoint, [(a, b)]∗ t = � a b tb a � in M2 (C) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It shows that the usual adjoint (conjugate-transpose) of [(a, b)]t is not contained “in Ht 2,” in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In particular, if t2 ̸= 1 ⇐⇒ either t ̸= 1 or t ̸= −1, in R, then [(a, b)]t /∈ Ht 2 in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Theorem 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The scale t ∈ R satisfies that t2 = 1 in R, if and only if the adjoint of every realization of a hypercomplex number Ht is contained in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', either t = 1, or t = −1⇐⇒ [ξ]∗ t ∈ Ht 2, ∀ξ ∈ Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' For an arbitrary scale t ∈ R, if (a, b) ∈ Ht, then [(a, b)]∗ t = � a b tb a � in M2 (C) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS 33 (⇒) Assume that either t = 1, or t = −1, equivalently, suppose t2 = 1 in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then [(a, b)]∗ t = � a b tb a � = � a t � b t � t2� b t � a � = � a t � b t � � b t � a � , contained in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, if either t = 1, or t = −1, then [(a, b)]∗ t ∈ Ht 2, for all (a, b) ∈ Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Moreover, in such a case, [(a, b)]∗ t = �� a, b t �� t in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2) (⇐) Assume now that t2 ̸= 1 in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then the adjoint [(a, b)]∗ t of [(a, b)]t is identical to the matrix, [(a, b)]∗ t = � a b tb a � in M2 (C) , which “can” be � a t � b t � t2 � b t � a � in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' However, by the assumption that t2 ̸= 1, the adjoint [(a, b)]∗ t is not contained in Ht 2, in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In particular, if b ̸= 0 in C, then the adjoint [(a, b)]∗ t /∈ Ht 2 in M2 (C), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', t2 ̸= 1 and b ̸= 0 in C =⇒ [(a, b)]∗ t ∈ (M2 (C) \\ Ht 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3) Therefore, the characterization (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1) holds by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Note that, if t = −1, then H−1 is the quaternions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' and if t = 1, then H1 is the bicomplex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The above theorem shows that, only when the scaled hypercomplex ring Ht is either the quaternions H−1, or the bicomplex numbers H1, the usual adjoint (∗) is closed on Ht 2, as a well-defined unary operation, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Free Probability on Ht In this section, we establish a universal free-probabilistic model on our t-scaled hypercomplex ring Ht, for “every” scale t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' First, recall that, on M2 (C), we have the usual trace tr, defined by tr �� a11 a12 a21 a22 �� = a11 + a22, for all � a11 a12 a21 a22 � ∈ M2 (C);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' and the normalized trace τ, τ = 1 2tr on M2 (C) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', we have two typical free-probabilistic models, (M2 (C) , tr) and (M2 (C) , τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 34 DANIEL ALPAY AND ILWOO CHO 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Free Probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' For more about free probability theory, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', [19] and [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let A be an noncommutative algebra over C, and ϕ : A → C, a linear functional on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then the pair (A, ϕ) is called a (noncommutative) free probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By definition, free probability spaces are the noncommutative version of classic measure spaces (X, µ) consisting of a set X and a measure µ on the σ- algebra of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' As in measure theory, the (noncommutative) free probability on (A, ϕ) is dictated by the linear functional ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Meanwhile, if (A, ϕ) is unital in the sense that (i) the unity 1A of A exists, and (ii) ϕ (1A) = 1, then it is called a unital free probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' These unital free probability spaces are the noncommutative analogue of classical probability spaces (Y, ρ) where the given measures ρ are the probability measures satisfying ρ (Y ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' If A is a topological algebra, and if ϕ is bounded (and hence, continuous under linearity), then the corresponding free probability space (A, ϕ) is said to be a topo- logical free probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Similarly, if A is a topological ∗-algebra equipped with the adjoint (∗), then the topological free probability space (A, ϕ) is said to be a topological (free) ∗-probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' More in detail, if A is a C∗-algebra, or a von Neumann algebra, or a Banach ∗-algebra, we call (A, ϕ), a C∗-probability space, respectively, a W ∗-probability space, respectively, a Banach ∗-probability space, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='. For our main purposes, we focus on C∗-probability spaces from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' If (A, ϕ) is a C∗-probability space, and a ∈ A, then the algebra-element a is said to be a free random variable of (A, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' For any arbitrarily fixed free random variables a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', as ∈ (A, ϕ) for s ∈ N, one can get the corresponding free distribution of a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', as, characterized by the joint free moments, ϕ � n � l=1 ari il � = ϕ � ar1 i1 ar2 i2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='arn in � , for all (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', in) ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', s}n and (r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', rn) ∈ {1, ∗}n, for all n ∈ N, where a∗ l are the adjoints of al, for all l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' For instance, if a ∈ (A, ϕ) is a free random variable, then the free distribution of a is fully characterized by the joint free moments of {a, a∗}, ϕ � n � l=1 arl � = ϕ (ar1ar2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='arn) , for all (r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', rn) ∈ {1, ∗}n, for all n ∈ N (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', [19] and [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, if a free random variable a ∈ (A, ϕ) is self-adjoint in the sense that: a∗ = a in A, then the free distribution of a is determined by the free-moment sequence, (ϕ (an))∞ n=1 = � ϕ (a) , ϕ � a2� , ϕ � a3� , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' � (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', [19] and [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Free-Probabilistic Models Induced by Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By identifying the t-scaled hypercomplex ring Ht and its realization Ht 2 as the same ring, we identify the t- scaled hypercomplex monoid H× t and its realization Ht× 2 as the same monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' As a subset in M2 (C), we define a subset, Ht× 2 (∗) def = � [ξ]∗ t ∈ M2 (C) : ξ ∈ H× t � , OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS 35 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1) Ht× 2 (∗) = �� a b tb a � ∈ M2 (C) : (a, b) ∈ H× t � , by the subset of all adjoints of realizations in H×t 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Indeed, [(a, b)]∗ t = � a tb b a �∗ = � a b tb a � in M2 (C) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' As we have seen in Section 4, the adjoint is not closed on Ht 2 in general, and hence, Ht× 2 (∗) ̸= Ht× 2 in M2 (C) , in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In particular, the scale t satisfies t2 ̸= 1 in R, if and only if the above non-equality holds in M2 (C), by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Now, let Ht× 2 (1, ∗) denote = Ht× 2 ∪ Ht× 2 (∗), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2) Ht× 2 (1, ∗) = �� a tb b a � , � a b tb a � : (a, b) ∈ H× t � , in M2 (C), set-theoretically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2), Ht× 2 (1, ∗) ⫌ Ht× 2 in M2 (C) , in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Define now the C∗-algebra Ht 2 by the C∗-subalgebra of M2 (C) generated by the set Ht× 2 (1, ∗) of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', Ht 2 denote = C∗ � Ht× 2 � def = C � Ht× 2 (1, ∗) � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3) in M2 (C), where C∗ (Z) means the C∗-subalgebra of B � C2� generated by the subset Z and their adjoints, and C[X] is the (pure-algebraic) algebra (over C) generated by a subset X of M2 (C), and Y means the operator-norm-topology closure of a subset Y of the operator algebra M2 (C) = B � C2� , which is a C∗- algebra over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Definition 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The C∗-algebra Ht 2 of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3), generated by the t-scaled hyper- complex monoid H× t monoid = Ht× 2 , is called the t-scaled(-hypercomplex)-monoidal C∗-algebra of H× t (or, of Ht).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Clearly, by the definition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3), the t-scaled-monoidal C∗-algebra Ht 2 is well- determined in M2 (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, the usual trace tr and the normalized trace τ on M2 (C) are well-defined on Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', we have two trivial free-probabilistic models of Ht 2, � Ht 2, tr � and � Ht 2, τ � , as C∗-probability spaces (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Note that such free-probabilistic structures are independent from the choice of the scales t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Observe that, if � al bl tbl al � ∈ Ht× 2 (∗) in Ht 2, for l = 1, 2, then � a1 b1 tb1 a1 � � a2 b2 tb2 a2 � = \uf8eb \uf8ed a1a2 + tb1b2 a1b2 + b1a2 t � b1a2 + a1b2 � tb1b2 + a1a2 \uf8f6 \uf8f8 , 36 DANIEL ALPAY AND ILWOO CHO identifying to be (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4) \uf8eb \uf8ed a1a2 + tb1b2 b1a2 + a1b2 t � b1a2 + a1b2 � a1a2 + tb1b2 \uf8f6 \uf8f8 in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Therefore, � a1 b1 tb1 a1 � � a2 b2 tb2 a2 � ∈ Ht× 2 (∗), too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', the matricial multiplication is closed on the set Ht× 2 (∗) of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2), by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In fact, under the closed-ness (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4), the algebraic pair, Ht× 2 (∗) denote = � Ht× 2 (∗), · � , forms a monoid with its identity I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, the generating set Ht× 2 (1, ∗) of the t- scaled-monoidal C∗-algebra Ht 2 is the set-theoretical union of two monoids Ht× 2 and Ht× 2 (∗), under the matricial multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Note, however, that the matricial multiplication is not closed on the generating set Ht× 2 (1, ∗) of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Indeed, if � a1 tb1 b1 a1 � ∈ Ht× 2 , � a2 b2 tb2 a2 � ∈ Ht× 2 (∗) in Ht 2, then � a1 tb1 b1 a1 � � a2 b2 tb2 a2 � = \uf8eb \uf8ed a1a2 + t2b1b2 a1b2 + ta2b1 a2b1 + ta1b2 b1b2 + a1a2 \uf8f6 \uf8f8 , and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5) � a2 b2 tb2 a2 � � a1 tb1 b1 a1 � = \uf8eb \uf8ed a1a2 + b1b2 tb1a2 + a1b2 ta1b2 + b1a2 t2b1b2 + a1a2 \uf8f6 \uf8f8 , in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' However, the resulted products of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5), contained in Ht 2, are not contained in Ht× 2 (1, ∗), in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5), one can realize that: (i) if A, B ∈ Ht× 2 , then AB ∈ Ht× 2 , (ii) if C, D ∈ Ht× 2 (∗), then CD ∈ Ht× 2 (∗), and (iii) if T, S ∈ Ht× 2 (1, ∗), then T S /∈ Ht× 2 (1, ∗), in general, as elements of the t-scaled-monoidal C∗-algebra Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Even though the non-closed rule (iii) is satisfied “on Ht 2 (1, ∗),” at least, we have a multiplication rule (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5) “in the C ∗-algebra Ht 2.” \x08 Assume that [(a, b)]t ∈ Ht× 2 in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then tr ([(a, b)]t) = a + a = 2Re (a) , and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='6) τ ([(a, b)]t) = 1 2tr ([(a, b)]t) = Re (a) , where Re (a) is the real part of a in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Similarly, if [(a, b)]∗ t ∈ Ht× 2 (∗) in Ht 2, then we have tr � [(a, b)]∗ t � = tr � a b tb a � = a + a = 2Re (a) , OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS 37 and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='7) τ � [(a, b)]∗ t � = 1 2 (2Re (a)) = Re (a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Remark that, since tr and τ are well-defined linear functional on the C∗-algebra Ht 2, they satisfy tr (T ∗) = tr (T ), and τ (T ∗) = τ (T ), for all T ∈ Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, the relation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='7) is well-verified, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Also, if [(a1, b1)]t , [(a2, b2)]∗ t ∈ Ht× 2 (1, ∗) in Ht 2, then tr � [(a1, b1)]t [(a2, b2)]∗ t � = tr \uf8eb \uf8ed \uf8eb \uf8ed a1a2 + t2b1b2 a1b2 + ta2b1 a2b1 + ta1b2 b1b2 + a1a2 \uf8f6 \uf8f8 \uf8f6 \uf8f8 by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5) = a1a2 + t2b1b2 + b1b2 + a1a2 = 2Re (a1a2) + t2b1b2 + b1b2, and similarly, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='8) tr � [(a1, b1)]∗ t [(a2, b2)]t � = 2Re (a1a2) + t2b1b2 + b1b2, and hence, τ � [(a1, b1)]t [(a2, b2)]∗ t � = Re (a1a2) + t2b1b2 + b1b2 2 , and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='9) τ � [(a1, b1)]∗ t [(a2, b2)]t � = Re (a1a2) + t2b1b2 + b1b2 2 , by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proposition 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let (a, b) , (al, bl) ∈ Ht, for l = 1, 2, and let A = [(a, b)]t and Al = [(al, bl)]t be the corresponding realizations of Ht 2, regarded as elements of the t-scaled-monoidal C∗-algebra Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then τ (A) = 1 2tr (A) = Re (a) = 1 2tr (A∗) = τ (A∗) , and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10) τ (A1A∗ 2) = 1 2tr (A1A∗ 2) = Re (a1a2) + t2b1b2 + b1b2 2 , and τ (A∗ 1A2) = 1 2tr (A∗ 1A2) = Re (a1a2) + t2b1b2 + b1b2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The joint free moments in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10) are proven by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='6), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='7), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='8) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The above computations in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10) provide a general way to compute free- distributional data, in particular, the joint free moments of matrices in the t-scaled- monoidal C∗-algebra Ht 2, up to the trace tr, and up to the normalized trace τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' And, they demonstrate that computing such free-distributional data is not easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, we will restrict our interests to a certain specific case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 38 DANIEL ALPAY AND ILWOO CHO 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Free Probability on (Ht 2, tr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In this section, we fix a scale t ∈ R, and the corresponding t-scaled-monoidal C∗-algebra Ht 2 generated by the t-scaled hyper- complex monoid H× t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let (Ht 2, tr) be the C∗-probability space with respect to the usual trace tr on Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Recall that if a scale t is negative, then the realization [ξ]t and the t-spectral form Σt (ξ) are similar “in Ht 2” by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='9), for all ξ ∈ Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It implies that the similarity “on Ht 2” is equivalent to the t-spectral relation on Ht by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Also, recall that if two matrices A and B are similar in Mn (C), for any n ∈ N, tr (A) = tr (B) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, if the realization [ξ]t and the t-spectral form Σt (ξ) are similar in Ht 2, then the free-moment computations would be much simpler than the computations of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Note again that if (a, b) ∈ Ht satisfies the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5), then tr ([(a, b)]t) = 2Re (a) = 2x = � x + i √ R � + � x − i √ R � = tr (Σt (a, b)) , where (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1) R = y2 − tu2 − tv2 in R, under RA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Even though the identical results hold in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1) (without simi- larity), if [(a, b)]t and Σt (a, b) are not similar in Ht 2, then tr ([(a, b)]n t ) ̸= tr ((Σt (a, b))n) , for some n ∈ N, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It implies that some (joint) free-moments of [(a, b)]t and those of Σt (a, b) are not identical, and hence, the free distributions of them are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Lemma 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Suppose the realization [(a, b)]t and the t-spectral form Σt (a, b) are similar in Ht 2 for (a, b) ∈ Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then tr ([(a, b)]n t ) = 2Re (σt (a, b)n) = tr �� [(a, b)]∗ t �n� (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2) for all n ∈ N, where σt (a, b) is the t-spectral value of (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Suppose (a, b) ∈ Ht satisfies the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then [(a, b)]t = � a tb b a � and Σt ((a, b)) = � σt (a, b) 0 0 σt (a, b) � , in Ht 2, where σt (a, b) = x + i � y2 − tu2 − tv2, under RA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Assume that [(a, b)]t and Σt ((a, b)) are similar in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then the matrices [(a, b)]n t and Σt ((a, b))n are similar in Ht 2, for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Indeed, if two elements A and B are similar in Ht 2, satisfying B = U −1AU in Ht 2, for an invertible element U ∈ Ht 2, then Bn = � U −1AU �n = U −1AnU in Ht 2, implying the similarity of An and Bn, for n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Thus, tr ([(a, b)]n t ) = tr (Σt ((a, b))n) , and tr (Σt ((a, b))n) = tr �� σt (a, b)n 0 0 σt (a, b)n �� , OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS 39 implying that tr ([(a, b)]n t ) = tr (Σt ((a, b))n) = 2Re (σt (a, b)n) , for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Therefore, the first equality in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Since tr is a well-defined linear functional on the C∗-algebra Ht 2, one has tr (A∗) = tr (A), for all A ∈ Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Since tr �� [(a, b)]∗ t �n� = tr � ([(a, b)]n t )∗� = tr ([(a, b)]n t ), one has tr �� [(a, b)]∗ t �n� = 2Re (σt (a, b)n) = 2Re (σt (a, b)n) , for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, the second equality in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2) holds, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Note that the formula (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2) holds true under the similarity assumption of the realization and the t-spectral form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Remark that every complex number w ∈ C is polar-decomposed to be w = |w| wo with wo ∈ T, uniquely, where T = {z ∈ C : |z| = 1} is the unit circle in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, all our t-spectral values σt (ξ) are polar-decomposed to be σt (ξ) = |σt (ξ)| σt (ξ)o with σt (ξ)o ∈ T, for all ξ ∈ Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In such a sense, we have that tr ([ξ]n t ) = 2 |σt (ξ)|n Re (σt (ξ)n o) , for all n ∈ N, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Corollary 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Suppose the realization [ξ]t and the t-spectral form Σt (ξ) are similar in Ht 2 for ξ ∈ Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then tr ([ξ]n t ) = 2 |σt (ξ)|n Re (σt (ξ)n o ) = tr �� [ξ]∗ t �n� , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3) for all n ∈ N, where σt (ξ) = |σt (ξ)| σt (ξ)o is the polar decomposition of σt (ξ), with σt (ξ)o ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The free-distributional data (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3) is immediately obtained by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2) under the polar decomposition of the t-spectral value σt (ξ) in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Assume again that a hypercomplex number (a, b) ∈ Ht satisfies our similarity assumption, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', T denote = [(a, b)]t and S denote = Σt ((a, b)) are similar in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then, for any (r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', rn) ∈ {1, ∗}n , for n ∈ N, the matrix n� l=1 T rl is similar to n� l=1 Srl in Ht 2 (and hence, in Ht 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Theorem 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let (a, b) ∈ Ht satisfy the similarity assumption that: T denote = [(a, b)]t and S denote = Σt ((a, b)) are similar in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' If σt (a, b) = rwo, polar decomposition, with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4) r = |σt (a, b)| and wo ∈ T, then 40 DANIEL ALPAY AND ILWOO CHO tr � n� l=1 T rl � = 2rnRe \uf8eb \uf8edw n � l=1 el o \uf8f6 \uf8f8 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5) for all (r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', rn) ∈ {1, ∗}n, for all n ∈ N, where el = � 1 if rl = 1 −1 if rl = ∗, for all l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Since the realization T and the t-spectral form S are assumed to be similar in Ht 2, their adjoints T ∗ and S∗ are similar in Ht× 2 (∗) ∪ {[(0, 0)]t};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' and hence, the matrix n� l=1 T rl and n� l=1 Srl are similar “in Ht 2.” Consider that S = \uf8eb \uf8ed σt (a, b) 0 0 σt (a, b) \uf8f6 \uf8f8 = � rwo 0 0 rwo � = r � wo 0 0 w−1 o � , under hypotheses, because z = 1 z = z−1 in T, whenever z ∈ T in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It shows that Sj = rj � wj o 0 0 w−j o � , for all j ∈ N ∪ {0} , and S∗ = r � wo 0 0 wo � = r � w−1 o 0 0 wo � , satisfying that (S∗)j = � Sj�∗ , for all j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It implies that, for any (r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', rn) ∈ {1, ∗}n, for n ∈ N, there exists (e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', en) ∈ {±1}n, such that el = � 1 if rl = 1 −1 if rl = ∗, for all l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', n, and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='6) n � l=1 Srl = rn \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed w n � l=1 el o 0 0 w − � n � l=1 el � o \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Thus, under our similarity assumption, tr � n � l=1 T rl � = tr � n � l=1 Srl � = rn \uf8eb \uf8edw n � l=1 el o + w − � n � l=1 el � o \uf8f6 \uf8f8 , implying that tr � n � l=1 T rl � = rn \uf8eb \uf8ed2Re \uf8eb \uf8edw n � l=1 el o \uf8f6 \uf8f8 \uf8f6 \uf8f8 , for all (r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', rn) ∈ {1, ∗}n, for all n ∈ N, where (e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', en) ∈ {±1}n satisfies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Therefore, under our similarity assumption and the polar decomposition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4), the free-distributional data (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS 41 By the above theorem, one immediately obtain the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Corollary 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let (a, b) ∈ Ht satisfy the similarity assumption that: T denote = [(a, b)]t and S denote = Σt ((a, b)) are similar in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' If σt (a, b) = rwo, polar decomposition, with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='7) r = |σt (a, b)| and wo ∈ T, then τ � n� l=1 T rl � = rnRe \uf8eb \uf8edw n � l=1 el o \uf8f6 \uf8f8 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='8) for all (r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', rn) ∈ {1, ∗}n, for all n ∈ N, where el = � 1 if rl = 1 −1 if rl = ∗, for all l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5), the free-distributional data (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='8) holds up to the normalized trace τ = 1 2tr on Ht 2, under (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Under our similarity assumption and the condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='7), the free-distributional data (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='8) fully characterizes the free distribution of [(a, b)]t ∈ Ht 2 in the C∗- probability space (Ht 2, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Corollary 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Suppose a given scale t is negative in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let (a, b) ∈ Ht, and let T denote = [(a, b)]t and S denote = Σt ((a, b)) in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' If σt (a, b) = rwo, polar decomposition, with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='9) r = |σt (a, b)| and wo ∈ T, then tr � n� l=1 T rl � = 2rnRe \uf8eb \uf8edw n � l=1 el o \uf8f6 \uf8f8 = 2τ � n� l=1 T rl � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10) for all (r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', rn) ∈ {1, ∗}n, for all n ∈ N, where el = � 1 if rl = 1 −1 if rl = ∗, for all l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In Theorem 48 and Corollary 49, we showed that if T and S are similar in Ht 2, then the free-distributional data (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10) holds under the condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='9), by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='8), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, it suffices to show that the realization T and the t-spectral form S are similar in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' However, since t < 0 in R, the matrices T and S are similar in Ht 2 by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 42 DANIEL ALPAY AND ILWOO CHO The above corollary shows that, if a given scale t is negative in R, then the free-distributional data (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10) fully characterizes the free distributions of the re- alizations [ξ]t in the t-scaled-monoidal C∗-algebra Ht 2 up to the usual trace tr, and the normalized trace τ, for “all” ξ ∈ Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In other words, it illustrates that, if t < 0 in R, then the free-distributional data on the C∗-probability spaces, � Ht 2, tr � and � Ht 2, τ � , are fully characterized by the spectra of hypercomplex numbers of Ht, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='9) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' But, if t ≥ 0, and hence, there are some hypercomplex numbers η of Ht whose realization and spectral form are not similar in Ht 2, then computing joint free mo- ments of [η]t in Ht 2 would not be easy e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' More Free-Distributional Data on (Ht 2, τ) for t < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In this section, a fixed scale t is automatically assumed to be negative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', t < 0 in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' At this moment, we emphasize that most main results of this section would hold even though t is not negative in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' However, we assume a given scale t is negative for convenience (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let Ht 2 be the t-scaled-monoidal C∗-algebra inducing a C∗-probability space (Ht 2, τ), where τ is the normalized trace on Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Since t is assumed to be negative in R, the realizations T = [η]t and the t-spectral forms S = Σt (η) are similar in Ht 2 by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='9), and hence, τ � n � l=1 T rl � = rnRe \uf8eb \uf8edw n � l=1 el o \uf8f6 \uf8f8 = τ � n � l=1 Srl � , by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5), where (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1) σt (η) = rwo ∈ C, polar decomposition, with r = |σt (η)| and wo ∈ T, for all (r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', rn) ∈ {1, ∗}n, where (e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', en) ∈ {±1}n satisfies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='6), for all n ∈ N, for “all” η ∈ Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' And the free-distributional data (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1) fully characterizes the free distribution of [η]t ∈ (Ht 2, τ), for all η ∈ Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In this section, we refine (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1) case-by-case, up to operator-theoretic properties of elements of (Ht 2, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Definition 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let A be a unital C∗-algebra with its unity 1A, and let T ∈ A, and T ∗ ∈ A, the adjoint of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (1) T is said to be self-adjoint, if T ∗ = T in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (2) T is a projection, if T ∗ = T = T 2 in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (3) T is normal, if T ∗T = T T ∗ in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (4) T is a unitary, if T ∗T = 1A = T T ∗ in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let (a, b) ∈ Ht, satisfying the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5), and T denote = [(a, b)]t ∈ Ht 2, as an element of (Ht 2, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then its adjoint, T ∗ = � a b tb a � ∈ Ht 2(∗), is well-defined in (Ht 2, τ), and the corresponding t-spectral form, S denote = Σt ((a, b)) = � w 0 0 w � ∈ Ht 2, OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS 43 is contained in (Ht 2, τ), where w is determined by RA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1, and w = σt (a, b) = x + i � y2 − tu2 − tv2 is the t-spectral value, uniqely polar-decomposed to be w = rwo with r = |σt (a, b)| and wo ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Assumption and Notation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (from below AN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1) From now on, if we say that “a given hypercomplex number (a, b) ∈ Ht satisfies AN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1,” then it means it has its realization denoted by T , its t-spectral form denoted by S, determined by the t-spectral value denoted by w, which is polar-decomposed to be w = rwo, as indicated in the very above paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' \x08 Let (a, b) ∈ Ht satisfy AN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then the self-adjointness of the realization T ∈ Ht 2 in Ht 2 says that T ∗ = T ⇐⇒ � a b tb a � = � a tb b a � , if and only if a = a and tb = b in C, if and only if (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2) a ∈ R and b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Especially, the equality b = 0 in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2) is obtained by our negative-scale assump- tion: t < 0 in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proposition 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let (a, b) ∈ Ht satisfy AN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then the realization T ∈ Ht 2 is self-adjoint in Ht 2, if and only if a ∈ R and b = 0 ⇐⇒ (a, b) = (Re (a) , 0) in Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The self-adjointness (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3) is shown by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The self-adjointness (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3) illustrates that the self-adjoint generating elements T ∈ Ht 2 of (Ht 2, τ) have their forms, T = � x 0 0 x � ∈ Ht 2 (1, ∗) with x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Remark and Observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The above self-adjointness characterization (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3) is obtained for the case where t < 0 in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' How about the other cases?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Generally, one has T is self-adjont in Ht 2, if and only if a = a and tb = b, like (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Thus one can verify that: (i) if t = 0, then T is self-adjoint, if and only if a ∈ R and b = 0, just like (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (ii) if t > 0 and t ̸= 1, then T is self-adjoint, if and only if a ∈ R and b = 0, just like (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' meanwhile, (iii) if t = 1 (equivalently, if (a, b) is a bicomplex number of H1), then T is self-adjoint in H1 2, if and only if a ∈ R, if and only if (a, b) = (Re (a) , b) in H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In summary, T is self-adjoint in Ht 2 ⇐⇒ (a, b) = (Re (a) , 0) in Ht, like (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3), whenever t ∈ R \\ {1}, meanwhile, T is self-adjoint in H1 2 ⇐⇒ (a, b) = (Re (a) , b) ∈ H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 44 DANIEL ALPAY AND ILWOO CHO \x08 Now, let (a, b) ∈ Ht, under AN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1 and our negative-scale assumption, satisfy the self-adjointness (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', it is actually (a, 0) with a ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then T = � a 0 0 a � = S in Ht 2 (1, ∗) , as an element of Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Theorem 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let (a, b) ∈ Ht satisfy AN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1, and assume that the realization T is self-adjoint in (Ht 2, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then τ � n� l=1 T rl � = τ (T n) = an in R, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4) for all (r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', rn) ∈ {1, ∗}n, for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By the self-adjointness (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3) of the realization T of (a, b) ∈ Ht, one has (a, b) = (a, 0) in Ht, with a ∈ R, and T = S = � a 0 0 a � = S∗ = T ∗ in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, τ � n � l=1 T rl � = τ (T n) = τ (Sn) = τ �� an 0 0 an �� , for all (r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', rn) ∈ {1, ∗}n, for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Therefore, the free-distributional data (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Similar to the above theorem, one can verify that: if t ∈ R\\{1}, then the free-distributional data (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4) holds for self-adjoint realizations T ∈ (Ht 2, τ) of (a, 0) ∈ Ht with a ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' \x08 By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3), the realization T of a hypercomplex number (a, b) ∈ Ht, satisfying AN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1, is self-adjoint, if and only if (a, b) = (a, 0) with a ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' And, by definition, such a self-adjoint matrix T can be a projection, if and only if it is idempotent in the sense that T 2 = T in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Observe that a self-adjoint realization T satisfies the above idempotence, if and only if T 2 = � a2 0 0 a2 � = � a 0 0 a � = T, if and only if (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5) a2 = a ⇐⇒ a = 0, or a = 1, in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proposition 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let (a, b) ∈ Ht satisfy AN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then the realization T is a projection, if and only if either T = I2, or T = O2 in Ht 2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='6) where I2 = [(1, 0)]t is the identity matrix, and O2 = [(0, 0)]t is the zero matrix of Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS 45 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The operator-equality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='6) holds in Ht 2 (and hence, in Ht 2) by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Like the above proposition, one can conclude that: if t ∈ R \\ {1}, then the realization T is a projection in Ht 2, if and only if it is either the identity matrix I2, or the zero matrix O2 of Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' How about the case where t = 1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' As we discussed above, T ∈ H1 2 is self-adjoint, if and only if (a, b) = (Re (a) , b) in H1, if and only if T = � x b b x � ∈ H1 2, and S = \uf8eb \uf8ed x + i √ −u2 − v2 0 0 x − i √ −u2 − v2 \uf8f6 \uf8f8 , implying that S = \uf8eb \uf8ed x − |b| 0 0 x + |b| \uf8f6 \uf8f8 in H1 2, under AN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Such a self-adjoint T is a projection, if and only if T 2 = T in H1 2, if and only if x2 + |b|2 = x and 2xb = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Thus if b = 0, then x ∈ {0, 1}, meanwhile, if b ̸= 0, then x2 + |b|2 = x and x = 1 2, ⇐⇒ x = 1 2 and 1 4 + |b|2 = 1 2, ⇐⇒ x = 1 2 and |b|2 = 1 4, if and only if (a, b) = �1 2, b � with |b|2 = 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It implies that T is a projection in H1 2, if and only if (a, b) = (0, 0) , or (a, b) = (1, 0) , or (a, b) = �1 2, b � with |b|2 = 1 4, in H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' \x08 The above proposition says that, under our negative-scale assumption, the only projections of Ht 2 induced by hypercomplex numbers of Ht are the identity ele- ment I2 = [(1, 0)]t, and the zero element O2 = [(0, 0)]t in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' For any unital C∗-probability spaces (A, ϕ), the unity 1A has its free distributions characterized by its free-moment sequence, (ϕ (1n A) = ϕ (1A))∞ n=1 = (1, 1, 1, 1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=');' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' and the free distribution of the zero element 0A is nothing but the zero-free distri- bution, characterized by the free-moment sequence, (ϕ (0n A) = ϕ (0A))∞ n=1 = (0, 0, 0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 46 DANIEL ALPAY AND ILWOO CHO Theorem 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let (a, b) ∈ Ht, satisfying AN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1, have its realization T ∈ Ht 2, which is a “non-zero” projection in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then τ (T n) = 1, ∀n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (In fact, this result holds true for all t ∈ R \\ {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Under hypothesis, the realization T ∈ Ht 2 is a projection in Ht 2, if and only if (a, b) = (1, 0), or (0, 0) in Ht, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Since T ∈ Ht 2 is assumed to a non-zero projection in Ht 2, we have (a, b) = (1, 0) in Ht, ⇐⇒ T = I2 = S in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Therefore, τ (T n) = τ (In 2 ) = 1, ∀n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (Note that it holds true for all t ∈ R \\ {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=') Let (a, b) ∈ Ht satisfy AN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1, and let T ∈ Ht 2 be the realization in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Observe that T ∗T = � a b tb a � � a tb b a � = \uf8eb \uf8ed |a|2 + |b|2 (t + 1) ab (t + 1) ab t2 |b|2 + |a|2 \uf8f6 \uf8f8 , and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='7) T T ∗ = � a tb b a � � a b tb a � = \uf8eb \uf8ed |a|2 + t2 |b|2 (t + 1) ab (t + 1) ab |b|2 + |a|2 \uf8f6 \uf8f8 , in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, the realization T of (a, b) is normal in Ht 2, if and only if |a|2 + t2 |b|2 = |a|2 + |b|2 and (t + 1) ab = (t + 1) ab, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='8) in C, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proposition 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let (a, b) ∈ Ht satisfy AN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then the realization T ∈ Ht 2 is normal in Ht 2, if and only if t2 |b|2 = |b|2 and (t + 1) ab = (t + 1) ab, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='9) in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In particular, if t = −1 (equivalently, if (a, b) ∈ H−1 is a quaternion), then T is normal in H−1 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' if t = 1, (equivalently, if (a, b) ∈ H1 is a bicomplex number), then T is normal in H1 2, if and only if either (a, b) = (Re (a) , b) or (a, b) = (a, 0) in H1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10) meanwhile, if t ∈ R \\ {±1}, then T is normal in Ht 2, if and only if b = 0 in C ⇐⇒ (a, b) = (a, 0) ∈ Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='11) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='8), the normality characterization (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='9) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='9), if t = −1 in R, and hence, if (a, b) ∈ H−1 is a quaternion, then the condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='9) is identified with |b|2 = |b|2 , and 0 = 0, which are the identities on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' These identities demonstrate that the realization of every quaternion is automatically normal in H−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS 47 Suppose t = 1 in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then the condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='9) is equivalent to |b|2 = |b|2 and 2ab = 2ab, if and only if either a = a in C ⇐⇒ (a, b) = (Re (a) , b) ∈ H1 (if b ̸= 0), or (a, b) = (a, 0) ∈ H1 (if b = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Thus, if t = 1, then T is normal, if and only if the condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Assume now that both t ̸= 1 and t ̸= −1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', suppose t2 ̸= 1 in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, the first condition of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='9) is identified with t2 |b|2 = |b|2 ⇐⇒ b = 0 in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, the second condition of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='9) automatically holds, since (t + 1) a · 0 = (t + 1) a · 0 ⇐⇒ 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Therefore, the realization T ∈ Ht 2 of (a, b) ∈ Ht is normal in Ht 2, if and only if (a, b) = (a, 0) in Ht, whenever t ∈ R \\ {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', the normality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='11) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The above proposition illustrates that: (i) the realizations of “all” quaternions are normal in H−1 2 , (ii) the realizations of bicomplex numbers are normal in H1 2, if and only if either (a, b) = (Re (a) , b), or (a, b) = (a, 0) in H1, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10), and (iii) the only realizations [(a, 0)]t are normal in Ht 2, whenever t ∈ R \\ {±1}, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Theorem 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let (a, b) ∈ Ht satisfy AN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='12) Suppose t = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then T is normal in H−1 2 , and its free distribution is characterized by the formula (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='13) Let t ∈ R \\ {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' If T is “non-zero” normal in Ht 2, then τ � n � l=1 T rl � = RnRe \uf8eb \uf8edW n � l=1 el o \uf8f6 \uf8f8 , with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='14) R = |a| and Wo = a |a| ∈ T, where el = � 1 if rl = 1 −1 if rl = ∗, for l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', n, for all (r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', rn) ∈ {1, ∗}n, for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The statement (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='12) holds by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Of course, if t < 0, and if T ∈ Ht 2, then the free-distributional data (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='14) holds by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10), because T and the t-spectral form S are similar in Ht 2 as elements of (Ht 2, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' However, in the statement (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='13), the normality works for all the scales t ∈ R \\ {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Assume that the realization T is a “non-zero,” “normal” element of Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then (a, b) = (a, 0) ∈ Ht, with a ̸= 0, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Therefore, T = � a 0 0 a � = S, 48 DANIEL ALPAY AND ILWOO CHO because σt (a, 0) = a in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', the realization T and the t-spectral form S are identical in Ht 2, implying the similarity of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, under AN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1, a = w denote = σt (a, 0) , polar-decomposed to be w = a = |a| � a |a| � ∈ C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', r = |a| and wo = a |a| under AN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Therefore, similar to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10), the free-distributional data (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='14) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Note that, in the proof of the statement (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='13), we did not use our negative- scale assumption for the cases where t < 0, but t ̸= −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Indeed, even though t ≥ 0, but t ̸= 1, the normality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='11) shows that the realization T is a diagonal matrix not affected by the scale t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, whatever scales t are given in R \\ {±1}, the free- distributional data (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='14) holds in (Ht 2, τ), under normality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then, how about the case where t = 1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Recall that if t = 1, then the realization T of (a, b) ∈ H1 is normal in H1 2, if and only if either (a, b) = (Re (a) , b) , if b ̸= 0, or (a, b) = (a, 0) , if b = 0, in H1, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, if (a, b) = (a, 0) in H1, the joint free moments of T are deter- mined similarly by the formula (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='14), by the identity (and hence, the similarity) of T and S (under AN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' However, if (a, b) = (Re (a) , b) with b ̸= 0, then we need a better tool than (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10) to compute the corresponding free-distributional data, because we cannot use our similarity technique (of Theorem 48) here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By the definition of the unitarity, if an element U of a C∗-algebra A is a unitary, then it is automatically normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', the unitarity implies the normality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let (a, b) ∈ Ht satisfy AN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1 with its realization T ∈ Ht 2 in (Ht 2, τ), and suppose it is a unitary in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By the assumption that T is a unitary in Ht 2, it is normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Assume first that t = −1 in R, and hence, (a, b) ∈ H−1 is a quaternion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then the realization T is automatically normal in Ht 2 by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Indeed, in this case, T = � a −b b a � with T ∗ = � a b −b a � = [(a, −b)]−1 , in H−1 2 , as elements of H−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, the normality is guaranteed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' T ∗T = \uf8eb \uf8ed |a|2 + |b|2 0 0 |a|2 + |b|2 \uf8f6 \uf8f8 = T T ∗, in H−1 2 , as elements of H−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' It shows that T is a unitary in H−1 2 , if and only if |a|2 + |b|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='15) Meanwhile, if t ∈ R \\ {±1} in R, then T is normal, if and only if (a, b) = (a, 0) in Ht by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='11), if and only if T = � a 0 0 a � ∈ Ht 2, OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS 49 which is identical (and hence, similar) to the t-spectral form S of (a, 0) in Ht 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' This normal element T is a unitary in Ht 2, if and only if T ∗T = I2 = T T ∗ ⇐⇒ � |a|2 0 0 |a|2 � = � 1 0 0 1 � , if and only if (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='16) |a|2 = 1 in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proposition 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let (a, b) ∈ Ht satisfy AN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='17) Let t = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then T is a unitary in Ht 2, if and only if |a|2 + |b|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='18) Let t ∈ R \\ {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then T is a unitary in Ht 2, if and only if |a|2 = 1 and b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The statements (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='17) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='18) hold by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='15) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='16), respec- tively, because a unitary realization T of (a, b) automatically satisfies the normality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Now, assume that t = 1, and let (a, b) ∈ H1 be a bicomplex number satisfying AN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10), the realization T ∈ H1 2 is normal in H1 2, if and only if either (a, b) = (Re (a) , b) , or (a, b) = (a, 0) , in H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, if (a, b) = (a, 0) in H1, then one obtains the unitarity that: T is a unitary in H1 2, if and only if |a|2 = 1, just like (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' if (a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' b) = (Re (a) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' b) = (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' b) in H1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' with b ̸= 0 in C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' then T is a unitary in H1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' if and only if � x b b x � � x b b x � = \uf8eb \uf8ed x2 + b2 2xRe (b) 2xRe (b) x2 + b2 \uf8f6 \uf8f8 = I2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' and � x b b x � � x b b x � = \uf8eb \uf8ed x2 + b2 2xRe (b) 2xRe (b) x2 + b2 \uf8f6 \uf8f8 = I2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' in H1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' if and only if x2 + b2 = x2 + b2 = 1 and 2xRe (b) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' if and only if b2 = b2 = 1 − x2 and 2xRe (b) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' if and only if b2 = 1 − x2 ∈ R and x = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' because b is assumed not to be zero in C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' if and only if x = 0 and b = ±1 in R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' if and only if T = � 0 1 1 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' or T = � 0 −1 −1 0 � in H1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' if and only if (a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' b) = (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' or (a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' b) = (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' −1) in H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 50 DANIEL ALPAY AND ILWOO CHO i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', if (a, b) = (Re (a) , b) in H1, then T is a unitary in H1 2, if and only if (a, b) = (0, 1) , or (a, b) = (0, −1), in H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In summary, the realization T ∈ H1 2 of a bicomplex number (a, b) ∈ H1 is a unitary in Ht 2, if and only if either (a, b) = (a, 0) with |a|2 = 1, or (a, b) = (0, 1) , or (a, b) = (0, −1), in H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' \x08 By the unitarity (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='17) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='18), one has the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Theorem 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let (a, b) ∈ Ht satisfy AN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='19) Suppose t = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' If T is a unitary in Ht 2, then its free distribution is characterized by the formula (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='10) with r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='20) Let t ∈ R \\ {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' If T is a unitary in Ht 2, then τ � n � l=1 T rl � = Re � a n � l=1 el � , with a ∈ T in C, where (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='21) el = � 1 if rl = 1 −1 if rl = ∗, for l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', n, for all (r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=', rn) ∈ {1, ∗}n, for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The statement (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='19) holds by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In particular, by the unitarity characterization (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='17), the free-distributional data in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='11) must have r = 1, since |σt (a, b)| = |w| denote = r = 1, under the similarity of T and S, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='13), if t ̸= ±1, then the free-distributional data (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='21) holds by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Indeed, under the unitarity of T, the formula (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='14) satisfies R = |a| = 1 and Wo = a ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Therefore, the joint free moments (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='21) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The above theorem characterizes the free distributions of unitary elements of (Ht 2, τ) induced by Ht, where t ∈ R \\ {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Suppose t = 1, and (a, b) ∈ H1 satisfies AN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' In the above Observation, we showed that the realization T ∈ H1 2 of (a, b) is a unitary, if and only if either (a, b) = (a, 0) with a ∈ T, or (a, b) = (0, 1) , or (a, b) = (0, −1), in H1, equivalently, either T = � a 0 0 a � with a ∈ T, or T = � 0 1 1 0 � , or T = � 0 −1 −1 0 � , OPERATORS INDUCED BY CERTAIN HYPERCOMPLEX SYSTEMS 51 in H1 2 (as an element of H1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Thus, if (a, b) = (a, 0) ∈ H1 with |a|2 = 1, then the free distribution of T is similarly characterized by the formula (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Meanwhile, if T = [(0, 1)]1, then T ∗ = T ∈ H1 2 ⊂ H1 2 (1, ∗) in H1 2, and T 2 = � 0 1 1 0 � � 0 1 1 0 � = � 1 0 0 1 � = I2, in H1 2, satisfying that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='22) (T n)∞ n=1 = (T, I2, T, I2, T, I2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=') ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' and, if T = [(0, −1)]1, then T ∗ = T ∈ H1 2 ⊂ H1 2 (1, ∗) in H1 2, and T 2 = � 0 −1 −1 0 � � 0 −1 −1 0 � = � 1 0 0 1 � = I2, in H1 2, satisfying that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='23) (T n)∞ n=1 = (T, I2, T, I2T, I2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=') .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Therefore, one obtains the following result in addition with Theorem 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Theorem 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Let (a, b) ∈ H1 be a bicomplex number satisfying AN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Then the realization T is a unitary in � H1 2, τ � , if and only if either (a, b) = (a, 0) , with |a|2 = 1, or (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='24) (a, b) = (0, 1) , or (a, b) = (0, −1) in H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='25) If (a, b) = (a, 0), with |a|2 = 1, in H1, then the free distribution of T is characterized by the formula (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='26) If either (a, b) = (0, 1), or (a, b) = (0, −1) in H1, then the free distribution of the unitary realization T is fully characterized by the free-moment sequence, (τ (T n))∞ n=1 = (0, 1, 0, 1, 0, 1, 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='27) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' By the very above Observation after Proposition 58, it is shown that the realization T ∈ H1 2 of a bicomplex number (a, b) ∈ H1 is a unitary in H1 2, if and only if the condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='24) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The statement (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='25) is shown similarly by the proof of the statement (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' So, the free-distributional data (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='21) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Now, if either T = [(0, 1)]1, or T = [(0, −1)]1 in H1 2, it is not only a unitary, but also a self-adjoint element of � H1 2, τ � , and hence, the free distribution of T is fully characterized by the free-moment sequence (τ (T n))∞ n=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' However, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='22) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='23), one immediately obtain the free-moment sequence (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Therefore, the statement (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content='26) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' The above theorem fully characterizes the free distributions of the unitaries of � H1 2, τ � induced by bicomplex numbers of H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' 52 DANIEL ALPAY AND ILWOO CHO References [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' Alpay, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFAT4oBgHgl3EQf3h4G/content/2301.08720v1.pdf'} +page_content=' E.' 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Then we prove that there exist graph manifolds virtually having no faithful +representations to the Seifert motion group. +1 +Introduction +In Kirby’s list of problems in low-dimensional topology [Kir97], Thurston proposed a question if +every finitely generated 3-manifold group G has a faithful representation in GL(4, R). However, +Button gave a counterexample [But14] which is a graph manifold made by gluing two Seifert man- +ifolds. The fundamental group of the graph manifold cannot be embedded in GL(n, K) for n ≤ 4 +and for any field K. +Then we can ask if every finitely generated 3-manifold group G has a faithful linear representa- +tion. According to the book [AFW15] written by Aschenbrenner, Friedl and Wilton, which gives a +detailed survey of 3-manifold groups, the isometry group of the following geometric 3-manifold are +subgroups of GL(4, R): spherical geometry, S2 × R, Euclidean geometry, Nil, Sol and hyperbolic +geometry . Moreover, the fundamental group of an H2 × R-manifold is a subgroup of GL(5, R) +[AFW15, p.50]. Hence, these seven geometric manifolds have faithful linear representations. How- +ever, the isometry group of � +SL(2, R) is not a linear group. With the fact that the fundamental +group of an S1 bundle over a surface is linear over Z [AFW15, p.43], Seifert geometry, whose model +space is � +SL(2, R), virtually has a faithful linear representation. However, we cannot fix the size of +the linear Lie group. +As for graph manifolds, which consists of Seifert manifolds, the bounded case has better prop- +erties. Result of Haglund and Wise [HW08][HW12] shows that if a 3-manifold admits a nonpositive +1 + +curvature, then its fundamental group can be embedded in a linear Lie group. +Kapovich and +Leeb proved that there exists a nonpositive curvature on a graph manifold with torus boundaries +[KL96]. Hence, a graph manifold with torus boundaries has a faithful linear representation. How- +ever, Kapovich and Leeb also gave an example of a closed graph manifold admitting no nonpositive +curvature. So, for closed graph manifolds, we do not know if the fundamental groups have faithful +linear representations. +Brooks and Goldman defined the Seifert volume by using a representation to the isometry +group of � +SL(2, R) for 3-manifolds [BG84]. +Therefore, we consider the representations of closed +graph manifold to the Seifert motion group, which is the identity component of the isometry group +of � +SL(2, R). We prove a theorem as the following: +Theorem 1.1. There exists a closed graph manifold virtually having no faithful representations to +Seifert motion group. In fact, there exists a closed graph manifold M such that for any finite cover +M ′ of M and any representation ρ : π1(M ′) → � +SL(2, R) ×Z R, ρ is not faithful restricted to any +Seifert block of M ′. +Here � +SL(2, R) ×Z R denotes the Seifert motion group. In addition, the theorem 1.1 holds for +closed graph manifolds satisfying the condition of strictly diagonally dominance (Definition 2.4). +This condition is inspired from Derbez, Liu and Wang’s paper [DLW20]. If the graph manifold has +nonempty torus boundaries, it can have a representation restricted on one submanifold is faithful. +We give an example of the graph manifold and construct the representation in Section 7. +To get the theorem 1.1 we prove a stronger version that there exist no virtually vertex faithful +representations (Definition 5). We suppose there is virtually a vertex faithful representation on a +closed graph manifold satisfying the condition of strictly diagonally dominance. Then the image of +a submanifold under the representation is an Abelian group. We list equations about the images of +fibers which are deduced form the topological structure of the graph manifold. The solution of the +equations contradicts with the assumption of the vertex faithful representation. So, we deduce the +result that this graph manifold virtually has no faithful representations. +The paper is organized as the following: +In section 2 we recall the relative definition of graph manifolds and introduce two important +topological invariants. In section 3 we introduce the Seifert motion group and classify its elements +in order to discuss representations. Then in section 4 we introduce JSJ characteristic finite covers +2 + +and give a property about topological invariants under this special finite cover. In section 5 we +mainly discuss the representations to the Seifert motion group. In section 6 we prove the Theorem +1.1 about the nonexistence of representation in the closed condition. In addition, we construct a +vertex faithful representation when a graph manifold has torus boundaries in section 7. +Acknowledgement. I would like to give my thanks to my supervisor Yi Liu for his guidance and +conversations. +2 +Preliminary +Let M be an orientable compact irreducible 3-manifold. There is a collection of disjoint incom- +pressible tori splits M into components which are either atoroidal or Seifert fibered. Moreover, +the minimal such collection is unique up to isotopy. This decomposition of 3-manifold is called +Jaco-Shalen-Johannson torus decomposition (JSJ decomposition) [JS79][Joh79]. +In the following we review the definitions, related properties and some topological invariants +of Seifert manifolds and graph manifolds. Definitions and notations mostly follows the paper of +Buyalo and Svetlov [BS04]. +2.1 +Seifert manifolds +A Seifert manifold is a 3-manifold foliated by circles. A useful definition according to Scott [Sco83] +is to define the Seifert manifold as a “circle bundle” over an orbifold. Let M be a Seifert manifold +and O be its base orbifold. There is a projection p : M → O and an exact sequence of fundamental +groups +{1} → K → π1(M) → π1(O) → {1}, +where K is a cyclic subgroup of π1(M) generated by a regular fiber. The group K is infinite cyclic +except the case where M is covered by S3. +Let the orbifold O has an underlying surface F and m singular points si with degree qi, 1 ≤ i ≤ +m. Then the Euler characteristic number of an orbifold is defined to be +χ(O) = χ(F) − +m +� +i=1 +� +1 − 1 +qi +� +. +3 + +If χ(O) < 0, then there is a finite cover of M having a hyperbolic structure on its base surface. The +fundamental group of the base surface can be embedded into PSL(2, R). In the following we only +consider the Seifert manifold having a base orbifold with a negative Euler characteristic number. +Hence, there is a finite cover of the Seifert manifold is a trivial circle bundle over a surface. In the +rest of this paper, the Seifert manifold is a trivial circle bundle without special mentions. +Waldhausen basis. +Let M be an oriented Seifert manifold with nonempty boundaries. Let W +be the set of boundary components. Consider an oriented fiber f of M and there is a homological +class fw ∈ H1(Tw; Q) corresponding to fiber f in each boundary torus Tw, w ∈ W. Then there is an +induced orientation from M to Tw. Hence, the intersection form (·, ·) : H1(Tw; Q)×H1(Tw; Q) → Q +can be defined. For each fw there is a zw ∈ H1(Tw; Q) satisfying (zw, fw) = 1. Here we modify the +notation of the intersection form which is different from the notation of Buyalo and Svetlov. +Definition 2.1. For each torus boundary Tw of a Seifert manifold M there is a zw such that +(zw, fw) = 1. Then a collection of elements {zw, fw}w∈W is called a Waldhausen basis of the Mv if +they satisfy the condition that z = � +w∈W zw lies in the kernel of the inclusion i∗ : H1(∂M; Q) → +H1(M; Q). +The Waldhausen basis is not unique. It is defined up to a transformation zw �→ zw + nwfw +where � +w∈W nw = 0. If M is a trivial circle bundle, then we can choose zw from H1(Tw; Z). This +choice of Waldhausen basis is equivalent to the choice of a trivialization of M = F × S1. +Framed Seifert manifolds. +Let M be a Seifert manifold and each boundary Tw has a fixed +element cw ∈ H1(Tw; Z) with (cw, fw) ̸= 0. Then M is called framed and the collection {cw}w∈W +is a framing of M. +For a framed Seifert manifold, there is a formula about its fiber and the framing. We state the +formula as a lemma. +Lemma 2.2. Let {cw}w∈W be a framing of a Seifert manifold M. Then +� +w∈W +1 +(cw, fw) · (iw)∗cw = k · f, +(1) +where (iw)∗ : H1(Tw; Z) → H1(M; Z) is the inclusion homomorphism induced by iw : Tw → Mv. +4 + +The homological class f ∈ H1(M; Z) represents the regular fiber, and (iw)∗fw = f. The number k +on the right hand is a rational number. +Proof. We choose a Waldhausen basis {zw, fw}w∈W for M. cw can be represented as cw = awfw + +bwzw. Since bw = (cw, fw) ̸= 0, +� +w∈W +1 +(cw, fw) · (iw)∗cw = +� +w∈W +1 +bw +· (iw)∗(awfw + bwzw) = +� � +w∈W +aw +bw +� +f. +Here the coefficient k = � +w∈W +aw +bw on the right hand is independent of the Waldhausen basis since +the left hand is independent of the choice. +2.2 +Graph manifolds +A graph manifold M is an irreducible 3-manifold with all components Seifert manifolds under the +JSJ decomposition. The graph manifold M is associated with a directed graph Γ(V, E) dual to the +JSJ decomposition. We call each component a Seifert block. The vertex set V of Γ is the set of +Seifert blocks and the set E of oriented edges of Γ is identified to boundaries of all blocks. If e ∈ E +is an edge directed form v1 to v2, then Tv1,v2 ⊂ ∂Mv1 and Tv2,v1 ⊂ ∂Mv2. Hence, e is identified to +an ordered pair (v1, v2). +If a graph Γ is connected, for two points v1, v2, Let P denotes the set of all paths L connecting +v1, v2. We define the length #L to be the number of edges in L. The metric on Γ is defined as +d(v1, v2) = min +L∈P #L. +Then we can define the distance between two Seifert blocks by their corresponding vertices as +d(Mv1, Mv2) = d(v1, v2). +We fix an orientation of a graph manifold M. Then a torus boundary of a Seifert block has an +induced orientation. Let Mv1, Mv2 be adjacent Seifert blocks. We denote the torus by Tv1,v2 with +the induced orientation from Mv1 dual to the directed edge e = (v1, v2). Then the orientation of +Tv2,v1 is opposite to the orientation of Tv1,v2 +A gluing map is a homeomorphism associated with a directed edge e = (v1, v2) maps Tv1,v2 to +5 + +Tv2,v1. If we choose {fv1, zv1} as a basis of π1(Tv1,v2) and {fv2, zv2} as a basis of π1(Tv2,v1), then +the gluing map induces an isomorphism on the fundamental groups. It can be represented as a 2×2 +matrix +gv1,v2 = + +a +b +c +d + + , +ad − bc = −1. +So we call the isomorphism gv1,v2 a gluing matrix. +In the following we introduce two topology invariants of graph manifolds referred to Buyalo and +Svetlov [BS04]. We make a few modifications by using the gluing matrix in definitions. +The intersection index. +Let Mv1 and Mv2 be two adjacent Seifert blocks in an oriented +graph manifold. +We use fv1,v2 to denote the homological class of the regular fiber of Mv1 in +H1(Tv1,v2; Z) and fv2,v1 to denote the fiber of Mv2 in H1(Tv2,v1; Z). By the gluing matrix gv1,v2, +the fiber fv1,v2 maps to an element in H1(Tv2,v1; Z). +Define +bv2,v1 = (fv2,v1, gv1,v2(fv1,v2)). +By definition of a graph manifold and its induced orientation, the integer number bv2,v1 satisfies +bv2,v1 = bv1,v2 ̸= 0. This integer bv2,v1 is called the intersection index of fibers fv2,v1 and fv1,v2. It +changes the sign when the orientation of one fiber fv2,v1 or fv1,v2 changes. +The charge. +For a Seifert block Mv, it has a natural framing {cw}w∈W by setting cw = gw,v(fw,v). +Then the coefficient of f on the right hand in the equation 1 is called the charge of Mv. Under a +chosen Waldhausen basis {fv,w, zv,w}w∈W of Mv, cw = gw,v(fw,v) = aw,vfv,w + bw,vzv,w. Then the +charge can be written as +kv = +� +w∈W +aw,v +bw,v +. +According to Lemma 2.2, the charge is independent of the choice of a Waldhausen basis. +But +aw,v +bw,v which is called the slope for the homological class fw,v is related to the Waldhausen basis +{fv,w, zv,w}w∈W . +If a Seifert block has a boundary T which is also the boundary of the whole graph manifold, +then this boundary T has no contribution to the charge of the Seifert block. We define its slope +always be zero in the calculation of the charge. +6 + +Strictly diagonally dominant graph manifold. +In the paper of Derbez, Liu and Wang [DLW20], they introduce the terminology, strictly diag- +onally dominance, to describe a property of graph manifolds. This terminology is original from the +theory of matrices. +If a n × n matrix A = (aij) satisfies |aii| > � +j̸=i |aij| for all i, then the matrix A is said to be +strictly diagonally dominant. +Proposition 2.3. If A is a strictly diagonally dominant matrix, then A is invertible. +Proof. Suppose A is not invertible, then there is a nonzero vector x such that Ax = 0. Let xi be the +element with maximum absolute value in x. Then there is an equation a1x1+· · ·+aixi+· · ·+anxn = +0. However, +������ +� +j̸=i +aijxj +������ +≤ +� +j̸=i +|aijxj| ≤ + +� +j̸=i +|aij| + + |xi| < |aiixi|. +This deduces a contradiction. Hence, A is invertible. +In the aspect of the graph manifolds, the definition is as the following. +Definition 2.4. For a Seifert block Mv, if the charge kv and all intersection numbers bv,w satisfying +|kv| > +� +w∈W +1 +|bv,w|, +(2) +in which W is the set of all adjacent blocks of Mv. Then this Seifert block is called strictly diagonally +dominant. If a graph manifold has every Seifert block strictly diagonally dominant, then it is called +a strictly diagonally dominant graph manifold. +Strictly diagonally dominance is a relation about the charges and intersection indices of a +graph manifold. In the following section 4 we will prove this relation still holds under a special +finite cover. +3 +Seifert Motion Group +Seifert geometry, or � +SL(2, R) geometry, is one of the eight 3-dimensional geometries as classified by +Thurston [Thu97]. We take Lie group � +SL(2, R) as the model of the Seifert geometry’s space. The +7 + +identity component of its isometry group can be identified with � +SL(2, R) ×Z R, which is called as +Seifert motion group. +Eisenbud, Hirsch and Neumann gave criteria for a Seifert manifold to admit a transverse foliation +[EHN81]. In their paper they studied the homomorphism π1(M) → � +SL(2, R). The name of Seifert +geometry is from Brooks and Goldman. They defined the Seifert volume by using a representation +to the Seifert motion group for 3-manifolds [BG84]. Then Derbez, Liu and Wang gave a formula +to compute the representation volume for a graph manifold with a representation to the Seifert +motion group [DLW20]. So, in this section we state the construction of the Seifert motion group +and classify elements in this group. +3.1 +Construction of Seifert motion group +In this part we use the notation from Derbez, Liu and Wang [DLW20]. Here � +SL(2, R) is the universal +cover of PSL(2, R). There is a map k : R → PSL(2, R), +k(r) = + +cos πr +− sin πr +sin πr +cos πr + + , r ∈ R. +Then there exists a lift ˜k : R → � +SL(2, R). The center Z(� +SL(2, R)) of � +SL(2, R) is the group {˜k(n), n ∈ +Z}, which projects to the identity in PSL(2, R). +The Seifert motion group is the quotient of a product group � +SL(2, R) × R by a subgroup +{(˜k(n), −n), n ∈ Z}. Then the Seifert motion group is denoted by � +SL(2, R) ×Z R. Its elements +can be denoted as g[s] for g ∈ � +SL(2, R) and s ∈ R. The multiplication rule is +g[s]g′[s′] = gg′[s + s′], +and there is a relation +g[s + n] = g˜k(n)[s], n ∈ Z. +3.2 +Classifying elements in Seifert motion group +Elements in SL(2,R) can be classified by their traces. Take a matrix A ∈ SL(2,R). If the absolute +value of the trace |Tr(A)| < 2, the matrix A is said to be elliptic. If |Tr(A)| > 2, it is said to be +8 + +hyperbolic. If |Tr(A)| = 2 and A is not the identity matrix or minus identity matrix, it is said to be +parabolic. Since PSL(2, R) = SL(2, R)/ ± I2, its elements can be classified by their representative +elements. This way can be checked that is well-defined. +Then we consider a natural projection pr : � +SL(2, R) ×Z R → PSL(2, R). The elements in the +Seifert motion group can be classified by their images. Let g ∈ � +SL(2, R) ×Z R, +• if pr(g) is the identity in PSL(2, R), we call it central type; +• if pr(g) is an elliptic element in PSL(2, R), we call it elliptic type; +• if pr(g) is a hyperbolic element in PSL(2, R), we call it hyperbolic type; +• if pr(g) is a parabolic element in PSL(2, R), we call it parabolic type. +Then there is a proposition about the subgroup C(g) = {h ∈ � +SL(2, R) ×Z R|gh = hg}. +Proposition 3.1. Let a, b be two elements in the Seifert motion group � +SL(2, R) ×Z R. +Then +ab = ba if and only if pr(a)pr(b) = pr(b)pr(a) in PSL(2, R). Hence, for a noncentral element +g ∈ � +SL(2, R) ×Z R, the subgroup C(g) is an Abelian group. +Proof. It is direct that ab = ba deduces pr(a)pr(b) = pr(b)pr(a) in PSL(2, R). +Conversely, if +pr(a)pr(b) = pr(b)pr(a), by the construction of the Seifert motion group, aba−1b−1 lies in the +center of � +SL(2, R) ×Z R. +For a, b ∈ � +SL(2, R) ×Z R, we have continuous paths starting from the identity element to a,b +respectively. Notice +(� +SL(2, R) ×Z R)×(� +SL(2, R) ×Z R) → � +SL(2, R) ×Z R, +(a, b) �→ aba−1b−1 +is a continuous map. However, the group {˜k(n), n ∈ Z}×ZR, the center of the Seifert motion group, +is discrete in the first component. So, the first component of aba−1b−1 is the identity element in +� +SL(2, R). Notice that the second component of aba−1b−1 is always zero. Hence, aba−1b−1 equal to +the identity element in � +SL(2, R) ×Z R. +If two nontrivial elements in PSL(2, R) are commutable, they can be diagonalized at the same +time and lie in the same type. Hence, for a noncentral element g ∈ � +SL(2, R) ×Z R, C(pr(g)) = +9 + +{pr(h) ∈ PSL(2, R)|pr(g)pr(h) = pr(h)pr(g)} is an Abelian group. So, C(g) is an Abelian group. +For elements in the Seifert motion group we have another classification with respect to the order +of their projective images. +Definition 3.2. If an element in Seifert motion group is projected to an element with finite order +in PSL(2, R), then we call it an element with projectively finite order. +If an element in Seifert motion group is projected to an element with infinite order in PSL(2, R), +then we call it an element with projectively infinite order. +Remark 3.3. Let f = ˜k(n) be a central element in � +SL(2, R) ×Z R. Consider an equation xm = +f, m ∈ Z. If n = lm + r, l, r ∈ Z, then ˜k(l)[ r +m] is a solution of the equation. However, ˜k(l)[ r +m] is +not a unique solution. We project the equation to PSL(2, R) then (pr(x))m = pr(f) = I2. So, any +solution of the equation xm = f is an element with projectively finite order. +4 +JSJ characteristic Finite Cover +In this section we discuss the finite covers of graph manifolds and the relation of two topological +invariants. If some finite cover of M has a property P, then M is said to have a property P virtually. +Sometimes we only consider the existence of representations. Therefore, we often pass to a finite +cover to prove theorems or construct representations. +According to Kapovich and Leeb’s result [KL98, Lemma 2.1], a graph manifold has a good +structure passing to a finite cover. We state the lemma as the following: +Lemma 4.1 (Kapovich, Leeb). Any graph manifold has an orientable finite cover where all Seifert +blocks are trivial circle bundle over a surface with genus greater than 1. +Furthermore, we can +arrange the intersection index of the fibers of adjacent Seifert blocks are ±1. +By the above lemma, passing to a finte cover each Seifert block Mv can be homeomorphic to +Fv × S1. In the following we assume the graph manifold satisfies the conditions in lemma 4.1. +Moreover, we consider the dual graph Γ is a nontrivial simple graph, which is a graph contains no +multiple edges and self-loops. Hence, every JSJ torus separates two distinct Seifert blocks from +each other. This condition can also be satisfied by passing to a finite cover. +10 + +Definition 4.2. Set p : M ′ → M be a regular finite cover of M. Let the set T be a union of JSJ +tori of M, the set T ′ be the preimage of T and n be a positive integer. If for each torus T of T and +each component T ′ of T ′ over T , the restriction p| : T ′ → T is a characteristic cover with index +n × n, then this finite cover p : M ′ → M is called a JSJ n-characteristic finite cover. +A regular finite cover p1 : M ′ → M is not necessarily a JSJ characteristic finite cover. However, +we can always find a JSJ characteristic finite cover p2 : M ′′ → M such that p2 factors through p1. +Hence, we can prove some properties on JSJ characteristic finite cover and then deduce to a general +case. For a JSJ characteristic finite cover, there is a property about topological invariants. +Proposition 4.3. Let M be a strictly diagonally dominant graph manifold, then any JSJ charac- +teristic finite cover of M is also strictly diagonally dominant. +Proof. Suppose p : M ′ → M is a JSJ characteristic finite cover. Take a Seifert block M ′ +v′ which is a +component of a Seifert block Mv’s preimage. Since this finite cover is JSJ characteristic, the slope +of each torus boundary is invariant. Then the covering degree of the fiber is fixed for each Seifert +block. Hence, the gluing matrices are also invariant under the covering. +Let W be the set of all adjacent block of Mv. If an edge e with ∂e = (v, w), w ∈ W has a +preimage p−1(e) connecting v′, we set ∂(p−1(e)) = {(v′, w′)|w′ ∈ W ′ +v′} in which W ′ +v′ is the set of +all Seifert blocks connecting v′ by one edge of p−1(e). For each torus T ′ +v′,w′ dual to (v′, w′), the +intersection number bv′,w′ = bv,w since the gluing maps have the same matrix representation for +T ′ +v′,w′ and Tv,w. Hence, the sum of slope of T ′ +v′,w′, w′ ∈ W ′ +v′ is just |W ′ +v′|, the size of W ′ +v′, times the +slope of Tv,w. Notice the size of W ′ +v′ is the same for all edges connecting v′. Since the degree of fiber +is fixed, the degree of JSJ characteristic cover for each torus is the same. Hence, the components +of preimage p−1(e) for each e is also the same. +Because Mv is strictly diagonally dominant, we have +|kv| > +� +w∈W +1 +|bv,w|. +11 + +As for M ′ +v′, the charge +kv′ = +� +w∈W +� +w′∈W ′ +v′ +av′,w′ +bv′,w′ = +� +w∈W +� +|W ′ +v′| × av′,w′ +bv′,w′ +� += |W ′ +v′| × +� +w∈W +av′,w′ +bv′,w′ = |W ′ +v′| × +� +w∈W +av,w +bv,w += |W ′ +v′| × kv. +Then +|kv′| = |W ′ +v′| × |kv| > |W ′ +v′| × +� +w∈W +1 +|bv,w| += +� +w∈W +� +|W ′ +v′| × +1 +|bv,w| +� += +� +w∈W +� +|W ′ +v′| × +1 +|bv′,w′| +� += +� +w∈W +� +w′∈W ′ +v′ +1 +|bv′,w′| +So the Seifert block M ′ +v′ is also strictly diagonally dominant. +Since the choice of vertex v is arbitrary, each Seifert block of M ′ is strictly diagonally dominant. +Hence, M ′ is a strictly diagonally dominant graph manifold. +5 +Representations to Seifert Motion Group +If a representation of a graph manifold is faithful, then it restricted on each Seifert block is faithful. +In convenience, we define a “local” faithful representation as the following. +Definition 5.1. If a representation restricted on a Seifert block dual to a vertex v is faithful, then +we call the representation a vertex faithful representation, and more precisely, faithful at v. +Sometimes we call a representation vertex faithful without indicating a vertex v. +In the following we discuss the representations restricted on each Seifert block and classify them +by the images in the Seifert motion group. Particularly, we list equations for Seifert blocks with +Abelian images. +12 + +5.1 +Faithful representations +Consider a Seifert block Mv which is a trivial circle bundle over a compact surface with genus g ≥ 2. +Suppose there is a faithful representation on Mv to the Seifert motion group. We have a conclusion +as the following: +Proposition 5.2. Let a Seifert block Mv be a trivial circle bundle over a compact surface Fv with +genus g ≥ 2. If Mv has a faithful representation ρ : π1(Mv) → � +SL(2, R) ×Z R, then the image of the +fiber of Mv is a nontrivial element of central type and the images of boundaries of the base surface +Fv are all noncentral elements with projectively infinite order. +Proof. Since Mv is a trivial circle bundle over a compact surface Fv with genus g ≥ 2, the funda- +mental group of Mv has a presentation as +π1(Mv) = +� +a1, b1, · · · , ag, bg, c1, · · · , cl +��� +g +� +i=1 +[ai, bi] = +l� +j=1 +cj +� +× ⟨f⟩. +Here [ai, bi] denotes aibia−1 +i b−1 +i . The integer g is the genus of the base surface and l is the number +of boundary components. +The image of the fiber of Mv projects to identity under the projection pr : � +SL(2, R) ×Z R → +PSL(2, R), otherwise the representation ρ is Abelian which contradicts the assumption that ρ is +faithful. Hence, the image of the fiber ρ(f) is a nontrivial central element. +As for the images of boundaries {cj} of the base surface Fv, they cannot lie in the center. If +one of them, ρ(cj), has projectively finite order, then there exists an integer n such that ρ(cj)n +lies in the center. Since ρ is faithful, (cj)n commute with every generator in π1(Mv). This result +contradicts the presentation of π1(Mv). Hence, the images of boundaries of base surface Fv are all +noncentral elements with projectively infinite order. +Corollary 5.3. If a representation ρ of a graph manifold M restricted on a Seifert block Mv is +faithful, then the adjacent blocks of Mv all have noncentral fibers with projectively infinite order. +Proof. By proposition 5.2, the image of the fiber of Mv is a nontrivial element of central type and +the images of boundaries of the base surface Fv are all noncentral elements with projectively infinite +order. +13 + +Since the intersection indices are nonzero integers, the gluing map induces a form of matrix +gv,w = +� a b +c d +� +with b ̸= 0. The fiber fw of an adjacent blocks can be written as fw = afv + bzv,w in +the fundamental group π1(Tv,w). Hence, the image ρ(fw) is a noncentral element with projectively +infinite order. +5.2 +Abelian representations +Let ρ be a representation of a Seifert block Mv to � +SL(2, R) ×Z R. If the image of the fundamental +group π1(Mv) under ρ is an Abelian group, the restricted representation ρ : π1(Mv) → � +SL(2, R)×ZR +is called an Abelian representation of Mv. Since the Abelian representation factors through the +homology group H1(Mv; Q), there is a Waldhausen basis to represent the boundaries of the base +surface. +Suppose Mv has an Abelian representation and it has torus boundary Tv,w, w ∈ W, where W is +the set of all adjacent Seifert blocks of Mv. We choose a Waldhausen basis {fv,w, zv,w}w∈W. Under +gluing matrix gw,v, the homological class fw,v ∈ H1(Tw,v; Z) has an image gw,v(fw,v) in H1(Tv,w; Z). +Notice H1(Tv,w; Z) has a basis fv,w, zv,w. Then fw,v can be represented by a linear combination +gw,v(fw,v) = aw,vfv,w + bw,vzv,w, aw,v, bw,v ∈ Z. +We define a nonzero integer +bv = +� +w∈W +bw,v. +Then we have +bv +bw,v +gw,v(fw,v) = bvaw,v +bw,v +fv,w + bvzv,w. +(3) +Since {zv,w} is in the Waldhausen basis and Mv has the charge kv which satisfies the equation +kv = +� +w∈W +aw,v +bw,v +. +Sum up on both sides for equation (3), then +� +w∈W +bv +bw,v +fw = bvkvfv. +(4) +In convenience, we ignore gluing matrices and inclusion maps. In addition, we denote fw,v by fw +as well as fv,w by fv. +14 + +Here we introduce the number bv in order to avoid discussing any n-th root of a central element +in � +SL(2, R) ×Z R. In the above equations we can find bv/bw,v is an integer, so it involves only group +multiplication. +5.3 +Abelian components +In this part, we firstly introduce some definitions in graph theory from Harary [Har69]. +Definition 5.4. Let Γ(V, E) be a graph, then Γ0(V0, E0) is called a subgraph of Γ if V0 and E0 are +subsets of V and E respectively. +Definition 5.5. Let Γ(V, E) be a graph. A vertex-induced subgraph Γ0(V0, E0), simply called “an +induced subgraph” of Γ, induced by the vertex set V0 is a graph with vertex set V0 and edge set E0 +consisting of those edges both of whose endpoints are in V0. +Let ρ be a representation of a graph manifold M and let Γ be a dual graph of M. Take Γ0 be +a connected subgraph of Γ. Then we use M0 to denote a submanifold consisting of Seifert blocks +corresponding to Γ0. If the image of the fundamental group π1(M0) under the representation ρ is +an Abelian group, then we call the connected subgraph Γ0 as an Abelian subgraph and call M0 +as an Abelian component. In addition, if the fibers of Seifert blocks in M0 are all noncentral type +under the representation ρ, then we call M0 the noncentral Abelian component and Γ0 noncentral +Abelian subgraph. +Definition 5.6. Let ρ be a representation of a graph manifold M. +The set of all noncentral +Abelian subgraphs has a partial order which is the inclusion of subgraphs. Then we define a maximal +noncentral Abelian subgraph which is not included by any Abelian subgraph except itself. Similarly, +we can define a maximal noncentral Abelian component. +The set of noncentral Abelian subgraphs may be an empty set under some representation. We +suppose the set of noncentral Abelian subgraphs is not empty under a representation. Then there +exists at least one maximal noncentral Abelian subgraphs. There are two propositions about the +maximal one. +Proposition 5.7. Let M0 be a maximal noncentral Abelian component under a representation ρ. +Then the adjacent Seifert blocks of M0 all have central fibers. +15 + +Proof. Suppose Mw is an adjacent Seifert block of M0 with a noncentral fiber fw. +Hence, the +fundamental group π1(Mw) is an Abelian group. Since ρ(fw) commutes with the images of all +fibers of Seifert blocks in M0, all elements of π1(Mw) commute with all elements of π1(M0). The +fundamental group of a submanifold consisting of M0 and Mw is an Abelian group. This result +contradicts with the definition of maximal noncentral Abelian component. Hence, the fiber of Mw +is of central type. +Proposition 5.8. Let M be a closed graph manifold with dual graph Γ(V, E). If Γ0(V0, E0) is a +maximal noncentral Abelian subgraph under a representation ρ, then there is a characteristic finite +cover p : M ′ → M such that the subgraph induced by vertex set V ′ +0 is a maximal noncentral Abelian +subgraph under the representation p ◦ ρ, where V ′ +0 is the preimage of V0. +Proof. If Γ0 is an induced subgraph, then it is a trivial case and the conclusion holds. Suppose +Γ0(V0, E0) is not an induced subgraph, and it has m fewer edges than a vertex-induced subgraph +ˆΓ0(V0, ˆE0) induced by V0. We denote m edges by ei and the corresponding torus by Ti. For each +Ti, there is a gluing matrix gi. Let M0 denote the corresponding submanifold of Γ0 and ˆ +M0 denote +the corresponding submanifold of ˆΓ0. If π1(M0) has a presentation ⟨S|R⟩. Then π1( ˆ +M0) is an HNN +extension of π1(M0). It has a presentation +⟨S, ti, 1 ≤ i ≤ m | R, tifit−1 +i += gi(fi), tizit−1 +i += gi(zi), 1 ≤ i ≤ m⟩, +where fi, zi are the generators of π1(Ti). +Consider the projection pr : � +SL(2, R)×Z R → PSL(2, R). Let α be an element of π1(M0). Notice +|Tr(pr(ρ(tiαt−1 +i +)))| = |Tr(pr(ρ(α)))|. +Hence, we have two possible cases, +pr(ρ(tiαt−1 +i )) = pr(ρ(α)) +(⋆) +or +pr(ρ(tiαt−1 +i )) = pr(ρ(α))−1. +(⋆⋆) +Since ρ(π1(M0)) is an Abelian group, by proposition 3.1, any element commuting with ρ(α) will +16 + +commute with all elements in ρ(π1(M0)). Hence, for each ti, it either satisfies the first relation (⋆) +for all elements in π1(M0) or satisfies the second relation (⋆⋆) for all elements in π1(M0). +Suppose there are n edges {ej, 1 ≤ j ≤ n} corresponding to {tj, 1 ≤ j ≤ n} where every tj +satisfies the second relation (⋆⋆). Notice each ρ(t2 +j) commutes with every element in ρ(π1(M0)). +Then we can construct a double cover satisfying the conclusion. +We state a construction of the cover on the level of graph, hence it is a JSJ characteristic finite +cover. Cut ej in Γ to get two “half edges” e+ +j , e− +j for each j and take a copy of the broken graph. +Then we glue e+ +j to the half edge ¯e− +j and glue e− +j to the half edge ¯e+ +j for each j, where ¯e+ +j , ¯e− +j are on +the copy of the broken graph. By this construction, we get a connected double cover Γ′ of the graph +Γ. The graph manifold M ′ dual to Γ′ is hence the double cover of M. We denote this covering +by p : M ′ → M. Then the subgraph Γ′ +0 induced by the vertex set V ′ +0 is a double cover of ˆΓ0. +The corresponding submanifold of Γ′ +0 is a noncentral Abelian component under the representation +p ◦ ρ. According to our construction, the fiber of Seifert blocks adjacent to Γ′ +0 are all central types. +Hence, the induced subgraph Γ′ +0 is a maximal noncentral Abelian subgraph. +Let a Seifert block Mv has a fiber with projectively infinite order. We set Mv contained in +a maximal noncentral Abelian components M0 and the corresponding subgraph Γ0 is an induced +subgraph. Let W denote the set of all adjacent Seifert blocks of Mv. Then we use WA to denote the +subset of W which contains adjacent Seifert blocks of Mv lying in the maximal noncentral Abelian +components M0 and use WC to denote the rest of elements in W which are not in M0. Then the +equation (4) can be written in other form, i.e. +bvkvfv − +� +w∈WA +bv +bw,v +fw = +� +w′∈WC +bv +bw′,v +fw′. +(5) +The fiber fw′ commute with fv. Moreover, the right-hand of equation (5) has all fibers with +images in the center of � +SL(2, R) ×Z R under the representation ρ by proposition 5.7. +If a maximal noncentral Abelian component has an induced corresponding subgraph under the +representation ρ, then we give a following definition. +Definition 5.9. Let M0 be a maximal noncentral Abelian component with an induced corresponding +subgraph. If M0 consists of n Seifert blocks, then for each Seifert block M0,i of M0, there is an +equation (5) associated to M0,i. Then we have n equations for the maximal Abelian component M0. +17 + +Denote the equations in form of matrices +e0fA = fC. +We call e0 the associated matrix to the maximal Abelian component M0 under the representation +ρ. +Let M0 be a maximal noncentral Abelian component with an induced corresponding subgraph. +If M0 consists of n Seifert blocks with fiber denoted by fi, 1 ≤ i ≤ n, we can write down the +associated matrix to the maximal noncentral Abelian component M0. For the i-th Seifert block in +M0, let W i +C denote the adjacent Seifert blocks not in M0 (W i +C may be empty). Then the matrix +equation is as the following: + + + + + + + + + + + + + + + + + + + + +b1k1 +β12 +· · · +β1,n +β21 +b2k2 +· · · +β2,n +... +... +... +... +βn,1 +βn,2 +· · · +bnkn + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +f1 +f2 +... +fn + + + + + + + + + + + + + + + + + + + + += + + + + + + + + + + + + + + + + + + + + + +� +w′∈W 1 +C +b1 +bw′,1 fw′ +� +w′∈W 2 +C +b2 +bw′,2 fw′ +... +� +w′∈W n +C +bn +bw′,n fw′ + + + + + + + + + + + + + + + + + + + + + +, +(6) +where βij(i ̸= j) is − bi +bj,i if the i-th Seifert block and the j-th Seifert block are adjacent. Otherwise, +βij is 0. +Since the image of π1(M0) is an Abelian group, we write the group multiplication in the form +of the addition. We can find that the entries of e0 are all integers and the diagonal elements of e0 +are in the form of biki. Moreover, all elements of the right-hand column are of central type. +18 + +6 +Virtually nonexistence of faithful representations in closed +case +In this section we prove the Theorem 1.1. In fact, we need to prove that a strictly diagonally +dominant closed graph manifold virtually has no vertex faithful representations to the Seifert motion +group. +Proof of Theorem 1.1. By lemma 4.1, there is a finite cover M1 of the graph manifold M such that +each Seifert block homeomorphic to a trivial circle bundle over a compact surface with genus greater +than 1. Then we can find a JSJ characteristic finite cover M ′ of M factors through M1 and every +Seifert block of M ′ is also a trivial circle bundle over a compact surface with genus greater than 1. +Suppose there exists a vertex faithful representation ρ′ : π1(M ′) → � +SL(2, R)×ZR which is faithful +at vertex v′. By corollary 5.3, the adjacent Seifert blocks of M ′ +v′ all have fibers of noncentral type +with images of projectively infinite order. Hence, each adjacent block has an Abelian representation +under ρ′. +Let M ′ +0 be a maximal noncentral Abelian component of M ′ which contains at least one adjacent +block of M ′ +v′. Then by proposition 5.8, there is JSJ characteristic double cover M ′′ such that the +preimage M ′′ +0 is a maximal noncentral Abelian component whose corresponding subgraph is an +induced subgraph. Let ρ′′ : π1(M ′′) → � +SL(2, R) ×Z R be the representation which is faithful at +vertex v′′ who is one component of the preimage of v′. +The graph manifold M ′′ is a JSJ characteristic double cover of M ′, and hence it is a JSJ +characteristic finite cover of M. Since M is strictly diagonally dominant, by proposition 4.3, M ′′ is +also strictly diagonally dominant. According to the definition 2.4 of a strictly diagonally dominant +graph manifold, each Seifert block of M ′′ is strictly diagonally dominant. Hence, M ′′ +0 also satisfies +the condition of strictly diagonally dominance. +Let M ′′ +u′′ be a Seifert block in M ′′ +0 . Then ρ′′ restricted to π1(M ′′ +u′′) is Abelian. We can write an +equation (5) for M ′′ +u′′, +bu′′ku′′fu′′ − +� +w′′ +A∈W ′′ +A +bu′′ +bw′′ +A,u′′ fw′′ +A = +� +w′′ +C∈W ′′ +C +bu′′ +bw′′ +C,u′′ fw′′ +C, +Here W ′′ +A denotes the set of adjacent blocks of M ′′ +u′′ which lie in M ′′ +0 and W ′′ +C denotes the set of +19 + +other adjacent blocks of M ′′ +u′′ which are not in M ′′ +0 . +Since M ′′ +u′′ is strictly diagonally dominant, we have an inequality +|bu′′ku′′| > +� +w′′ +A∈W ′′ +A +|bu′′| +|bw′′ +A,u′′| + +� +w′′ +C∈W ′′ +C +|bu′′| +|bw′′ +C,u′′| ≥ +� +w′′ +A∈W ′′ +A +|bu′′| +|bw′′ +A,u′′|. +Notice that the left-hand is the absolute value of a diagonal element bu′′ku′′ in the associated matrix +e′′ +0 to the submanifold M ′′ +0 . The right-hand is the sum of absolute value of other entries which lie in +the same row with bu′′ku′′. For each row of the associated matrix e′′ +0 , the corresponding inequality +holds. Then the associated matrix e′′ +0 to M ′′ +0 is strictly diagonally dominant. Hence, it is invertible. +Consider the equation +e′′ +0 f ′′ +A = f ′′ +C. +We have +(e′′ +0 )∗e′′ +0 f ′′ +A = (e′′ +0 )∗f ′′ +C, +det(e′′ +0 )f ′′ +A = (e′′ +0 )∗f ′′ +C, +where (e′′ +0 )∗ is the adjoint matrix of e′′ +0 and det(e′′ +0 ) is the determinant of the matrix e′′ +0 . +The entries of the matrix e′′ +0 are all integers, so are the entries of the adjoint matrix (e′′ +0 )∗. +The images of fibers are all in an Abelian group, the equation has solutions. By remark 3.3, the +elements of any solution all have projectively finite order. This conclusion contradicts the result +that the fibers of adjacent blocks of M ′′ +v′′ are noncentral elements with projectively infinite order. +So, the vertex faithful representation ρ′′ does not exist. +As for a general finite cover M2, there is a JSJ characteristic finite cover M3 which factors +through M2. Then M3 is strictly diagonally dominant. There is no vertex faithful representations +on M3. Hence, there is no vertex faithful representations on M2. So, a closed strictly diagonally +dominant graph manifold M virtually has no vertex faithful representations. +As a deduced result, a closed strictly diagonally dominant graph manifold M virtually has no +faithful representations to the Seifert motion group. +In another point of view, the proof of Theorem 1.1 is extending a representation from a Seifert +block to the whole graph manifold. So, we can state in a form of a corollary. +Corollary 6.1. Let M be a strictly diagonally dominant graph manifold. There is a representation +20 + +ρ on a Seifert block Mv is faithful to the Seifert motion group. Then the representation ρ virtually +cannot be extended to the whole graph manifold M. +7 +Existence of a vertex faithful representation in bounded +case +If a graph manifold has torus boundaries, then there is a vertex faithful representation on it. In +other word, the faithful representation of a Seifert block can be extended to a representation on the +whole graph manifold. The representation is not necessarily faithful on the whole graph manifold. +In the following we give a construction for one class of graph manifolds with nonempty boundary. +Theorem 7.1. Let M be a graph manifold whose dual graph Γ is a tree and Each Seifert block +dual to end vertex of the tree has at least one torus boundary. If one Seifert block has a faithful +representation to the Seifert motion group, then the representation can be extended to the whole +graph manifold M. +Proof. Suppose there is a faithful representation ρ0 on a Seifert block Mv of the graph manifold +M. Then there is a direction on Γ starting from v to all the ends. +Let the representation denoted by ρ : π1(M) → � +SL(2, R)×ZR. Let W denote all adjacent blocks +of Mv. The images of a homological basis {fw,v, zw,v} of each torus Tw,v are determined by a gluing +matrix Av,w. Then we set the image of π1(Mw) to be Abelian. So the base surface boundaries of +Mw have all images in a group C(ρ(fw)) = {h ∈ � +SL(2, R) ×Z R | ρ(fw)h = hρ(fw)}. We choose +elements from C(ρ(fw)) such that they form a Waldhausen basis for Mw. Hence, each adjacent +block of Mv has an Abelian representation. +Suppose the representation ρ has been extended to all Seifert blocks with distance less than n +to Mv. When n = 2, we have done it in the last paragraph. If a Seifert block Mu with distance n +to Mv, it is adjacent to a unique Seifert block Mt with distance n − 1 to Mv. The basis {fu,t, zu,t} +of the torus Tu,t is determined by the gluing matrix gt,u from Tt,u to Tu,t, + +fu,t +zu,t + + = gt,u + +ft,u +zt,u + + . +Hence, the image of the fiber fu is determined by the inclusion map. +21 + +Since the graph is a tree with all end points corresponding to Seifert manifolds meeting the +boundary of M. Mu, has at least one torus boundary is free. Then we can choose elements in +the group C(ρ(fu,t)) ∩ C(ρ(zu,t)) to get a Waldhausen basis of Mu. Hence, the representation ρ +restricted on π1(Mu) is Abelian. +By induction, we can extend the representation ρ0 to ρ on the whole graph manifold such that +ρ restricted on π1(Mv) is faithful. +References +[AFW15] Matthias Aschenbrenner, Stefan Friedl, and Henry Wilton. +3-manifold groups. +EMS +Series of Lectures in Mathematics. European Mathematical Society (EMS), Z¨urich, 2015. +[BG84] +Robert Brooks and William Goldman. +Volumes in Seifert space. +Duke Math. J., +51(3):529–545, 1984. +[BS04] +S. V. Buyalo and P. V. Svetlov. Topological and geometric properties of graph manifolds. +Algebra i Analiz, 16(2):3–68, 2004. +[But14] +Jack O. Button. A 3-manifold group which is not four dimensional linear. J. Pure Appl. +Algebra, 218(9):1604–1619, 2014. +[DLW20] Pierre Derbez, Yi Liu, and Shicheng Wang. Volume of Seifert representations for graph +manifolds and their finite covers. arXiv e-prints, page arXiv:2006.14770, June 2020. +[EHN81] David Eisenbud, Ulrich Hirsch, and Walter Neumann. Transverse foliations of Seifert +bundles and self-homeomorphism of the circle. Comment. Math. Helv., 56(4):638–660, +1981. +[Har69] +Frank Harary. Graph theory. +Addison-Wesley Publishing Co., Reading, Mass.-Menlo +Park, Calif.-London, 1969. +[HW08] +Fr´ed´eric Haglund and Daniel T. Wise. Special cube complexes. Geom. Funct. Anal., +17(5):1551–1620, 2008. +[HW12] +Fr´ed´eric Haglund and Daniel T. Wise. A combination theorem for special cube complexes. +Ann. of Math. (2), 176(3):1427–1482, 2012. +[Joh79] +Klaus Johannson. Homotopy equivalences of 3-manifolds with boundaries, volume 761 of +Lecture Notes in Mathematics. Springer, Berlin, 1979. +[JS79] +William Jaco and Peter B. Shalen. Seifert fibered spaces in 3-manifolds. In Geometric +topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), pages 91–99. Academic +Press, New York-London, 1979. +22 + +[Kir97] +Rob Kirby. Problems in low-dimensional topology. In Geometric topology (Athens, GA, +1993), volume 2 of AMS/IP Stud. Adv. Math., pages 35–473. Amer. Math. Soc., Provi- +dence, RI, 1997. +[KL96] +Michael Kapovich and Bernhard Leeb. Actions of discrete groups on nonpositively curved +spaces. Math. Ann., 306(2):341–352, 1996. +[KL98] +Michael Kapovich and Bernhard Leeb. 3-manifold groups and nonpositive curvature. +Geom. Funct. Anal., 8(5):841–852, 1998. +[Sco83] +Peter Scott. The geometries of 3-manifolds. Bull. London Math. Soc., 15(5):401–487, +1983. +[Thu97] +William P. Thurston. Three-dimensional geometry and topology. Vol. 1, volume 35 of +Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1997. Edited +by Silvio Levy. +23 + diff --git a/a9A0T4oBgHgl3EQfGP_A/content/tmp_files/load_file.txt b/a9A0T4oBgHgl3EQfGP_A/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ce9c66b473eef74372244ab2e863841da31e774c --- /dev/null +++ b/a9A0T4oBgHgl3EQfGP_A/content/tmp_files/load_file.txt @@ -0,0 +1,628 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf,len=627 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='02045v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='GT] 5 Jan 2023 Virtual representations for closed graph manifolds and Seifert geometry YAO FAN Abstract In this paper, we mainly discuss the representations of closed graph manifolds to the Seifert motion group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then we prove that there exist graph manifolds virtually having no faithful representations to the Seifert motion group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' 1 Introduction In Kirby’s list of problems in low-dimensional topology [Kir97], Thurston proposed a question if every finitely generated 3-manifold group G has a faithful representation in GL(4, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' However, Button gave a counterexample [But14] which is a graph manifold made by gluing two Seifert man- ifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The fundamental group of the graph manifold cannot be embedded in GL(n, K) for n ≤ 4 and for any field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then we can ask if every finitely generated 3-manifold group G has a faithful linear representa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' According to the book [AFW15] written by Aschenbrenner, Friedl and Wilton, which gives a detailed survey of 3-manifold groups, the isometry group of the following geometric 3-manifold are subgroups of GL(4, R): spherical geometry, S2 × R, Euclidean geometry, Nil, Sol and hyperbolic geometry .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Moreover, the fundamental group of an H2 × R-manifold is a subgroup of GL(5, R) [AFW15, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Hence, these seven geometric manifolds have faithful linear representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' How- ever, the isometry group of � SL(2, R) is not a linear group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' With the fact that the fundamental group of an S1 bundle over a surface is linear over Z [AFW15, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='43], Seifert geometry, whose model space is � SL(2, R), virtually has a faithful linear representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' However, we cannot fix the size of the linear Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' As for graph manifolds, which consists of Seifert manifolds, the bounded case has better prop- erties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Result of Haglund and Wise [HW08][HW12] shows that if a 3-manifold admits a nonpositive 1 curvature, then its fundamental group can be embedded in a linear Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Kapovich and Leeb proved that there exists a nonpositive curvature on a graph manifold with torus boundaries [KL96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Hence, a graph manifold with torus boundaries has a faithful linear representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' How- ever, Kapovich and Leeb also gave an example of a closed graph manifold admitting no nonpositive curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' So, for closed graph manifolds, we do not know if the fundamental groups have faithful linear representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Brooks and Goldman defined the Seifert volume by using a representation to the isometry group of � SL(2, R) for 3-manifolds [BG84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Therefore, we consider the representations of closed graph manifold to the Seifert motion group, which is the identity component of the isometry group of � SL(2, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' We prove a theorem as the following: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' There exists a closed graph manifold virtually having no faithful representations to Seifert motion group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' In fact, there exists a closed graph manifold M such that for any finite cover M ′ of M and any representation ρ : π1(M ′) → � SL(2, R) ×Z R, ρ is not faithful restricted to any Seifert block of M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Here � SL(2, R) ×Z R denotes the Seifert motion group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' In addition, the theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='1 holds for closed graph manifolds satisfying the condition of strictly diagonally dominance (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' This condition is inspired from Derbez, Liu and Wang’s paper [DLW20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If the graph manifold has nonempty torus boundaries, it can have a representation restricted on one submanifold is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' We give an example of the graph manifold and construct the representation in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' To get the theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='1 we prove a stronger version that there exist no virtually vertex faithful representations (Definition 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' We suppose there is virtually a vertex faithful representation on a closed graph manifold satisfying the condition of strictly diagonally dominance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then the image of a submanifold under the representation is an Abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' We list equations about the images of fibers which are deduced form the topological structure of the graph manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The solution of the equations contradicts with the assumption of the vertex faithful representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' So, we deduce the result that this graph manifold virtually has no faithful representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The paper is organized as the following: In section 2 we recall the relative definition of graph manifolds and introduce two important topological invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' In section 3 we introduce the Seifert motion group and classify its elements in order to discuss representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then in section 4 we introduce JSJ characteristic finite covers 2 and give a property about topological invariants under this special finite cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' In section 5 we mainly discuss the representations to the Seifert motion group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' In section 6 we prove the Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='1 about the nonexistence of representation in the closed condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' In addition, we construct a vertex faithful representation when a graph manifold has torus boundaries in section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' I would like to give my thanks to my supervisor Yi Liu for his guidance and conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' 2 Preliminary Let M be an orientable compact irreducible 3-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' There is a collection of disjoint incom- pressible tori splits M into components which are either atoroidal or Seifert fibered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Moreover, the minimal such collection is unique up to isotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' This decomposition of 3-manifold is called Jaco-Shalen-Johannson torus decomposition (JSJ decomposition) [JS79][Joh79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' In the following we review the definitions, related properties and some topological invariants of Seifert manifolds and graph manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Definitions and notations mostly follows the paper of Buyalo and Svetlov [BS04].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='1 Seifert manifolds A Seifert manifold is a 3-manifold foliated by circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' A useful definition according to Scott [Sco83] is to define the Seifert manifold as a “circle bundle” over an orbifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let M be a Seifert manifold and O be its base orbifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' There is a projection p : M → O and an exact sequence of fundamental groups {1} → K → π1(M) → π1(O) → {1}, where K is a cyclic subgroup of π1(M) generated by a regular fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The group K is infinite cyclic except the case where M is covered by S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let the orbifold O has an underlying surface F and m singular points si with degree qi, 1 ≤ i ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then the Euler characteristic number of an orbifold is defined to be χ(O) = χ(F) − m � i=1 � 1 − 1 qi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' 3 If χ(O) < 0, then there is a finite cover of M having a hyperbolic structure on its base surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The fundamental group of the base surface can be embedded into PSL(2, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' In the following we only consider the Seifert manifold having a base orbifold with a negative Euler characteristic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Hence, there is a finite cover of the Seifert manifold is a trivial circle bundle over a surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' In the rest of this paper, the Seifert manifold is a trivial circle bundle without special mentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Waldhausen basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let M be an oriented Seifert manifold with nonempty boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let W be the set of boundary components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Consider an oriented fiber f of M and there is a homological class fw ∈ H1(Tw;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Q) corresponding to fiber f in each boundary torus Tw, w ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then there is an induced orientation from M to Tw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Hence, the intersection form (·, ·) : H1(Tw;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Q)×H1(Tw;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Q) → Q can be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' For each fw there is a zw ∈ H1(Tw;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Q) satisfying (zw, fw) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Here we modify the notation of the intersection form which is different from the notation of Buyalo and Svetlov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' For each torus boundary Tw of a Seifert manifold M there is a zw such that (zw, fw) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then a collection of elements {zw, fw}w∈W is called a Waldhausen basis of the Mv if they satisfy the condition that z = � w∈W zw lies in the kernel of the inclusion i∗ : H1(∂M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Q) → H1(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The Waldhausen basis is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' It is defined up to a transformation zw �→ zw + nwfw where � w∈W nw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If M is a trivial circle bundle, then we can choose zw from H1(Tw;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' This choice of Waldhausen basis is equivalent to the choice of a trivialization of M = F × S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Framed Seifert manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let M be a Seifert manifold and each boundary Tw has a fixed element cw ∈ H1(Tw;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Z) with (cw, fw) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then M is called framed and the collection {cw}w∈W is a framing of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' For a framed Seifert manifold, there is a formula about its fiber and the framing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' We state the formula as a lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let {cw}w∈W be a framing of a Seifert manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then � w∈W 1 (cw, fw) · (iw)∗cw = k · f, (1) where (iw)∗ : H1(Tw;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Z) → H1(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Z) is the inclusion homomorphism induced by iw : Tw → Mv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' 4 The homological class f ∈ H1(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Z) represents the regular fiber, and (iw)∗fw = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The number k on the right hand is a rational number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' We choose a Waldhausen basis {zw, fw}w∈W for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' cw can be represented as cw = awfw + bwzw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Since bw = (cw, fw) ̸= 0, � w∈W 1 (cw, fw) · (iw)∗cw = � w∈W 1 bw (iw)∗(awfw + bwzw) = � � w∈W aw bw � f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Here the coefficient k = � w∈W aw bw on the right hand is independent of the Waldhausen basis since the left hand is independent of the choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='2 Graph manifolds A graph manifold M is an irreducible 3-manifold with all components Seifert manifolds under the JSJ decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The graph manifold M is associated with a directed graph Γ(V, E) dual to the JSJ decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' We call each component a Seifert block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The vertex set V of Γ is the set of Seifert blocks and the set E of oriented edges of Γ is identified to boundaries of all blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If e ∈ E is an edge directed form v1 to v2, then Tv1,v2 ⊂ ∂Mv1 and Tv2,v1 ⊂ ∂Mv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Hence, e is identified to an ordered pair (v1, v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If a graph Γ is connected, for two points v1, v2, Let P denotes the set of all paths L connecting v1, v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' We define the length #L to be the number of edges in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The metric on Γ is defined as d(v1, v2) = min L∈P #L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then we can define the distance between two Seifert blocks by their corresponding vertices as d(Mv1, Mv2) = d(v1, v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' We fix an orientation of a graph manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then a torus boundary of a Seifert block has an induced orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let Mv1, Mv2 be adjacent Seifert blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' We denote the torus by Tv1,v2 with the induced orientation from Mv1 dual to the directed edge e = (v1, v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then the orientation of Tv2,v1 is opposite to the orientation of Tv1,v2 A gluing map is a homeomorphism associated with a directed edge e = (v1, v2) maps Tv1,v2 to 5 Tv2,v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If we choose {fv1, zv1} as a basis of π1(Tv1,v2) and {fv2, zv2} as a basis of π1(Tv2,v1), then the gluing map induces an isomorphism on the fundamental groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' It can be represented as a 2×2 matrix gv1,v2 = \uf8eb \uf8eda b c d \uf8f6 \uf8f8 , ad − bc = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' So we call the isomorphism gv1,v2 a gluing matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' In the following we introduce two topology invariants of graph manifolds referred to Buyalo and Svetlov [BS04].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' We make a few modifications by using the gluing matrix in definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The intersection index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let Mv1 and Mv2 be two adjacent Seifert blocks in an oriented graph manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' We use fv1,v2 to denote the homological class of the regular fiber of Mv1 in H1(Tv1,v2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Z) and fv2,v1 to denote the fiber of Mv2 in H1(Tv2,v1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' By the gluing matrix gv1,v2, the fiber fv1,v2 maps to an element in H1(Tv2,v1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Define bv2,v1 = (fv2,v1, gv1,v2(fv1,v2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' By definition of a graph manifold and its induced orientation, the integer number bv2,v1 satisfies bv2,v1 = bv1,v2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' This integer bv2,v1 is called the intersection index of fibers fv2,v1 and fv1,v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' It changes the sign when the orientation of one fiber fv2,v1 or fv1,v2 changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' For a Seifert block Mv, it has a natural framing {cw}w∈W by setting cw = gw,v(fw,v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then the coefficient of f on the right hand in the equation 1 is called the charge of Mv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Under a chosen Waldhausen basis {fv,w, zv,w}w∈W of Mv, cw = gw,v(fw,v) = aw,vfv,w + bw,vzv,w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then the charge can be written as kv = � w∈W aw,v bw,v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' According to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='2, the charge is independent of the choice of a Waldhausen basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' But aw,v bw,v which is called the slope for the homological class fw,v is related to the Waldhausen basis {fv,w, zv,w}w∈W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If a Seifert block has a boundary T which is also the boundary of the whole graph manifold, then this boundary T has no contribution to the charge of the Seifert block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' We define its slope always be zero in the calculation of the charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' 6 Strictly diagonally dominant graph manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' In the paper of Derbez, Liu and Wang [DLW20], they introduce the terminology, strictly diag- onally dominance, to describe a property of graph manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' This terminology is original from the theory of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If a n × n matrix A = (aij) satisfies |aii| > � j̸=i |aij| for all i, then the matrix A is said to be strictly diagonally dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If A is a strictly diagonally dominant matrix, then A is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Suppose A is not invertible, then there is a nonzero vector x such that Ax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let xi be the element with maximum absolute value in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then there is an equation a1x1+· · ·+aixi+· · ·+anxn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' However, ������ � j̸=i aijxj ������ ≤ � j̸=i |aijxj| ≤ \uf8eb \uf8ed� j̸=i |aij| \uf8f6 \uf8f8 |xi| < |aiixi|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' This deduces a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Hence, A is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' In the aspect of the graph manifolds, the definition is as the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' For a Seifert block Mv, if the charge kv and all intersection numbers bv,w satisfying |kv| > � w∈W 1 |bv,w|, (2) in which W is the set of all adjacent blocks of Mv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then this Seifert block is called strictly diagonally dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If a graph manifold has every Seifert block strictly diagonally dominant, then it is called a strictly diagonally dominant graph manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Strictly diagonally dominance is a relation about the charges and intersection indices of a graph manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' In the following section 4 we will prove this relation still holds under a special finite cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' 3 Seifert Motion Group Seifert geometry, or � SL(2, R) geometry, is one of the eight 3-dimensional geometries as classified by Thurston [Thu97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' We take Lie group � SL(2, R) as the model of the Seifert geometry’s space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The 7 identity component of its isometry group can be identified with � SL(2, R) ×Z R, which is called as Seifert motion group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Eisenbud, Hirsch and Neumann gave criteria for a Seifert manifold to admit a transverse foliation [EHN81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' In their paper they studied the homomorphism π1(M) → � SL(2, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The name of Seifert geometry is from Brooks and Goldman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' They defined the Seifert volume by using a representation to the Seifert motion group for 3-manifolds [BG84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then Derbez, Liu and Wang gave a formula to compute the representation volume for a graph manifold with a representation to the Seifert motion group [DLW20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' So, in this section we state the construction of the Seifert motion group and classify elements in this group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='1 Construction of Seifert motion group In this part we use the notation from Derbez, Liu and Wang [DLW20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Here � SL(2, R) is the universal cover of PSL(2, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' There is a map k : R → PSL(2, R), k(r) = \uf8eb \uf8edcos πr − sin πr sin πr cos πr \uf8f6 \uf8f8 , r ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then there exists a lift ˜k : R → � SL(2, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The center Z(� SL(2, R)) of � SL(2, R) is the group {˜k(n), n ∈ Z}, which projects to the identity in PSL(2, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The Seifert motion group is the quotient of a product group � SL(2, R) × R by a subgroup {(˜k(n), −n), n ∈ Z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then the Seifert motion group is denoted by � SL(2, R) ×Z R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Its elements can be denoted as g[s] for g ∈ � SL(2, R) and s ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The multiplication rule is g[s]g′[s′] = gg′[s + s′], and there is a relation g[s + n] = g˜k(n)[s], n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='2 Classifying elements in Seifert motion group Elements in SL(2,R) can be classified by their traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Take a matrix A ∈ SL(2,R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If the absolute value of the trace |Tr(A)| < 2, the matrix A is said to be elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If |Tr(A)| > 2, it is said to be 8 hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If |Tr(A)| = 2 and A is not the identity matrix or minus identity matrix, it is said to be parabolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Since PSL(2, R) = SL(2, R)/ ± I2, its elements can be classified by their representative elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' This way can be checked that is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then we consider a natural projection pr : � SL(2, R) ×Z R → PSL(2, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The elements in the Seifert motion group can be classified by their images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let g ∈ � SL(2, R) ×Z R, if pr(g) is the identity in PSL(2, R), we call it central type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' if pr(g) is an elliptic element in PSL(2, R), we call it elliptic type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' if pr(g) is a hyperbolic element in PSL(2, R), we call it hyperbolic type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' if pr(g) is a parabolic element in PSL(2, R), we call it parabolic type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then there is a proposition about the subgroup C(g) = {h ∈ � SL(2, R) ×Z R|gh = hg}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let a, b be two elements in the Seifert motion group � SL(2, R) ×Z R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then ab = ba if and only if pr(a)pr(b) = pr(b)pr(a) in PSL(2, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Hence, for a noncentral element g ∈ � SL(2, R) ×Z R, the subgroup C(g) is an Abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' It is direct that ab = ba deduces pr(a)pr(b) = pr(b)pr(a) in PSL(2, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Conversely, if pr(a)pr(b) = pr(b)pr(a), by the construction of the Seifert motion group, aba−1b−1 lies in the center of � SL(2, R) ×Z R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' For a, b ∈ � SL(2, R) ×Z R, we have continuous paths starting from the identity element to a,b respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Notice (� SL(2, R) ×Z R)×(� SL(2, R) ×Z R) → � SL(2, R) ×Z R, (a, b) �→ aba−1b−1 is a continuous map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' However, the group {˜k(n), n ∈ Z}×ZR, the center of the Seifert motion group, is discrete in the first component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' So, the first component of aba−1b−1 is the identity element in � SL(2, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Notice that the second component of aba−1b−1 is always zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Hence, aba−1b−1 equal to the identity element in � SL(2, R) ×Z R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If two nontrivial elements in PSL(2, R) are commutable, they can be diagonalized at the same time and lie in the same type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Hence, for a noncentral element g ∈ � SL(2, R) ×Z R, C(pr(g)) = 9 {pr(h) ∈ PSL(2, R)|pr(g)pr(h) = pr(h)pr(g)} is an Abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' So, C(g) is an Abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' For elements in the Seifert motion group we have another classification with respect to the order of their projective images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If an element in Seifert motion group is projected to an element with finite order in PSL(2, R), then we call it an element with projectively finite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If an element in Seifert motion group is projected to an element with infinite order in PSL(2, R), then we call it an element with projectively infinite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let f = ˜k(n) be a central element in � SL(2, R) ×Z R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Consider an equation xm = f, m ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If n = lm + r, l, r ∈ Z, then ˜k(l)[ r m] is a solution of the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' However, ˜k(l)[ r m] is not a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' We project the equation to PSL(2, R) then (pr(x))m = pr(f) = I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' So, any solution of the equation xm = f is an element with projectively finite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' 4 JSJ characteristic Finite Cover In this section we discuss the finite covers of graph manifolds and the relation of two topological invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If some finite cover of M has a property P, then M is said to have a property P virtually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Sometimes we only consider the existence of representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Therefore, we often pass to a finite cover to prove theorems or construct representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' According to Kapovich and Leeb’s result [KL98, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='1], a graph manifold has a good structure passing to a finite cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' We state the lemma as the following: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='1 (Kapovich, Leeb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Any graph manifold has an orientable finite cover where all Seifert blocks are trivial circle bundle over a surface with genus greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Furthermore, we can arrange the intersection index of the fibers of adjacent Seifert blocks are ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' By the above lemma, passing to a finte cover each Seifert block Mv can be homeomorphic to Fv × S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' In the following we assume the graph manifold satisfies the conditions in lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Moreover, we consider the dual graph Γ is a nontrivial simple graph, which is a graph contains no multiple edges and self-loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Hence, every JSJ torus separates two distinct Seifert blocks from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' This condition can also be satisfied by passing to a finite cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' 10 Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Set p : M ′ → M be a regular finite cover of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let the set T be a union of JSJ tori of M, the set T ′ be the preimage of T and n be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If for each torus T of T and each component T ′ of T ′ over T , the restriction p| : T ′ → T is a characteristic cover with index n × n, then this finite cover p : M ′ → M is called a JSJ n-characteristic finite cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' A regular finite cover p1 : M ′ → M is not necessarily a JSJ characteristic finite cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' However, we can always find a JSJ characteristic finite cover p2 : M ′′ → M such that p2 factors through p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Hence, we can prove some properties on JSJ characteristic finite cover and then deduce to a general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' For a JSJ characteristic finite cover, there is a property about topological invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let M be a strictly diagonally dominant graph manifold, then any JSJ charac- teristic finite cover of M is also strictly diagonally dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Suppose p : M ′ → M is a JSJ characteristic finite cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Take a Seifert block M ′ v′ which is a component of a Seifert block Mv’s preimage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Since this finite cover is JSJ characteristic, the slope of each torus boundary is invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then the covering degree of the fiber is fixed for each Seifert block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Hence, the gluing matrices are also invariant under the covering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let W be the set of all adjacent block of Mv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If an edge e with ∂e = (v, w), w ∈ W has a preimage p−1(e) connecting v′, we set ∂(p−1(e)) = {(v′, w′)|w′ ∈ W ′ v′} in which W ′ v′ is the set of all Seifert blocks connecting v′ by one edge of p−1(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' For each torus T ′ v′,w′ dual to (v′, w′), the intersection number bv′,w′ = bv,w since the gluing maps have the same matrix representation for T ′ v′,w′ and Tv,w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Hence, the sum of slope of T ′ v′,w′, w′ ∈ W ′ v′ is just |W ′ v′|, the size of W ′ v′, times the slope of Tv,w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Notice the size of W ′ v′ is the same for all edges connecting v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Since the degree of fiber is fixed, the degree of JSJ characteristic cover for each torus is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Hence, the components of preimage p−1(e) for each e is also the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Because Mv is strictly diagonally dominant, we have |kv| > � w∈W 1 |bv,w|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' 11 As for M ′ v′, the charge kv′ = � w∈W � w′∈W ′ v′ av′,w′ bv′,w′ = � w∈W � |W ′ v′| × av′,w′ bv′,w′ � = |W ′ v′| × � w∈W av′,w′ bv′,w′ = |W ′ v′| × � w∈W av,w bv,w = |W ′ v′| × kv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then |kv′| = |W ′ v′| × |kv| > |W ′ v′| × � w∈W 1 |bv,w| = � w∈W � |W ′ v′| × 1 |bv,w| � = � w∈W � |W ′ v′| × 1 |bv′,w′| � = � w∈W � w′∈W ′ v′ 1 |bv′,w′| So the Seifert block M ′ v′ is also strictly diagonally dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Since the choice of vertex v is arbitrary, each Seifert block of M ′ is strictly diagonally dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Hence, M ′ is a strictly diagonally dominant graph manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' 5 Representations to Seifert Motion Group If a representation of a graph manifold is faithful, then it restricted on each Seifert block is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' In convenience, we define a “local” faithful representation as the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If a representation restricted on a Seifert block dual to a vertex v is faithful, then we call the representation a vertex faithful representation, and more precisely, faithful at v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Sometimes we call a representation vertex faithful without indicating a vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' In the following we discuss the representations restricted on each Seifert block and classify them by the images in the Seifert motion group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Particularly, we list equations for Seifert blocks with Abelian images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' 12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='1 Faithful representations Consider a Seifert block Mv which is a trivial circle bundle over a compact surface with genus g ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Suppose there is a faithful representation on Mv to the Seifert motion group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' We have a conclusion as the following: Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let a Seifert block Mv be a trivial circle bundle over a compact surface Fv with genus g ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If Mv has a faithful representation ρ : π1(Mv) → � SL(2, R) ×Z R, then the image of the fiber of Mv is a nontrivial element of central type and the images of boundaries of the base surface Fv are all noncentral elements with projectively infinite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Since Mv is a trivial circle bundle over a compact surface Fv with genus g ≥ 2, the funda- mental group of Mv has a presentation as π1(Mv) = � a1, b1, · · · , ag, bg, c1, · · · , cl ��� g � i=1 [ai, bi] = l� j=1 cj � × ⟨f⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Here [ai, bi] denotes aibia−1 i b−1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The integer g is the genus of the base surface and l is the number of boundary components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The image of the fiber of Mv projects to identity under the projection pr : � SL(2, R) ×Z R → PSL(2, R), otherwise the representation ρ is Abelian which contradicts the assumption that ρ is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Hence, the image of the fiber ρ(f) is a nontrivial central element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' As for the images of boundaries {cj} of the base surface Fv, they cannot lie in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If one of them, ρ(cj), has projectively finite order, then there exists an integer n such that ρ(cj)n lies in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Since ρ is faithful, (cj)n commute with every generator in π1(Mv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' This result contradicts the presentation of π1(Mv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Hence, the images of boundaries of base surface Fv are all noncentral elements with projectively infinite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If a representation ρ of a graph manifold M restricted on a Seifert block Mv is faithful, then the adjacent blocks of Mv all have noncentral fibers with projectively infinite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' By proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='2, the image of the fiber of Mv is a nontrivial element of central type and the images of boundaries of the base surface Fv are all noncentral elements with projectively infinite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' 13 Since the intersection indices are nonzero integers, the gluing map induces a form of matrix gv,w = � a b c d � with b ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The fiber fw of an adjacent blocks can be written as fw = afv + bzv,w in the fundamental group π1(Tv,w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Hence, the image ρ(fw) is a noncentral element with projectively infinite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='2 Abelian representations Let ρ be a representation of a Seifert block Mv to � SL(2, R) ×Z R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If the image of the fundamental group π1(Mv) under ρ is an Abelian group, the restricted representation ρ : π1(Mv) → � SL(2, R)×ZR is called an Abelian representation of Mv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Since the Abelian representation factors through the homology group H1(Mv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Q), there is a Waldhausen basis to represent the boundaries of the base surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Suppose Mv has an Abelian representation and it has torus boundary Tv,w, w ∈ W, where W is the set of all adjacent Seifert blocks of Mv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' We choose a Waldhausen basis {fv,w, zv,w}w∈W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Under gluing matrix gw,v, the homological class fw,v ∈ H1(Tw,v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Z) has an image gw,v(fw,v) in H1(Tv,w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Notice H1(Tv,w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Z) has a basis fv,w, zv,w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then fw,v can be represented by a linear combination gw,v(fw,v) = aw,vfv,w + bw,vzv,w, aw,v, bw,v ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' We define a nonzero integer bv = � w∈W bw,v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then we have bv bw,v gw,v(fw,v) = bvaw,v bw,v fv,w + bvzv,w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' (3) Since {zv,w} is in the Waldhausen basis and Mv has the charge kv which satisfies the equation kv = � w∈W aw,v bw,v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Sum up on both sides for equation (3), then � w∈W bv bw,v fw = bvkvfv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' (4) In convenience, we ignore gluing matrices and inclusion maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' In addition, we denote fw,v by fw as well as fv,w by fv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' 14 Here we introduce the number bv in order to avoid discussing any n-th root of a central element in � SL(2, R) ×Z R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' In the above equations we can find bv/bw,v is an integer, so it involves only group multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='3 Abelian components In this part, we firstly introduce some definitions in graph theory from Harary [Har69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let Γ(V, E) be a graph, then Γ0(V0, E0) is called a subgraph of Γ if V0 and E0 are subsets of V and E respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let Γ(V, E) be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' A vertex-induced subgraph Γ0(V0, E0), simply called “an induced subgraph” of Γ, induced by the vertex set V0 is a graph with vertex set V0 and edge set E0 consisting of those edges both of whose endpoints are in V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let ρ be a representation of a graph manifold M and let Γ be a dual graph of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Take Γ0 be a connected subgraph of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then we use M0 to denote a submanifold consisting of Seifert blocks corresponding to Γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If the image of the fundamental group π1(M0) under the representation ρ is an Abelian group, then we call the connected subgraph Γ0 as an Abelian subgraph and call M0 as an Abelian component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' In addition, if the fibers of Seifert blocks in M0 are all noncentral type under the representation ρ, then we call M0 the noncentral Abelian component and Γ0 noncentral Abelian subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let ρ be a representation of a graph manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The set of all noncentral Abelian subgraphs has a partial order which is the inclusion of subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then we define a maximal noncentral Abelian subgraph which is not included by any Abelian subgraph except itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Similarly, we can define a maximal noncentral Abelian component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The set of noncentral Abelian subgraphs may be an empty set under some representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' We suppose the set of noncentral Abelian subgraphs is not empty under a representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then there exists at least one maximal noncentral Abelian subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' There are two propositions about the maximal one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let M0 be a maximal noncentral Abelian component under a representation ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then the adjacent Seifert blocks of M0 all have central fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' 15 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Suppose Mw is an adjacent Seifert block of M0 with a noncentral fiber fw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Hence, the fundamental group π1(Mw) is an Abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Since ρ(fw) commutes with the images of all fibers of Seifert blocks in M0, all elements of π1(Mw) commute with all elements of π1(M0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The fundamental group of a submanifold consisting of M0 and Mw is an Abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' This result contradicts with the definition of maximal noncentral Abelian component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Hence, the fiber of Mw is of central type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let M be a closed graph manifold with dual graph Γ(V, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If Γ0(V0, E0) is a maximal noncentral Abelian subgraph under a representation ρ, then there is a characteristic finite cover p : M ′ → M such that the subgraph induced by vertex set V ′ 0 is a maximal noncentral Abelian subgraph under the representation p ◦ ρ, where V ′ 0 is the preimage of V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If Γ0 is an induced subgraph, then it is a trivial case and the conclusion holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Suppose Γ0(V0, E0) is not an induced subgraph, and it has m fewer edges than a vertex-induced subgraph ˆΓ0(V0, ˆE0) induced by V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' We denote m edges by ei and the corresponding torus by Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' For each Ti, there is a gluing matrix gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let M0 denote the corresponding submanifold of Γ0 and ˆ M0 denote the corresponding submanifold of ˆΓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If π1(M0) has a presentation ⟨S|R⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then π1( ˆ M0) is an HNN extension of π1(M0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' It has a presentation ⟨S, ti, 1 ≤ i ≤ m | R, tifit−1 i = gi(fi), tizit−1 i = gi(zi), 1 ≤ i ≤ m⟩, where fi, zi are the generators of π1(Ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Consider the projection pr : � SL(2, R)×Z R → PSL(2, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let α be an element of π1(M0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Notice |Tr(pr(ρ(tiαt−1 i )))| = |Tr(pr(ρ(α)))|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Hence, we have two possible cases, pr(ρ(tiαt−1 i )) = pr(ρ(α)) (⋆) or pr(ρ(tiαt−1 i )) = pr(ρ(α))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' (⋆⋆) Since ρ(π1(M0)) is an Abelian group, by proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='1, any element commuting with ρ(α) will 16 commute with all elements in ρ(π1(M0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Hence, for each ti, it either satisfies the first relation (⋆) for all elements in π1(M0) or satisfies the second relation (⋆⋆) for all elements in π1(M0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Suppose there are n edges {ej, 1 ≤ j ≤ n} corresponding to {tj, 1 ≤ j ≤ n} where every tj satisfies the second relation (⋆⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Notice each ρ(t2 j) commutes with every element in ρ(π1(M0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then we can construct a double cover satisfying the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' We state a construction of the cover on the level of graph, hence it is a JSJ characteristic finite cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Cut ej in Γ to get two “half edges” e+ j , e− j for each j and take a copy of the broken graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then we glue e+ j to the half edge ¯e− j and glue e− j to the half edge ¯e+ j for each j, where ¯e+ j , ¯e− j are on the copy of the broken graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' By this construction, we get a connected double cover Γ′ of the graph Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The graph manifold M ′ dual to Γ′ is hence the double cover of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' We denote this covering by p : M ′ → M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then the subgraph Γ′ 0 induced by the vertex set V ′ 0 is a double cover of ˆΓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The corresponding submanifold of Γ′ 0 is a noncentral Abelian component under the representation p ◦ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' According to our construction, the fiber of Seifert blocks adjacent to Γ′ 0 are all central types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Hence, the induced subgraph Γ′ 0 is a maximal noncentral Abelian subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let a Seifert block Mv has a fiber with projectively infinite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' We set Mv contained in a maximal noncentral Abelian components M0 and the corresponding subgraph Γ0 is an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let W denote the set of all adjacent Seifert blocks of Mv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then we use WA to denote the subset of W which contains adjacent Seifert blocks of Mv lying in the maximal noncentral Abelian components M0 and use WC to denote the rest of elements in W which are not in M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then the equation (4) can be written in other form, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' bvkvfv − � w∈WA bv bw,v fw = � w′∈WC bv bw′,v fw′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' (5) The fiber fw′ commute with fv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Moreover, the right-hand of equation (5) has all fibers with images in the center of � SL(2, R) ×Z R under the representation ρ by proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If a maximal noncentral Abelian component has an induced corresponding subgraph under the representation ρ, then we give a following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let M0 be a maximal noncentral Abelian component with an induced corresponding subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If M0 consists of n Seifert blocks, then for each Seifert block M0,i of M0, there is an equation (5) associated to M0,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then we have n equations for the maximal Abelian component M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' 17 Denote the equations in form of matrices e0fA = fC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' We call e0 the associated matrix to the maximal Abelian component M0 under the representation ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let M0 be a maximal noncentral Abelian component with an induced corresponding subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If M0 consists of n Seifert blocks with fiber denoted by fi, 1 ≤ i ≤ n, we can write down the associated matrix to the maximal noncentral Abelian component M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' For the i-th Seifert block in M0, let W i C denote the adjacent Seifert blocks not in M0 (W i C may be empty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then the matrix equation is as the following: \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed b1k1 β12 · · β1,n β21 b2k2 · · β2,n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' βn,1 βn,2 · · bnkn \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed f1 f2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' fn \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed � w′∈W 1 C b1 bw′,1 fw′ � w′∈W 2 C b2 bw′,2 fw′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' � w′∈W n C bn bw′,n fw′ \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , (6) where βij(i ̸= j) is − bi bj,i if the i-th Seifert block and the j-th Seifert block are adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Otherwise, βij is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Since the image of π1(M0) is an Abelian group, we write the group multiplication in the form of the addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' We can find that the entries of e0 are all integers and the diagonal elements of e0 are in the form of biki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Moreover, all elements of the right-hand column are of central type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' 18 6 Virtually nonexistence of faithful representations in closed case In this section we prove the Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' In fact, we need to prove that a strictly diagonally dominant closed graph manifold virtually has no vertex faithful representations to the Seifert motion group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' By lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='1, there is a finite cover M1 of the graph manifold M such that each Seifert block homeomorphic to a trivial circle bundle over a compact surface with genus greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then we can find a JSJ characteristic finite cover M ′ of M factors through M1 and every Seifert block of M ′ is also a trivial circle bundle over a compact surface with genus greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Suppose there exists a vertex faithful representation ρ′ : π1(M ′) → � SL(2, R)×ZR which is faithful at vertex v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' By corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='3, the adjacent Seifert blocks of M ′ v′ all have fibers of noncentral type with images of projectively infinite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Hence, each adjacent block has an Abelian representation under ρ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let M ′ 0 be a maximal noncentral Abelian component of M ′ which contains at least one adjacent block of M ′ v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then by proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='8, there is JSJ characteristic double cover M ′′ such that the preimage M ′′ 0 is a maximal noncentral Abelian component whose corresponding subgraph is an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let ρ′′ : π1(M ′′) → � SL(2, R) ×Z R be the representation which is faithful at vertex v′′ who is one component of the preimage of v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The graph manifold M ′′ is a JSJ characteristic double cover of M ′, and hence it is a JSJ characteristic finite cover of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Since M is strictly diagonally dominant, by proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='3, M ′′ is also strictly diagonally dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' According to the definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='4 of a strictly diagonally dominant graph manifold, each Seifert block of M ′′ is strictly diagonally dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Hence, M ′′ 0 also satisfies the condition of strictly diagonally dominance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let M ′′ u′′ be a Seifert block in M ′′ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then ρ′′ restricted to π1(M ′′ u′′) is Abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' We can write an equation (5) for M ′′ u′′, bu′′ku′′fu′′ − � w′′ A∈W ′′ A bu′′ bw′′ A,u′′ fw′′ A = � w′′ C∈W ′′ C bu′′ bw′′ C,u′′ fw′′ C, Here W ′′ A denotes the set of adjacent blocks of M ′′ u′′ which lie in M ′′ 0 and W ′′ C denotes the set of 19 other adjacent blocks of M ′′ u′′ which are not in M ′′ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Since M ′′ u′′ is strictly diagonally dominant, we have an inequality |bu′′ku′′| > � w′′ A∈W ′′ A |bu′′| |bw′′ A,u′′| + � w′′ C∈W ′′ C |bu′′| |bw′′ C,u′′| ≥ � w′′ A∈W ′′ A |bu′′| |bw′′ A,u′′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Notice that the left-hand is the absolute value of a diagonal element bu′′ku′′ in the associated matrix e′′ 0 to the submanifold M ′′ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The right-hand is the sum of absolute value of other entries which lie in the same row with bu′′ku′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' For each row of the associated matrix e′′ 0 , the corresponding inequality holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then the associated matrix e′′ 0 to M ′′ 0 is strictly diagonally dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Hence, it is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Consider the equation e′′ 0 f ′′ A = f ′′ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' We have (e′′ 0 )∗e′′ 0 f ′′ A = (e′′ 0 )∗f ′′ C, det(e′′ 0 )f ′′ A = (e′′ 0 )∗f ′′ C, where (e′′ 0 )∗ is the adjoint matrix of e′′ 0 and det(e′′ 0 ) is the determinant of the matrix e′′ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The entries of the matrix e′′ 0 are all integers, so are the entries of the adjoint matrix (e′′ 0 )∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The images of fibers are all in an Abelian group, the equation has solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' By remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='3, the elements of any solution all have projectively finite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' This conclusion contradicts the result that the fibers of adjacent blocks of M ′′ v′′ are noncentral elements with projectively infinite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' So, the vertex faithful representation ρ′′ does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' As for a general finite cover M2, there is a JSJ characteristic finite cover M3 which factors through M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then M3 is strictly diagonally dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' There is no vertex faithful representations on M3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Hence, there is no vertex faithful representations on M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' So, a closed strictly diagonally dominant graph manifold M virtually has no vertex faithful representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' As a deduced result, a closed strictly diagonally dominant graph manifold M virtually has no faithful representations to the Seifert motion group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' In another point of view, the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='1 is extending a representation from a Seifert block to the whole graph manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' So, we can state in a form of a corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let M be a strictly diagonally dominant graph manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' There is a representation 20 ρ on a Seifert block Mv is faithful to the Seifert motion group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then the representation ρ virtually cannot be extended to the whole graph manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' 7 Existence of a vertex faithful representation in bounded case If a graph manifold has torus boundaries, then there is a vertex faithful representation on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' In other word, the faithful representation of a Seifert block can be extended to a representation on the whole graph manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The representation is not necessarily faithful on the whole graph manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' In the following we give a construction for one class of graph manifolds with nonempty boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let M be a graph manifold whose dual graph Γ is a tree and Each Seifert block dual to end vertex of the tree has at least one torus boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If one Seifert block has a faithful representation to the Seifert motion group, then the representation can be extended to the whole graph manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Suppose there is a faithful representation ρ0 on a Seifert block Mv of the graph manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then there is a direction on Γ starting from v to all the ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let the representation denoted by ρ : π1(M) → � SL(2, R)×ZR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Let W denote all adjacent blocks of Mv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The images of a homological basis {fw,v, zw,v} of each torus Tw,v are determined by a gluing matrix Av,w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then we set the image of π1(Mw) to be Abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' So the base surface boundaries of Mw have all images in a group C(ρ(fw)) = {h ∈ � SL(2, R) ×Z R | ρ(fw)h = hρ(fw)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' We choose elements from C(ρ(fw)) such that they form a Waldhausen basis for Mw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Hence, each adjacent block of Mv has an Abelian representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Suppose the representation ρ has been extended to all Seifert blocks with distance less than n to Mv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' When n = 2, we have done it in the last paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' If a Seifert block Mu with distance n to Mv, it is adjacent to a unique Seifert block Mt with distance n − 1 to Mv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' The basis {fu,t, zu,t} of the torus Tu,t is determined by the gluing matrix gt,u from Tt,u to Tu,t, \uf8eb \uf8edfu,t zu,t \uf8f6 \uf8f8 = gt,u \uf8eb \uf8edft,u zt,u \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Hence, the image of the fiber fu is determined by the inclusion map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' 21 Since the graph is a tree with all end points corresponding to Seifert manifolds meeting the boundary of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Mu, has at least one torus boundary is free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Then we can choose elements in the group C(ρ(fu,t)) ∩ C(ρ(zu,t)) to get a Waldhausen basis of Mu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' Hence, the representation ρ restricted on π1(Mu) is Abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' By induction, we can extend the representation ρ0 to ρ on the whole graph manifold such that ρ restricted on π1(Mv) is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' References [AFW15] Matthias Aschenbrenner, Stefan Friedl, and Henry Wilton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' 3-manifold groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' EMS Series of Lectures in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' European Mathematical Society (EMS), Z¨urich, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} +page_content=' [BG84] Robert Brooks and William Goldman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9A0T4oBgHgl3EQfGP_A/content/2301.02045v1.pdf'} 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b/eNFJT4oBgHgl3EQfSyxE/content/tmp_files/2301.11501v1.pdf.txt @@ -0,0 +1,1648 @@ +arXiv:2301.11501v1 [eess.SP] 27 Jan 2023 +1 +Practical Frequency-Hopping MIMO Joint Radar +Communications: Design and Experiment +Jiangtao Liu, Kai Wu, Tao Su and J. Andrew Zhang +Abstract—Joint radar and communications (JRC) can realize +two radio frequency (RF) functions using one set of resources, +greatly saving hardware, energy and spectrum for wireless +systems needing both functions. Frequency-hopping (FH) MIMO +radar is a popular candidate for JRC, as the achieved commu- +nication symbol rate can greatly exceed radar pulse repetition +frequency. However, practical transceiver imperfections can fail +many existing theoretical designs. In this work, we unveil for +the first time the non-trivial impact of hardware imperfections +on FH-MIMO JRC and analytically model the impact. We also +design new waveforms and, accordingly, develop a low-complexity +algorithm to jointly estimate the hardware imperfections of +unsynchronized receiver. Moreover, employing low-cost software- +defined radios and commercial off-the-shelf (COTS) products, +we build the first FH-MIMO JRC experiment platform with +radar and communications simultaneously validated over the +air. Corroborated by simulation and experiment results, the +proposed designs achieves high performances for both radar and +communications. +Index +Terms—Joint +radar +and +communications +(JRC), +frequency-hopping (FH) MIMO radar, sampling timing off- +set (STO), carrier frequency offset (CFO), front-end errors, +software-defined radio (SDR), commercial off-the-shelf (COTS) +I. INTRODUCTION +The proliferation of wireless systems has caused severe +spectrum congestion and scarcity worldwide. To alleviate +the issue, joint radar and communications (JRC) has been +identified as a promising solution [1], [2]. By sharing wave- +form, spectrum frequency, hardware and signal processing +modules, JRC can substantially improve cost, energy and spec- +tral efficiency of wireless systems that require both sensing +and communications functions [3]. One of the major JRC +designs is radar-centric by integrating data communications +into existing radar platforms [4]. Such design is also referred +as dual-function radar-communication (DFRC) in the open +literature [5]. +Initial DFRC works, e.g., [6]–[8], employ the linear +frequency-modulated (LFM) signal-based pulsed radars given +their wide applicability in radar community. In general, these +works [6]–[8] employ the frequency modulation rate, e.g., pos- +itive and negative, to convey one communication symbol per +radar repetition time (PRT). To increase the communication +symbol rate, more recent DFRC designs lean toward using +MIMO radars due to their rich degree of freedom (DoF) in +waveform design. For example, beam patterns of a MIMO +Jiangtao Liu and Tao Su are with National Laboratory of Radar Signal +Processing, Xidian University, Xi’an Shaanxi 710071, China (e-mail: jiang- +taoliu@xidian.edu.cn; sutao@xidian.edu.cn) +Kai Wu and J. Andrew Zhang are with the Global Big Data Technologies +Centre, University of Technology Sydney, NSW 2007, Australia (e-mail: +kai.wu@uts.edu.au; andrew.zhang@uts.edu.au) +radar is optimized to exploit sidelobes to conduct communica- +tion modulations, e.g., phase shift keying (PSK) and amplitude +shift keying [9], [10]. The MIMO radar waveform has also +been optimized to conduct non-conventional modulations, e.g., +code shift keying [11] and waveform shuffling [12]. Despite +that more information bits can be carried per symbol (com- +pared with initial LFM-based DFRC designs), these works [9]– +[12] still embed one information symbol over one or several +radar pulses. Thus, their achieved symbol rate is still limited +by the radar pulse repetition frequency (PRF), the reciprocal +of PRT. +Recently, frequency-hopping (FH) MIMO (FH-MIMO) +radar has attracted extensive interest in DFRC designs [4], +[13]–[21]. Compared with other pulsed MIMO radars, FH- +MIMO radar further divides each pulse into multiple sub- +pulses, also called hops, enabling the communication symbol +rate to exceed radar PRF [4]. Moreover, FH-MIMO radars also +provide new DoF for information modulation, e.g., the combi- +nations of hopping frequencies [17] and also the permutations +[21]. +However, as in sole wireless communications, the effective +demodulation of FH-MIMO DFRC generally requires accurate +channel estimation as well as transceiver time/frequency syn- +chronization. Communication training for FH-MIMO DFRC +is only studied by a few works in the literature. In [18], +[19], the estimations of communication channel and sampling +timing offset (STO) are studied for FH-MIMO DFRC with a +single antenna receiver equipped at the communication user +end (UE). In [20], the deep fading issue that can severely +degrade DFRC performance is identified and solved by in- +troducing multi-antenna receiver for UE. Novel waveforms +and methods are also designed to estimate channel and timing +offset. Despite the effectiveness of these designs [18]–[20], +they ignored the carrier frequency offset (CFO) and other +hardware errors, e.g., inconsistency of transceiver front-ends. +In this work, we develop a practical FH-MIMO DFRC +scheme by comprehensively treating all hardware errors, chan- +nel estimation, as well as time and frequency synchronization. +Using software defined ratio (SDR) platforms and commercial +off-the-shelf (COTS) products, we build an FH-MIMO DFRC +experiment platform with both radar and communications +functions. Moreover, we carry out over-the-air experiments +outdoors and indoors, validating the effectiveness of the pro- +posed designs and analysis in real-life scenarios. The main +contributions and results are summarized as follows. +1) We investigate the impact of practical hardware errors +on FH-MIMO DFRC, including STO, CFO and front-end +errors (FEE). Here, FEE includes the coupled errors from +radio frequency (RF) chains and antennas on both radar + +2 +transmitter and communication receiver sides. We model +these errors and unveil their non-trivial impact on FH- +MIMO DFRC. To the best of our knowledge, this is the +first time all these hardware errors are jointly considered +for FH-MIMO DFRC. +2) We design new DFRC waveforms by introducing moderate +changes to conventional FH-MIMO radar waveforms. We +also develop a low-complexity algorithm jointly estimating +STO, CFO and FEE at a communication receiver. More- +over, we identify some useful features of the impact of +STO, CFO and FEE under the proposed waveforms, and +exploit the features to further improve the accuracy of +estimating these practical errors. +3) We build a first FH-MIMO DFRC experiment platform +based on Xilinx Zynq SDR [22] and ADI’s FMCOMMS3 +RF board [23]. We also conduct first over-the-air experi- +ments, performing both radar and communications in the +same time. Using the proposed FH-MIMO DFRC wave- +forms, the radar sensing results highly match the sensing +scenario, as extracted from a high-resolution satellite map. +This manifests that the proposed waveform design has a +minimal impact on radar sensing. Moreover, we process +the experiment data collected at a communication receiver +using the proposed estimation methods. The achieved com- +munications performance is greatly improved over prior art +that does not consider all hardware errors as we do. +We underline that, though our work is focused on FH- +MIMO DFRC, the design and analysis has the potential +to serve DFRC based on other radars and communications +systems. This is because the hardware errors considered in +this work, namely STO, CFO and FEE, are common to most, +if not all, wireless systems. +We also remark that most FH-MIMO DFRC, as well +as other radars-based DFRC, have mainly been performed +through theoretical analysis and simulations. Only a few works +have illustrated DFRC through prototypes or proof-of-concept +platforms. In [24], the communications function of the FH- +MIMO DFRC using differential PSK modulations is imple- +mented using the universal software radio peripheral (USRP). +With a focus on validating the communication feasibility, the +work employs single-antenna transmitter and receiver, and +makes them synchronized. In contrast, we consider a more +practical case in our work with far separated transmitter and +receiver which are not physically synchronized. In [25], a pro- +totype is developed to demonstrate a spatial modulation-based +DFRC scheme. In [26], a low-complexity proof-of-concept +platform named JCR70 was developed for all-digital joint +communications radar at a carrier frequency of 73 GHz and a +bandwidth of 2 GHz. These works [25], [26] employ specially +designed hardwares for specific DFRC schemes. Despite a lack +of generality, they are pioneers in respective areas. Moreover, +we notice that, since around 2009, there has been a constant +interest in using SDR platforms to perform communication +waveform-based radar sensing [27]–[29]. These works provide +great guidance in designing proof-of-concept prototypes based +on SDR platforms. Nevertheless, they mainly focus on using +communication waveforms for sensing, while we deal with a +different problem of using radar signals for communications. +The rest of the paper is organized as follows. Section II +provides the signal model of FH-MIMO DFRC and introduces +how information is embedded in the DFRC. Section III first +illustrates the impact of practical hardware errors on the FH- +MIMO DFRC and then develops new waveforms and methods +to estimate and remove those errors. Section IV builds an FH- +MIMO DFRC experiment platform and shows simulation and +experiment results. Section V concludes the work. +II. SIGNAL MODEL OF FH-MIMO DFRC +This section briefly describes the principle of FH-MIMO +radar-based DFRC, including how the radar works and how +data communications is performed by reusing the radar wave- +form. +A. FH-MIMO Radar +The FH-MIMO radar considered here is a pulse-based or- +thogonal MIMO radar. It uses separate but cohrent transceiver +arrays to achieve the extended array aperture. It also em- +ploys the fast frequency hopping, namely, each radar pulse +is divided into mutiple sub-pulses, i.e., hops, and the fre- +quency changes over hops and antennas. Let B denote the +radar bandwidth. The frequency band is divided into K +sub-bands. The baseband frequency of the k-th sub-band is +fk = (⌊− K +2 ⌋+k)B +K +(k = 0, 1, · · · , K − 1). The baseband +frequency of the h-th hop at antenna m is denoted by fhm +which can take fk (∀k ∈ [0, K − 1]). Denote the total number +of hops in a radar pulse as H. Then the signal transmitted by +antenna m in a radar pulse can be given by +sm(t) = e−j2πfhmt, 0 ≤ t ≤ T + hT, h = 0, · · · , H − 1, +(1) +where T denotes the time duration of a hop (sub-pulse). To +facilitate DFRC, we employ the following constraints [14], +fhm ̸= fhm′ (∀m ̸= m′, ∀h), +BT/K ∈ I+, +(2) +where I+ denotes the set of positive integers. As a result of the +above constraints, the signals transmitted by the M antennas +at any hop are orthogonal, i.e., +� T +0 shm(t)s∗ +hm′ (t)dt = 0 given +∀m ̸= m′. An FH-MIMO radar receiving processing scheme +will be presented in Section III-D. +B. FH-MIMO DFRC +With reference to the DFRC scheme introduced in [18]. The +communication information can be conveyed by two ways. +First, the transmitted signal in each hop and antenna can be +multiplied by a PSK symbol, as denoted by ej̟hm, where +̟hm ∈ ΩJ (J ≥ 1) and ΩJ = +� +0, 2π +2J , · · · , 2π(2J−1) +2J +� +is +a PSK constellation with the modulation order J. Second, +the combination of the hopping frequencies at each hop is +also used for conveying information, which is referred to as +frequency hopping code selection (FHCS) [17]. In particular, +given K radar sub-bands and M transmitter antennas, there +can be CM +K numbers of combinations when selecting M out + +3 +of K sub-bands. FHCS uses these combinations to convey +information bits whose maximum number is +� +log2(CM +K ) +� +. +For simplicity, we use a single-antenna communication +receiver to illustrate information demodulation in FH-MIMO +DFRC. The communication-received signal at hop h is +sh(t) = +M−1 +� +m=0 +βhmej̟hme−j2πfhmt + v(t), +(3) +where βhm is the complex gain between m-th transmitter +antenna of the radar and the communication receiver, and +v(t) denotes AWGN, and M denotes the number of transmit +antennas. +To demodulate information symbols, we need to estimate +{fhm ∀m} and ̟hm (∀m). Given the constraint (2), the +former can be estimated by detecting the strongest M peaks +in the Fourier transform of yh(t). In contrast, ̟hm (∀m) is +more difficult to estimate, as we need to know βhm first. +Their estimations are studied in [18] under ideal conditions +that no timing or frequency offset exists between the radar +transmitter and communication receiver. In [19], timing offset +is considered; however, frequency offset is not yet. Moreover, +array calibration error has not been considered for FH-MIMO +DFRC so far. These non-ideal conditions are investigated in +the following. +III. PRACTICAL FH-MIMO DFRC DESIGN +In this section, we first investigate the impact of practical +transceiver errors on FH-MIMO DFRC. Then we design +waveforms and propose novel methods to estimate and remove +the errors. +A. Impact of Practical Transceiver Errors +The clock asynchrony between the radar transmitter and a +communication receiver can cause STO and CFO. Let ∆ω +denote the CFO. It changes over time slowly and hence can +be treated as a fixed value here. Different from CFO, STO +accumulates, and its impact varies fast over time. Let ∆t0 de- +note the initial STO and ∆Ts be the sampling time difference +between the radar transmitter and the communication. Then at +the h-hop of the i-th PRT, the accumulated STO can be given +by +∆tih = ∆t0 + (iNp + hNh)∆Ts, +(4) +where Np (Nh) is the number of samples in a PRT (hop). +Based on (3), the communication-received signal in the h- +th hop and i-th PRT, with CFO and STO included, can be +expressed as +sih(t) = +M−1 +� +m=0 +βihm(ωihm)rect +�t − iTp − hT +T +� +× +ej(ωihm+∆ω)(t+∆tih)ej̟ihm, +(5) +where an additional subscript (·)i is used to indicate the PRT +index and rect( x +T ) is the rectangular function that takes one +for x ∈ [0, T ] and zero elsewhere. Here, Tp and T denote the +time of a PRT and a hop, respectively. Different from βhm in +(3), βihm(ωihm) in (5) is a function of ωihm to account for +other frequency dependent gains caused by the radio frequency +chains of different antennas. +Calculating the Fourier transform of sihm(t) at ω = ωihm, +we obtain +Sihm +(a) += +� T +0 +sihm(˜t)e−jωihm(˜t+iTp+hT )d˜t +(b) +≈ +� T +0 +βihm(ωihm) +ej(ωihm+∆ω)(˜t+iTp+hT +∆tih)e̟ihme−jωihm(˜t+iTp+hT)d˜t +(c) +≈ A(∆ω)βihm(ωihm)ej̟ihmejωihm(∆t0+(iNp+hNh)∆Ts)× +ej∆ω(iTp+hT )ej∆ω(iNp+hNh)∆Ts +(6) +where the substitution ˜t = t−iTp−hT is performed to get +(a) += +with the integral variable changed from t to ˜t; the expression +of sihm(t) given in (5) is plugged in +(a) += to get +(b) +≈; and +(c) +≈ +is obtained by replacing ∆tih with its expression given in (4) +and by taking ej∆ω∆t0 ≈ 1. Note that the integral over ˜t yields +A(∆ω) = T sin∆ωT +2 +� �∆ωT +2 +� +ej ∆ωT +2 . +(7) +Moreover, the approximation +(b) +≈ is because we have neglected +the Fourier transforms of the signals from other antennas. +It is obvious from (6) that STO, as indicated by ∆tih +and CFO, as represented by ∆ω, have non-trivial impact on +communication demodulation. In order to estimate the PSK +symbol ̟ihm, all the other phases need to be estimated and +suppressed first. This is studied next. In particular, we start +with developing the demodulation methods, during which we +shall introduce some conditions on the waveform to enable the +new methods. Then, we translate those conditions to waveform +design. +B. Proposed Demodulation Method +From (6), we obtain the following, when ωihm = 0 and +̟ihm = 0, +˜Sihm = Sihm |ωihm=0= A(∆ω)βihm(0)× +ej∆ω(iTp+hT )ej∆ω(iNp+hNh)∆Ts. +(8) +Note that i, h and m are indexes of PRT, hop and antennas, +respectively. The result in (8) facilitates the estimation of ∆ω, +as detailed below. +First, similar to ˜Sihm, we can set ω(i+1)hm = 0 and obtain +˜S(i+1)hm = S(i+1)hm |ω(i+1)hm=0= A(∆ω)β(i+1)hm(0)× +ej∆ω((i+1)Tp+hT )ej∆ω((i+1)Np+hNh)∆Ts. +(9) +Then, taking the ratio between ˜S(i+1)hm and ˜Sihm leads to +˜S(i+1)hm +˜Sihm += β(i+1)hm(0) +βihm(0) +ej∆ω(Tp+Np∆Ts) ≈ ej∆ωTp, +(10) +where the approximation is because Tp +≫ Np∆Ts and +βihm(0) ≈ β(i+1)hm(0). The validity of the first condition +is illustrated in A. For the second, it is because the channel is +approximately unchanged in a single PRT with a short time + +4 +duration, e.g., 40 µs to be validated in our experiment. From +(10), we can estimate CFO as +� +∆ω = arg +� ˜S(i+1)nm +˜Sihm +� � +Tp. +(11) +Note that ∆ω and ω are the same multiples of ∆fCLK +and fCLK, respectively. Here, ∆fCLK denotes the clock offset +and fCLK is the nominal clock frequency. The ratio between +∆fCLK/fCLK is often called the clock stability. Therefore, +with � +∆ω attained, we can estimate the clock stability, as given +by +�ρ = � +∆ω +� +ω, +(12) +where ω denotes the nominal local oscillator angular fre- +quency. Based on (24) in Appendix A, we can further estimate +STO ∆Ts as +� +∆Ts = −�ρ +� � +f t +s(1 − �ρ) +� +, +(13) +where f t +s is the sampling frequency at the transmitter. With +∆Ts estimated, we see from (4) that ∆tih is also partially +estimated. +From (6), we see that the remaining unknowns that hinder +communication demodulation, i.e., the estimation of ̟ihm, +is βihm(ωihm)ejωihm∆t0. The coupling of the two terms +makes their individual estimates difficult to obtain. Thus, +we consider their joint estimation. To do so, we introduce +˘Si1h1m (∀i1, ∀h1 ̸= h) which is obtained by taking ̟i1h1m = +0 and ωi1h1m = 2πk/K in (6). Different from ˜Sihm given in +(8) with the zero hopping frequency, ˘Si1h1m is obtained under +non-zero hopping frequency. Moreover, they can be attained +under the same PRT with i1 = i, but they are always obtained +in different hops, i.e., h1 ̸= h. +Assuming the above conditions are satisfied, let us check the +ratio between ˘Si1h1m and ˜Si1hm, where the latter is obtained +by taking i = i1 in (8). The ratio can be expressed as +dmk = +˘Si1h1m +˜Si1hm += C βi1h1m( 2πk +K ) +βi1hm(0) +ej 2πk∆t0 +K +, k = 1, · · · , K − 1 +(14) +s.t. C = ej +2πk(i1Np+h1Nh)∆Ts +K +ej∆ω(h1−h)T ej∆ω(h1−h)Nh∆Ts. +Similarly, let us further construct the ratio between ˘Si2h2m and +˜Si2hm with i2 ̸= i1 and h2 ̸= h. After some basic calculations, +we attain +˘Si2h2m +˜Si2hm += d′ +kDej̟i2h2m, s.t. d′ +k = C βi2h2m +� 2πk +K +� +βi2hm(0) +ej 2πk∆t0 +K +, +D = ej +2πk((i2−i1)Np+(h2−h1)Nh)∆Ts +K +ej∆ω(h2−h1)T × +ej∆ω(h2−h1)Nh∆Ts, +(15) +where C is given in (14), and h1 in D is due to the inclusion of +C in d′ +k. Note that D can be estimated based on the estimates +obtained in (11) and (13). Assuming that d′ +k is known for the +moment, we can then estimate ̟i2h2m as +�̟i2h2m = arg +� +˘Si2h2m +d′ +kD ˜Si2hm +� +s.t. ωi2h2m = 2πk/K. (16) +Our next question is how to know d′ +k. Comparing (14) and +(15), we see that d′ +k has a very similar form to dmk. In fact, +they are approximately the same, as ensured by the following +lemma. +Lemma 1: Provided that |(i1 − i2)Tp| is smaller than the +stable time of the transceiver front-ends, we have d′ +k = dmk, +where Tp is the PRT duration. +Proof: From (14) and (15), we see that d′ +k and dmk are +almost the same other than some differences in the subscripts +of the β·(·) terms. As illustrated in the texts below (5), β·(·) +is the composite impact of the channel response and the +complex gains of the transceiver front-ends. In the same radar +pulse, the channel response can be seen fixed. Thus, the two +ratios +βi1h1m( 2πk +K ) +βi1hm(0) +and +βi2h2m( 2πk +K ) +βi2hm(0) +are only dependent on the +complex gains of transceiver front-ends. As a result, the ratios +are the same if i1 and i2 satisfy the condition stated in the +lemma. We notice that in modern transceivers, front-ends are +generally stable in a contiguous operation, i.e., a whole course +of running after a system is powered on. This is also validated +through our experiments, as to be presented in Section IV-C. +Algorithm 1 Proposed FH-MIMO DFRC Scheme +Input: M (radar transmitter antenna number), H (the number of hops +in a radar pulse), T (hop duration), Tp (PRT duration), K (the number +of sub-bands), B (radar bandwidth; also the sampling frequency), +sih(t) given in (5) (the time-domain communication-received signal) +1) For each i in Sx = +� +(0, 1, · · · , K − 1) + xK (∀x ≥ 0) +� +: +a) Take the Fourier transform of sih(t); +b) Identify the largest M peaks, yielding Sihm given in (6); +c) For ∀m, calculate dmk given in (14) by taking i1 = i, h1 = +m + 1 (due to (D2)) and h = m (due to (D1)); +d) For ∀h2, m, Calculate +˘Si2h2m +˜ +Si2hm +in (15) by taking i2 = i and +h = m (due to (D1)); +2) Estimate � +∆ω as in (11), where h = m based on (D1); +3) Estimate � +∆Ts jointly using (12) and (13); +4) For ∀i ∈ Sx, ∀m, ∀h ̸= m or (m + 1): +a) If ωihm = 2πk/K, set i1 = xK + k, h1 = m + 1, i2 = i and +h2 = h; +b) Estimate D based on (15); +c) Estimate ̟ihm based on (16) with h = m taken for ˜Si2hm in +the denominator; +C. Novel DFRC Waveform Designs +We have shown above that under certain conditions imposed +on FH-MIMO waveforms, we can suppress unknown channel +and hardware errors to estimate communication symbols. +Those conditions can be ensured through proper waveform +designs, as illustrated below. +From (8) to (13), we can see the importance of the zero +baseband frequency, i.e., k = 0 for some ωihm. Since different +antennas have distinct channel responses and front-end gains, +we need to ensure that each antenna takes the zero baseband +frequency at least once. Achieving this will require at least +M hops, since different hopping frequencies are required to +used for different antennas in the same hop, as enforced in +(2). Thus, our first waveform design can be established as +Design 1 (D1): ωihm = 0 and ̟ihm = 0 at h = m, given ∀i. + +5 +From Lemma 1, we know that dmk needs to be computed +for estimating d′ +k. From (14), we see that dmk is obtained +under ωihm = 2πk/K and ̟ihm = 0. Thus, to avoid heavily +changing the originally radar waveform, we adopt the follow- +ing waveform design to calculate dmk (k = 0, 1, · · · , K − 1) +over K different PRTs: +Design 2 (D2): ωihm = 2π ⟨i⟩K/K and ̟ihm = 0 at h = +m + 1, given H ≥ M + 1, where ⟨i⟩K denotes the modulo-K +of i. +Note that h = m + 1 is because h = m has been occupied in +Design 1. +The two designs are sufficient for effective communication +demodulation in a practical FH-MIMO DFRC with hardware +errors and unknown channels. For clarity, we summarize +the whole procedure in Algorithm 1. While most steps in +Algorithm 1 are straightforward based on the illustrations +in this section, we provide some more notes on several key +steps. From Step 1), we see that every consecutive K PRTs +are jointly used for communication demodulation. The main +reason is that (D2) only allows us to estimate one dmk per PRT. +While this design is not a must, it introduces minimal changes +to the primary radar function. In Step 1b), identifying M +peaks from the Fourier transform result is not hard; however, +assigning the peaks to the M radar transmitter antennas. Thus, +we employ another waveform constraint [19] +ωihm < ωihm′ ∀i, h and ∀m′ > m. +(17) +To implement the constraint, we let the FH-MIMO radar +select its hopping frequencies randomly, then simply re-order +the frequencies in ascending order, and assign them to the +antennas, one each. A nice feature was disclosed in [18], +stating that the above re-ordering does not change the range +ambiguity function of the underlying FH-MIMO radar. In Step +2) of Algorithm 1, the estimates � +∆ω obtained under different +i’s can be averaged to improve the estimation performance. +Then Step 3) can be performed based on the improved � +∆ω to +get a more accurate � +∆Ts. +We remark that the FH-MIMO DFRC design is radar-centric +in this work; namely, we seek to introduce only minimal +changes to the radar yet facilitating effective communications +in the presence of hardware errors. The waveform design +illustrated above only requires a few hops over antennas to use +assigned hopping frequencies, in contrast to random selection +in the original radar. Thus, we expect the introduced waveform +design has little impact on the radar function. This will be +validated in Section IV. The significance of our design to +communications is that the practical hardware errors are, for +the first time, modeled and effectively suppressed for FH- +MIMO DFRC. Without considering these inevitable pratical +errors, communications performance can be rather poor in +practice; this will be demonstrated in Section IV through over- +the-air experiments. +D. FH-MIMO Radar Receiving Processing +Here, we briefly describe an FH-MIMO radar processing +scheme. It will be performed in simulations and experi- +ments. Let si(t) = [si1(t), si2(t), · · · , siM(t)]T collect the +signals transmitted by the M antennas in the i-th PRT, +where sim(t) (∀m) is obtained by replacing 2πfhm in +(1) by ωihm designed in Section III-C. Also, let yi(t) = +[yi1(t), yi2(t), · · · , yiN(t)]T denote the signals received by the +N receiver antennas. Denote the steering vectors of the trans- +mitter and receiver arrays by at(θ) and ar(θ), respectively. +Being co-located, they have the same direction. Considering +a single target for simplicity, the radar echo signal in the i-th +PRT can be given by +yi(t) = δar(θ)aT +t (θ)si(t − τ) + wi(t), +s.t. t ∈ (i − 1)TPRT + [HT, TPRT], +(18) +where τ represents the target echo delay, δ denotes the target +scattering coefficient, wi(t) collects additive white Gaussian +noises (AWGNs), and TPRT is the time duration of a PRT. As +mentioned earlier, the FH-MIMO of interest is a pulsed radar. +Thus, there shall be no valid echo signal during the radar +transmission, as the receiver is either turned off or saturated +with invalid echo signals. This explains the starting time of +each PRT given in (18). The common steps of the radar +receiving processing, as will be performed in simulations and +experiments, are described below. +Matched filtering is a typical first step of pulsed radar signal +processing [30]. The filter coefficients are s∗ +im(−t) (m = +0, 1, · · · , M − 1), where “()∗” takes conjugate. As each +antenna receives a combination of all transmitted signals, +yin(t) (∀n) needs to pass each of the M filters, where yin(t) +denotes the n-th entry of yi(t) given in (18). The matched +filtering result can be written as, +˜yip(t) = yin(t) ⊛ s∗ +m(−t), p = (n − 1)M + m, +(19) +where ⊛ calculates the linear convolution. +Moving target detection (MTD) is often performed after the +matched filter. For ˜yip(t) (∀p, t), a Fourier transform can be +performed over i, leading to the so-called range-Doppler map +(RDM), +˜Yfp(t) = +I−1 +� +i=0 +˜yip(t)e−j2πfiTPRT, +(20) +where f and t span the Doppler and range dimensions, +respectively. +Target detection can be performed based on �P −1 +p=0 | ˜Yfp(t)|, +where the incoherent accumulation over p (indexing spatial +channels) is performed as we do not have angle information +yet; otherwise, the coherent beamforming can be performed. +The constant false-alarm rate (CFAR) detector has been popu- +larly employed for target detection and hence will be used for +our simulation and experiment. Interested readers are referred +to [30] for the details of the CFAR detector. An intuitive +simulation tutorial is also available at [31]. Let (f ∗, t∗) denote +the location of a target. Extracting the signal at each RDM, +we obtain +z = +� +˜Yf ∗0(t∗), ˜Yf ∗1(t∗), · · · , ˜Yf ∗(P −1)(t∗) +�T +, +(21) +where P = MN according to (19). +Angle estimation is carried out using z. Based on (18), +the steering vector depicting the spatial information in z can + +6 + FMCOMMS3 +PA0 +PA1 +LNA0 +LNA1 + +FMCOMMS3 +TX0 +TX1 +RX0 +RX1 +RX0 +ZedBoard Radio Hardware +ZC706 Radio Hardware +Fig. 1: The schematic diagram of the testing system +be written as ˜a(θ) = ar(θ) ⊗ at(θ), where ⊗ denote the +Kronecker product. In practice, the radio frequency chains as- +sociated with transmitter and receiver antennas always present +some differences, as depicted by et and er, respectively. +Incorporating them, the steering vector can be revised as +˘a(θ) = ˜a(θ) ⊙ (er ⊗ et). Therefore, a naive angle estimate +can be obtained as +ˆθ = argmaxθl=2πl/L |˘a(θl)Hz|2, l = 0, 1, · · · , L − 1 (22) +where L can take a relatively large value for a fine spatial +resolution. More advancing methods, such as the multiple +signal classification (MUSIC) [32] and the DFT interpolation- +based methods [33], can be employed for more accurate angle +estimation accuracy. +IV. OVER-THE-AIR FH-MIMO DFRC EXPERIMENTS +In this section, we perform over-the-air experiments to +validate the proposed FH-MIMO DFRC scheme. +A. Schematic Diagram and Experiment Platform +The schematic diagram of our testing system is shown in +Fig. 1. We employ the Xilinx Zynq software-defined radio +(SDR) ZC706 [34] and ZedBoard [35] to build the FH-MIMO +radar and communication receiver, respectively. Both SDRs are +equipped with RF FPGA mezzanine card (RF FMC) boards, +FMCOMMS3 [23] in specific. Each FMCOMMS3 supports +two transmitting and two receiving RF chains with the RF +range of 70 MHz ∼ 6 GHz and a baseband frequency range +of 200 KHz ∼ 56 MHz. A 40 MHz oscillator with the stability +of 10 ppm is used by each FMCOMMS3 [36]. Note that +both SDRs are supported by MATLAB [22]. Thus, they are +connected to host computers, where MATLABs are installed +and used to program and control the SDRs independently. +For the FH-MIMO radar, we use MATLAB to generate +the base-band signals of the two transmitting antennas for +a CPI and download them to ZC706 once through Ethernet. +The SDR is configured to cyclically transmit the signals. The +two radar receiving channels are configured to capture echo +signal in a consecutive time of 204.8 ms, corresponding to +8, 192, 000 samples. Also note that the maximum number of +samples that can be transferred in one capture is 8, 388, 608, +A +Power +C +PA +G +Rx Antenna +E +ZC706 +Antenna Front +B +Tx horn +Anchor +Target +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +D +FMCOMMS3 +F +LNA +E +C +F +A +D +G +B +Ethernet +cable +Fig. 2: The established FH-MIMO radar, where the host computer is not +shown but connected with the SDR through an Ethernet cable. +TABLE I: Parameters of the Established FH-MIMO Radar +Variable +Parameter +Value +- +Central frequency +5.5GHz +- +power amplifier gain +43dB +M +Number of horn antennas +2 +- +horn antenna gain +20.39dB +- +horizontal beamwidth of horn antenna +16◦ +- +vertical beamwidth of horn antenna +15.5◦ +- +Maximum transmit power of PA +20W +N +Number of radar receiver antennas +12 +- +Receiving antenna element gain +13dB +- +Horizontal beamwidth of microstrip antenna +120◦ +- +Vertical beamwidth of microstrip antenna +20◦ +- +Low noise amplifier gain +23dB +which is a limitation of the employed SDR [22]. Note that +the echo capture needs to be triggered in MATLAB of the +host computer. The communication receiver, as configured +similar to the radar receiver, also needs a triggering signal +from the MATLAB connected to the SDR. Next, we provide +more elaborations on radar and communication sub-systems. +1) Radar subsystem: Based on ZC706, we build an FH- +MIMO radar platform, as shown in Fig. 2. A host computer +(not shown in the figure) is connected to the SDR through an +Ethernet cable (which is shown on the lower left side). The RF +board, FMCOMMS3, is underlain by AD9361 which has the +maximum output power of 6.5 dBm at 5.5 GHz [37], where +5.5 GHz is the carrier frequency used in our experiments. +Moreover, we use external power amplifiers (PAs) to increase +the radar transmission power. The maximum output power of +the employed PA is 20 W, i,e., 43 dBm. By controlling the +output power of AD9361, the transmission power is fixed at 2 +W in the sequential experiments. Moreover, two identical horn +antennas are also used for radar transmission, each connected +to a RF chain. For radar receiving, a microstrip uniform linear +array of 12 antennas is used. The antenna spacing is half-the- +wavelength at the center frequency of 5.5 GHz. Two low-noise +amplifiers (LNAs) are used, one for each receiving RF chain +on FMCOMMS3. Key Parameters of the above components +are listed in Table I. + +7 +TABLE II: Parameters of the FH-MIMO Radar +Variable +Parameter +Value +B +the Signal Bandwidth +20MHz +fk +radar sub-band baseband frequency +−10 : 1 : 9 MHz +T +hop duration +1µs +H +number of hops per pulse +5 +Tp +PRT +40µs +Nc +number of PRTs per CPI +128 +fs +sampling frequency +40 MHz +Np +number of samples per PRT +1600(= fsTp) +Several notes are given below. First, AD9361 has its gain +adjustable in −89.75 ∼ 0 dB for the transmitting RF chain +and 0 ∼ 61 dB for receiving RF chain. Thus, together with +the gains from other components, see Table I, the maximum +transmitting and receiving power gain of the FH-MIMO radar +built in Fig. 2 can be 43(= 0 + 43) dB and 84(= 23 + 61) +dB, respectively. Second, during experiments, we place the +two horn antennas 6λ apart from each other. According to +the MIMO radar processing illustrated in Section III-D, the +way how the transceiver arrays are placed leads to a virtual +array of 2 × 12 = 24 antennas. Third, limited by the number +of receiving RF chains on FMCOMMS3, the time division +multiplexing (TDM) MIMO is employed to achieve the above- +mentioned virtual array. As shown in Fig. 2, the receiving +antenna array has 12 elements, each connected to an SMA +port. However, we can see from Fig. 3, there are only two +receiving RF chains. Thus, we collect echo signals from the +12 antennas in six consecutive data captures. In the n-th (n = +0, 1, · · · , 5) capture, the two receiving SMAs are connected +to the n-th and (n + 6)-th antenna element shown in Fig. +2. Interested readers are referred to [38] for more details on +TDM-MIMO radars. +Based on the hardware features illustrated in Section IV-A, +we set the parameters of the FH-MIMO radar, as given in +Table II. As the considered FH-MIMO radar is a pulsed radar, +the receiver channel suffers from strong self-interference when +the transmitter works, leading to a blind zone of CHT/2 +(in meter), where C denotes the microwave speed, T is a +hop duration and H is the hop number. Given the limited +link budget, see Table I, the maximum measurable distance +of the radar platform would be very limited. Thus, we want +to keep the blind zone small as well. To do so, we set +T = 1µs and H = 5. This leads to a blind zone of 750 +m. Other parameters in Table II are straightforward based on +the descriptions therein. +2) Communications Subsystem: It is built on the SDR Zed- +Board, FMCOMMS3 and an 8 dBi omnidirectional antenna, as +shown in Fig. 3. As only downlink communication is consid- +ered in this work, the communication subsystem only receives, +hence much simpler compared with the radar system. We do +not use external LNA for the communication subsystem. Thus, +its receiving power gain ranges in 0 ∼ 61 dB, solely dependent +on AD9361 of FMCOMMS3. +Based on the parameters given in Table II, we can calculate +the data rate of the radar-enabled communications. As illus- +trated in Section II-B, the combinations of hopping frequencies +are used as communication data symbols. Given K = 20 sub- +C +Zedboard +B +FMCOMMS3 +A +Antenna +TX1 +RX1 +TX0 +RX0 +A +B +C +Ethernet +cable +Fig. 3: the Communication Subsystem +Fig. 4: The 50 random targets for simulation. +-55 -47 -39 -31 -23 +10-1 +100 +101 +102 +deg +N +T +-55 -47 -39 -31 -23 +SNR +100 +101 +102 +103 +m +-55 -47 -39 -31 -23 +10-1 +100 +101 +102 +103 +m/s +Fig. 5: Root mean squared error (RMSE) of angle, distance and velocity +estimations under different SNR, where “N” and “T” represent new and +traditional FH-MIMO radar waveforms, respectively. +bands and M = 2 transmitting antennas, we have C2 +20 = 190. +Considering the integer number of bits, out of 190 combina- +tions, 128 numbers of combinations can be used to convey +7 bits per radar hop. Given H = 5, using the combinations +convey a total of 7 × 5 = 35 bits per PRT. Moreover, PSK is +also employed for information demodulation, one symbol per +hop and antenna. Thus, Considering an x-bit PSK modulation. +The overall number of bits conveyed by PSK is xMH = 10x +per PRF. In summary, the communication data rate is +(35 + 10x)/Tp = (0.875 + 0.25x) Mbps. +(23) +For x = 1, 2, 3 and 4, the data rate is 1.125, 1.375, 1.625 and +1.875 Mbps, respectively. +B. Simulation Analysis +Before performing experiments, simulations are carried out +to validate the proposed FH-MIMO DFRC. We start with +validating the radar performance, where the radar is configured +as per Table II. As for sensing scenario, we set 50 targets with +random speeds, distances and angles which are uniformly dis- +tributed in [−170, 170] m/s, [750, 4, 185] m and [−4, 4] deg, +respectively. To evaluate the root mean squared error (RMSE) +of parameter estimations, we perform 100 independent trials, + +pa +0 +-100 +-200 +-340 +0 +1000 +2000 +3000 +40 +Distance (m).85 +80 +00 +5000 +6000 +75340 +200908 +-5 +0 +5 +10 +15 +SNR +10-5 +10-4 +10-3 +10-2 +10-1 +100 +BER +P_0.5_16 +P_0.5_8 +P_1_16 +P_1_8 +F_0.5_16 +F_0.5_8 +F_1_16 +F_1_8 +Fig. 6: Simulation results of BER curve versus SNR +with target parameters randomly generated over trials. For each +trial, we perform the radar processing illustrated in Section +III-D for target detection and estimation. Both the conventional +FH-MIMO radar in Section II-A and the one modified for +DFRC in Section III-C are simulated for a comparison. +Fig. 4 plots a snapshot of a range-Doppler map (RDM) +obtained in one trial. We can see many strong points scattering +over the range-Doppler domain, each point representing a +target. As illustrated in Section III-D, CFAR is performed +based on an RDM. Then the delay and Doppler bins of the +detected targets are used to estimate their parameters. Fig. 5 +plots the RMSEs of angle, distance and velocity averaged +over all targets and trials. We see that the RMSEs of all +parameter estimates first decrease and then converge, as SNR +increases. This complies with general understanding, and the +convergence is due to the quantized distance, velocity and +angle grids used during the estimation; see Section III-D. +More importantly, we see from Fig. 5 that the traditional +and new FH-MIMO radar waveforms lead to almost the +same estimation performance. This validates that the proposed +waveform designs for DFRC only incur minimal changes to +the underlying FH-MIMO radar. +Next, we demonstrate the communication performance of +FH-MIMO DFRC employing the metric of bit error rate +(BER). FHCS and PSK, as illustrated in Section II-B, are +simulated. Most radar configurations in Table III are used. +But here, we also consider two different hop duration, i.e., +0.5µs and 1µs. As for the PSK, we simulate 8PSK and +16PSK. In this simulation, we let the communication receiver +know the channel responses. Thus, the results here provide +a performance lower bound of the experiment results to be +presented shortly. +Fig. 6 plots the BER performance of different modulations +under different settings. We see that FHCS generally has +lower BER than PSK modulations. This is consistent with +previous works, e.g., [18]. We also see that when the hop +duration doubles, both FHCS and PSK achieve better BER +performance. This is because the demodulation SNR increases +with the hop duration. For FHCS, we see from Fig. 6 that +the modulation order does not affect its BER performance. +This is due to the way FHCS is demodulated. In particular, +as illustrated in Section II-B, we only need to identify DFT +peaks for demodulating FHCS, where the phases of peaks are +irrelevant. +C. Experimental Results +Employing the hardware platforms illustrated in Section +IV-A, we perform over-the-air experiments. As shown in Fig.2, +the radar transceiver is placed on top of a building with +the height of about 20 m. Fig. 7 plots the radar imaging +results, where the observation distance is up to 3 km from +the radar and the angular region is [−30◦, 30◦] around the +normal direction of the radar. To plot the 2D radar imaging, +the zero-th Doppler channel of the MTD result, as obtained +in (20), is extracted in each CPI. Then, the beamforming, as +illustrated in (18), is performed to scan the angular region with +a step of 1◦. +To calibrate the radar transceiver arrays, we use a known +target (i.e., target B in Fig. 7) to calculate the array calibration +coefficients, i.e., er ⊗ et given in (22). In particular, we place +the radar transceiver in such a way that target B is in the +normal direction of the radar. Since the target distance is +known, we extract the signal of the p-th (p = 0, 1, · · · , P −1) +virtual spatial channel, i.e., the signal given in (20), at the +known distance and zero-th Doppler bin. Given the radar +configuration in Table II, we have P(= 24) virtual channels. +Ideally, the extracted signals should be the same. But, as +affected by the array calibration errors, their values can be +distinct. Thus, we use the signal of the first virtual spatial +channel as a reference, and all other extracted signals are +normalized to the reference one, leading to the array calibra- +tion vector. In fact, in our experiment, we do not have the +facilities to calibrate the radar transceiver arrays. Using the +above anchor-based method, we are able to obtain a relatively +good calibrated array, as demonstrated by the high match +between the measured and map-illustrated targets in Fig. 7. +In the figure, we superimpose the radar imaging above a +satellite map of the observed area, where the map is obtained +from [39]. We see from Fig. 7 that the strong signals in the +radar imaging match well with the objects observed on the +map. The distance profiles at the three selected angels further +highlight most pinpointed targets. This illustrates the effective- +ness of the above-mentioned array calibration. More impor- +tantly, this validates that the proposed waveform modifications +for FH-MIMO radar to accommodate data communications do +not obviously affect the primary radar function. +Next, we illustrate the communications performance. In the +first set of experiments (i.e., Figs. 7, 8 and 9), we place +the communication receiver, as illustrated in Fig. 3, about +ten meters behind the radar transmitter antennas (which are +placed outdoor as shown in Fig. 2). That is, the communication +receiver is in the line-of-sight (LoS) posterior views of radar +transmitter antennas. The proposed communication demodula- +tion method, as summarized in Algorithm 1, is performed on +collected experiment data. +Fig. 8(a) plots the CFO estimate obtained each pair of two +consecutive PRTs; see (11), since there are 128 PRTs, there + +9 +-30 +-25 +-20 +-15 +-10 +-5 +Radar +A +B +D +E +F +G +H +I +J +K +L +C +Normalized Power (dB) +Distance (m) +Fig. 7: A two-dimensional space-distance radar imaging using the proposed FH-MIMO DFRC waveform and the hardware platform built in Section IV-A. +The zero-Doppler channel is observed, and hence only static targets are shown in the radar imaging. Representative targets are pinpointed based on the latest +satellite map of the location [39]. The distances profiles of three angles covering most pinpointed targets are separately plotted in the upper-right sub-figure. +The lower-right sub-figure is a trimmed version of Fig. 2, giving a glimpse on the radar platform and the surrounding environment. +0 +50 +100 +PRT NO. +9 +9.5 +10 +CFO estimate (KHz) +Tx0 +Tx1 +m0 +m1 +(a) +0 +50 +100 +PRT NO. +-40 +-20 +0 +20 +40 +d +Tx0 +Tx1 +(b) +Fig. 8: The CFO estimate, i.e., � +∆ω, is illustrated in (a); the estimate of +arg{dmk} in (b), where � +∆ω is obtained in (11) and dmk in (14). In a +CPI, the value of k cyclic increases with the PRTs, and it can be known from +(14) that for a k, arg{dmk} increases with the PRT index. +are 127 results. We see that there is a non-negligible CFO +between the radar transmitter and the communication receiver. +Moreover, we also see that the two communication receivers +have different CFOs. In addition, we see that the CFO estimate +changes over time but stays around approximately a fixed value +(m0 and m1 in Fig. 8(a)). This validates the slow-varying +feature of CFO in a certain coherent processing period. This +also confirms that we can average the estimates of CFO over a +suitable time period to obtain a more accurate estimation. Fig. +8(b) plots the estimate of dmk. We see that dmk changes over +PRTs in the way depicted by (14). Recall that, in the proposed +waveform design, the i-th PRT estimates dmk for k = (i)K−1, +where ()K−1 denotes modulo-(K − 1). +Fig. 9 provides the scatter plots of the demodulated com- +munication symbols, where 8PSK is given in the first row +and 16PSK is in the second. Three demodulation methods +(a) +(b) +(c) +(d) +(e) +(f) +Fig. 9: Illustrating demodulation performance, where the top row is for 8PSK +and the second row is for 16PSK. The first column is the demodulation +results without considering frequency-dependent transceiver gains, which +approximates the results obtained in [33]. The middle column is based on +the proposed design given in Section III. The third column further averages +the estimates of D over a CPI at the same frequency. +are performed based on the same collected experiment data. +The first method, leading to the results in the first column, +neglects the frequency-dependent transceiver gains, which +is essentially the method in [19]. The second method, as +summarized in Algorithm 1, leads to the results in the middle +column. Following the overall steps of Algorithm 1, the third +method, further averages D obtained in Step 4b) of Algorithm +1 in a CPI, thus improving the estimate of �̟i2h2m and then + +T000500030-50TOBDCK8 +CEH10 +Rx +Tx +Fig. 10: Indoor BER Test Scenario +0 +5 +10 +15 +20 +Gain Control (dB) +0 +5 +10 +15 +20 +Actual Gain (dB) +0.5_16 +0.5_8 +1_16 +1_8 +Fig. 11: actual gain versus receive gain +demodulation performance. From Fig. 9, we see that neglect- +ing frequency-dependent transceiver gains can substantially +degrade the communication performance of FH-MIMO DFRC. +This validates the critical importance of the proposed de- +signs which specifically account for the frequency-dependent +transceiver gains. From the middle column of Fig. 9, we +see that the proposed waveform and demodulation method +can achieve relatively good communication performance. By +further averaging arg{D} over time, where D is given in (15), +we can further improve the estimation accuracy of ̟i2h2m and +hence the communication performance, as manifested in the +third column of Fig. 9. +In the second set of experiment for validating commu- +nication performances, we observe the BER performance +under different SNRs, as simulated in Fig. 6. To precisely +control demodulation SNRs, we use two SDRs indoors, one +transmitting and one receiving, as illustrated in Fig. 10. The +transmitting SDR is the one used for radar transmitter, but is +equipped with two omni-directional antennas (with 12 dBi). +During the experiment, we set the gains of the two transmitting +RF chains as 0 dB and adjust the gains of the receiving RF +chains to achieve different demodulation SNRs. Fig. 11 plots +the actual gain under different gain control. The actual gain +is estimated by averaging the power of the received signal +which is normalized to the estimated gain under the 0 dB +gain control. +Fig. 12 plots the BER versus SNR, where 800 CPIs are +collected, corresponding to 1, 024, 000 demodulation results. +We see that the trends of all BER curves match well with +what observed in Fig. 6. This validates the proposed design +in practical DFRC scenarios. The curves are not as smooth as +the simulated ones, mainly because the gain control curve is +not ideally linear and stable during the collection of four sets +of data, as shown in Fig. 11. +V. CONCLUSION +In this work, a practical FH-MIMO DFRC is developed +comprehensively treating all practically inevitable hardware +errors, including STO, CFO and front-end imperfections of +0 +5 +10 +15 +20 +Receive Gain (dB) +10-6 +10-5 +10-4 +10-3 +10-2 +10-1 +100 +BER +P_0.5_16 +P_0.5_8 +P_1_16 +P_1_8 +F_0.5_16 +F_0.5_8 +F_1_16 +F_1_8 +Fig. 12: Curve of BER versus receive gain. Since the linearity of the actual +gain curve is not very good, as shown in Fig. 11, there is a certain error in +the BER result of the actual test. +transceivers. We model these errors and analyze their impacts +on FH-MIMO DFRC. Moreover, we design new waveforms +and develop a low-complexity algorithm jointly estimating all +hardware errors at a communication receiver. In addition, we +build an FH-MIMO JRC experiment platform employing low- +cost SDR and COST products that are popular in IoT system +designs. Outdoor and indoor experiments are conducted using +the platform. Applying the proposed designs on the collected +experiment data achieves high performances for both radar and +communications. +APPENDIX +As a radar PRT, Tp = NpT t +s , where T t +s denotes the sampling +time at the radar transmitter. Thus, to compare Tp and Np∆Ts, +we need to show that ∆Ts ≪ T t +s . +Since ∆Ts is the sampling time difference between the radar +transmitter and the communication receiver, we can perform +the following calculation, +∆Ts = T t +s − T r +s = f r +s − f t +s +f ts f rs += +−∆fs +fs +t(fs +t − ∆fs) = +−ρ +f ts (1 − ρ), +(24) +where we use the superscripts (·)t and (·)r to differentiate the +variables between the transmitter and receiver, respectively; +fs = 1/Ts denotes the sampling frequency; ∆fs = f t +s − f r +s ; +and ρ = ∆fs/f t +s. The reason of introducing ρ is that it can +be linked with the clock stability at the radar transmitter. +Specifically, we have +ρ = ∆fs/f t +s = ∆fCLK/f t +CLK, +(25) +where f t +CLK is the transmitter clock frequency, f t +s is a multiple +of f t +CLK, and ∆fs is the same multiple of ∆fCLK. The typical +value of ρ is about tens of parts per million (ppm). 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Rao, Tdm mimo radar, https://www.ti.com/lit/an/swra554a/swra554a.pdf, +july 2018. +[39] GaoDe, Map. https://ditu.amap.com/. + diff --git a/eNFJT4oBgHgl3EQfSyxE/content/tmp_files/load_file.txt b/eNFJT4oBgHgl3EQfSyxE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3d4840bfaad73ae3b069f2ef6549f71aa6060946 --- /dev/null +++ b/eNFJT4oBgHgl3EQfSyxE/content/tmp_files/load_file.txt @@ -0,0 +1,939 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf,len=938 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='11501v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='SP] 27 Jan 2023 1 Practical Frequency-Hopping MIMO Joint Radar Communications: Design and Experiment Jiangtao Liu, Kai Wu, Tao Su and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Andrew Zhang Abstract—Joint radar and communications (JRC) can realize two radio frequency (RF) functions using one set of resources, greatly saving hardware, energy and spectrum for wireless systems needing both functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Frequency-hopping (FH) MIMO radar is a popular candidate for JRC, as the achieved commu- nication symbol rate can greatly exceed radar pulse repetition frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' However, practical transceiver imperfections can fail many existing theoretical designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' In this work, we unveil for the first time the non-trivial impact of hardware imperfections on FH-MIMO JRC and analytically model the impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' We also design new waveforms and, accordingly, develop a low-complexity algorithm to jointly estimate the hardware imperfections of unsynchronized receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Moreover, employing low-cost software- defined radios and commercial off-the-shelf (COTS) products, we build the first FH-MIMO JRC experiment platform with radar and communications simultaneously validated over the air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Corroborated by simulation and experiment results, the proposed designs achieves high performances for both radar and communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Index Terms—Joint radar and communications (JRC), frequency-hopping (FH) MIMO radar, sampling timing off- set (STO), carrier frequency offset (CFO), front-end errors, software-defined radio (SDR), commercial off-the-shelf (COTS) I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' INTRODUCTION The proliferation of wireless systems has caused severe spectrum congestion and scarcity worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' To alleviate the issue, joint radar and communications (JRC) has been identified as a promising solution [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' By sharing wave- form, spectrum frequency, hardware and signal processing modules, JRC can substantially improve cost, energy and spec- tral efficiency of wireless systems that require both sensing and communications functions [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' One of the major JRC designs is radar-centric by integrating data communications into existing radar platforms [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Such design is also referred as dual-function radar-communication (DFRC) in the open literature [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Initial DFRC works, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=', [6]–[8], employ the linear frequency-modulated (LFM) signal-based pulsed radars given their wide applicability in radar community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' In general, these works [6]–[8] employ the frequency modulation rate, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=', pos- itive and negative, to convey one communication symbol per radar repetition time (PRT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' To increase the communication symbol rate, more recent DFRC designs lean toward using MIMO radars due to their rich degree of freedom (DoF) in waveform design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' For example, beam patterns of a MIMO Jiangtao Liu and Tao Su are with National Laboratory of Radar Signal Processing, Xidian University, Xi’an Shaanxi 710071, China (e-mail: jiang- taoliu@xidian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' sutao@xidian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='cn) Kai Wu and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Andrew Zhang are with the Global Big Data Technologies Centre, University of Technology Sydney, NSW 2007, Australia (e-mail: kai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='wu@uts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='au;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' andrew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='zhang@uts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='au) radar is optimized to exploit sidelobes to conduct communica- tion modulations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=', phase shift keying (PSK) and amplitude shift keying [9], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The MIMO radar waveform has also been optimized to conduct non-conventional modulations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=', code shift keying [11] and waveform shuffling [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Despite that more information bits can be carried per symbol (com- pared with initial LFM-based DFRC designs), these works [9]– [12] still embed one information symbol over one or several radar pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Thus, their achieved symbol rate is still limited by the radar pulse repetition frequency (PRF), the reciprocal of PRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Recently, frequency-hopping (FH) MIMO (FH-MIMO) radar has attracted extensive interest in DFRC designs [4], [13]–[21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Compared with other pulsed MIMO radars, FH- MIMO radar further divides each pulse into multiple sub- pulses, also called hops, enabling the communication symbol rate to exceed radar PRF [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Moreover, FH-MIMO radars also provide new DoF for information modulation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=', the combi- nations of hopping frequencies [17] and also the permutations [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' However, as in sole wireless communications, the effective demodulation of FH-MIMO DFRC generally requires accurate channel estimation as well as transceiver time/frequency syn- chronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Communication training for FH-MIMO DFRC is only studied by a few works in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' In [18], [19], the estimations of communication channel and sampling timing offset (STO) are studied for FH-MIMO DFRC with a single antenna receiver equipped at the communication user end (UE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' In [20], the deep fading issue that can severely degrade DFRC performance is identified and solved by in- troducing multi-antenna receiver for UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Novel waveforms and methods are also designed to estimate channel and timing offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Despite the effectiveness of these designs [18]–[20], they ignored the carrier frequency offset (CFO) and other hardware errors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=', inconsistency of transceiver front-ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' In this work, we develop a practical FH-MIMO DFRC scheme by comprehensively treating all hardware errors, chan- nel estimation, as well as time and frequency synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Using software defined ratio (SDR) platforms and commercial off-the-shelf (COTS) products, we build an FH-MIMO DFRC experiment platform with both radar and communications functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Moreover, we carry out over-the-air experiments outdoors and indoors, validating the effectiveness of the pro- posed designs and analysis in real-life scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The main contributions and results are summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 1) We investigate the impact of practical hardware errors on FH-MIMO DFRC, including STO, CFO and front-end errors (FEE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Here, FEE includes the coupled errors from radio frequency (RF) chains and antennas on both radar 2 transmitter and communication receiver sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' We model these errors and unveil their non-trivial impact on FH- MIMO DFRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' To the best of our knowledge, this is the first time all these hardware errors are jointly considered for FH-MIMO DFRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 2) We design new DFRC waveforms by introducing moderate changes to conventional FH-MIMO radar waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' We also develop a low-complexity algorithm jointly estimating STO, CFO and FEE at a communication receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' More- over, we identify some useful features of the impact of STO, CFO and FEE under the proposed waveforms, and exploit the features to further improve the accuracy of estimating these practical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 3) We build a first FH-MIMO DFRC experiment platform based on Xilinx Zynq SDR [22] and ADI’s FMCOMMS3 RF board [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' We also conduct first over-the-air experi- ments, performing both radar and communications in the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Using the proposed FH-MIMO DFRC wave- forms, the radar sensing results highly match the sensing scenario, as extracted from a high-resolution satellite map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' This manifests that the proposed waveform design has a minimal impact on radar sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Moreover, we process the experiment data collected at a communication receiver using the proposed estimation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The achieved com- munications performance is greatly improved over prior art that does not consider all hardware errors as we do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' We underline that, though our work is focused on FH- MIMO DFRC, the design and analysis has the potential to serve DFRC based on other radars and communications systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' This is because the hardware errors considered in this work, namely STO, CFO and FEE, are common to most, if not all, wireless systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' We also remark that most FH-MIMO DFRC, as well as other radars-based DFRC, have mainly been performed through theoretical analysis and simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Only a few works have illustrated DFRC through prototypes or proof-of-concept platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' In [24], the communications function of the FH- MIMO DFRC using differential PSK modulations is imple- mented using the universal software radio peripheral (USRP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' With a focus on validating the communication feasibility, the work employs single-antenna transmitter and receiver, and makes them synchronized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' In contrast, we consider a more practical case in our work with far separated transmitter and receiver which are not physically synchronized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' In [25], a pro- totype is developed to demonstrate a spatial modulation-based DFRC scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' In [26], a low-complexity proof-of-concept platform named JCR70 was developed for all-digital joint communications radar at a carrier frequency of 73 GHz and a bandwidth of 2 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' These works [25], [26] employ specially designed hardwares for specific DFRC schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Despite a lack of generality, they are pioneers in respective areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Moreover, we notice that, since around 2009, there has been a constant interest in using SDR platforms to perform communication waveform-based radar sensing [27]–[29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' These works provide great guidance in designing proof-of-concept prototypes based on SDR platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Nevertheless, they mainly focus on using communication waveforms for sensing, while we deal with a different problem of using radar signals for communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Section II provides the signal model of FH-MIMO DFRC and introduces how information is embedded in the DFRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Section III first illustrates the impact of practical hardware errors on the FH- MIMO DFRC and then develops new waveforms and methods to estimate and remove those errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Section IV builds an FH- MIMO DFRC experiment platform and shows simulation and experiment results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Section V concludes the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' SIGNAL MODEL OF FH-MIMO DFRC This section briefly describes the principle of FH-MIMO radar-based DFRC, including how the radar works and how data communications is performed by reusing the radar wave- form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' FH-MIMO Radar The FH-MIMO radar considered here is a pulse-based or- thogonal MIMO radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' It uses separate but cohrent transceiver arrays to achieve the extended array aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' It also em- ploys the fast frequency hopping, namely, each radar pulse is divided into mutiple sub-pulses, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=', hops, and the fre- quency changes over hops and antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Let B denote the radar bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The frequency band is divided into K sub-bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The baseband frequency of the k-th sub-band is fk = (⌊− K 2 ⌋+k)B K (k = 0, 1, · · · , K − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The baseband frequency of the h-th hop at antenna m is denoted by fhm which can take fk (∀k ∈ [0, K − 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Denote the total number of hops in a radar pulse as H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Then the signal transmitted by antenna m in a radar pulse can be given by sm(t) = e−j2πfhmt, 0 ≤ t ≤ T + hT, h = 0, · · · , H − 1, (1) where T denotes the time duration of a hop (sub-pulse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' To facilitate DFRC, we employ the following constraints [14], fhm ̸= fhm′ (∀m ̸= m′, ∀h), BT/K ∈ I+, (2) where I+ denotes the set of positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' As a result of the above constraints, the signals transmitted by the M antennas at any hop are orthogonal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=', � T 0 shm(t)s∗ hm′ (t)dt = 0 given ∀m ̸= m′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' An FH-MIMO radar receiving processing scheme will be presented in Section III-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' FH-MIMO DFRC With reference to the DFRC scheme introduced in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The communication information can be conveyed by two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' First, the transmitted signal in each hop and antenna can be multiplied by a PSK symbol, as denoted by ej̟hm, where ̟hm ∈ ΩJ (J ≥ 1) and ΩJ = � 0, 2π 2J , · · · , 2π(2J−1) 2J � is a PSK constellation with the modulation order J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Second, the combination of the hopping frequencies at each hop is also used for conveying information, which is referred to as frequency hopping code selection (FHCS) [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' In particular, given K radar sub-bands and M transmitter antennas, there can be CM K numbers of combinations when selecting M out 3 of K sub-bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' FHCS uses these combinations to convey information bits whose maximum number is � log2(CM K ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' For simplicity, we use a single-antenna communication receiver to illustrate information demodulation in FH-MIMO DFRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The communication-received signal at hop h is sh(t) = M−1 � m=0 βhmej̟hme−j2πfhmt + v(t), (3) where βhm is the complex gain between m-th transmitter antenna of the radar and the communication receiver, and v(t) denotes AWGN, and M denotes the number of transmit antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' To demodulate information symbols, we need to estimate {fhm ∀m} and ̟hm (∀m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Given the constraint (2), the former can be estimated by detecting the strongest M peaks in the Fourier transform of yh(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' In contrast, ̟hm (∀m) is more difficult to estimate, as we need to know βhm first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Their estimations are studied in [18] under ideal conditions that no timing or frequency offset exists between the radar transmitter and communication receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' In [19], timing offset is considered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' however, frequency offset is not yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Moreover, array calibration error has not been considered for FH-MIMO DFRC so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' These non-ideal conditions are investigated in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' PRACTICAL FH-MIMO DFRC DESIGN In this section, we first investigate the impact of practical transceiver errors on FH-MIMO DFRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Then we design waveforms and propose novel methods to estimate and remove the errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Impact of Practical Transceiver Errors The clock asynchrony between the radar transmitter and a communication receiver can cause STO and CFO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Let ∆ω denote the CFO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' It changes over time slowly and hence can be treated as a fixed value here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Different from CFO, STO accumulates, and its impact varies fast over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Let ∆t0 de- note the initial STO and ∆Ts be the sampling time difference between the radar transmitter and the communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Then at the h-hop of the i-th PRT, the accumulated STO can be given by ∆tih = ∆t0 + (iNp + hNh)∆Ts, (4) where Np (Nh) is the number of samples in a PRT (hop).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Based on (3), the communication-received signal in the h- th hop and i-th PRT, with CFO and STO included, can be expressed as sih(t) = M−1 � m=0 βihm(ωihm)rect �t − iTp − hT T � × ej(ωihm+∆ω)(t+∆tih)ej̟ihm, (5) where an additional subscript (·)i is used to indicate the PRT index and rect( x T ) is the rectangular function that takes one for x ∈ [0, T ] and zero elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Here, Tp and T denote the time of a PRT and a hop, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Different from βhm in (3), βihm(ωihm) in (5) is a function of ωihm to account for other frequency dependent gains caused by the radio frequency chains of different antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Calculating the Fourier transform of sihm(t) at ω = ωihm, we obtain Sihm (a) = � T 0 sihm(˜t)e−jωihm(˜t+iTp+hT )d˜t (b) ≈ � T 0 βihm(ωihm) ej(ωihm+∆ω)(˜t+iTp+hT +∆tih)e̟ihme−jωihm(˜t+iTp+hT)d˜t (c) ≈ A(∆ω)βihm(ωihm)ej̟ihmejωihm(∆t0+(iNp+hNh)∆Ts)× ej∆ω(iTp+hT )ej∆ω(iNp+hNh)∆Ts (6) where the substitution ˜t = t−iTp−hT is performed to get (a) = with the integral variable changed from t to ˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' the expression of sihm(t) given in (5) is plugged in (a) = to get (b) ≈;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' and (c) ≈ is obtained by replacing ∆tih with its expression given in (4) and by taking ej∆ω∆t0 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Note that the integral over ˜t yields A(∆ω) = T sin∆ωT 2 � �∆ωT 2 � ej ∆ωT 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' (7) Moreover, the approximation (b) ≈ is because we have neglected the Fourier transforms of the signals from other antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' It is obvious from (6) that STO, as indicated by ∆tih and CFO, as represented by ∆ω, have non-trivial impact on communication demodulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' In order to estimate the PSK symbol ̟ihm, all the other phases need to be estimated and suppressed first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' This is studied next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' In particular, we start with developing the demodulation methods, during which we shall introduce some conditions on the waveform to enable the new methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Then, we translate those conditions to waveform design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Proposed Demodulation Method From (6), we obtain the following, when ωihm = 0 and ̟ihm = 0, ˜Sihm = Sihm |ωihm=0= A(∆ω)βihm(0)× ej∆ω(iTp+hT )ej∆ω(iNp+hNh)∆Ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' (8) Note that i, h and m are indexes of PRT, hop and antennas, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The result in (8) facilitates the estimation of ∆ω, as detailed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' First, similar to ˜Sihm, we can set ω(i+1)hm = 0 and obtain ˜S(i+1)hm = S(i+1)hm |ω(i+1)hm=0= A(∆ω)β(i+1)hm(0)× ej∆ω((i+1)Tp+hT )ej∆ω((i+1)Np+hNh)∆Ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' (9) Then, taking the ratio between ˜S(i+1)hm and ˜Sihm leads to ˜S(i+1)hm ˜Sihm = β(i+1)hm(0) βihm(0) ej∆ω(Tp+Np∆Ts) ≈ ej∆ωTp, (10) where the approximation is because Tp ≫ Np∆Ts and βihm(0) ≈ β(i+1)hm(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The validity of the first condition is illustrated in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' For the second, it is because the channel is approximately unchanged in a single PRT with a short time 4 duration, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=', 40 µs to be validated in our experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' From (10), we can estimate CFO as � ∆ω = arg � ˜S(i+1)nm ˜Sihm � � Tp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' (11) Note that ∆ω and ω are the same multiples of ∆fCLK and fCLK, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Here, ∆fCLK denotes the clock offset and fCLK is the nominal clock frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The ratio between ∆fCLK/fCLK is often called the clock stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Therefore, with � ∆ω attained, we can estimate the clock stability, as given by �ρ = � ∆ω � ω, (12) where ω denotes the nominal local oscillator angular fre- quency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Based on (24) in Appendix A, we can further estimate STO ∆Ts as � ∆Ts = −�ρ � � f t s(1 − �ρ) � , (13) where f t s is the sampling frequency at the transmitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' With ∆Ts estimated, we see from (4) that ∆tih is also partially estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' From (6), we see that the remaining unknowns that hinder communication demodulation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=', the estimation of ̟ihm, is βihm(ωihm)ejωihm∆t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The coupling of the two terms makes their individual estimates difficult to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Thus, we consider their joint estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' To do so, we introduce ˘Si1h1m (∀i1, ∀h1 ̸= h) which is obtained by taking ̟i1h1m = 0 and ωi1h1m = 2πk/K in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Different from ˜Sihm given in (8) with the zero hopping frequency, ˘Si1h1m is obtained under non-zero hopping frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Moreover, they can be attained under the same PRT with i1 = i, but they are always obtained in different hops, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=', h1 ̸= h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Assuming the above conditions are satisfied, let us check the ratio between ˘Si1h1m and ˜Si1hm, where the latter is obtained by taking i = i1 in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The ratio can be expressed as dmk = ˘Si1h1m ˜Si1hm = C βi1h1m( 2πk K ) βi1hm(0) ej 2πk∆t0 K , k = 1, · · · , K − 1 (14) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' C = ej 2πk(i1Np+h1Nh)∆Ts K ej∆ω(h1−h)T ej∆ω(h1−h)Nh∆Ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Similarly, let us further construct the ratio between ˘Si2h2m and ˜Si2hm with i2 ̸= i1 and h2 ̸= h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' After some basic calculations, we attain ˘Si2h2m ˜Si2hm = d′ kDej̟i2h2m, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' d′ k = C βi2h2m � 2πk K � βi2hm(0) ej 2πk∆t0 K , D = ej 2πk((i2−i1)Np+(h2−h1)Nh)∆Ts K ej∆ω(h2−h1)T × ej∆ω(h2−h1)Nh∆Ts, (15) where C is given in (14), and h1 in D is due to the inclusion of C in d′ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Note that D can be estimated based on the estimates obtained in (11) and (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Assuming that d′ k is known for the moment, we can then estimate ̟i2h2m as �̟i2h2m = arg � ˘Si2h2m d′ kD ˜Si2hm � s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' ωi2h2m = 2πk/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' (16) Our next question is how to know d′ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Comparing (14) and (15), we see that d′ k has a very similar form to dmk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' In fact, they are approximately the same, as ensured by the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Lemma 1: Provided that |(i1 − i2)Tp| is smaller than the stable time of the transceiver front-ends, we have d′ k = dmk, where Tp is the PRT duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Proof: From (14) and (15), we see that d′ k and dmk are almost the same other than some differences in the subscripts of the β·(·) terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' As illustrated in the texts below (5), β·(·) is the composite impact of the channel response and the complex gains of the transceiver front-ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' In the same radar pulse, the channel response can be seen fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Thus, the two ratios βi1h1m( 2πk K ) βi1hm(0) and βi2h2m( 2πk K ) βi2hm(0) are only dependent on the complex gains of transceiver front-ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' As a result, the ratios are the same if i1 and i2 satisfy the condition stated in the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' We notice that in modern transceivers, front-ends are generally stable in a contiguous operation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=', a whole course of running after a system is powered on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' This is also validated through our experiments, as to be presented in Section IV-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Algorithm 1 Proposed FH-MIMO DFRC Scheme Input: M (radar transmitter antenna number), H (the number of hops in a radar pulse), T (hop duration), Tp (PRT duration), K (the number of sub-bands), B (radar bandwidth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' also the sampling frequency), sih(t) given in (5) (the time-domain communication-received signal) 1) For each i in Sx = � (0, 1, · · · , K − 1) + xK (∀x ≥ 0) � : a) Take the Fourier transform of sih(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' b) Identify the largest M peaks, yielding Sihm given in (6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' c) For ∀m, calculate dmk given in (14) by taking i1 = i, h1 = m + 1 (due to (D2)) and h = m (due to (D1));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' d) For ∀h2, m, Calculate ˘Si2h2m ˜ Si2hm in (15) by taking i2 = i and h = m (due to (D1));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 2) Estimate � ∆ω as in (11), where h = m based on (D1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 3) Estimate � ∆Ts jointly using (12) and (13);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 4) For ∀i ∈ Sx, ∀m, ∀h ̸= m or (m + 1): a) If ωihm = 2πk/K, set i1 = xK + k, h1 = m + 1, i2 = i and h2 = h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' b) Estimate D based on (15);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' c) Estimate ̟ihm based on (16) with h = m taken for ˜Si2hm in the denominator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Novel DFRC Waveform Designs We have shown above that under certain conditions imposed on FH-MIMO waveforms, we can suppress unknown channel and hardware errors to estimate communication symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Those conditions can be ensured through proper waveform designs, as illustrated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' From (8) to (13), we can see the importance of the zero baseband frequency, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=', k = 0 for some ωihm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Since different antennas have distinct channel responses and front-end gains, we need to ensure that each antenna takes the zero baseband frequency at least once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Achieving this will require at least M hops, since different hopping frequencies are required to used for different antennas in the same hop, as enforced in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Thus, our first waveform design can be established as Design 1 (D1): ωihm = 0 and ̟ihm = 0 at h = m, given ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 5 From Lemma 1, we know that dmk needs to be computed for estimating d′ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' From (14), we see that dmk is obtained under ωihm = 2πk/K and ̟ihm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Thus, to avoid heavily changing the originally radar waveform, we adopt the follow- ing waveform design to calculate dmk (k = 0, 1, · · · , K − 1) over K different PRTs: Design 2 (D2): ωihm = 2π ⟨i⟩K/K and ̟ihm = 0 at h = m + 1, given H ≥ M + 1, where ⟨i⟩K denotes the modulo-K of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Note that h = m + 1 is because h = m has been occupied in Design 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The two designs are sufficient for effective communication demodulation in a practical FH-MIMO DFRC with hardware errors and unknown channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' For clarity, we summarize the whole procedure in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' While most steps in Algorithm 1 are straightforward based on the illustrations in this section, we provide some more notes on several key steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' From Step 1), we see that every consecutive K PRTs are jointly used for communication demodulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The main reason is that (D2) only allows us to estimate one dmk per PRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' While this design is not a must, it introduces minimal changes to the primary radar function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' In Step 1b), identifying M peaks from the Fourier transform result is not hard;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' however, assigning the peaks to the M radar transmitter antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Thus, we employ another waveform constraint [19] ωihm < ωihm′ ∀i, h and ∀m′ > m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' (17) To implement the constraint, we let the FH-MIMO radar select its hopping frequencies randomly, then simply re-order the frequencies in ascending order, and assign them to the antennas, one each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' A nice feature was disclosed in [18], stating that the above re-ordering does not change the range ambiguity function of the underlying FH-MIMO radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' In Step 2) of Algorithm 1, the estimates � ∆ω obtained under different i’s can be averaged to improve the estimation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Then Step 3) can be performed based on the improved � ∆ω to get a more accurate � ∆Ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' We remark that the FH-MIMO DFRC design is radar-centric in this work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' namely, we seek to introduce only minimal changes to the radar yet facilitating effective communications in the presence of hardware errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The waveform design illustrated above only requires a few hops over antennas to use assigned hopping frequencies, in contrast to random selection in the original radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Thus, we expect the introduced waveform design has little impact on the radar function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' This will be validated in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The significance of our design to communications is that the practical hardware errors are, for the first time, modeled and effectively suppressed for FH- MIMO DFRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Without considering these inevitable pratical errors, communications performance can be rather poor in practice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' this will be demonstrated in Section IV through over- the-air experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' FH-MIMO Radar Receiving Processing Here, we briefly describe an FH-MIMO radar processing scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' It will be performed in simulations and experi- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Let si(t) = [si1(t), si2(t), · · · , siM(t)]T collect the signals transmitted by the M antennas in the i-th PRT, where sim(t) (∀m) is obtained by replacing 2πfhm in (1) by ωihm designed in Section III-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Also, let yi(t) = [yi1(t), yi2(t), · · · , yiN(t)]T denote the signals received by the N receiver antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Denote the steering vectors of the trans- mitter and receiver arrays by at(θ) and ar(θ), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Being co-located, they have the same direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Considering a single target for simplicity, the radar echo signal in the i-th PRT can be given by yi(t) = δar(θ)aT t (θ)si(t − τ) + wi(t), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' t ∈ (i − 1)TPRT + [HT, TPRT], (18) where τ represents the target echo delay, δ denotes the target scattering coefficient, wi(t) collects additive white Gaussian noises (AWGNs), and TPRT is the time duration of a PRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' As mentioned earlier, the FH-MIMO of interest is a pulsed radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Thus, there shall be no valid echo signal during the radar transmission, as the receiver is either turned off or saturated with invalid echo signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' This explains the starting time of each PRT given in (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The common steps of the radar receiving processing, as will be performed in simulations and experiments, are described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Matched filtering is a typical first step of pulsed radar signal processing [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The filter coefficients are s∗ im(−t) (m = 0, 1, · · · , M − 1), where “()∗” takes conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' As each antenna receives a combination of all transmitted signals, yin(t) (∀n) needs to pass each of the M filters, where yin(t) denotes the n-th entry of yi(t) given in (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The matched filtering result can be written as, ˜yip(t) = yin(t) ⊛ s∗ m(−t), p = (n − 1)M + m, (19) where ⊛ calculates the linear convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Moving target detection (MTD) is often performed after the matched filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' For ˜yip(t) (∀p, t), a Fourier transform can be performed over i, leading to the so-called range-Doppler map (RDM), ˜Yfp(t) = I−1 � i=0 ˜yip(t)e−j2πfiTPRT, (20) where f and t span the Doppler and range dimensions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Target detection can be performed based on �P −1 p=0 | ˜Yfp(t)|, where the incoherent accumulation over p (indexing spatial channels) is performed as we do not have angle information yet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' otherwise, the coherent beamforming can be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The constant false-alarm rate (CFAR) detector has been popu- larly employed for target detection and hence will be used for our simulation and experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Interested readers are referred to [30] for the details of the CFAR detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' An intuitive simulation tutorial is also available at [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Let (f ∗, t∗) denote the location of a target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Extracting the signal at each RDM, we obtain z = � ˜Yf ∗0(t∗), ˜Yf ∗1(t∗), · · · , ˜Yf ∗(P −1)(t∗) �T , (21) where P = MN according to (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Angle estimation is carried out using z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Based on (18), the steering vector depicting the spatial information in z can 6 FMCOMMS3 PA0 PA1 LNA0 LNA1 FMCOMMS3 TX0 TX1 RX0 RX1 RX0 ZedBoard Radio Hardware ZC706 Radio Hardware Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 1: The schematic diagram of the testing system be written as ˜a(θ) = ar(θ) ⊗ at(θ), where ⊗ denote the Kronecker product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' In practice, the radio frequency chains as- sociated with transmitter and receiver antennas always present some differences, as depicted by et and er, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Incorporating them, the steering vector can be revised as ˘a(θ) = ˜a(θ) ⊙ (er ⊗ et).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Therefore, a naive angle estimate can be obtained as ˆθ = argmaxθl=2πl/L |˘a(θl)Hz|2, l = 0, 1, · · · , L − 1 (22) where L can take a relatively large value for a fine spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' More advancing methods, such as the multiple signal classification (MUSIC) [32] and the DFT interpolation- based methods [33], can be employed for more accurate angle estimation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' OVER-THE-AIR FH-MIMO DFRC EXPERIMENTS In this section, we perform over-the-air experiments to validate the proposed FH-MIMO DFRC scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Schematic Diagram and Experiment Platform The schematic diagram of our testing system is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' We employ the Xilinx Zynq software-defined radio (SDR) ZC706 [34] and ZedBoard [35] to build the FH-MIMO radar and communication receiver, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Both SDRs are equipped with RF FPGA mezzanine card (RF FMC) boards, FMCOMMS3 [23] in specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Each FMCOMMS3 supports two transmitting and two receiving RF chains with the RF range of 70 MHz ∼ 6 GHz and a baseband frequency range of 200 KHz ∼ 56 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' A 40 MHz oscillator with the stability of 10 ppm is used by each FMCOMMS3 [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Note that both SDRs are supported by MATLAB [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Thus, they are connected to host computers, where MATLABs are installed and used to program and control the SDRs independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' For the FH-MIMO radar, we use MATLAB to generate the base-band signals of the two transmitting antennas for a CPI and download them to ZC706 once through Ethernet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The SDR is configured to cyclically transmit the signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The two radar receiving channels are configured to capture echo signal in a consecutive time of 204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='8 ms, corresponding to 8, 192, 000 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Also note that the maximum number of samples that can be transferred in one capture is 8, 388, 608, A Power C PA G Rx Antenna E ZC706 Antenna Front B Tx horn Anchor Target 0 1 2 3 4 5 6 7 8 9 10 11 D FMCOMMS3 F LNA E C F A D G B Ethernet cable Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 2: The established FH-MIMO radar, where the host computer is not shown but connected with the SDR through an Ethernet cable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' TABLE I: Parameters of the Established FH-MIMO Radar Variable Parameter Value Central frequency 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='5GHz power amplifier gain 43dB M Number of horn antennas 2 horn antenna gain 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='39dB horizontal beamwidth of horn antenna 16◦ vertical beamwidth of horn antenna 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='5◦ Maximum transmit power of PA 20W N Number of radar receiver antennas 12 Receiving antenna element gain 13dB Horizontal beamwidth of microstrip antenna 120◦ Vertical beamwidth of microstrip antenna 20◦ Low noise amplifier gain 23dB which is a limitation of the employed SDR [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Note that the echo capture needs to be triggered in MATLAB of the host computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The communication receiver, as configured similar to the radar receiver, also needs a triggering signal from the MATLAB connected to the SDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Next, we provide more elaborations on radar and communication sub-systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 1) Radar subsystem: Based on ZC706, we build an FH- MIMO radar platform, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' A host computer (not shown in the figure) is connected to the SDR through an Ethernet cable (which is shown on the lower left side).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The RF board, FMCOMMS3, is underlain by AD9361 which has the maximum output power of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='5 dBm at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='5 GHz [37], where 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='5 GHz is the carrier frequency used in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Moreover, we use external power amplifiers (PAs) to increase the radar transmission power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The maximum output power of the employed PA is 20 W, i,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=', 43 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' By controlling the output power of AD9361, the transmission power is fixed at 2 W in the sequential experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Moreover, two identical horn antennas are also used for radar transmission, each connected to a RF chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' For radar receiving, a microstrip uniform linear array of 12 antennas is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The antenna spacing is half-the- wavelength at the center frequency of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='5 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Two low-noise amplifiers (LNAs) are used, one for each receiving RF chain on FMCOMMS3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Key Parameters of the above components are listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 7 TABLE II: Parameters of the FH-MIMO Radar Variable Parameter Value B the Signal Bandwidth 20MHz fk radar sub-band baseband frequency −10 : 1 : 9 MHz T hop duration 1µs H number of hops per pulse 5 Tp PRT 40µs Nc number of PRTs per CPI 128 fs sampling frequency 40 MHz Np number of samples per PRT 1600(= fsTp) Several notes are given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' First, AD9361 has its gain adjustable in −89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='75 ∼ 0 dB for the transmitting RF chain and 0 ∼ 61 dB for receiving RF chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Thus, together with the gains from other components, see Table I, the maximum transmitting and receiving power gain of the FH-MIMO radar built in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 2 can be 43(= 0 + 43) dB and 84(= 23 + 61) dB, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Second, during experiments, we place the two horn antennas 6λ apart from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' According to the MIMO radar processing illustrated in Section III-D, the way how the transceiver arrays are placed leads to a virtual array of 2 × 12 = 24 antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Third, limited by the number of receiving RF chains on FMCOMMS3, the time division multiplexing (TDM) MIMO is employed to achieve the above- mentioned virtual array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 2, the receiving antenna array has 12 elements, each connected to an SMA port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' However, we can see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 3, there are only two receiving RF chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Thus, we collect echo signals from the 12 antennas in six consecutive data captures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' In the n-th (n = 0, 1, · · · , 5) capture, the two receiving SMAs are connected to the n-th and (n + 6)-th antenna element shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Interested readers are referred to [38] for more details on TDM-MIMO radars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Based on the hardware features illustrated in Section IV-A, we set the parameters of the FH-MIMO radar, as given in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' As the considered FH-MIMO radar is a pulsed radar, the receiver channel suffers from strong self-interference when the transmitter works, leading to a blind zone of CHT/2 (in meter), where C denotes the microwave speed, T is a hop duration and H is the hop number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Given the limited link budget, see Table I, the maximum measurable distance of the radar platform would be very limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Thus, we want to keep the blind zone small as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' To do so, we set T = 1µs and H = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' This leads to a blind zone of 750 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Other parameters in Table II are straightforward based on the descriptions therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 2) Communications Subsystem: It is built on the SDR Zed- Board, FMCOMMS3 and an 8 dBi omnidirectional antenna, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' As only downlink communication is consid- ered in this work, the communication subsystem only receives, hence much simpler compared with the radar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' We do not use external LNA for the communication subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Thus, its receiving power gain ranges in 0 ∼ 61 dB, solely dependent on AD9361 of FMCOMMS3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Based on the parameters given in Table II, we can calculate the data rate of the radar-enabled communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' As illus- trated in Section II-B, the combinations of hopping frequencies are used as communication data symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Given K = 20 sub- C Zedboard B FMCOMMS3 A Antenna TX1 RX1 TX0 RX0 A B C Ethernet cable Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 3: the Communication Subsystem Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 4: The 50 random targets for simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 55 -47 -39 -31 -23 10-1 100 101 102 deg N T 55 -47 -39 -31 -23 SNR 100 101 102 103 m 55 -47 -39 -31 -23 10-1 100 101 102 103 m/s Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 5: Root mean squared error (RMSE) of angle, distance and velocity estimations under different SNR, where “N” and “T” represent new and traditional FH-MIMO radar waveforms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' bands and M = 2 transmitting antennas, we have C2 20 = 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Considering the integer number of bits, out of 190 combina- tions, 128 numbers of combinations can be used to convey 7 bits per radar hop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Given H = 5, using the combinations convey a total of 7 × 5 = 35 bits per PRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Moreover, PSK is also employed for information demodulation, one symbol per hop and antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Thus, Considering an x-bit PSK modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The overall number of bits conveyed by PSK is xMH = 10x per PRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' In summary, the communication data rate is (35 + 10x)/Tp = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='875 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='25x) Mbps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' (23) For x = 1, 2, 3 and 4, the data rate is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='125, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='375, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='625 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='875 Mbps, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Simulation Analysis Before performing experiments, simulations are carried out to validate the proposed FH-MIMO DFRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' We start with validating the radar performance, where the radar is configured as per Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' As for sensing scenario, we set 50 targets with random speeds, distances and angles which are uniformly dis- tributed in [−170, 170] m/s, [750, 4, 185] m and [−4, 4] deg, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' To evaluate the root mean squared error (RMSE) of parameter estimations, we perform 100 independent trials, pa 0 100 200 340 0 1000 2000 3000 40 Distance (m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='85 80 00 5000 6000 75340 200908 5 0 5 10 15 SNR 10-5 10-4 10-3 10-2 10-1 100 BER P_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='5_16 P_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='5_8 P_1_16 P_1_8 F_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='5_16 F_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='5_8 F_1_16 F_1_8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 6: Simulation results of BER curve versus SNR with target parameters randomly generated over trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' For each trial, we perform the radar processing illustrated in Section III-D for target detection and estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Both the conventional FH-MIMO radar in Section II-A and the one modified for DFRC in Section III-C are simulated for a comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 4 plots a snapshot of a range-Doppler map (RDM) obtained in one trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' We can see many strong points scattering over the range-Doppler domain, each point representing a target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' As illustrated in Section III-D, CFAR is performed based on an RDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Then the delay and Doppler bins of the detected targets are used to estimate their parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 5 plots the RMSEs of angle, distance and velocity averaged over all targets and trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' We see that the RMSEs of all parameter estimates first decrease and then converge, as SNR increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' This complies with general understanding, and the convergence is due to the quantized distance, velocity and angle grids used during the estimation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' see Section III-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' More importantly, we see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 5 that the traditional and new FH-MIMO radar waveforms lead to almost the same estimation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' This validates that the proposed waveform designs for DFRC only incur minimal changes to the underlying FH-MIMO radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Next, we demonstrate the communication performance of FH-MIMO DFRC employing the metric of bit error rate (BER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' FHCS and PSK, as illustrated in Section II-B, are simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Most radar configurations in Table III are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' But here, we also consider two different hop duration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='5µs and 1µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' As for the PSK, we simulate 8PSK and 16PSK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' In this simulation, we let the communication receiver know the channel responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Thus, the results here provide a performance lower bound of the experiment results to be presented shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 6 plots the BER performance of different modulations under different settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' We see that FHCS generally has lower BER than PSK modulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' This is consistent with previous works, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=', [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' We also see that when the hop duration doubles, both FHCS and PSK achieve better BER performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' This is because the demodulation SNR increases with the hop duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' For FHCS, we see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 6 that the modulation order does not affect its BER performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' This is due to the way FHCS is demodulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' In particular, as illustrated in Section II-B, we only need to identify DFT peaks for demodulating FHCS, where the phases of peaks are irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Experimental Results Employing the hardware platforms illustrated in Section IV-A, we perform over-the-air experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='2, the radar transceiver is placed on top of a building with the height of about 20 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 7 plots the radar imaging results, where the observation distance is up to 3 km from the radar and the angular region is [−30◦, 30◦] around the normal direction of the radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' To plot the 2D radar imaging, the zero-th Doppler channel of the MTD result, as obtained in (20), is extracted in each CPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Then, the beamforming, as illustrated in (18), is performed to scan the angular region with a step of 1◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' To calibrate the radar transceiver arrays, we use a known target (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=', target B in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 7) to calculate the array calibration coefficients, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=', er ⊗ et given in (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' In particular, we place the radar transceiver in such a way that target B is in the normal direction of the radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Since the target distance is known, we extract the signal of the p-th (p = 0, 1, · · · , P −1) virtual spatial channel, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=', the signal given in (20), at the known distance and zero-th Doppler bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Given the radar configuration in Table II, we have P(= 24) virtual channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Ideally, the extracted signals should be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' But, as affected by the array calibration errors, their values can be distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Thus, we use the signal of the first virtual spatial channel as a reference, and all other extracted signals are normalized to the reference one, leading to the array calibra- tion vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' In fact, in our experiment, we do not have the facilities to calibrate the radar transceiver arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Using the above anchor-based method, we are able to obtain a relatively good calibrated array, as demonstrated by the high match between the measured and map-illustrated targets in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' In the figure, we superimpose the radar imaging above a satellite map of the observed area, where the map is obtained from [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' We see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 7 that the strong signals in the radar imaging match well with the objects observed on the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The distance profiles at the three selected angels further highlight most pinpointed targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' This illustrates the effective- ness of the above-mentioned array calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' More impor- tantly, this validates that the proposed waveform modifications for FH-MIMO radar to accommodate data communications do not obviously affect the primary radar function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Next, we illustrate the communications performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' In the first set of experiments (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=', Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 7, 8 and 9), we place the communication receiver, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 3, about ten meters behind the radar transmitter antennas (which are placed outdoor as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' That is, the communication receiver is in the line-of-sight (LoS) posterior views of radar transmitter antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The proposed communication demodula- tion method, as summarized in Algorithm 1, is performed on collected experiment data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 8(a) plots the CFO estimate obtained each pair of two consecutive PRTs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' see (11), since there are 128 PRTs, there 9 30 25 20 15 10 5 Radar A B D E F G H I J K L C Normalized Power (dB) Distance (m) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 7: A two-dimensional space-distance radar imaging using the proposed FH-MIMO DFRC waveform and the hardware platform built in Section IV-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The zero-Doppler channel is observed, and hence only static targets are shown in the radar imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Representative targets are pinpointed based on the latest satellite map of the location [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The distances profiles of three angles covering most pinpointed targets are separately plotted in the upper-right sub-figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The lower-right sub-figure is a trimmed version of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 2, giving a glimpse on the radar platform and the surrounding environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 0 50 100 PRT NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='5 10 CFO estimate (KHz) Tx0 Tx1 m0 m1 (a) 0 50 100 PRT NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 40 20 0 20 40 d Tx0 Tx1 (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 8: The CFO estimate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=', � ∆ω, is illustrated in (a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' the estimate of arg{dmk} in (b), where � ∆ω is obtained in (11) and dmk in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' In a CPI, the value of k cyclic increases with the PRTs, and it can be known from (14) that for a k, arg{dmk} increases with the PRT index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' are 127 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' We see that there is a non-negligible CFO between the radar transmitter and the communication receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Moreover, we also see that the two communication receivers have different CFOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' In addition, we see that the CFO estimate changes over time but stays around approximately a fixed value (m0 and m1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 8(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' This validates the slow-varying feature of CFO in a certain coherent processing period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' This also confirms that we can average the estimates of CFO over a suitable time period to obtain a more accurate estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 8(b) plots the estimate of dmk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' We see that dmk changes over PRTs in the way depicted by (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Recall that, in the proposed waveform design, the i-th PRT estimates dmk for k = (i)K−1, where ()K−1 denotes modulo-(K − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 9 provides the scatter plots of the demodulated com- munication symbols, where 8PSK is given in the first row and 16PSK is in the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Three demodulation methods (a) (b) (c) (d) (e) (f) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 9: Illustrating demodulation performance, where the top row is for 8PSK and the second row is for 16PSK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The first column is the demodulation results without considering frequency-dependent transceiver gains, which approximates the results obtained in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The middle column is based on the proposed design given in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The third column further averages the estimates of D over a CPI at the same frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' are performed based on the same collected experiment data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The first method, leading to the results in the first column, neglects the frequency-dependent transceiver gains, which is essentially the method in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The second method, as summarized in Algorithm 1, leads to the results in the middle column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Following the overall steps of Algorithm 1, the third method, further averages D obtained in Step 4b) of Algorithm 1 in a CPI, thus improving the estimate of �̟i2h2m and then T000500030-50TOBDCK8 CEH10 Rx Tx Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 10: Indoor BER Test Scenario 0 5 10 15 20 Gain Control (dB) 0 5 10 15 20 Actual Gain (dB) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='5_16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='5_8 1_16 1_8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 11: actual gain versus receive gain demodulation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 9, we see that neglect- ing frequency-dependent transceiver gains can substantially degrade the communication performance of FH-MIMO DFRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' This validates the critical importance of the proposed de- signs which specifically account for the frequency-dependent transceiver gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' From the middle column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 9, we see that the proposed waveform and demodulation method can achieve relatively good communication performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' By further averaging arg{D} over time, where D is given in (15), we can further improve the estimation accuracy of ̟i2h2m and hence the communication performance, as manifested in the third column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' In the second set of experiment for validating commu- nication performances, we observe the BER performance under different SNRs, as simulated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' To precisely control demodulation SNRs, we use two SDRs indoors, one transmitting and one receiving, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The transmitting SDR is the one used for radar transmitter, but is equipped with two omni-directional antennas (with 12 dBi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' During the experiment, we set the gains of the two transmitting RF chains as 0 dB and adjust the gains of the receiving RF chains to achieve different demodulation SNRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 11 plots the actual gain under different gain control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The actual gain is estimated by averaging the power of the received signal which is normalized to the estimated gain under the 0 dB gain control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 12 plots the BER versus SNR, where 800 CPIs are collected, corresponding to 1, 024, 000 demodulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' We see that the trends of all BER curves match well with what observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' This validates the proposed design in practical DFRC scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The curves are not as smooth as the simulated ones, mainly because the gain control curve is not ideally linear and stable during the collection of four sets of data, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' CONCLUSION In this work, a practical FH-MIMO DFRC is developed comprehensively treating all practically inevitable hardware errors, including STO, CFO and front-end imperfections of 0 5 10 15 20 Receive Gain (dB) 10-6 10-5 10-4 10-3 10-2 10-1 100 BER P_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='5_16 P_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='5_8 P_1_16 P_1_8 F_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='5_16 F_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='5_8 F_1_16 F_1_8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 12: Curve of BER versus receive gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Since the linearity of the actual gain curve is not very good, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 11, there is a certain error in the BER result of the actual test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' We model these errors and analyze their impacts on FH-MIMO DFRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Moreover, we design new waveforms and develop a low-complexity algorithm jointly estimating all hardware errors at a communication receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' In addition, we build an FH-MIMO JRC experiment platform employing low- cost SDR and COST products that are popular in IoT system designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Outdoor and indoor experiments are conducted using the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Applying the proposed designs on the collected experiment data achieves high performances for both radar and communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' APPENDIX As a radar PRT, Tp = NpT t s , where T t s denotes the sampling time at the radar transmitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Thus, to compare Tp and Np∆Ts, we need to show that ∆Ts ≪ T t s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Since ∆Ts is the sampling time difference between the radar transmitter and the communication receiver, we can perform the following calculation, ∆Ts = T t s − T r s = f r s − f t s f ts f rs = −∆fs fs t(fs t − ∆fs) = −ρ f ts (1 − ρ), (24) where we use the superscripts (·)t and (·)r to differentiate the variables between the transmitter and receiver, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' fs = 1/Ts denotes the sampling frequency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' ∆fs = f t s − f r s ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' and ρ = ∆fs/f t s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The reason of introducing ρ is that it can be linked with the clock stability at the radar transmitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Specifically, we have ρ = ∆fs/f t s = ∆fCLK/f t CLK, (25) where f t CLK is the transmitter clock frequency, f t s is a multiple of f t CLK, and ∆fs is the same multiple of ∆fCLK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' The typical value of ρ is about tens of parts per million (ppm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Substituting the value into (24), we obtain ∆Ts ≈ 10−4T t s ≪ T t s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' 11 REFERENCES [1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Cui, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' Jing, 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+page_content=' [39] GaoDe, Map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content=' https://ditu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='amap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} +page_content='com/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFJT4oBgHgl3EQfSyxE/content/2301.11501v1.pdf'} diff --git a/edE3T4oBgHgl3EQfHQkd/vector_store/index.faiss b/edE3T4oBgHgl3EQfHQkd/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..87084feb53b4f2e2e173329582f148a93cc42e99 --- /dev/null +++ b/edE3T4oBgHgl3EQfHQkd/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:db1300a2af01c2bae9fb6b75af860334af322fbf4ab35a7422d94c885f657acf 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Deep neural networks (DNNs) are a state-of-the-art technol- +ogy, capable of outstanding performance in many key tasks. However, +it is challenging to integrate DNNs into safety-critical systems, such as +those in the aerospace or automotive domains, due to the risk of ad- +versarial inputs: slightly perturbed inputs that can cause the DNN to +make grievous mistakes. Adversarial inputs have been shown to plague +even modern DNNs; and so the risks they pose must be measured and +mitigated to allow the safe deployment of DNNs in safety-critical sys- +tems. Here, we present a novel and scalable tool called gRoMA, which +uses a statistical approach for formally measuring the global categorial +robustness of a DNN — i.e., the probability of randomly encountering +an adversarial input for a specific output category. Our tool operates +on pre-trained, black-box classification DNNs. It randomly generates in- +put samples that belong to an output category of interest, measures the +DNN’s susceptibility to adversarial inputs around these inputs, and then +aggregates the results to infer the overall global robustness of the DNN up +to some small bounded error. For evaluation purposes, we used gRoMA +to measure the global robustness of the widespread Densenet DNN model +over the CIFAR10 dataset, and our results exposed significant gaps in +the robustness of the different output categories. This experiment demon- +strates the scalability of the new approach, and showcases its potential +for allowing DNNs to be deployed within critical systems of interest. +Keywords: Neural networks · adversarial robustness · probabilistic ver- +ification. +1 +Introduction +Deep neural networks (DNNs) have become fundamental components in many +applications that perform classification [18,2]. Empirically, DNNs often outper- +form traditional software, and even humans [26,30]. Nevertheless, DNNs have a +significant drawback: they are notoriously susceptible to small input perturba- +tions, called adversarial inputs [10], which can cause them to produce erroneous +outputs. These adversarial inputs are one of the causes that delay the adoption +[*] Both authors contributed equally. +arXiv:2301.02288v1 [cs.LG] 5 Jan 2023 + +2 +N. Levy et al. +of DNNs in safety-critical domains, such as in aerospace [15], autonomous ve- +hicles [19], and medical devices [11]. In these domains, systems are required to +meet a high bar of dependability. While strict guidelines exist for certifying that +hand-crafted software meets these standards (e.g., the DO-178 standard [9] in +the aerospace industry), no such guidelines currently exist for certifying systems +incorporating DNNs. Although various regulatory agencies have identified this +gap and created work groups and road-maps to address it [8], certifying DNN +robustness remains an open problem. +The formal methods community has begun addressing this need, by devis- +ing methods for rigorously quantifying the local robustness of a DNN; i.e., its +robustness to adversarial inputs around a specific point within the input space. +However, to the best of our knowledge, there exist no scalable tools that can mea- +sure the global categorial robustness of a DNN; i.e., the aggregated robustness of +all points within the input space that belong to a category of interest. +Here, we present the gRoMA tool, which measures the probabilistic global +categorial robustness (PGCR) of a given DNN. The DNN is treated as a black +box: gRoMA makes no assumptions, e.g., about the Lipschitz continuity of the +DNN, the kinds of activation functions it uses, or its internal topology. Instead, +gRoMA uses and extends the recently proposed RoMA algorithm [21] for mea- +suring local robustness. gRoMA repeatedly invokes this algorithm on a random +collection of samples, drawn to represent a specific output category of interest; +and then aggregates the results in order to compute a global robustness score +for this category, across the entire input space. As a result, gRoMA is highly +scalable, typically taking only a few minutes to run, even for large networks. +Further, the tool formally computes an error bound for the estimated PGCR +scores, using Hoeffding’s inequality. Thus, gRoMA’s results can be used in the +certification process for components of critical systems. +For evaluation purposes, we focused on a Densenet DNN [14], trained on the +CIFAR10 dataset [18]; and then measured the network’s global robustness using +100 arbitrary images for each of the CIFAR10 categories. gRoMA successfully +computed the global robustness scores for these categories, demonstrating, e.g., +that the airplane category is significantly more robust than other categories. To +the best of our knowledge, this is the first time that a scalable tool for computing +the global categorial robustness of large DNNs has been presented. +2 +DNNs and Adversarial Robustness +Neural Networks. A DNN N : Rn → Rm maps input ⃗x ∈ Rn to output +⃗y ∈ Rm. In classifier DNNs, which are our subject matter here, ⃗y is interpreted +as a vector of confidence scores, one for each of m possible labels. We say that N +classifies ⃗x as label l iff arg max(⃗y) = l, i.e., when y’s l’th entry has the highest +score. We use L to denote the set of all possible labels, L = {1, . . . , m}. +Local Adversarial Robustness. The local adversarial robustness of N around +input ⃗x is a measure of how sensitive N is to small perturbations around ⃗x [3]: + +gRoMA: a Tool for Measuring Deep Neural Networks Global Robustness +3 +Definition 1. A DNN N is ϵ-locally-robust at input point ⃗x0 iff +∀⃗x. +||⃗x − ⃗x0||∞ ≤ ϵ ⇒ arg max(N(⃗x)) = arg max(N( ⃗x0)) +Intuitively, Definition 1 states that the network assigns to ⃗x the same label that +it assigns to ⃗x0, for input ⃗x that is within an ϵ-ball around ⃗x0. Larger values of +ϵ imply a larger ball around ⃗x0, and consequently — higher robustness. +There are two main drawbacks in Definition 1: (i) it considers a single input +point in potentially vast input space, and knowing that DNN N is ϵ-locally- +robust at ⃗x0 does not imply that it is also robust around other points; and (ii) +it assumes that DNN robustness is consistent across categories, although it has +already been observed that it is possible for some categories to be more robust +than others [21]. To overcome these drawbacks, the notion of global categorial +robustness has been proposed [16,25]: +Definition 2. A DNN N is (ϵ, δ)-globally-robust in input region D iff +∀ ⃗x1, ⃗x2 ∈ D. +|| ⃗x1 − ⃗x2||∞ ≤ ϵ ⇒ ∀l ∈ L. |N( ⃗x1)[l] − N( ⃗x2)[l]| < δ +Intuitively, Definition 2 states that for every two inputs ⃗x1 and ⃗x2 that are +at most ϵ apart, there are no spikes greater than δ in the confidence scores that +the DNN assigns to each of the labels. +Definitions 1 and 2 are Boolean in nature: given ϵ and δ, the DNN is either +robust, or it is not. However, in real-world settings, safety-critical systems can +still be determined to be sufficiently robust if the likelihood of encountering +adversarial inputs is sufficiently low [20]. Moreover, as we discuss later, it is +often useful to measure robustness per output category. To address this, we +propose to compute real-valued, probabilistic global categorial robustness scores: +Definition 3. Let N be a DNN, let l ∈ L be an output label, and let I be a finite +set of labeled data that represents the input space for N. The (ϵ, δ)-PGCR score +for N with respect to l and I, denoted pgcrδ,ϵ(N, l, I), is defined as: +pgcrδ,ϵ(N, l, I) ≜ P ⃗ +x1∈I,|| ⃗ +x1− ⃗ +x2||∞≤ϵ[|N( ⃗x1)[l] − N( ⃗x2)[l]| < δ] +Intuitively, the definition captures the probability that for an input ⃗x1 drawn +randomly from I, and for an additional input ⃗x2 drawn randomly so that it is at +most ϵ apart from ⃗x1, inputs ⃗x1 and ⃗x2 will be assigned confidence scores that +differ by at most δ for the label l. +3 +Statistical Vs. Precise Approaches +Measuring the local adversarial robustness of DNNs has received significant at- +tention in recent years. Two notable approaches for addressing it include: +(i) Formal-verification approaches [24,28,17], which apply constraint solving and +abstract interpretation to efficiently reason about a DNN’s robustness. These +approaches can precisely determine whether a DNN is robust, but typically +afford limited scalability and operate on white-box DNNs. + +4 +N. Levy et al. +(i) Statistical approaches, which evaluate the probability of encountering adver- +sarial inputs. These approaches often need to balance between scalability and +accuracy, with prior work [27,23,13,7] typically leaning towards scalability. +Recently, we introduced the robustness measurement and assessment (RoMA) +algorithm, which is a statistical method that allows balancing between scalability +and accuracy. RoMA works by sampling representative samples of input around +an input point of interest; measuring the confidence scores assigned by the DNN +to incorrect labels on each of these input points; and then using this information +to compute the probability of encountering an input on which the confidence +score for the incorrect label will be high enough to result in misclassification. +This final step is performed by using properties of the normal distribution [21]. +RoMA handles black-box DNNs, without any a priori assumptions; but it can +only measure local, as opposed to global, robustness. +Fewer approaches have been proposed for measuring global adversarial ro- +bustness. Initial work formulated and defined the concept [15,23], pointing out +that global robustness is difficult to check or compute compared to local robust- +ness. More recently, there has been an attempt to use formal verification to check +global adversarial robustness [29]; however, due to the reliance on verification, +this approach has limited scalability and requires a white-box DNN, with spe- +cific activation functions. Another approach [25] redefines global robustness as +an expectation of the maximal safe radius over a test data set, and then pro- +poses an approximate method for computing lower and upper bounds on the +network’s robustness. However, this modified definition is inconsistent with the +common definition of robustness, and focuses on estimating worst-case behavior +— as opposed to average behavior, which is often a more relevant measure [23]. +It is difficult to accurately compute local robustness, let alone global robust- +ness. Consequently, we argue that statistical approaches are a promising vector +for addressing this challenge. +4 +Introducing the gRoMA Tool +Algorithm 1 gRoMA(N, I, l, n, ϵ, δ) +1: ⃗X := drawSamples(N, I, l, n) +2: for i := 1 to n do +3: +if ( N(X[i]) = l ) then +4: +⃗ +plr[i] := RoMA(X[i], ϵ, δ) +5: +end if +6: end for +7: pgcr := aggregate( ⃗ +plr) +8: e := computeError(pgcr, ⃗ +plr, ⃗X) +9: return (pgcr,e) +Algorithm. The high-level flow of +gRoMA implementing Definition 3 is +described in Alg. 1. The inputs to +gRoMA are: (i) a network N; (ii) I, a +finite set of labeled data that repre- +sents the input space, to draw sam- +ples from; (iii) a label l; (iv) n, the +number of representative samples of +inputs classified as l to use; and (v) ϵ +and δ, which determine the allowed +perturbation sizes and difference in +confidence scores, as per Definition 3. gRoMA’s output consists of the computed +pgcrδ,ϵ(N, l, I) score and an error term e, both specific to l. Our method guaran- +tees that, with some high, predefined probability, the distance of the computed +pgcr value from its true value is at most e. + +gRoMA: a Tool for Measuring Deep Neural Networks Global Robustness +5 +In line 1, gRoMA begins by creating a vector, ⃗X, of perturbed inputs — +by drawing from I, at random, n samples of inputs labeled as l. Next, for each +correctly classified sample (line 3), gRoMA computes the sample’s probabilistic +local robustness (plr) score using RoMA [21] (line 4). Finally, gRoMA applies +statistical aggregation (line 7) to compute the pgcr score and the error bound +(line 8); and these two values are then returned on line 9. +gRoMA is modular in the sense that any aggregation method (line 7) and +error computation method (line 8) can be used. There are several suitable tech- +niques in the statistics literature for both tasks, a thorough discussion of which +is beyond our scope here. We focus here on a few straightforward mechanisms +for these tasks, which we describe next. +For score aggregation, we used the numerical average of the local robustness +scores computed for the individual input samples. Additional approaches include +computing a median score and more complex methods, e.g., methods based +on normal distribution properties [5], maximum likelihood methods, Bayesian +computations, and others. For computing the PGCR score’s probabilistic error +bound, we used Hoeffding’s Inequality [12], which provides an upper bound on +the likelihood that a predicted value will deviate from its expected value by more +than some specified amount. +Implementation. We implemented gRoMA as a Tensorflow framework [1]. In- +ternally, it uses the Google Colab [4] tools, and accepts input DNNs in Keras +H5 format [6]. The gRoMA tool is relatively simple, and can be easily extended +and customized to support, e.g., multiple input distributions of interest, vari- +ous methods for computing aggregated robustness scores and probabilistic error +bounds, and also to accept additional DNN input formats. gRoMA is available +online [22]. +5 +Evaluation +Setup and Configuration. We evaluated gRoMA on a Densenet model [14], +trained on the CIFAR10 dataset [18] with a standard 200-epoch training period. +We measured the global categorial robustness of each output category using 100 +images drawn arbitrarily, representing the set I. +These images contained varied angles, lighting conditions and resolutions. +Consistently with prior work [21], we set ϵ to 0.04 and δ to 0.07, so that the +PGCR scores would provide high assurances for a reasonable perturbation size. +Due to our desire to check the approach’s applicability to the aerospace indus- +try, we paid special attention to the airplane category — where we used images +of Airbus A320-200 commercial airplanes, either airborne or on the ground. The +images are available online, alongside our code and dependencies [22]. +Next, for each output category, we used RoMA to compute the probabilistic +local robustness (plr) score for each input sample. We configured gRoMA to use +the numerical average as the aggregation method; and for assessing the error of +gRoMA, we used Hoeffding’s inequality [12]. Specifically, we aimed for a max- +imum expected error value of 5%, which appears to be a generally acceptable + +6 +N. Levy et al. +error value when calculating a DNN’s robustness [13]; and used Hoeffding’s in- +equality to calculate the probability that the error is higher than this value. This +was achieved by setting the upper and lower bounds of the plr values to be plus +and minus five standard deviations of the plr values. +Results. The complete evaluation took less than twenty-one minutes for each +category, using a Google Colab [4] machine. The various global robustness scores +for each category, and the calculated probabilistic error appear in Fig. 1. In +the evaluation, the Airplane category obtained a categorial robustness score of +99.91% — the highest among all categories. The maximum probability of error +larger than 5% of the estimators is in the Ship category, being less than 0.2% — +which is extremely low. +0.000% +0.000% +0.002% +0.086% +0.008% +0.000% +0.000% +0.025% +0.159% +0.107% +99.30% +99.40% +99.50% +99.60% +99.70% +99.80% +99.90% +100.00% +0.00% +0.02% +0.04% +0.06% +0.08% +0.10% +0.12% +0.14% +0.16% +0.18% +Error Bound +PGCR +Error Bound +Global Categorial Robustness Score (PGCR) +Fig. 1: PGCR scores, per category, for all CIFAR10 categories (blue); and the +corresponding statistical errors (yellow). +The PGCR scores calculated above are compatible with previous research on +local adversarial robustness, which indicates that different categories may obtain +different robustness scores [21]. +6 +Conclusion and Future Work +We introduced here the notion of PGCR, and presented the gRoMA tool for +probabilistically measuring the global categorial robustness of DNNs, e.g., cal- +culating the pgcrϵ,δ score — which we regard as a step towards formally quantify- +ing the safety and reliability of DNNs, and their incorporation into safety-critical +applications. While the local adversarial robustness of DNNs has been studied +extensively, we know of no other scalable tool that can measure categorial global +DNN robustness. Moving forward, we plan to empirically validate the accuracy +of gRoMA using different input distributions and sampling methods that rep- +resent diverse input spaces; and also to extend it to additional types of DNNs, +such as regression networks, and others. + +gRoMA: a Tool for Measuring Deep Neural Networks Global Robustness +7 +References +1. M. Abadi, A. Agarwal, et al. TensorFlow: Large-Scale Machine Learning on Het- +erogeneous Systems, 2015. https://www.tensorflow.org/. +2. A. Al-Saffar, H. Tao, and M. Talab. +Review of Deep Convolution Neural Net- +work in Image Classification. In 2017 Int. Conf. on Radar, Antenna, Microwave, +Electronics, and Telecommunications (ICRAMET), pages 26–31, 2017. +3. O. Bastani, Y. Ioannou, L. Lampropoulos, D. Vytiniotis, A. Nori, and A. 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IEEE Access, 7:9893–9902, 2019. + diff --git a/fNE0T4oBgHgl3EQfXQAP/content/tmp_files/load_file.txt b/fNE0T4oBgHgl3EQfXQAP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b2cdd3aa8dcfaf4db1ed0af5d811094c01e503b8 --- /dev/null +++ b/fNE0T4oBgHgl3EQfXQAP/content/tmp_files/load_file.txt @@ -0,0 +1,499 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf,len=498 +page_content='gRoMA: a Tool for Measuring Deep Neural Networks Global Robustness Natan Levy1∗, Raz Yerushalmi1,2∗, and Guy Katz1 1 The Hebrew University of Jerusalem, Jerusalem, Israel {natan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='levy1,g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='katz}@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='huji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='il 2 The Weizmann Institute of Science, Rehovot, Israel raz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='yerushalmi@weizmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='il Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Deep neural networks (DNNs) are a state-of-the-art technol- ogy, capable of outstanding performance in many key tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' However, it is challenging to integrate DNNs into safety-critical systems, such as those in the aerospace or automotive domains, due to the risk of ad- versarial inputs: slightly perturbed inputs that can cause the DNN to make grievous mistakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Adversarial inputs have been shown to plague even modern DNNs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' and so the risks they pose must be measured and mitigated to allow the safe deployment of DNNs in safety-critical sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Here, we present a novel and scalable tool called gRoMA, which uses a statistical approach for formally measuring the global categorial robustness of a DNN — i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=', the probability of randomly encountering an adversarial input for a specific output category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Our tool operates on pre-trained, black-box classification DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' It randomly generates in- put samples that belong to an output category of interest, measures the DNN’s susceptibility to adversarial inputs around these inputs, and then aggregates the results to infer the overall global robustness of the DNN up to some small bounded error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' For evaluation purposes, we used gRoMA to measure the global robustness of the widespread Densenet DNN model over the CIFAR10 dataset, and our results exposed significant gaps in the robustness of the different output categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' This experiment demon- strates the scalability of the new approach, and showcases its potential for allowing DNNs to be deployed within critical systems of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Keywords: Neural networks · adversarial robustness · probabilistic ver- ification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' 1 Introduction Deep neural networks (DNNs) have become fundamental components in many applications that perform classification [18,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Empirically, DNNs often outper- form traditional software, and even humans [26,30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Nevertheless, DNNs have a significant drawback: they are notoriously susceptible to small input perturba- tions, called adversarial inputs [10], which can cause them to produce erroneous outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' These adversarial inputs are one of the causes that delay the adoption [*] Both authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='02288v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='LG] 5 Jan 2023 2 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Levy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' of DNNs in safety-critical domains, such as in aerospace [15], autonomous ve- hicles [19], and medical devices [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' In these domains, systems are required to meet a high bar of dependability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' While strict guidelines exist for certifying that hand-crafted software meets these standards (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=', the DO-178 standard [9] in the aerospace industry), no such guidelines currently exist for certifying systems incorporating DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Although various regulatory agencies have identified this gap and created work groups and road-maps to address it [8], certifying DNN robustness remains an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' The formal methods community has begun addressing this need, by devis- ing methods for rigorously quantifying the local robustness of a DNN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=', its robustness to adversarial inputs around a specific point within the input space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' However, to the best of our knowledge, there exist no scalable tools that can mea- sure the global categorial robustness of a DNN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=', the aggregated robustness of all points within the input space that belong to a category of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Here, we present the gRoMA tool, which measures the probabilistic global categorial robustness (PGCR) of a given DNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' The DNN is treated as a black box: gRoMA makes no assumptions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=', about the Lipschitz continuity of the DNN, the kinds of activation functions it uses, or its internal topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Instead, gRoMA uses and extends the recently proposed RoMA algorithm [21] for mea- suring local robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' gRoMA repeatedly invokes this algorithm on a random collection of samples, drawn to represent a specific output category of interest;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' and then aggregates the results in order to compute a global robustness score for this category, across the entire input space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' As a result, gRoMA is highly scalable, typically taking only a few minutes to run, even for large networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Further, the tool formally computes an error bound for the estimated PGCR scores, using Hoeffding’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Thus, gRoMA’s results can be used in the certification process for components of critical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' For evaluation purposes, we focused on a Densenet DNN [14], trained on the CIFAR10 dataset [18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' and then measured the network’s global robustness using 100 arbitrary images for each of the CIFAR10 categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' gRoMA successfully computed the global robustness scores for these categories, demonstrating, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=', that the airplane category is significantly more robust than other categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' To the best of our knowledge, this is the first time that a scalable tool for computing the global categorial robustness of large DNNs has been presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' 2 DNNs and Adversarial Robustness Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' A DNN N : Rn → Rm maps input ⃗x ∈ Rn to output ⃗y ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' In classifier DNNs, which are our subject matter here, ⃗y is interpreted as a vector of confidence scores, one for each of m possible labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' We say that N classifies ⃗x as label l iff arg max(⃗y) = l, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=', when y’s l’th entry has the highest score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' We use L to denote the set of all possible labels, L = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' , m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Local Adversarial Robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' The local adversarial robustness of N around input ⃗x is a measure of how sensitive N is to small perturbations around ⃗x [3]: gRoMA: a Tool for Measuring Deep Neural Networks Global Robustness 3 Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' A DNN N is ϵ-locally-robust at input point ⃗x0 iff ∀⃗x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' ||⃗x − ⃗x0||∞ ≤ ϵ ⇒ arg max(N(⃗x)) = arg max(N( ⃗x0)) Intuitively, Definition 1 states that the network assigns to ⃗x the same label that it assigns to ⃗x0, for input ⃗x that is within an ϵ-ball around ⃗x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Larger values of ϵ imply a larger ball around ⃗x0, and consequently — higher robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' There are two main drawbacks in Definition 1: (i) it considers a single input point in potentially vast input space, and knowing that DNN N is ϵ-locally- robust at ⃗x0 does not imply that it is also robust around other points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' and (ii) it assumes that DNN robustness is consistent across categories, although it has already been observed that it is possible for some categories to be more robust than others [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' To overcome these drawbacks, the notion of global categorial robustness has been proposed [16,25]: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' A DNN N is (ϵ, δ)-globally-robust in input region D iff ∀ ⃗x1, ⃗x2 ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' || ⃗x1 − ⃗x2||∞ ≤ ϵ ⇒ ∀l ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' |N( ⃗x1)[l] − N( ⃗x2)[l]| < δ Intuitively, Definition 2 states that for every two inputs ⃗x1 and ⃗x2 that are at most ϵ apart, there are no spikes greater than δ in the confidence scores that the DNN assigns to each of the labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Definitions 1 and 2 are Boolean in nature: given ϵ and δ, the DNN is either robust, or it is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' However, in real-world settings, safety-critical systems can still be determined to be sufficiently robust if the likelihood of encountering adversarial inputs is sufficiently low [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Moreover, as we discuss later, it is often useful to measure robustness per output category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' To address this, we propose to compute real-valued, probabilistic global categorial robustness scores: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Let N be a DNN, let l ∈ L be an output label, and let I be a finite set of labeled data that represents the input space for N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' The (ϵ, δ)-PGCR score for N with respect to l and I, denoted pgcrδ,ϵ(N, l, I), is defined as: pgcrδ,ϵ(N, l, I) ≜ P ⃗ x1∈I,|| ⃗ x1− ⃗ x2||∞≤ϵ[|N( ⃗x1)[l] − N( ⃗x2)[l]| < δ] Intuitively, the definition captures the probability that for an input ⃗x1 drawn randomly from I, and for an additional input ⃗x2 drawn randomly so that it is at most ϵ apart from ⃗x1, inputs ⃗x1 and ⃗x2 will be assigned confidence scores that differ by at most δ for the label l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' 3 Statistical Vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Precise Approaches Measuring the local adversarial robustness of DNNs has received significant at- tention in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Two notable approaches for addressing it include: (i) Formal-verification approaches [24,28,17], which apply constraint solving and abstract interpretation to efficiently reason about a DNN’s robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' These approaches can precisely determine whether a DNN is robust, but typically afford limited scalability and operate on white-box DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' 4 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Levy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' (i) Statistical approaches, which evaluate the probability of encountering adver- sarial inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' These approaches often need to balance between scalability and accuracy, with prior work [27,23,13,7] typically leaning towards scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Recently, we introduced the robustness measurement and assessment (RoMA) algorithm, which is a statistical method that allows balancing between scalability and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' RoMA works by sampling representative samples of input around an input point of interest;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' measuring the confidence scores assigned by the DNN to incorrect labels on each of these input points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' and then using this information to compute the probability of encountering an input on which the confidence score for the incorrect label will be high enough to result in misclassification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' This final step is performed by using properties of the normal distribution [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' RoMA handles black-box DNNs, without any a priori assumptions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' but it can only measure local, as opposed to global, robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Fewer approaches have been proposed for measuring global adversarial ro- bustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Initial work formulated and defined the concept [15,23], pointing out that global robustness is difficult to check or compute compared to local robust- ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' More recently, there has been an attempt to use formal verification to check global adversarial robustness [29];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' however, due to the reliance on verification, this approach has limited scalability and requires a white-box DNN, with spe- cific activation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Another approach [25] redefines global robustness as an expectation of the maximal safe radius over a test data set, and then pro- poses an approximate method for computing lower and upper bounds on the network’s robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' However, this modified definition is inconsistent with the common definition of robustness, and focuses on estimating worst-case behavior — as opposed to average behavior, which is often a more relevant measure [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' It is difficult to accurately compute local robustness, let alone global robust- ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Consequently, we argue that statistical approaches are a promising vector for addressing this challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' 4 Introducing the gRoMA Tool Algorithm 1 gRoMA(N, I, l, n, ϵ, δ) 1: ⃗X := drawSamples(N, I, l, n) 2: for i := 1 to n do 3: if ( N(X[i]) = l ) then 4: ⃗ plr[i] := RoMA(X[i], ϵ, δ) 5: end if 6: end for 7: pgcr := aggregate( ⃗ plr) 8: e := computeError(pgcr, ⃗ plr, ⃗X) 9: return (pgcr,e) Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' The high-level flow of gRoMA implementing Definition 3 is described in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' The inputs to gRoMA are: (i) a network N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' (ii) I, a finite set of labeled data that repre- sents the input space, to draw sam- ples from;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' (iii) a label l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' (iv) n, the number of representative samples of inputs classified as l to use;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' and (v) ϵ and δ, which determine the allowed perturbation sizes and difference in confidence scores, as per Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' gRoMA’s output consists of the computed pgcrδ,ϵ(N, l, I) score and an error term e, both specific to l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Our method guaran- tees that, with some high, predefined probability, the distance of the computed pgcr value from its true value is at most e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' gRoMA: a Tool for Measuring Deep Neural Networks Global Robustness 5 In line 1, gRoMA begins by creating a vector, ⃗X, of perturbed inputs — by drawing from I, at random, n samples of inputs labeled as l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Next, for each correctly classified sample (line 3), gRoMA computes the sample’s probabilistic local robustness (plr) score using RoMA [21] (line 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Finally, gRoMA applies statistical aggregation (line 7) to compute the pgcr score and the error bound (line 8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' and these two values are then returned on line 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' gRoMA is modular in the sense that any aggregation method (line 7) and error computation method (line 8) can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' There are several suitable tech- niques in the statistics literature for both tasks, a thorough discussion of which is beyond our scope here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' We focus here on a few straightforward mechanisms for these tasks, which we describe next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' For score aggregation, we used the numerical average of the local robustness scores computed for the individual input samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Additional approaches include computing a median score and more complex methods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=', methods based on normal distribution properties [5], maximum likelihood methods, Bayesian computations, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' For computing the PGCR score’s probabilistic error bound, we used Hoeffding’s Inequality [12], which provides an upper bound on the likelihood that a predicted value will deviate from its expected value by more than some specified amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' We implemented gRoMA as a Tensorflow framework [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' In- ternally, it uses the Google Colab [4] tools, and accepts input DNNs in Keras H5 format [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' The gRoMA tool is relatively simple, and can be easily extended and customized to support, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=', multiple input distributions of interest, vari- ous methods for computing aggregated robustness scores and probabilistic error bounds, and also to accept additional DNN input formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' gRoMA is available online [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' 5 Evaluation Setup and Configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' We evaluated gRoMA on a Densenet model [14], trained on the CIFAR10 dataset [18] with a standard 200-epoch training period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' We measured the global categorial robustness of each output category using 100 images drawn arbitrarily, representing the set I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' These images contained varied angles, lighting conditions and resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Consistently with prior work [21], we set ϵ to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='04 and δ to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='07, so that the PGCR scores would provide high assurances for a reasonable perturbation size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Due to our desire to check the approach’s applicability to the aerospace indus- try, we paid special attention to the airplane category — where we used images of Airbus A320-200 commercial airplanes, either airborne or on the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' The images are available online, alongside our code and dependencies [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Next, for each output category, we used RoMA to compute the probabilistic local robustness (plr) score for each input sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' We configured gRoMA to use the numerical average as the aggregation method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' and for assessing the error of gRoMA, we used Hoeffding’s inequality [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Specifically, we aimed for a max- imum expected error value of 5%, which appears to be a generally acceptable 6 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Levy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' error value when calculating a DNN’s robustness [13];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' and used Hoeffding’s in- equality to calculate the probability that the error is higher than this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' This was achieved by setting the upper and lower bounds of the plr values to be plus and minus five standard deviations of the plr values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' The complete evaluation took less than twenty-one minutes for each category, using a Google Colab [4] machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' The various global robustness scores for each category, and the calculated probabilistic error appear in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' In the evaluation, the Airplane category obtained a categorial robustness score of 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='91% — the highest among all categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' The maximum probability of error larger than 5% of the estimators is in the Ship category, being less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='2% — which is extremely low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='000% 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='159% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='107% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='30% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='40% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='50% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='60% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='70% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='80% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='90% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='00% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='00% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='02% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='04% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='06% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='08% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='10% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='12% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='14% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='16% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='18% Error Bound PGCR Error Bound Global Categorial Robustness Score (PGCR) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' 1: PGCR scores, per category, for all CIFAR10 categories (blue);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' and the corresponding statistical errors (yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' The PGCR scores calculated above are compatible with previous research on local adversarial robustness, which indicates that different categories may obtain different robustness scores [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' 6 Conclusion and Future Work We introduced here the notion of PGCR, and presented the gRoMA tool for probabilistically measuring the global categorial robustness of DNNs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=', cal- culating the pgcrϵ,δ score — which we regard as a step towards formally quantify- ing the safety and reliability of DNNs, and their incorporation into safety-critical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' While the local adversarial robustness of DNNs has been studied extensively, we know of no other scalable tool that can measure categorial global DNN robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Moving forward, we plan to empirically validate the accuracy of gRoMA using different input distributions and sampling methods that rep- resent diverse input spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' and also to extend it to additional types of DNNs, such as regression networks, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' gRoMA: a Tool for Measuring Deep Neural Networks Global Robustness 7 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Abadi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' Agarwal, et al.' metadata={'source': 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Complex Human Activity Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} +page_content=' IEEE Access, 7:9893–9902, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfXQAP/content/2301.02288v1.pdf'} diff --git a/hNE0T4oBgHgl3EQf6gJU/content/tmp_files/2301.02764v1.pdf.txt b/hNE0T4oBgHgl3EQf6gJU/content/tmp_files/2301.02764v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d37eed76aaed82149321b69c74912cb8b8e9d67c --- /dev/null +++ b/hNE0T4oBgHgl3EQf6gJU/content/tmp_files/2301.02764v1.pdf.txt @@ -0,0 +1,2636 @@ +LAGA: A Learning Adaptive Genetic Algorithm for Earth +Electromagnetic Satellite Scheduling Problem +Yanjie Song1, Jie Chun1, Qinwen Yang2, Junwei Ou1, Lining Xing3,∗, Yingwu Chen1 +Abstract +Earth electromagnetic exploration satellites are widely used in many fields due to their wide +detection range and high detection sensitivity. The complex environment and the prolif- +erating number of satellites make management a primary issue. +We propose a learning +adaptive genetic algorithm (LAGA) for the earth electromagnetic satellite scheduling prob- +lem (EESSP). Control parameters are vital for evolutionary algorithms, and their sensitivity +to the problem makes tuning parameters usually require a lot of effort. In the LAGA, we +use a GRU artificial neural network model to control the parameters of variation operators. +The GRU model can utilize online information to achieve adaptive adjustment of the param- +eters during population search. Moreover, a policy gradient-based reinforcement learning +method is designed to update the GRU network parameters. By using an adaptive evolu- +tion mechanism in the algorithm, the LAGA can autonomously select crossover operators. +Furthermore, a heuristic initialization method, an elite strategy, and a local search method +are adopted in the LAGA to enhance the overall performance. The proposed algorithm can +obtain a more optimal solution on the EESSP through sufficient experimental validations +compared to the state-of-the-art algorithms. +Keywords: +reinforcement learning; learning adaptive ; control parameters; electromagnetic +exploration satellite scheduling problem; genetic algorithm; GRU +1. Introduction +In recent years, satellite technology develops rapidly and has changed our lives to a +great extent. During the development process, the surge in user demands and the number +of satellites has made the management of earth electromagnetic satellite (EES) resources +∗Corresponding author +Email addresses: songyj_2017@163.com (Yanjie Song), chunjie0720@163.com (Jie Chun), +yangqq7160@gmail.com (Qinwen Yang), junweiou@163.com (Junwei Ou), lnxing@xidian.edu.cn (Lining +Xing), ywchen@nudt.edu.cn (Yingwu Chen) +1College of Systems Engineering, National University of Defense Technology, Changsha, Hunan, China, +410073 +2School of Computer Science and Engineering, North Minzu University, Yinchuan, China, 750021 +3School of Electronic Engineering, Xidian University, Xi’an, China, 710126 +Preprint submitted to Elsevier +January 10, 2023 +arXiv:2301.02764v1 [cs.NE] 7 Jan 2023 + +more challenging [1, 2]. +EES is an artificial earth satellite equipped with antennas and +signal receiving equipment that can acquire surface electromagnetic signal data. The ability +to meet users’ various detection needs and to obtain the corresponding data promptly is +critical for the control of satellites. The earth electromagnetic satellite scheduling problem +(EESSP) is proposed in response to satellite management challenges. Specifically, the EESSP +is to obtain appropriate plans for a series of EESs under the condition of satisfying various +constraints. As one of the categories of satellite task scheduling problems, the EESSP also +requires reasonable model construction and smart algorithm design to obtain satisfactory +results. +The EESSP has a series of earth electromagnetic satellites and tasks to schedule. Each +electromagnetic detection satellite flies in a fixed orbit, which makes the area they can fly +through limited. +The detection range of the satellite antenna is limited by its antenna +aperture, which is fixed at design time. As for the task, it can only be detected if it is within +the detection range of an EES [3]. In addition, several other settings and satellite operating +conditions extremely limit the number of tasks that can be accomplished. To solve the +EESSP, a scientific mathematical model and an efficient solution algorithm are required to +construct. +Among many solution algorithms, the genetic algorithm (GA) is a meta-heuristic al- +gorithm based on population search with good applications in multi-class combinatorial +optimization problems such as satellite task scheduling problems, vehicle path planning, +and workshop scheduling [4]-[6]. In the GA, crossover and mutation are two of the most +essential evolution operators. Whether these two operators can be efficiently combined to +complete the search to obtain the optimal solution is a key point for the algorithm design. +Crossover implies exploration in the whole solution space, while mutation implies exploita- +tion in a local space. The control parameters controlling the crossover and mutation are +of great importance to find the optimal solution. In general, the crossover probability (de- +noted as CR) of GA is in the interval of [0.7,0.99], and the mutation probability (denoted as +MR) is in the interval of [0.01,0.3]. In the process of solving a combinatorial optimization +problem, a good algorithm keeps exploring new spaces as much as possible in the initial +stage. In contrast, after a certain number of searches, the algorithm should exploit a certain +space. Along with this process, the values of these two control parameters differ in different +problem scenario settings. It is difficult to determine a specific value. In this paper, we +propose a novel evolutionary algorithm that combines reinforcement learning and a genetic +algorithm framework. It is meaningful that this algorithm can work in various scenarios for +solving the EESSP and finding the optimal solution with the smallest possible parameter +tuning cost. The well-performing algorithm can adaptively adjust the control parameters +according to the scenarios’ features. +Among the existing studies on the satellite scheduling problem, the EESSP is still in its +infancy, while the earth observation satellite scheduling problem (EOSSP) has been studied +in many ways. Compared to the EOSSP, the EESSP does not need to consider cloud cover, +resolution, and other factors but additionally needs to consider a series of constraints closely +related to electromagnetic detection such as detection mode, bandwidth, and polarization +mode [7]. The proposed model and method used to solve the EOSSP is a good guide for the +2 + +EESSP. The mixed-integer programming model is a classical model form for constructing +mathematical models of EOSSP in many studies [8]-[10]. Chen et al. proposed a conflict +metric and analyzed the interdependence of time windows to construct a mixed-integer +programming model [8]. +Zhu et al. +used a directed acyclic graph to represent feasible +observation task plans [9]. Valicka et al. proposed an extended two-stage and three-stage +stochastic mixed-integer scheduling model by considering cloud uncertainty [10]. Some other +model forms, such as quadratic scheduling models and graphical models, have also been used +by researchers [11, 12]. Specific detailed research progress about the EOSSP model in detail +is described in [13]. +Due to the high complexity and difficulty in solving the satellite task scheduling prob- +lem, the exact solution algorithm is only able to solve the small-scale problem and is time- +consuming. Small-scale scenarios tend to exist only in theoretical problems, while practical +application problems are commonly large-scale or super-scale scenarios. Therefore, exact +solution algorithms have difficulty finding satisfactory results within a limited time. Heuris- +tic and evolutionary algorithms, on the other hand, can effectively overcome the bottleneck +of the exponential computation time explosion. To find satisfactory task plans, a series of +evolutionary algorithms such as genetic algorithm [14], ant colony algorithm [15], modal al- +gorithm [16], and other improved forms based on simple evolutionary algorithms have been +proposed. Among these algorithms, genetic algorithm has wide application in satellite task +scheduling problems because of good global search performance and fast convergence. Chen +et al. considered the high computational cost of satellite task scheduling problems and de- +signed a population perturbation mechanism in a genetic algorithm. This mechanism can +improve the ability of the algorithm to find the optimal solution [17]. Li et al. used a new +encoding method in a genetic algorithm [18]. Individual coding is used to determine based +on the ground station ID to achieve a reduction in computational complexity during task +scheduling. +Most of the studies related to optimization algorithms focus on the algorithm design +itself, and not enough consideration is given to the factors that affect the performance of the +algorithm. Control parameters have an impact on the search performance of evolutionary +algorithms. +Parameter control varies slightly in different literature. +In [19], parameter +control without learning and parameter control based on learning are classified according to +whether the information used is offline or online. Reference control methods can likewise +be classified as deterministic, dynamic, adaptive, and hybrid control methods combining +multiple methods depending on whether and how they vary [20]. +Sun et al. +proposed +a learning adaptive differential evolutionary algorithm for solving numerical optimization +problems [19]. This algorithm adopts a policy gradient method to train the long and short +memory networks to get good control parameter settings. While there are many factors +affecting the model effect in LSTM and the training period is quite long. +In addition to evolutionary algorithms, reinforcement learning (RL) methods are also +trying to solve the EOSSP. Huang et al. treated the EOSSP problem as a Markov deci- +sion process in continuous time and constructed a reinforcement learning algorithm based +on policy gradient [21]. Wei et al. proposed a reinforcement learning algorithm based on +parameter transfer to solve the multi-objective EOSSP [22]. An encoding-decoding based +3 + +network model was used to obtain the solution to the decomposed subproblem. Lam et al. +learned a heuristic algorithm structure by reinforcement learning to achieve that some sub- +sequent tasks can be selected after a given part of the task solution [23]. Ren et al. designed +a block encoding reinforcement learning training algorithm to solve the Agile EOSSP [24]. +He et al. used a Markov decision process to complete the assignment of observation tasks, +then a specific execution plan was obtained using a dynamic scheduling approach [25]. +The related studies using reinforcement learning methods are few and show a trend of +attention to this type of solution method. Hybridization of reinforcement learning methods +and evolutionary algorithms is a new idea of algorithm design that effectively combines the +respective advantages of both methods [26, 27]. Although there have been many successful +practices of combining reinforcement learning with evolutionary algorithms in numerical +optimization problems [19, 28] and combinatorial optimization problems [29]-[31], few studies +have been done to solve satellite task scheduling problems using this idea [32]. To address the +shortcomings of traditional evolutionary algorithms for new problem scenarios that require +the algorithm to run repeatedly to adjust the control parameters consuming a large number +of computational resources, we propose a novel learning adaptive genetic algorithm, named +LAGA. In the algorithm, a GRU model is used to obtain the control parameters, and the +model parameters are trained using a reinforcement learning method. After training, the +GRU can obtain reasonable control parameters which can promote the search process of +LAGA. The main contributions of this paper are as follows. +1. We construct a mixed-integer programming model for the EESSP. The goal is to find +the task plan with the highest detection profit. In the model, constraints such as satel- +lite abilities and task execution requirements are considered, and transition times between +executing two tasks are treated in the form of maximization functions. +2. We propose an evolutionary algorithm based on adaptive learning. This algorithm +treats the population evolution process as a time series and uses a GRU model to obtain +the control parameters of the evolution operation based on online information prediction. +A policy gradient-based reinforcement learning training method is given for optimizing the +GRU parameters. In the LAGA, a heuristic initialization method is designed to generate +high-quality initial populations. An elite strategy and a local search method are also used +to accelerate the convergence of the algorithm during an iterative search. The numerical +experimental results of each evaluation dimension are combined to show that the proposed +algorithm can obtain plans with high detection yield and improve the overall performance +of the satellite system. +This paper is organized as follows. The second part introduces the description of EESSP +and the mathematical model. The third part introduces the genetic algorithm and reinforce- +ment learning methods based on adaptive learning. The fourth part verifies the performance +of the proposed algorithm through several experiments. The fifth part introduces conclusions +obtained from the study and the future research directions. +4 + +2. Model +In this section, we describe the EESSP in detail and analyze its difficulties first. Af- +ter that, the variables, assumption conditions, objective functions, constraints, and other +elements used to construct the mathematical model are given. +2.1. Problem Description +Within a certain scheduling horizon, a series of EES need to develop task plans for each +satellite to accomplish the detection tasks proposed by users. Working capabilities, task +requirements, and other circumstances of EESs need to consider in the plan. +Each task execution needs to meet the users’ requirements. Specifically, the geographic +location of the task to be detected, the length of the detection time, and the time range +in which the detection are all set before. A signal cannot be detected all the time. It can +only be detected if a satellite has visible time windows and the task is performed within +one visible time window. In addition, detection activities need to be conducted within the +detection time range and within the required detection time length. The angle between the +antenna and the task changes during the motion in the detectable range over the satellite +task. If the angle is too large, the task detection will not be completed. +When an EES just executed one detection task, it cannot immediately detect another +task. The satellite needs to wait for the satellite payload after a series of working parameters +configuration adjustments before meeting the next detection task parameters configuration +requirements. Satellite payload parameter configuration adjustment includes detection mode +adjustment, bandwidth adjustment, frequency band adjustment, etc. The parameter con- +figuration adjustment will make part of the time window resources unavailable, and the +scarcity of resources will increase the difficulty of developing the plan. +The main difficulty of the EESSP problem is over-subscription. +On the other hand, +over-subscription arises due to limited resource capacity, where large-scale tasks exceed the +upper limit of what satellite resources can accomplish in a given time horizon [33]. Several +tasks need to be weighed against the need to add to the plan because each time a new task +is added to the plan it may face one or several tasks that need to be discarded from the plan. +The single observation satellite task scheduling problem has been proven to be an NP-Hard +problem [34]. Compared with it, the EESSP considers more factors and faces more complex +constraints. As a result of all of the above, the EESSP is more challenging. Therefore, +we need to accurately construct the model and design an efficient algorithm for solving the +problem. +2.2. Symbols and Variables +This section introduces the variables and symbols involved in the mathematical model. +Sat :the set of detection satellites, Ns = |Sat|, si denotes satellite i; +Oi: the set of orbits of the satellite i; +ϑi: the maximum angle that can be detected by the satellite i; +T: the set of detection tasks, Nt = |T|, taskj denotes the detection task j; +dj: the required detection time length of the task j; +5 + +restj: the earliest allowable detection time required for task j; +rletj: the latest allowable completion time required for task j; +θmax +j +: the maximum allowable detection angle for task j; +pj: the profit that can be obtained from the successful completion of task j; +TW: the set of time windows, Ntw = |TW|; +twijko: the time window k of the task j in on orbit o for satellite i; +evtijko: the earliest visible time of the task j in the time window k on orbit o for satellite +i; +lvtijko: the latest visible time of the task j in the time window k on orbit o for satellite +i; +t: the moment of satellite flight; +θt +ij: angle between the satellite’s antenna and task j at moment t; +F m +i : function for the detection mode transition time of satellite i; +F b +i : function for the bandwidth setting transition time of satellite i; +F f +i : function for the frequency setting transition time of satellite i; +trm +ijj′: detection mode transition time of satellite i between task j and task j; +trb +ijj′: bandwidth mode transition time of satellite i between task j and task j′; +trf +ijj′: frequency transition time of satellite i between task j and task j′; +trijj′: transition time of satellite i between task j and task j′; +I: a very large integer; +Decision variables: +xijko : whether satellite i performs task j within kth time window on orbit o, if it is +done, xijko = 1; otherwise, xijko = 0; +stijo : the start time of the satellite i to perform the task j on orbit o. +2.3. Mathematical Model +In this section, a mathematical model is constructed. First, the assumptions of the model +are introduced. +Assumptions: +1. +The task is covered by a single satellite detection, without the need for multiple +repetitions. +2. The detection task can be carried out at most once, without considering multiple +repetitions. +3. The impact of satellite sequestration and energy on satellite detection activities is not +considered. +4. Equal value of the profit obtained by the satellite from performing the task at any +moment in the time window. +5. +The detection tasks to be performed by the satellite are determined in advance +before scheduling, and there will be no adjustment of the task performance requirements +during scheduling as well as during the execution of the satellite’s task, such as early or late +completion of the task requirements, temporary cancellation of the task. +6. The satellite can operate normally throughout the whole planning time horizon. +6 + +We give the calculation method for the transition time of satellite i between tasks j and +j′ at first. It can be calculated as follows: +trm +ijj′ = F m +i +� +rm +j , rm +j′ +� +(1) +trb +ijj′ = F b +i +� +rb +j, rb +j′ +� +(2) +trf +ijj′ = F f +i +� +rf +j , rf +j′ +� +(3) +trijj′ = max +� +0, trm +ijj′, trb +ijj′, trf +ijj′ +� +(4) +where rm +j , rb +j, rf +j represent detection mode setting requirement, bandwidth setting re- +quirement for task, frequency setting requirement for task j repectively. +The transition time is the max time of the time required to convert the satellite param- +eters. +Our goal is to find a sequence of tasks that can be executed in the solution space with +high detection profits. Therefore, the objective function is to maximize the profit of the task +plan. The objective function is calculated as shown below. +Objective function: +� +i∈S +� +j∈J +� +k∈TW +� +o∈Oi +pj · xijko +(5) +where pj denotes the profit that can be obtained by the successful completion of task j, +xijko denotes whether satellite i performs task j within kth time window of orbit o. +Constraints: +• The task is to be executed within the required time range. +stijo ≤ restj · xijko, i ∈ Sat, j ∈ T, k ∈ TW, o ∈ Oi +(6) +(stijo + dj) · xijko ≤ rletj, i ∈ Sat, j ∈ T, k ∈ TW, o ∈ Oi +(7) +• The task needs to be executed within the time window in which it can be detected. +stijo ≤ evtijko · xijko, i ∈ Sat, j ∈ T, k ∈ TW, o ∈ Oi +(8) +(stijo + dj) · xijko ≤ evtijko, i ∈ Sat, j ∈ T, k ∈ TW, o ∈ Oi +(9) +• The angle between the satellite detection and the task needs to be less than the +maximum allowable angular requirement. +7 + +θt +ij · xijko ≤ min +� +ϑi, θmax +j +� +, i ∈ Sat, j ∈ T, k ∈ TW, o ∈ Oi, t ∈ [stijo, stijo + durj] +(10) +• Each task can only be detected at most once. +� +i∈S +� +k∈TW +� +o∈Oi +xijko ≤ 1, i ∈ Sat, j ∈ T, k ∈ TW, o ∈ Oi +(11) +• The transition between two tasks needs to meet the transition time requirement. +(stijo + durj) · xijko + trijj′ ≤ stij′o + I · (1 − xij′k′o) , +j ̸= j′, i ∈ Sat, j, j′ ∈ T, o ∈ Oi, k, k′ ∈ TW +(12) +• Decision variables need to be valued in the corresponding ranges. +xijko ∈ {0, 1} , i ∈ Sat, j ∈ T, k ∈ TW, o ∈ Oi +(13) +stijo ∈ N, i ∈ Sat, j ∈ T, k ∈ TW, o ∈ Oi +(14) +3. The Proposed Method +3.1. Embedding GRU in the GA Framework +Figure 1: GRU unit structure +8 + ++ +Zt +tanh +0 +0 +NtThe crossover and mutation in population evolution can be regarded as a random time +series with obvious time-dependent properties. The control parameters affect the way and +probability of the population evolution. So these two operators play a crucial role in whether +the algorithm can find the optimal solution. +Therefore, we adopt a nonlinear equation +prediction idea to obtain parameter configurations that match the search pattern. From +another perspective, the changes in the control parameters during the population evolution +can also be considered a time series. To solve time series-related problems, the recurrent +neural network is the primary choice. The population evolution information includes both +current information and historical information. +The network model is expected to pay +more attention to the information related to the most recent evolutionary generations and +less attention to the information obtained from the earlier time evolution. Based on the +time-dependent characteristic of online information, we use a GRU model to obtain useful +information to facilitate GA search. GRU is a classical recurrent neural network (RNN) +model proposed by Cho, which effectively solves the problems of long memory dependence +and gradient explosion compared to traditional recurrent neural network models [35, 36]. +Moreover, the GRU model requires fewer parameters and fewer training times compared +to the classical long-short memory network (LSTM) model [37]. Each GRU model may +consist of a series of GRU units, which effectively capture the interrelationship of data in +the temporal dimension through the combination of units. The gate structure is a unique +information flow regulation structure for LSTM and GRU because the information memory +gate is omitted from the GRU cell structure. To describe GRU more intuitively, we give its +specific structure in Figure 1. In each GRU unit, the update gate and reset gate are used to +achieve a good prediction of the time series. The following equations can describe the GRU. +hj +t = +� +1 − zj +t +� +hj +t−1 + zj +t ˜hj +t +(15) +zj +t = σ(Wzxt + Uzht−1)j +(16) +˜hj +t = tanh (Wxt + U (rt ⊙ ht−1))j +(17) +rj +t = σ(Wrxt + Urht−1)j +(18) +σ (z) = +1 +1 + e−z +(19) +tanh (z) = ez − e−z +ez + e−z +(20) +where hj +t−1 and xt denote the input of GRU, zj +t denotes the update gate, rj +t denotes a set +of reset gates, ⊙ denotes the multiplication of the corresponding element positions of the +matrix, σ (·) denotes the sigmoid activation function, and tanh (·) is the tanh function. +In the practical application of the GRU model, the complete network model consists of +two types of neural network structures, the GRU unit, and the fully connected network. The +full-connected network is used after the GRU unit to further process the data stream, and +the complete data stream of the GRU network model is as follows: First, the input data is +processed by several GRU units. Then, several fully connected network layers are used to +further process the data. After that, the Softmax function is used to obtain the outputs. +9 + +To simplify the intermediate process, GRU can be abbreviated as: +Ht = GRU(St, Ht−1, WG) +(21) +where WG denotes the parameter of GRU units. And the fully connected network layer +can be abbreviated as: +CRt = Linear(Ht, Wc, bc) +(22) +MRt = Linear(Ht, Wm, bm) +(23) +where Wc, Wm denotes network parameters and bc, bm denotes bias. +According to the above network model, the current state value St is processed as input +data to fit a combination of control parameters that will help the population search. We +denote Ωt = [CRt, MRt] and use such parameters for the population evolution of the gener- +ation t. Once a set of parameters is obtained, the solution space can be searched and mined +using such parameters. The complete data flow can be expressed as follows: +Ωt, Ht = GRU(St, Ht−1, W) +(24) +Figure 2: GRU network model embedding GA in generation t +After obtaining the complete data flow of the GRU model, the process of embedding the +GRU model in the evolution process of the GA generation population can be given. As shown +in Figure 2, the symbol S in the figure represents the selection operator, FE represents the +fitness evaluation, CO represents the crossover operator, and MO represents the mutation +operator. The state information St is the input of the GRU model. The crossover probability +CRt and the mutation probability MRt required for the current population evolution are +obtained through the output action of the neural network. These two probabilities can have +a significant impact on population search. +10 + +Pt +CR, +CO +FE +S +GRU +S +[+} +MO +MR; +F3.2. Reinforcement Learning for Evolutionary Search +A reinforcement learning method is proposed to allow GRU to update the network model +parameters according to the population evolution process. This method can obtain a param- +eter configuration that is more helpful for population search. We construct the population +evolution process of the algorithm as a Markov decision process (MDP). The RL method +allows the agent to choose the appropriate action to obtain a high reward in a dynamic +environment. State, action, reward, and transition constitute the main elements of the RL +method. The main components of the RL method are status, action, reward, and state tran- +sition. Each major component of MDP will be introduced in detail in this section, followed +by an introduction to the policy gradient training method. +3.2.1. State +The state is the input of the GRU network model, which allows the agent to select +appropriate actions for the evolution process of LAGA. To provide plenty of features and +information for choosing actions, the state should accurately reflect the current population +evolution information. In our proposed algorithm, the set of states S denotes a set of states +constitutes. S can be described as follows. +S = {S0, S1, ..., St, ...} +(25) +where St denotes the agent state value at time step t. St consists of a set of representations +of task attributes and population evolution information. The details of the specific attribute +values of the state are as follows. +St = +� +yt +j = (dj, pj, tsj, lj) |j = 1, 2, ..., Nt +� +(26) +where dj is the task duration, pj is the task profit, lj denotes the cumulative arrangement +of the task j in the current population, and its value indicates the number of successful +placements. tsj denotes the range of the indicated time interval, which can be calculated as +follows: +tsj = rletj − restj +(27) +3.2.2. Action +Action is a core part of the MDP. The actions of reinforcement learning are distinctly dif- +ferent in the continuous and discrete action space. Since the reinforcement learning method +belongs to the selection of actions on the continuous action space. Therefore, we obtain +the probability density function of action At under state St based on the parameters θ of +the policy. Assuming that the action probability values obey a Gaussian distribution, the +equation of policy is as follows. +π (At | St) = N +� +At | GRU (St; W) , σ2� += +1 +√ +2πσ +� +St; ˜θ +� exp +� +� +� +� +� +− +� +At − µ +� +St; ˜θ +��2 +2σ2 +� +St; ˜θ +� +� +� +� +� +� +(28) +11 + +where W denotes the parameter of the GRU model and ˜θ = +� +˜θµ, ˜θσ +�T +is the parameter +of the strategy, which is obtained by fitting the neural network model. +The probability density function corresponding to the action is obtained by the input +state. The two control parameters we need are obeyed by this obtained function, and the +required control parameter values can be obtained by sampling according to this probability +density function. +3.2.3. Reward +In our case, the reward should reflect the effect of the population evolution after the +action is taken by the population evolution. As an evolutionary algorithm, the change in +the optimal local solution that can be found for the contemporary population can effectively +reflect the effect of population evolution. Therefore, we use the percentage improvement of +the current population’s optimal fitness function value compared to the previous generation +population’s optimal fitness function value. And when the population becomes worse instead +of finding a better task plan, the reward is updated in the form of penalty. The equation of +reward is as follows. +Rt = f best +t +− f best +t−1 +f best +t−1 +(29) +where f best +t +denotes the optimal fitness function value in the current population and f best +t−1 +denotes the optimal fitness function value in the last generation population. +A better population search performance is implied by higher reward values, which can +also effectively reflect the influence that the control parameters produced on the population +evolution process. +3.2.4. +Transition +The state transition records the agent state changes, actions taken, and rewards obtained. +Since we use the policy gradient method to train the network model, it does not matter +whether the state transition can be recorded or not. This is because the policy gradient +approach uses gradient descent to optimize subsequent strategies directly for the desired +reward. Although state shifts do not have an impact on strategy updates, the triple of +(St, At, Rt) needs to be stored for updating the network model. +After introducing the meaning of each element of the MDP process, the accuracy of the +policy taken by the agent needs to be improved by a reinforcement learning training method. +We introduce the reinforcement learning training method based on policy gradients in the +next section. +3.2.5. Policy Gradient Training Method +The training effect of the GRU network model can make the LAGA have a great impact +on the solution of the EESSP. The algorithm control parameters need to find a suitable +combination scheme on the continuous space. +The policy gradient method, one of the +typical reinforcement learning methods, can effectively cope with the training method of +12 + +model parameter optimization on the continuous space. Here, the policy gradient method +optimizes model parameters by trajectory sampling of a batch. Since the LAGA uses a +population to search the solution space, the batch can be replaced by the population size. +In our study, the policy gradient approach updates network model parameters by per- +forming a gradient descent on the objective function of the reward. The equation used in +the training of the network is as follows: +θt+1 = θt + α∇θL (θt) +(30) +where ∇θL (θt) is the gradient of the reward function and α is the learning rate. ∇θL (θt) +can be further expressed as: +∇θL (θt) = Eτ∼θt +� T +� +t′=0 +∇θ log πθ (At |St) +T +� +t=t′ +rt +� +(31) +Before training the network model, the objective function of reward should be found and +the trajectory should be determined. According to the MDP process constructed by the +population evolution process, a trajectory can be formed from states and actions. When +the number of trajectories is reached at Ntra, the GRU network model parameters are +updated according to the agent states, actions, and rewards using the Eq. 25-29. We use a +REINFORCE Monte Carlo method, then the expectation of ∇Lθ(θ) can be approximated +by sampling Ntra trajectories to obtain: +∇Lθ(θ) ≈ 1 +L +L +� +i=1 +r +� +τ i� T−1 +� +t=0 +∇θ ln π +� +A(i) +t += a(i) +t +| S(i) +t += s(i) +t +� +(32) +where at +i denotes the value of the action belonging to the ith trajectory at time step t. +After that, we can train the network model parameters in this way. The pseudo-code of +the policy gradient method is shown in the Algorithm 1. +As shown in Algorithm 1, the policy gradient training method repeats multiple epochs +for each problem scenario. One epoch needs to obtain the control parameters (line 9) based +on the state values. The states need to be obtained by computing the task scheduling results +and combining them with other features. The LAGA generates new populations (line 10) +and computes the reward (line 12) by population evolution. After completing a series of +trajectories of the algorithm runs, the network model parameters W (line 14) are updated +using the back propagation method. +Algorithm training by multiple epochs allows the GRU model to obtain a set of parameter +configurations that achieve good prediction results. The trained network model is then used +in the LAGA presented in the subsequent section to provide the two control parameters +MRt and CRt for the evolution of the tth generation population. In the next section, we +present the use of the GRU model in the proposed algorithm in combination with other +population search strategies to develop suitable task plans for EESs. +13 + +Algorithm 1: Policy Gradient Training Method +Input: max epoch Epoch, population size Np, the number of trajectories Ntra, +learning rate α, max time step TSmax. +Output: Updated W. +1 Initialize the GRU model parameters W; +2 for epoch = 1 → Epoch do +3 +for tra = 1 → Ntra do +4 +Set t ← 1, H0 = 0; +5 +Initialize LAGA parameters; +6 +P ←Generate an initial population randomly; +7 +while t ≤ TSmax do +8 +Get the latest state St; +9 +Ωt = [CRt, MRt] ←Generate control parameters by GRU (St, Ht−1, W); +10 +Pt ←Population Evolution by LAGA(Pt−1, Ωt, CO, MO); +11 +Ft ←Calculate the fitness function value of the new population(Pt); +12 +Rt ←Calculate reward using Eq. 29; +13 +t ← t + 1 +14 +Update W using Eq. 30 and Eq. 32; +3.3. Learning Adaptive Genetic Algorithm +The LAGA applies reinforcement learning methods to the genetic algorithm framework, +allowing the algorithm to learn the useful information obtained from the population evolu- +tion. An artificial neural network model after training is adopted to provide decision support +for the control parameters. The effective configuration of the control parameters allows the +LAGA to find the optimal solution more effectively. At the initial stage of the search, the +algorithm should focus on exploring new solution spaces. In contract continuously. When +the search reaches a certain stage, it is worthwhile to focus on a small search area. While +using the network model to obtain the parameters, an adaptive crossover approach is used +in the genetic algorithm. Adaptive crossover can effectively ensure the generalization ability +of the algorithm and allow the algorithm to easily select operators that are conducive to +finding better solutions. To further improve the search performance of the algorithm, we +also design a population initialization method and a local search method. The quality of the +initialized population will largely affect the population search performance, while the local +search can improve the exploitation performance of the algorithm in the local search space. +Figure 3 illustrates the overall framework of the LAGA. Compared with the traditional +genetic algorithm, the proposed algorithm has several differences. First, the control param- +eters are no longer set artificially by the designer, but are obtained in a nonlinear predictive +manner by the artificial neural network model. This is an improvement that we have made +to the algorithm. +Second, the proposed algorithm proposes several improvement strate- +gies, including population generation, population evolution, and search strategies. In the +subsequent sections, we introduce the specific processes of the algorithm, initialization of +14 + +population, fitness evaluation, individual selection, population evolution, elite strategy, and +local search. After that, we also analyze the time complexity of the LAGA. +Figure 3: The overall framework of LAGA +3.3.1. Algorithm Overall Process +The learning adaptive genetic algorithm uses a reinforcement learning method to train +the GRU model, which can provide appropriate control parameters for population evolution +operators. A genetic algorithm framework usually contains steps such as initialization of the +population, individual selection, fitness evaluation, and variation. Before each generation +of population search, the LAGA uses a trained artificial neural network model, and the +two main control parameters are obtained based on online information. The adjustment +of parameters allows the algorithm to choose an appropriate search method based on en- +vironment information. It contributes to finding the optimal solution. The algorithm also +15 + +Start +Heuristic rules +Initialization +Elite strategy +N +Y +Update the best +Fitness evaluation +individual +Generate control +GRU model +Local search? +parameters +N +Y +Offspring Generation +Local Search +Selection +Randomly choose +Adaptive +two locations +Crossover +Mutation +Swap +N +Y +Stop? +Continue? +N +Y +Enduses adaptive crossover, an individual evolution operator that selects the better operator +based on search performance. The crossover operators dynamically adjust the probability +of operators by evaluating the iterative search. Details about the population evolution are +given in Section 3.3.5. It is worth mentioning that the algorithm can also adjust the search +strategy according to the population search performance, moving from exploration within +the whole solution space to the exploitation of local space to discover better solutions. +The pseudo-code of the LAGA is shown in Algorithm 2. +As expressed in Algorithm 2, the LAGA first generates a population of Np individuals. +The initial population is obtained by a heuristic initialization method (line 2). After com- +pleting the evaluation of the initial population fitness and algorithm starts a population +search iteratively. In the search process, an elite strategy (lines 20-22) and a local search +method are used (lines 25-29) in addition to the crossover and mutation operators that con- +tain the genetic algorithm. The optimal solution found by the search will become the final +detection task plan after the algorithm search is completed. +In the subsequent sections, we present the main steps of the LAGA, the population +evolution operators, and some improvement strategies to enhance the search performance. +3.3.2. Initialization +The population evolution is based on the initial population, and a series of selection and +variation operators are performed to obtain a high-quality plan. Population initialization +strives to get individuals located in a good position in the search space while ensuring +diversity. Such an initialization approach can significantly reduce the number of searches +required to find a good task plan. So we design a population initialization method in the +algorithm that combines heuristics with randomization. The heuristic rule is donated as +UPF. UPF rule is described as follows and the equation for the unit profit of the task(upj) +is shown in Eq. 33. +Heuristic rule: Calculate unit profits of tasks and generate an individual according to +the index value from highest to lowest. +upj = pj/dj +(33) +where pj is the profit of task j and dj is the required detection time. +If every individual in the population follows the above heuristic rule will make the chro- +mosome structure between individuals highly similar, which will not facilitate the search. +Therefore, to ensure the diversity of individuals in the population, we use a parameter +η. Through the setting of this parameter, some genes within individuals are added to the +chromosome randomly. The parameter η denotes the proportion of chromosomes gener- +ated according to the heuristic rule within an individual. This part of the task generates a +chromosome, ordered according to the heuristic rule. The rest (1-η) proportion of genes is +inserted into the existing chromosome in random positions to form a complete individual. +The pseudo-code of the initialization method is shown in Algorithm 3. +As expressed in the Algorithm 3, a certain proportion of chromosomes for each individ- +ual within the population is generated according to the heuristic rule (Line 4), while the +16 + +Algorithm 2: LAGA +Input: population size Np, GRU model parameters W, task set T, time window +set TW, control parameter of elite strategy Thre1, control parameter of +local search Thre2 +Output: Solution +1 Initialize algorithm parameters; +2 P ←Generate the initial population by Algorithm 3 and calculate the fitness; +3 Set t ← 1, tri1 ← 0, tri2 ← 0; +4 while termination criterion is not met do +5 +Get the latest state St; +6 +Ωt = [CRt, MRt] , Ht ←Generate control parameters by GRU (St, Ht−1, W); +7 +for i = 1 → Np do +8 +indi ←Roulette chooses individuals(P); +9 +if rand () < CRt then +10 +Perform adaptive crossover(indi); +11 +if rand () < MRt then +12 +Perform mutation(indi); +13 +local best indi, local best ←Evaluate the fitness function value(P); +14 +Update the scores of crossover operators; +15 +if local best > gobal best then +16 +gobal best ← local best; +17 +gobal best indi ← local best indi; +18 +else +19 +tri1 ← tri1 + 1; +20 +//Elite Strategy; +21 +if tri1 < Thre1 and gobal best! = local best then +22 +local best indi ← gobal best indi; +23 +if local best < temp local best then +24 +tri2 ← tri2 + 1; +25 +//Local Search; +26 +if tri2 == Thre2 then +27 +new indi ← Local search using Algorithm 4; +28 +P ← Update population by replace the worst individual by new indi; +29 +tri2 ← 0; +30 +t ← t + 1; +31 +temp local best ← local best; +17 + +Algorithm 3: Initialization +Input: population size Np, task set T +Output: initial Population P0 +1 Set η ← 0, PI = []; +2 for i = 1 → Np do +3 +η ← i/(Np + 1); +4 +indii ← Select tasks using UPF(T, η); +5 +PI ←Generate a set of tasks to be inserted; +6 +while PI ̸= ∅ do +7 +task ← Random select a task from PI; +8 +indii ← Select a position randomly and insert task; +9 +Remove the task from PI; +remaining part generates a task set (Line 5) and uses a random approach to select genes +(Line 7) and insert them to the chromosome. After generating the initial population, the +population search will be carried out. +3.3.3. Fitness Evaluation +The purpose of fitness evaluation is to allow the LAGA to identify the parent individuals +of the change operation from the population based on individual performance. Our objective +function in the EEESSP problem is to maximize the profit of the detection task sequence. +Therefore, the evaluation of individual fitness is obtained in the same way as the calculation +of the objective function. The specific calculation is shown in Eq. 5. The results of the +fitness evaluation will also support the computation of the reward value for reinforcement +learning. +3.3.4. Individual Selection +Individual selection selects individuals from the population according to a certain strat- +egy. Then, offspring will generate based on the selected individual. Individual selection is +usually done by roulette, k-tournament, etc. In our algorithm, we use a roulette method to +select individuals from the population for subsequent evolution. The equation for roulette +selection of individuals is as follows. +¯pi = +fi +� +i∈P +fi +(34) +where fi denotes the fitness value of individual i. +After Selecting the individual for variation, it will perform crossover or mutation to gen- +erate an offspring. Variation is vital to finding the optimal solution. A detailed description +of variation is given in the subsequent section. +18 + +3.3.5. Variation +The variation consists of two population evolution operators: crossover and mutation. +The difference between crossover and mutation is the degree of chromosome change within +an individual. +We restrict the population search by controlling the parameters so that +the algorithm achieves efficient use of exploration and exploitation throughout the search +process. This search method allows the algorithm to find the optimal or as close to the +optimal solution as possible. The control parameters for the probability of crossover and +mutation are adapted to the problem scenarios. Other factors, such as the length of the +selected gene fragment within an individual, and the way of variation, can also affect the +population evolution. +In the following, the specific evolution operators of crossover and +mutation are described. +Crossover is one of the most frequent population evolution operators of genetic algorithms +in the search process, attempting to explore the entire solution space. The offspring produced +by using this operator will be significantly different from the parent individual. We propose +an adaptive crossover operation in the LAGA. This adaptive crossover operator initializes +the same score for each crossover rule when initializing the algorithm parameters. Afterward, +weights are determined based on the crossover rules’ scores. A random choice approach is +used to select a rule for crossover based on weights. This approach is extremely similar to +individual selection. The equation for calculating the weights is as follows. +ˆpi = +soci +� +i∈R +soci +(35) +where ˆpi denotes the probability of the ith crossover rule, soci denotes the score of the +ith crossover rule, and R denotes the set of crossover rules. +The crossover rules used in crossover operators are divided into three types, two-point +crossover rule, multi-point crossover rule, and fragment flipping crossover rule, respectively. +In the following, a detailed description of the rules is given. +Figure 4: Two point crossover +Two-point crossover rule: Two equal-length gene fragments are obtained from two dif- +ferent positions of the parent. Without changing their internal order, these two fragments +exchange positions to generate offspring. +19 + +Parent +1 +5 +2 +3 +4 +6 +Offspring +1 +3 +4 +7 +5 +2 +6Figure 5: Multi-point crossover +Multi-point crossover rule: Multiple genes are selected from the parent, and a new gene +fragment is formed according to the relative order in the parent. Then, the fragment is +inserted into the remaining part at a random position to generate offspring. +Fragment flipping crossover rule: Select the start and end points of a gene fragment +from the parent. A new fragment is created in the order from the back to the front of the +selected fragment. The new fragment is placed in the same position as the parent to generate +offspring. +Figure 6: Fragment flipping crossover +These three crossovers affect the chromosome within individuals to different degrees. How- +ever, it is difficult to indicate which crossover operator is significant. Each population evolu- +tion is expected to have the desired effect as much as possible. Therefore, after determining +the crossover used and executing the corresponding evolution, the algorithm updates the +score of the used crossover operator according to the individual fitness change. The score of +the updated value is determined according to the search performance. If the fitness value +of the offspring is increasing compared to the fitness value of the parent, the score is in- +creased by µ1; otherwise, the score is increased by µ2. When a certain number of times +is reached, the weights of the crossover rules will be updated according to the latest score +value according to the Eq. 35. +Compared to crossover, the mutation is simple. The mutation is done by double point +swapping, i.e., two genes are randomly selected from the chromosome of a parent and their +positions are swapped to obtain an offspring. +20 + +Parent +1 +5 +2 +7 +3 +4 +6 +Offspring +5 +7 +2 +4 +9 +3Parent +1 +5 +2 +7 +3 +4 +6 +Offspring +1 +5 +3 +7 +2 +4 +63.3.6. Elite Strategy +The elite strategy is designed to improve the convergence performance of the algorithm. +After each generation of population search is completed, we choose to add the best individ- +ual found through the search directly to the offspring population. An elite individual can +effectively improve the convergence speed, allowing the population search to find higher- +quality task plans quickly. When the population search reaches the threshold Thre1, the +elite strategy is no longer used. +3.3.7. Local Search +Local search (LS) is a way to find the local optimum within a certain solution space, +which can often play a crucial role when the search is not effective in the entire solution +space. We use a 2-opt local search operation, which is a simple and efficient method to +update the neighborhood structure. In the 2-opt, two genes are randomly selected from +the best individual found in the population search so far, and the positions are exchanged +to generate a new individual. A fitness evaluation and comparison process will determine +whether to continue the local search. Local search needs to effectively balance with global +search. If too many times searches are done, it will make the algorithm solution fall into +local optimization without jumping out. Therefore, when there is no further improvement +in the individual fitness value, the local search stage should end and the algorithm returns +to the population search stage. The pseudo-code of the local search is shown in Algorithm +4. +Algorithm 4: Local Search +Input: gobal best individual gobal best indi, fitness of gobal best individual +gobal best +Output: new individual new indi +1 while termination criterion is not met do +2 +gene1, gene2 ← Random select two genes from gobal best indi; +3 +new indi ← Swap two gene positions to generate a new individual; +4 +fitness ← Calculate the fitness function value (new indi); +5 +if fitness > gobal best then +6 +gobal best ← fitness; +7 +else +8 +Loop While; +As expressed in algorithm 4, two genes are randomly selected from the chromosome of the +best individual in the population search (line 2), and the positions are exchanged to obtain +offspring (line 3). After that, the value of the fitness function is evaluated to determine +whether to continue the local search process or to return to the population search (lines +5-8). The local search algorithm will update the best individual for the population search +at the end of the algorithm. +21 + +3.3.8. Termination Conditions +The LAGA ends the algorithm search after a certain search stage and outputs the optimal +solution found as the final task plan. In the proposed algorithm, we use the maximum +number of fitness evaluations (MFE) as the algorithm termination condition. The algorithm +ends when the number of fitness calculations is equal to the maximum number of evaluations. +3.3.9. Complexity Analysis +In the LAGA, the complexity of the GRU model is O(Batch ∗ |T|2 ∗ d), where d denotes +the number of features. The complexity of GA algorithm population search generation is +O(Batch ∗ |T| ∗ |TW|), and the complexity of the local search is O(|T| ∗ |TW|). +Since +|TW| >> d. So the time complexity of the LAGA is O(Batch ∗ |T| ∗ |TW|). While batch +uses population P instead, the time complexity can also be rewritten as O(|P|∗|T|∗|TW|). +4. Experiment +To verify the effectiveness of the LAGA, we design a series of experiments and use +three state-of-the-art algorithms for comparison. In addition, we also examine whether the +strategies used in the algorithm can improve the search performance. +4.1. Experimental Setup +Experimental environment: The experiments in this study are done on a desktop com- +puter with Intel(R) Core(TM) I7-7700 3.6 GHz CPU, 16 GB RAM, and NVIDIA GeForce +2070Ti. The coding environment is Python 3.9.7 + Numpy 1.22.3 + Pytorch 1.11.0 (Cuda +v11.3). +Comparison algorithms: Three algorithms were used to verify the problem-solving per- +formance of the proposed algorithm. We selected a population perturbation and elimina- +tion strategy based genetic algorithm (GA-PE) [17], improved adaptive large neighborhood +search algorithm (ALNS-I) [38], and artificial bee colony algorithm (ABC) [39], respectively. +GA-PE: A perturbation and elimination strategy is used in the genetic algorithm frame- +work to improve the convergence speed of the algorithm. the GA-PE method has effectively +solved the multi-satellite TT&C scheduling problem. +ALNS-I: The improved adaptive large neighborhood search algorithm designs the corre- +sponding destroy and repair method for EOSSP based on the adaptive large neighborhood +search framework and can obtain high-quality observation plans. +ABC: Artificial bee colony algorithm uses three types of bees: employed bees, onlooker +bees, and scout bees to achieve exploration and exploitation of the search space for combi- +natorial optimization problems. +Experimental scenarios: Scenarios with different task scales are used to evaluate the +scheduling performance of the algorithm in all aspects. Since there is no public test set for +the EESSP, we use a random approach to generate a series of scenarios. To represent the +scene concisely, we use the ”A-B” format to represent a scenario. In ”A-B”, ”A” denotes +the number of tasks, and ”B” denotes the number of the scenario of task-scale ”A”. In the +experiment, the tasks are sized from 100 to 1000. For the convenience of differentiation, +22 + +we artificially divide the cases into small-scale sets (denoted as Set I), medium-scale sets +(denoted as Set II), and large-scale sets (denoted as Set III). +Algorithm parameter settings: The number of fitness evaluations for all algorithms is set +to 5000. In LAGA, the population size Np is set to 10, the number of trajectories Ntra is +set to 10, the learning rate α is set to 0.001, the initial score of each crossover rule is set to +50, µ1 is set to 30, µ2 is set to 10, Thre1 is set to 2000, Thre2 is set to 20. The parameter +settings of the other comparison state-of-the-art algorithms were kept consistent with those +in related studies. +Evaluation metrics: All algorithms run 30 times in each scenario to evaluate algorithms’ +performance. The optimal profit (denoted as Best), the mean profit (denoted as Mean), +and the standard deviation (denoted as Std.) are set as evaluation indicators. Based on the +data obtained, the Wilcoxon rank-sum hypothesis test (denoted as WR) is used to analyze +whether the differences between the results obtained by different algorithms are significant. +In terms of the algorithm convergence performance, we evaluate it through convergence +curves. Furthermore, we analyze the effect of the strategy used in the proposed algorithm. +4.2. Results and Analysis +Table 1: Scheduling Results for Set I +Instance +LAGA +GA-PE +ALNS-I +ABC +Best +Mean +Std. +Best +Mean +Std. +WR +Best +Mean +Std. +WR +Best +Mean +Std. +WR +100-1 +891 +891.00 +0.00 +891 +891.00 +0.00 += +891 +891.00 +0.00 += +891 +891.00 +0.00 += +100-2 +805 +805.00 +0.00 +805 +805.00 +0.00 += +805 +805.00 +0.00 += +805 +805.00 +0.00 += +100-3 +806 +806.00 +0.00 +806 +806.00 +0.00 += +806 +806.00 +0.00 += +806 +806.00 +0.00 += +100-4 +819 +819.00 +0.00 +819 +819.00 +0.00 += +819 +817.80 +2.73 +- +819 +819.00 +0.00 += +100-5 +820 +820.00 +0.00 +820 +820.00 +0.00 += +820 +820.00 +0.00 += +820 +820.00 +0.00 += +100-6 +822 +822.00 +0.00 +822 +822.00 +0.00 += +822 +822.00 +0.00 += +822 +822.00 +0.00 += +200-1 +1550 +1542.70 +4.65 +1544 +1530.63 +5.57 +- +1539 +1529.47 +5.66 +- +1544 +1534.87 +4.71 +- +200-2 +1666 +1648.03 +7.91 +1628 +1605.33 +8.61 +- +1621 +1604.83 +9.01 +- +1624 +1607.90 +6.34 +- +200-3 +1615 +1608.27 +3.49 +1601 +1586.57 +5.63 +- +1596 +1585.07 +5.54 +- +1606 +1588.47 +6.05 +- +200-4 +1677 +1671.57 +4.55 +1672 +1662.90 +4.82 +- +1675 +1660.50 +5.61 +- +1680 +1664.47 +5.08 +- +200-5 +1535 +1523.70 +4.33 +1526 +1516.13 +4.80 +- +1530 +1516.33 +5.65 +- +1528 +1519.50 +5.60 +- +200-6 +1589 +1581.93 +5.35 +1570 +1553.17 +7.21 +- +1566 +1553.13 +5.92 +- +1573 +1558.23 +6.41 +- +300-1 +2166 +2147.47 +8.56 +2041 +2013.50 +11.42 +- +2037 +2012.10 +10.66 +- +2065 +2020.80 +12.34 +- +300-2 +2216 +2199.67 +8.09 +2110 +2081.83 +11.02 +- +2106 +2081.17 +11.14 +- +2104 +2084.23 +8.70 +- +300-3 +2211 +2202.20 +4.21 +2107 +2088.00 +9.31 +- +2114 +2083.97 +9.40 +- +2124 +2092.37 +8.82 +- +300-4 +2205 +2194.03 +7.42 +2094 +2078.30 +8.30 +- +2100 +2074.63 +11.63 +- +2105 +2084.10 +7.78 +- +300-5 +2161 +2145.63 +7.70 +2051 +2030.83 +7.51 +- +2057 +2033.80 +9.83 +- +2066 +2040.23 +7.90 +- +300-6 +2153 +2134.20 +10.46 +2035 +1994.63 +15.91 +- +2049 +1999.20 +16.41 +- +2030 +2007.77 +10.55 +- ++/-/= +0/12/6 +0/12/6 +0/12/6 +Firstly, we verify the performance of the algorithms in Set I. As shown in Table 1, +finding the optimal solution in a 100 task-scale scenario is not difficult for all algorithms +used in the experiments. Moreover, the majority of the algorithms’ search performance is +stable, and only ALNS-I still has some gap between the average and optimal values of the +profits obtained in the scenario 100-4. Starting from scenarios with a 200 task scale, the +proposed algorithm can find better optimal values than the compared algorithms. And the +WD indicator shows that the LAGA differs significantly from the other algorithms in most +of the scenarios. +The LAGA performs as well in Set II as in Set I. As shown in Table 2, we can see +that the best performance is obtained except for some scenarios where the results are not +the best in terms of stability. Moreover, the gap between algorithms increases significantly +23 + +Table 2: Scheduling Results for Set II +Instance +LAGA +GA-PE +ALNS-I +ABC +Best +Mean +Std. +Best +Mean +Std. +WR +Best +Mean +Std. +WR +Best +Mean +Std. +WR +400-1 +3096 +3092.00 +2.78 +3096 +3087.20 +4.80 +- +3096 +3088.10 +3.96 +- +3096 +3090.40 +3.67 +- +400-2 +3252 +3238.97 +4.53 +3236 +3223.27 +5.87 +- +3237 +3223.67 +5.97 +- +3239 +3230.07 +6.55 +- +400-3 +3305 +3290.40 +7.06 +3284 +3257.40 +10.24 +- +3270 +3254.60 +9.43 +- +3284 +3266.43 +8.36 +- +400-4 +3097 +3096.37 +1.50 +3097 +3089.03 +5.62 +- +3097 +3087.17 +5.47 +- +3097 +3092.17 +3.91 +- +400-5 +3059 +3049.30 +5.05 +3058 +3040.43 +7.21 +- +3051 +3038.40 +6.12 +- +3055 +3044.87 +5.31 +- +400-6 +3115 +3105.63 +4.23 +3107 +3091.27 +5.26 +- +3107 +3090.60 +6.25 +- +3103 +3095.97 +4.58 +- +500-1 +4061 +4037.67 +10.01 +3946 +3915.50 +15.43 +- +3946 +3913.00 +12.76 +- +3965 +3932.80 +13.16 +- +500-2 +3735 +3724.10 +7.57 +3650 +3618.37 +13.56 +- +3641 +3620.90 +9.42 +- +3673 +3635.07 +13.15 +- +500-3 +3894 +3876.57 +8.02 +3774 +3745.93 +12.38 +- +3768 +3743.57 +10.29 +- +3785 +3758.10 +11.96 +- +500-4 +3987 +3968.80 +9.67 +3924 +3890.30 +12.26 +- +3923 +3888.10 +14.09 +- +3922 +3902.20 +9.41 +- +500-5 +3831 +3810.67 +8.06 +3740 +3694.00 +14.10 +- +3718 +3688.20 +13.67 +- +3740 +3702.13 +13.87 +- +500-6 +3896 +3881.77 +8.48 +3841 +3815.93 +11.00 +- +3835 +3816.63 +8.93 +- +3850 +3828.50 +10.33 +- +600-1 +4549 +4354.17 +12.93 +4399 +4339.70 +20.39 +- +4366 +4334.43 +15.22 +- +4380 +4526.23 +13.61 +- +600-2 +4577 +4549.37 +10.51 +4344 +4316.00 +13.55 +- +4386 +4318.17 +18.61 +- +4374 +4334.03 +13.97 +- +600-3 +4361 +4341.60 +9.89 +4197 +4164.73 +14.68 +- +4197 +4161.73 +13.67 +- +4205 +4180.00 +12.98 +- +600-4 +4575 +4535.70 +14.11 +4347 +4308.70 +19.11 +- +4335 +4310.67 +13.23 +- +4354 +4329.97 +13.19 +- +600-5 +4553 +4526.77 +12.48 +4325 +4256.63 +19.46 +- +4313 +4265.70 +18.49 +- +4302 +4274.57 +14.75 +- +600-6 +4580 +4559.20 +10.11 +4350 +4296.10 +19.45 +- +4342 +4294.30 +22.75 +- +4347 +4315.77 +15.64 +- ++/-/= +0/18/0 +0/18/0 +0/18/0 +compared with results in Set I. In the 600 task-scale scenarios, the gap between the optimal +and average profits of the LAGA and other comparison algorithms can reach about two +hundred. +Table 3: Scheduling Results for Set III +Instance +LAGA +GA-PE +ALNS-I +ABC +Best +Mean +Std. +Best +Mean +Std. +WR +Best +Mean +Std. +WR +Best +Mean +Std. +WR +800-1 +6494 +6481.20 +6.57 +6480 +6457.77 +9.26 +- +6477 +6455.57 +10.74 +- +6475 +6463.80 +6.19 +- +800-2 +6389 +6378.27 +6.59 +6371 +6351.33 +7.15 +- +6375 +6355.70 +9.71 +- +6379 +6364.03 +7.43 +- +800-3 +6462 +6455.97 +4.36 +6449 +6438.57 +6.95 +- +6462 +6441.17 +8.27 +- +6462 +6447.17 +5.34 +- +800-4 +6379 +6367.43 +7.06 +6374 +6349.20 +10.33 +- +6360 +6348.07 +7.65 +- +6374 +6355.57 +8.87 +- +800-5 +6361 +6353.77 +4.35 +6361 +6347.33 +5.57 +- +6358 +6347.43 +6.22 +- +6358 +6351.77 +4.12 +- +800-6 +6103 +6100.03 +3.22 +6103 +6095.03 +4.44 +- +6103 +6095.67 +4.33 +- +6103 +6099.80 +2.76 +- +900-1 +7018 +7007.47 +6.57 +6980 +6963.53 +10.37 +- +6997 +6967.73 +12.33 +- +6991 +6976.67 +7.49 +- +900-2 +7303 +7287.40 +7.15 +7253 +7217.40 +10.89 +- +7246 +7216.30 +13.14 +- +7246 +7230.23 +8.23 +- +900-3 +7006 +6992.50 +10.36 +6942 +6920.07 +13.52 +- +6964 +6915.23 +15.90 +- +6955 +6936.97 +10.33 +- +900-4 +7070 +7051.10 +8.88 +7013 +6984.80 +14.08 +- +7020 +6983.87 +15.60 +- +7016 +6999.60 +9.77 +- +900-5 +7352 +7339.97 +7.08 +7330 +7299.20 +14.40 +- +7341 +7304.10 +12.27 +- +7335 +7317.00 +8.27 +- +900-6 +7193 +7175.77 +8.24 +7154 +7121.97 +11.29 +- +7164 +7130.27 +17.50 +- +7169 +7145.23 +11.16 +- +1000-1 +8047 +8024.27 +12.15 +7918 +7883.97 +15.21 +- +7925 +7880.90 +18.35 +- +7929 +7902.97 +12.46 +- +1000-2 +8027 +8004.53 +9.17 +7878 +7830.87 +21.51 +- +7881 +7835.27 +19.63 +- +7885 +7853.40 +12.99 +- +1000-3 +7803 +7778.63 +11.96 +7664 +7618.47 +18.07 +- +7676 +7622.53 +19.42 +- +7683 +7643.80 +17.35 +- +1000-4 +7926 +7894.03 +11.60 +7693 +7647.40 +14.92 +- +7683 +7640.23 +20.93 +- +7711 +7665.87 +19.66 +- +1000-5 +7976 +7952.07 +12.44 +7868 +7846.97 +12.29 +- +7886 +7849.33 +15.74 +- +7894 +7865.60 +13.77 +- +1000-6 +8061 +8044.77 +9.07 +7925 +7898.33 +14.10 +- +7945 +7898.20 +17.13 +- +7972 +7920.70 +18.18 +- ++/-/= +0/18/0 +0/18/0 +0/18/0 +After using Set II to validate algorithms, the task scale is further increased and the results +in Set III are shown in Table 3. The experiments on the algorithm’s solving ability by large- +scale scenarios can well reflect the balanced performance of the algorithm’s exploration and +exploitation as well as its application prospects. It is not difficult to find that the LAGA +still performs well in Set III. This is due to the good combination of positive population +exploration and local space exploitation. Due to the large solution space, the local search +can often be of great help in improving the solution quality. +To observe the solution performance of the algorithm more intuitively, we present the +results of the average performance of the 400 task-scale scenarios in the form of a bar +chart. As shown in Fig. 7(a)-7(f), the average profit obtained by the proposed algorithm is +24 + +significantly higher than the other three algorithms. And the search performance of the other +three algorithms is close. From the paired statistical tests, the results obtained by LAGA +are significantly different from the state-of-the-art algorithms at the level of p < 0.001. +(a) Average performance of sce- +nario 400-1 +(b) Average performance of sce- +nario 400-2 +(c) Average performance of sce- +nario 400-3 +(d) Average performance of sce- +nario 400-4 +(e) Average performance of sce- +nario 400-5 +(f) Average performance of sce- +nario 400-6 +Figure 7: Average performance in different scenarios +In addition to statistical analysis of algorithm differences, convergence performance is +another significant evaluation criterion for search algorithms. We analyze convergence curves +for 800 and 1000 task-scale scenarios. As shown in Fig. 8(a)-8(l). All these four algorithms +have good global search capability and can exploit solutions in the local space. The heuristic +population initialization method used in the LAGA algorithm has a positive effect on finding +the optimal solution, and this initialization method has outstanding performance in 1000 +task-scale scenarios. After the search process starts, the LAGA has a fast convergence speed +and can start a new search through the search strategy after the convergence encounters a +bottleneck. +Then, we compare the results using the algorithm with the algorithm that uses a random +way to generate the initial population (denoted as LAGA/RI). This experiment can check +the effectiveness of the improved approach adopted by the algorithm. +25 + +2400 +*** +2250 +2100 +Profit +1950 +1800 +1650 +1500 +LAGA GA-PEALNS-I ABC2400 +*** +2250 +2100 +Profit +1950 +F +1800 +1650 +1500 +LAGA GA-PEALNS-I ABC2400 +*** +2250 +2100 +Profit +1950 +F +1800 +1650 +1500 +LAGA GA-PEALNS-I ABC2400 +*** +2250 +2100 +Profit +1950 +F +1800 +1650 +1500 +LAGA GA-PEALNS-I ABC2400 +*** +2250 +2100 +Profit +1950 +1800 +1650 +1500 +LAGA GA-PEALNS-I ABC2400 +*** +2250 +2100 +Profit +1950 +1800 +1650 +1500 +LAGA GA-PEALNS-I ABC(a) Convergence curves of sce- +nario 800-1 +(b) Convergence curves of sce- +nario 800-2 +(c) Convergence curves of sce- +nario 800-3 +(d) Convergence curves of sce- +nario 800-4 +(e) Convergence curves of sce- +nario 800-5 +(f) Convergence curves of sce- +nario 800-6 +(g) Convergence curves of sce- +nario 1000-1 +(h) Convergence curves of sce- +nario 1000-2 +(i) Convergence curves of sce- +nario 1000-3 +(j) Convergence curves of sce- +nario 1000-4 +(k) Convergence curves of sce- +nario 1000-5 +(l) Convergence curves of sce- +nario 1000-6 +Figure 8: Convergence curves of 800 and 1000 task-scale scenarios +26 + +6500 +LAGA- +GA-PE +ALNS-I- +ABC +6480 +6460 +Profit +6440 +6420 +6400 +6380 +0 +1000 +2000 +3000 +4000 +5000 +Evaluation Times6400 +LAGA- +GA-PE +ALNS-I- +ABC +6380 +Profit +6360 +6340 +6320 +6300 +0 +1000 +2000 +3000 +4000 +5000 +Evaluation Times6480 +LAGA- +GA-PE +ALNS-I- +ABC +6460 +6440 +Profit +6420 +6400 +6380 +6360 +0 +1000 +2000 +3000 +4000 +5000 +Evaluation TimesLAGA- +GA-PE +ALNS-I- +ABC +6400 +6350 +Profit +6300 +6250 +6200 +0 +1000 +2000 +3000 +4000 +5000 +Evaluation TimesLAGA- +GA-PE +ALNS-I- +ABC +6360 +6340 +Profit +6320 +6300 +6280 +0 +1000 +2000 +3000 +4000 +5000 +Evaluation TimesLAGA- +GA-PE +ALNS-I- +ABC +6120 +6100H +6080 +Profit +6060 +6040 +6020F +6000 +0 +1000 +2000 +3000 +4000 +5000 +Evaluation Times8100 +LAGA- +GA-PE +ALNS-- +ABC +8000 +Profit +7900 +7800 +7700 +0 +1000 +2000 +3000 +4000 +5000 +Evaluation Times8050 +LAGA- +GA-PE +ALNS-I- +ABC +8000 +7950 +Profit +7900 +7850 +7800 +7750 +7700 +0 +1000 +2000 +3000 +4000 +5000 +Evaluation Times7900 +LAGA- +GA-PE +ALNS-I- +ABC +7800 +Profit +7700 +7600 +7500 +7400 +0 +1000 +2000 +3000 +4000 +5000 +Evaluation TimesLAGA- +GA-PE +ALNS-I- +ABC +8000 +7900 +7800 +Profit +7700 +7600 +7500 +7400 +0 +1000 +2000 +3000 +4000 +5000 +Evaluation Times8000 +LAGA- +GA-PE +ALNS-I- +ABC +7950 +7900 +Profit +7850 +7800 +7750 +7700 +0 +1000 +2000 +3000 +4000 +5000 +Evaluation Times8150 +LAGA- +GA-PE +ALNS-I- +ABC +8100 +8050 +8000 +Profit +7950 +7900 +7850 +7800 +7750 +0 +1000 +2000 +3000 +4000 +5000 +Evaluation Times(a) Results under 300 task-scale scenarios +(b) Results under 600 task-scale scenario +(c) Results under 800 task-scale scenarios +(d) Results under 1000 task-scale scenarios +Figure 9: Results of different population initialization methods in different scenarios +27 + +2400 +LAGA +LAGA/RI +2100 +Profit +1800 +1500 +1200 +300-1 300-2 300-3 300-4 300-5 300-6 +InstanceLAGA +LAGA/RI +4600 +4500 +4400 +Profit +4300 +4200 +4100 +4000 +600-1 +600-2 600-3 600-4 600-5 600-6 +Instance6500 +LAGA +LAGARI +6400 +Profit +6300 +6200 +6100 +6000 +800-1 +800-2 800-3 800-4 800-5 800-6 +InstanceLAGA LAGA/RI +8100 +7950 +Profit +7800 +7650 +7500 +1000-11000-2 1000-3 1000-4 1000-5 1000-6 +InstanceThe results for 300, 600, 800, and 1000 task-scale scenarios are shown in Figure 9(a)- +9(d). From the results, it can be seen that the use of the heuristic initialization method +can effectively improve the search performance of the LAGA. This initialization method can +effectively utilize the knowledge and improve the effectiveness of the algorithm in finding +solutions while maintaining the diversity of populations. It is clear that as the task-scale +increases, the role of the heuristic population initialization method tends to diminish and +then increase. This is because the complexity of the problem depends more on the search +process and solutions obtained by initialization are not decentralized throughout the solution +space. Then, when the search space is large enough, knowledge again drives the population +search in a good direction. In summary, the above experimental results show that the LAGA +performs significantly better than other state-of-the-art algorithms in solving the EESSP. +From several metric aspects, the GA framework combined with an artificial neural network +model and various strategies designed is effective. +5. Conclusion +For the earth electromagnetic satellite scheduling problem (EESSP), we construct a +mixed-integer programming model and design a learning adaptive genetic algorithm. The +LAGA effectively combines the respective advantages of evolutionary algorithms and arti- +ficial neural networks. Combined with the characteristics of population optimization, we +adopt a policy gradient-based reinforcement learning training method to train the GRU +model. The algorithm in this paper also uses a series of improvement strategies besides +artificial neural networks to make the algorithm search more efficient. The elite strategy al- +lows the population search to have better convergence performance at the beginning. While +focusing on the global search, we design a local search method to find the optimal local solu- +tion. The improvement of the algorithm effectively balances the exploration and exploitation +in the population search and makes it easier to find a satisfactory satellite detection plan. +Through extensive experiments, we verify that using adaptive learning methods to adjust +the parameter configuration, which is based on the information obtained from the search, +can allow the genetic algorithm to obtain better plans. +In the future, we will consider other reinforcement learning methods and other ways of +combining algorithms, such as using reinforcement learning methods to generate solutions +or select appropriate operators for population evolution. Reinforcement learning can also be +used to generate offspring population by selecting individuals from the parent population. +As for EESSP, some more complex situations deserve to be considered in the model design +phase, such as some uncertain environmental factors or possible equipment emergencies. +6. Declaration of Competing Interest +The authors declare that they have no known competing financial interests or personal +relationships that could have appeared to influence the work reported in this paper. +28 + +7. Acknowledgements +This work was supported by the National Natural Science Foundation of China (71901213, +72001212), the Special Projects in Key Fields of Universities in Guangdong (2021ZDZX1019). +Jie Chun and Yanjie Song contribute equally to this article. +Thanks to the editor and reviewers for their valuable comments. +References +[1] Shen, X., Zhang, X., Yuan, S., Wang, L., Cao, J., Huang, J., & Dai, J. (2018). 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A reinforcement-learning-based +evolutionary algorithm using solution space clustering for multimodal optimization problems. In 2021 +IEEE Congress on Evolutionary Computation (CEC) (pp. 1938-1945). IEEE. +[33] Jiang, Z. Z., Feng, G., Yi, Z., & Guo, X. (2021). Service-oriented manufacturing: A literature review +and future research directions. Frontiers of Engineering Management, 1-18. +[34] Barbulescu, L., Watson, J. P., Whitley, L. D., & Howe, A. E. (2004). Scheduling space–ground com- +munications for the air force satellite control network. Journal of Scheduling, 7(1), 7-34. +[35] Cho, K., van Merri¨enboer, B., Bahdanau, D., & Bengio, Y. (2014, October). On the Properties of Neural +Machine Translation: Encoder–Decoder Approaches. In Proceedings of SSST-8, Eighth Workshop on +Syntax, Semantics and Structure in Statistical Translation (pp. 103-111). +[36] Chung, J., Gulcehre, C., Cho, K., & Bengio, Y. (2014). Empirical evaluation of gated recurrent neural +networks on sequence modeling. In NIPS 2014 Workshop on Deep Learning, December 2014. +[37] Hochreiter, S., & Schmidhuber, J. (1997). Long short-term memory. Neural computation, 9(8), 1735- +1780. +[38] Chen, Y., Chen, M., Wen, J., Chen, Y., & Xiang, W. (2020). An Adaptive Large Neighborhood Search +30 + +Algorithm for the Satellite Data Transmission Scheduling Problem. International Journal of Aerospace +Engineering, 2020. +[39] Karaboga, D. (2010). Artificial bee colony algorithm. Scholarpedia, 5(3), 6915. +31 + diff --git a/hNE0T4oBgHgl3EQf6gJU/content/tmp_files/load_file.txt b/hNE0T4oBgHgl3EQf6gJU/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8c309070f7f488b0fd8e49df5cc27f2e72ce4a5b --- /dev/null +++ b/hNE0T4oBgHgl3EQf6gJU/content/tmp_files/load_file.txt @@ -0,0 +1,1734 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf,len=1733 +page_content='LAGA: A Learning Adaptive Genetic Algorithm for Earth Electromagnetic Satellite Scheduling Problem Yanjie Song1, Jie Chun1, Qinwen Yang2, Junwei Ou1, Lining Xing3,∗, Yingwu Chen1 Abstract Earth electromagnetic exploration satellites are widely used in many fields due to their wide detection range and high detection sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The complex environment and the prolif- erating number of satellites make management a primary issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' We propose a learning adaptive genetic algorithm (LAGA) for the earth electromagnetic satellite scheduling prob- lem (EESSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Control parameters are vital for evolutionary algorithms, and their sensitivity to the problem makes tuning parameters usually require a lot of effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In the LAGA, we use a GRU artificial neural network model to control the parameters of variation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The GRU model can utilize online information to achieve adaptive adjustment of the param- eters during population search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Moreover, a policy gradient-based reinforcement learning method is designed to update the GRU network parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' By using an adaptive evolu- tion mechanism in the algorithm, the LAGA can autonomously select crossover operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Furthermore, a heuristic initialization method, an elite strategy, and a local search method are adopted in the LAGA to enhance the overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The proposed algorithm can obtain a more optimal solution on the EESSP through sufficient experimental validations compared to the state-of-the-art algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Keywords: reinforcement learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' learning adaptive ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' control parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' electromagnetic exploration satellite scheduling problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' genetic algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' GRU 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Introduction In recent years, satellite technology develops rapidly and has changed our lives to a great extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' During the development process, the surge in user demands and the number of satellites has made the management of earth electromagnetic satellite (EES) resources ∗Corresponding author Email addresses: songyj_2017@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='com (Yanjie Song), chunjie0720@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='com (Jie Chun), yangqq7160@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='com (Qinwen Yang), junweiou@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='com (Junwei Ou), lnxing@xidian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='cn (Lining Xing), ywchen@nudt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='cn (Yingwu Chen) 1College of Systems Engineering, National University of Defense Technology, Changsha, Hunan, China, 410073 2School of Computer Science and Engineering, North Minzu University, Yinchuan, China, 750021 3School of Electronic Engineering, Xidian University, Xi’an, China, 710126 Preprint submitted to Elsevier January 10, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='02764v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='NE] 7 Jan 2023 more challenging [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' EES is an artificial earth satellite equipped with antennas and signal receiving equipment that can acquire surface electromagnetic signal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The ability to meet users’ various detection needs and to obtain the corresponding data promptly is critical for the control of satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The earth electromagnetic satellite scheduling problem (EESSP) is proposed in response to satellite management challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Specifically, the EESSP is to obtain appropriate plans for a series of EESs under the condition of satisfying various constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' As one of the categories of satellite task scheduling problems, the EESSP also requires reasonable model construction and smart algorithm design to obtain satisfactory results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The EESSP has a series of earth electromagnetic satellites and tasks to schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Each electromagnetic detection satellite flies in a fixed orbit, which makes the area they can fly through limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The detection range of the satellite antenna is limited by its antenna aperture, which is fixed at design time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' As for the task, it can only be detected if it is within the detection range of an EES [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In addition, several other settings and satellite operating conditions extremely limit the number of tasks that can be accomplished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' To solve the EESSP, a scientific mathematical model and an efficient solution algorithm are required to construct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Among many solution algorithms, the genetic algorithm (GA) is a meta-heuristic al- gorithm based on population search with good applications in multi-class combinatorial optimization problems such as satellite task scheduling problems, vehicle path planning, and workshop scheduling [4]-[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In the GA, crossover and mutation are two of the most essential evolution operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Whether these two operators can be efficiently combined to complete the search to obtain the optimal solution is a key point for the algorithm design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Crossover implies exploration in the whole solution space, while mutation implies exploita- tion in a local space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The control parameters controlling the crossover and mutation are of great importance to find the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In general, the crossover probability (de- noted as CR) of GA is in the interval of [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='7,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='99], and the mutation probability (denoted as MR) is in the interval of [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='01,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In the process of solving a combinatorial optimization problem, a good algorithm keeps exploring new spaces as much as possible in the initial stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In contrast, after a certain number of searches, the algorithm should exploit a certain space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Along with this process, the values of these two control parameters differ in different problem scenario settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' It is difficult to determine a specific value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In this paper, we propose a novel evolutionary algorithm that combines reinforcement learning and a genetic algorithm framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' It is meaningful that this algorithm can work in various scenarios for solving the EESSP and finding the optimal solution with the smallest possible parameter tuning cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The well-performing algorithm can adaptively adjust the control parameters according to the scenarios’ features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Among the existing studies on the satellite scheduling problem, the EESSP is still in its infancy, while the earth observation satellite scheduling problem (EOSSP) has been studied in many ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Compared to the EOSSP, the EESSP does not need to consider cloud cover, resolution, and other factors but additionally needs to consider a series of constraints closely related to electromagnetic detection such as detection mode, bandwidth, and polarization mode [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The proposed model and method used to solve the EOSSP is a good guide for the 2 EESSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The mixed-integer programming model is a classical model form for constructing mathematical models of EOSSP in many studies [8]-[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' proposed a conflict metric and analyzed the interdependence of time windows to construct a mixed-integer programming model [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' used a directed acyclic graph to represent feasible observation task plans [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Valicka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' proposed an extended two-stage and three-stage stochastic mixed-integer scheduling model by considering cloud uncertainty [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Some other model forms, such as quadratic scheduling models and graphical models, have also been used by researchers [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Specific detailed research progress about the EOSSP model in detail is described in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Due to the high complexity and difficulty in solving the satellite task scheduling prob- lem, the exact solution algorithm is only able to solve the small-scale problem and is time- consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Small-scale scenarios tend to exist only in theoretical problems, while practical application problems are commonly large-scale or super-scale scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Therefore, exact solution algorithms have difficulty finding satisfactory results within a limited time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Heuris- tic and evolutionary algorithms, on the other hand, can effectively overcome the bottleneck of the exponential computation time explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' To find satisfactory task plans, a series of evolutionary algorithms such as genetic algorithm [14], ant colony algorithm [15], modal al- gorithm [16], and other improved forms based on simple evolutionary algorithms have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Among these algorithms, genetic algorithm has wide application in satellite task scheduling problems because of good global search performance and fast convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' considered the high computational cost of satellite task scheduling problems and de- signed a population perturbation mechanism in a genetic algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' This mechanism can improve the ability of the algorithm to find the optimal solution [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' used a new encoding method in a genetic algorithm [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Individual coding is used to determine based on the ground station ID to achieve a reduction in computational complexity during task scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Most of the studies related to optimization algorithms focus on the algorithm design itself, and not enough consideration is given to the factors that affect the performance of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Control parameters have an impact on the search performance of evolutionary algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Parameter control varies slightly in different literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In [19], parameter control without learning and parameter control based on learning are classified according to whether the information used is offline or online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Reference control methods can likewise be classified as deterministic, dynamic, adaptive, and hybrid control methods combining multiple methods depending on whether and how they vary [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' proposed a learning adaptive differential evolutionary algorithm for solving numerical optimization problems [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' This algorithm adopts a policy gradient method to train the long and short memory networks to get good control parameter settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' While there are many factors affecting the model effect in LSTM and the training period is quite long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In addition to evolutionary algorithms, reinforcement learning (RL) methods are also trying to solve the EOSSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' treated the EOSSP problem as a Markov deci- sion process in continuous time and constructed a reinforcement learning algorithm based on policy gradient [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' proposed a reinforcement learning algorithm based on parameter transfer to solve the multi-objective EOSSP [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' An encoding-decoding based 3 network model was used to obtain the solution to the decomposed subproblem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Lam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' learned a heuristic algorithm structure by reinforcement learning to achieve that some sub- sequent tasks can be selected after a given part of the task solution [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' designed a block encoding reinforcement learning training algorithm to solve the Agile EOSSP [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' used a Markov decision process to complete the assignment of observation tasks, then a specific execution plan was obtained using a dynamic scheduling approach [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The related studies using reinforcement learning methods are few and show a trend of attention to this type of solution method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Hybridization of reinforcement learning methods and evolutionary algorithms is a new idea of algorithm design that effectively combines the respective advantages of both methods [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Although there have been many successful practices of combining reinforcement learning with evolutionary algorithms in numerical optimization problems [19, 28] and combinatorial optimization problems [29]-[31], few studies have been done to solve satellite task scheduling problems using this idea [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' To address the shortcomings of traditional evolutionary algorithms for new problem scenarios that require the algorithm to run repeatedly to adjust the control parameters consuming a large number of computational resources, we propose a novel learning adaptive genetic algorithm, named LAGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In the algorithm, a GRU model is used to obtain the control parameters, and the model parameters are trained using a reinforcement learning method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' After training, the GRU can obtain reasonable control parameters which can promote the search process of LAGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The main contributions of this paper are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' We construct a mixed-integer programming model for the EESSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The goal is to find the task plan with the highest detection profit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In the model, constraints such as satel- lite abilities and task execution requirements are considered, and transition times between executing two tasks are treated in the form of maximization functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' We propose an evolutionary algorithm based on adaptive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' This algorithm treats the population evolution process as a time series and uses a GRU model to obtain the control parameters of the evolution operation based on online information prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' A policy gradient-based reinforcement learning training method is given for optimizing the GRU parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In the LAGA, a heuristic initialization method is designed to generate high-quality initial populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' An elite strategy and a local search method are also used to accelerate the convergence of the algorithm during an iterative search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The numerical experimental results of each evaluation dimension are combined to show that the proposed algorithm can obtain plans with high detection yield and improve the overall performance of the satellite system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The second part introduces the description of EESSP and the mathematical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The third part introduces the genetic algorithm and reinforce- ment learning methods based on adaptive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The fourth part verifies the performance of the proposed algorithm through several experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The fifth part introduces conclusions obtained from the study and the future research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Model In this section, we describe the EESSP in detail and analyze its difficulties first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Af- ter that, the variables, assumption conditions, objective functions, constraints, and other elements used to construct the mathematical model are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Problem Description Within a certain scheduling horizon, a series of EES need to develop task plans for each satellite to accomplish the detection tasks proposed by users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Working capabilities, task requirements, and other circumstances of EESs need to consider in the plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Each task execution needs to meet the users’ requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Specifically, the geographic location of the task to be detected, the length of the detection time, and the time range in which the detection are all set before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' A signal cannot be detected all the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' It can only be detected if a satellite has visible time windows and the task is performed within one visible time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In addition, detection activities need to be conducted within the detection time range and within the required detection time length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The angle between the antenna and the task changes during the motion in the detectable range over the satellite task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' If the angle is too large, the task detection will not be completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' When an EES just executed one detection task, it cannot immediately detect another task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The satellite needs to wait for the satellite payload after a series of working parameters configuration adjustments before meeting the next detection task parameters configuration requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Satellite payload parameter configuration adjustment includes detection mode adjustment, bandwidth adjustment, frequency band adjustment, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The parameter con- figuration adjustment will make part of the time window resources unavailable, and the scarcity of resources will increase the difficulty of developing the plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The main difficulty of the EESSP problem is over-subscription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' On the other hand, over-subscription arises due to limited resource capacity, where large-scale tasks exceed the upper limit of what satellite resources can accomplish in a given time horizon [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Several tasks need to be weighed against the need to add to the plan because each time a new task is added to the plan it may face one or several tasks that need to be discarded from the plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The single observation satellite task scheduling problem has been proven to be an NP-Hard problem [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Compared with it, the EESSP considers more factors and faces more complex constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' As a result of all of the above, the EESSP is more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Therefore, we need to accurately construct the model and design an efficient algorithm for solving the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Symbols and Variables This section introduces the variables and symbols involved in the mathematical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Sat :the set of detection satellites, Ns = |Sat|, si denotes satellite i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Oi: the set of orbits of the satellite i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' ϑi: the maximum angle that can be detected by the satellite i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' T: the set of detection tasks, Nt = |T|, taskj denotes the detection task j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' dj: the required detection time length of the task j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 5 restj: the earliest allowable detection time required for task j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' rletj: the latest allowable completion time required for task j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' θmax j : the maximum allowable detection angle for task j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' pj: the profit that can be obtained from the successful completion of task j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' TW: the set of time windows, Ntw = |TW|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' twijko: the time window k of the task j in on orbit o for satellite i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' evtijko: the earliest visible time of the task j in the time window k on orbit o for satellite i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' lvtijko: the latest visible time of the task j in the time window k on orbit o for satellite i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' t: the moment of satellite flight;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' θt ij: angle between the satellite’s antenna and task j at moment t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' F m i : function for the detection mode transition time of satellite i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' F b i : function for the bandwidth setting transition time of satellite i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' F f i : function for the frequency setting transition time of satellite i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' trm ijj′: detection mode transition time of satellite i between task j and task j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' trb ijj′: bandwidth mode transition time of satellite i between task j and task j′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' trf ijj′: frequency transition time of satellite i between task j and task j′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' trijj′: transition time of satellite i between task j and task j′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' I: a very large integer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Decision variables: xijko : whether satellite i performs task j within kth time window on orbit o, if it is done, xijko = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' otherwise, xijko = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' stijo : the start time of the satellite i to perform the task j on orbit o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Mathematical Model In this section, a mathematical model is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' First, the assumptions of the model are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Assumptions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The task is covered by a single satellite detection, without the need for multiple repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The detection task can be carried out at most once, without considering multiple repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The impact of satellite sequestration and energy on satellite detection activities is not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Equal value of the profit obtained by the satellite from performing the task at any moment in the time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The detection tasks to be performed by the satellite are determined in advance before scheduling, and there will be no adjustment of the task performance requirements during scheduling as well as during the execution of the satellite’s task, such as early or late completion of the task requirements, temporary cancellation of the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The satellite can operate normally throughout the whole planning time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 6 We give the calculation method for the transition time of satellite i between tasks j and j′ at first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' It can be calculated as follows: trm ijj′ = F m i � rm j , rm j′ � (1) trb ijj′ = F b i � rb j, rb j′ � (2) trf ijj′ = F f i � rf j , rf j′ � (3) trijj′ = max � 0, trm ijj′, trb ijj′, trf ijj′ � (4) where rm j , rb j, rf j represent detection mode setting requirement, bandwidth setting re- quirement for task, frequency setting requirement for task j repectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The transition time is the max time of the time required to convert the satellite param- eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Our goal is to find a sequence of tasks that can be executed in the solution space with high detection profits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Therefore, the objective function is to maximize the profit of the task plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The objective function is calculated as shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Objective function: � i∈S � j∈J � k∈TW � o∈Oi pj · xijko (5) where pj denotes the profit that can be obtained by the successful completion of task j, xijko denotes whether satellite i performs task j within kth time window of orbit o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Constraints: The task is to be executed within the required time range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' stijo ≤ restj · xijko, i ∈ Sat, j ∈ T, k ∈ TW, o ∈ Oi (6) (stijo + dj) · xijko ≤ rletj, i ∈ Sat, j ∈ T, k ∈ TW, o ∈ Oi (7) The task needs to be executed within the time window in which it can be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' stijo ≤ evtijko · xijko, i ∈ Sat, j ∈ T, k ∈ TW, o ∈ Oi (8) (stijo + dj) · xijko ≤ evtijko, i ∈ Sat, j ∈ T, k ∈ TW, o ∈ Oi (9) The angle between the satellite detection and the task needs to be less than the maximum allowable angular requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 7 θt ij · xijko ≤ min � ϑi, θmax j � , i ∈ Sat, j ∈ T, k ∈ TW, o ∈ Oi, t ∈ [stijo, stijo + durj] (10) Each task can only be detected at most once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' � i∈S � k∈TW � o∈Oi xijko ≤ 1, i ∈ Sat, j ∈ T, k ∈ TW, o ∈ Oi (11) The transition between two tasks needs to meet the transition time requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' (stijo + durj) · xijko + trijj′ ≤ stij′o + I · (1 − xij′k′o) , j ̸= j′, i ∈ Sat, j, j′ ∈ T, o ∈ Oi, k, k′ ∈ TW (12) Decision variables need to be valued in the corresponding ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' xijko ∈ {0, 1} , i ∈ Sat, j ∈ T, k ∈ TW, o ∈ Oi (13) stijo ∈ N, i ∈ Sat, j ∈ T, k ∈ TW, o ∈ Oi (14) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The Proposed Method 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Embedding GRU in the GA Framework Figure 1: GRU unit structure 8 + Zt tanh 0 0 NtThe crossover and mutation in population evolution can be regarded as a random time series with obvious time-dependent properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The control parameters affect the way and probability of the population evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' So these two operators play a crucial role in whether the algorithm can find the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Therefore, we adopt a nonlinear equation prediction idea to obtain parameter configurations that match the search pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' From another perspective, the changes in the control parameters during the population evolution can also be considered a time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' To solve time series-related problems, the recurrent neural network is the primary choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The population evolution information includes both current information and historical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The network model is expected to pay more attention to the information related to the most recent evolutionary generations and less attention to the information obtained from the earlier time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Based on the time-dependent characteristic of online information, we use a GRU model to obtain useful information to facilitate GA search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' GRU is a classical recurrent neural network (RNN) model proposed by Cho, which effectively solves the problems of long memory dependence and gradient explosion compared to traditional recurrent neural network models [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Moreover, the GRU model requires fewer parameters and fewer training times compared to the classical long-short memory network (LSTM) model [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Each GRU model may consist of a series of GRU units, which effectively capture the interrelationship of data in the temporal dimension through the combination of units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The gate structure is a unique information flow regulation structure for LSTM and GRU because the information memory gate is omitted from the GRU cell structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' To describe GRU more intuitively, we give its specific structure in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In each GRU unit, the update gate and reset gate are used to achieve a good prediction of the time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The following equations can describe the GRU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' hj t = � 1 − zj t � hj t−1 + zj t ˜hj t (15) zj t = σ(Wzxt + Uzht−1)j (16) ˜hj t = tanh (Wxt + U (rt ⊙ ht−1))j (17) rj t = σ(Wrxt + Urht−1)j (18) σ (z) = 1 1 + e−z (19) tanh (z) = ez − e−z ez + e−z (20) where hj t−1 and xt denote the input of GRU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' zj t denotes the update gate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' rj t denotes a set of reset gates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' ⊙ denotes the multiplication of the corresponding element positions of the matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' σ (·) denotes the sigmoid activation function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' and tanh (·) is the tanh function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In the practical application of the GRU model, the complete network model consists of two types of neural network structures, the GRU unit, and the fully connected network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The full-connected network is used after the GRU unit to further process the data stream, and the complete data stream of the GRU network model is as follows: First, the input data is processed by several GRU units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Then, several fully connected network layers are used to further process the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' After that, the Softmax function is used to obtain the outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 9 To simplify the intermediate process, GRU can be abbreviated as: Ht = GRU(St, Ht−1, WG) (21) where WG denotes the parameter of GRU units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' And the fully connected network layer can be abbreviated as: CRt = Linear(Ht, Wc, bc) (22) MRt = Linear(Ht, Wm, bm) (23) where Wc, Wm denotes network parameters and bc, bm denotes bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' According to the above network model, the current state value St is processed as input data to fit a combination of control parameters that will help the population search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' We denote Ωt = [CRt, MRt] and use such parameters for the population evolution of the gener- ation t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Once a set of parameters is obtained, the solution space can be searched and mined using such parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The complete data flow can be expressed as follows: Ωt, Ht = GRU(St, Ht−1, W) (24) Figure 2: GRU network model embedding GA in generation t After obtaining the complete data flow of the GRU model, the process of embedding the GRU model in the evolution process of the GA generation population can be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' As shown in Figure 2, the symbol S in the figure represents the selection operator, FE represents the fitness evaluation, CO represents the crossover operator, and MO represents the mutation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The state information St is the input of the GRU model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The crossover probability CRt and the mutation probability MRt required for the current population evolution are obtained through the output action of the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' These two probabilities can have a significant impact on population search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 10 Pt CR, CO FE S GRU S [+} MO MR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' F3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Reinforcement Learning for Evolutionary Search A reinforcement learning method is proposed to allow GRU to update the network model parameters according to the population evolution process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' This method can obtain a param- eter configuration that is more helpful for population search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' We construct the population evolution process of the algorithm as a Markov decision process (MDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The RL method allows the agent to choose the appropriate action to obtain a high reward in a dynamic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' State, action, reward, and transition constitute the main elements of the RL method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The main components of the RL method are status, action, reward, and state tran- sition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Each major component of MDP will be introduced in detail in this section, followed by an introduction to the policy gradient training method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' State The state is the input of the GRU network model, which allows the agent to select appropriate actions for the evolution process of LAGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' To provide plenty of features and information for choosing actions, the state should accurately reflect the current population evolution information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In our proposed algorithm, the set of states S denotes a set of states constitutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' S can be described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' S = {S0, S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=', St, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='} (25) where St denotes the agent state value at time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' St consists of a set of representations of task attributes and population evolution information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The details of the specific attribute values of the state are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' St = � yt j = (dj, pj, tsj, lj) |j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=', Nt � (26) where dj is the task duration, pj is the task profit, lj denotes the cumulative arrangement of the task j in the current population, and its value indicates the number of successful placements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' tsj denotes the range of the indicated time interval, which can be calculated as follows: tsj = rletj − restj (27) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Action Action is a core part of the MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The actions of reinforcement learning are distinctly dif- ferent in the continuous and discrete action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Since the reinforcement learning method belongs to the selection of actions on the continuous action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Therefore, we obtain the probability density function of action At under state St based on the parameters θ of the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Assuming that the action probability values obey a Gaussian distribution, the equation of policy is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' π (At | St) = N � At | GRU (St;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' W) , σ2� = 1 √ 2πσ � St;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' ˜θ � exp � � � � � − � At − µ � St;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' ˜θ ��2 2σ2 � St;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' ˜θ � � � � � � (28) 11 where W denotes the parameter of the GRU model and ˜θ = � ˜θµ, ˜θσ �T is the parameter of the strategy, which is obtained by fitting the neural network model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The probability density function corresponding to the action is obtained by the input state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The two control parameters we need are obeyed by this obtained function, and the required control parameter values can be obtained by sampling according to this probability density function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Reward In our case, the reward should reflect the effect of the population evolution after the action is taken by the population evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' As an evolutionary algorithm, the change in the optimal local solution that can be found for the contemporary population can effectively reflect the effect of population evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Therefore, we use the percentage improvement of the current population’s optimal fitness function value compared to the previous generation population’s optimal fitness function value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' And when the population becomes worse instead of finding a better task plan, the reward is updated in the form of penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The equation of reward is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Rt = f best t − f best t−1 f best t−1 (29) where f best t denotes the optimal fitness function value in the current population and f best t−1 denotes the optimal fitness function value in the last generation population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' A better population search performance is implied by higher reward values, which can also effectively reflect the influence that the control parameters produced on the population evolution process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Transition The state transition records the agent state changes, actions taken, and rewards obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Since we use the policy gradient method to train the network model, it does not matter whether the state transition can be recorded or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' This is because the policy gradient approach uses gradient descent to optimize subsequent strategies directly for the desired reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Although state shifts do not have an impact on strategy updates, the triple of (St, At, Rt) needs to be stored for updating the network model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' After introducing the meaning of each element of the MDP process, the accuracy of the policy taken by the agent needs to be improved by a reinforcement learning training method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' We introduce the reinforcement learning training method based on policy gradients in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Policy Gradient Training Method The training effect of the GRU network model can make the LAGA have a great impact on the solution of the EESSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The algorithm control parameters need to find a suitable combination scheme on the continuous space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The policy gradient method, one of the typical reinforcement learning methods, can effectively cope with the training method of 12 model parameter optimization on the continuous space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Here, the policy gradient method optimizes model parameters by trajectory sampling of a batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Since the LAGA uses a population to search the solution space, the batch can be replaced by the population size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In our study, the policy gradient approach updates network model parameters by per- forming a gradient descent on the objective function of the reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The equation used in the training of the network is as follows: θt+1 = θt + α∇θL (θt) (30) where ∇θL (θt) is the gradient of the reward function and α is the learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' ∇θL (θt) can be further expressed as: ∇θL (θt) = Eτ∼θt � T � t′=0 ∇θ log πθ (At |St) T � t=t′ rt � (31) Before training the network model, the objective function of reward should be found and the trajectory should be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' According to the MDP process constructed by the population evolution process, a trajectory can be formed from states and actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' When the number of trajectories is reached at Ntra, the GRU network model parameters are updated according to the agent states, actions, and rewards using the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 25-29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' We use a REINFORCE Monte Carlo method, then the expectation of ∇Lθ(θ) can be approximated by sampling Ntra trajectories to obtain: ∇Lθ(θ) ≈ 1 L L � i=1 r � τ i� T−1 � t=0 ∇θ ln π � A(i) t = a(i) t | S(i) t = s(i) t � (32) where at i denotes the value of the action belonging to the ith trajectory at time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' After that, we can train the network model parameters in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The pseudo-code of the policy gradient method is shown in the Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' As shown in Algorithm 1, the policy gradient training method repeats multiple epochs for each problem scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' One epoch needs to obtain the control parameters (line 9) based on the state values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The states need to be obtained by computing the task scheduling results and combining them with other features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The LAGA generates new populations (line 10) and computes the reward (line 12) by population evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' After completing a series of trajectories of the algorithm runs, the network model parameters W (line 14) are updated using the back propagation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Algorithm training by multiple epochs allows the GRU model to obtain a set of parameter configurations that achieve good prediction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The trained network model is then used in the LAGA presented in the subsequent section to provide the two control parameters MRt and CRt for the evolution of the tth generation population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In the next section, we present the use of the GRU model in the proposed algorithm in combination with other population search strategies to develop suitable task plans for EESs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 13 Algorithm 1: Policy Gradient Training Method Input: max epoch Epoch, population size Np, the number of trajectories Ntra, learning rate α, max time step TSmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Output: Updated W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 1 Initialize the GRU model parameters W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 2 for epoch = 1 → Epoch do 3 for tra = 1 → Ntra do 4 Set t ← 1, H0 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 5 Initialize LAGA parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 6 P ←Generate an initial population randomly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 7 while t ≤ TSmax do 8 Get the latest state St;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 9 Ωt = [CRt, MRt] ←Generate control parameters by GRU (St, Ht−1, W);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 10 Pt ←Population Evolution by LAGA(Pt−1, Ωt, CO, MO);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 11 Ft ←Calculate the fitness function value of the new population(Pt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 12 Rt ←Calculate reward using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 29;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 13 t ← t + 1 14 Update W using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 30 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Learning Adaptive Genetic Algorithm The LAGA applies reinforcement learning methods to the genetic algorithm framework, allowing the algorithm to learn the useful information obtained from the population evolu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' An artificial neural network model after training is adopted to provide decision support for the control parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The effective configuration of the control parameters allows the LAGA to find the optimal solution more effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' At the initial stage of the search, the algorithm should focus on exploring new solution spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In contract continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' When the search reaches a certain stage, it is worthwhile to focus on a small search area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' While using the network model to obtain the parameters, an adaptive crossover approach is used in the genetic algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Adaptive crossover can effectively ensure the generalization ability of the algorithm and allow the algorithm to easily select operators that are conducive to finding better solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' To further improve the search performance of the algorithm, we also design a population initialization method and a local search method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The quality of the initialized population will largely affect the population search performance, while the local search can improve the exploitation performance of the algorithm in the local search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Figure 3 illustrates the overall framework of the LAGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Compared with the traditional genetic algorithm, the proposed algorithm has several differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' First, the control param- eters are no longer set artificially by the designer, but are obtained in a nonlinear predictive manner by the artificial neural network model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' This is an improvement that we have made to the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Second, the proposed algorithm proposes several improvement strate- gies, including population generation, population evolution, and search strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In the subsequent sections, we introduce the specific processes of the algorithm, initialization of 14 population, fitness evaluation, individual selection, population evolution, elite strategy, and local search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' After that, we also analyze the time complexity of the LAGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Figure 3: The overall framework of LAGA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Algorithm Overall Process The learning adaptive genetic algorithm uses a reinforcement learning method to train the GRU model, which can provide appropriate control parameters for population evolution operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' A genetic algorithm framework usually contains steps such as initialization of the population, individual selection, fitness evaluation, and variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Before each generation of population search, the LAGA uses a trained artificial neural network model, and the two main control parameters are obtained based on online information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The adjustment of parameters allows the algorithm to choose an appropriate search method based on en- vironment information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' It contributes to finding the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The algorithm also 15 Start Heuristic rules Initialization Elite strategy N Y Update the best Fitness evaluation individual Generate control GRU model Local search?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' parameters N Y Offspring Generation Local Search Selection Randomly choose Adaptive two locations Crossover Mutation Swap N Y Stop?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Continue?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' N Y Enduses adaptive crossover, an individual evolution operator that selects the better operator based on search performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The crossover operators dynamically adjust the probability of operators by evaluating the iterative search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Details about the population evolution are given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' It is worth mentioning that the algorithm can also adjust the search strategy according to the population search performance, moving from exploration within the whole solution space to the exploitation of local space to discover better solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The pseudo-code of the LAGA is shown in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' As expressed in Algorithm 2, the LAGA first generates a population of Np individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The initial population is obtained by a heuristic initialization method (line 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' After com- pleting the evaluation of the initial population fitness and algorithm starts a population search iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In the search process, an elite strategy (lines 20-22) and a local search method are used (lines 25-29) in addition to the crossover and mutation operators that con- tain the genetic algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The optimal solution found by the search will become the final detection task plan after the algorithm search is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In the subsequent sections, we present the main steps of the LAGA, the population evolution operators, and some improvement strategies to enhance the search performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Initialization The population evolution is based on the initial population, and a series of selection and variation operators are performed to obtain a high-quality plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Population initialization strives to get individuals located in a good position in the search space while ensuring diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Such an initialization approach can significantly reduce the number of searches required to find a good task plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' So we design a population initialization method in the algorithm that combines heuristics with randomization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The heuristic rule is donated as UPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' UPF rule is described as follows and the equation for the unit profit of the task(upj) is shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Heuristic rule: Calculate unit profits of tasks and generate an individual according to the index value from highest to lowest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' upj = pj/dj (33) where pj is the profit of task j and dj is the required detection time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' If every individual in the population follows the above heuristic rule will make the chro- mosome structure between individuals highly similar, which will not facilitate the search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Therefore, to ensure the diversity of individuals in the population, we use a parameter η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Through the setting of this parameter, some genes within individuals are added to the chromosome randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The parameter η denotes the proportion of chromosomes gener- ated according to the heuristic rule within an individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' This part of the task generates a chromosome, ordered according to the heuristic rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The rest (1-η) proportion of genes is inserted into the existing chromosome in random positions to form a complete individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The pseudo-code of the initialization method is shown in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' As expressed in the Algorithm 3, a certain proportion of chromosomes for each individ- ual within the population is generated according to the heuristic rule (Line 4), while the 16 Algorithm 2: LAGA Input: population size Np, GRU model parameters W, task set T, time window set TW, control parameter of elite strategy Thre1, control parameter of local search Thre2 Output: Solution 1 Initialize algorithm parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 2 P ←Generate the initial population by Algorithm 3 and calculate the fitness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 3 Set t ← 1, tri1 ← 0, tri2 ← 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 4 while termination criterion is not met do 5 Get the latest state St;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 6 Ωt = [CRt, MRt] , Ht ←Generate control parameters by GRU (St, Ht−1, W);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 7 for i = 1 → Np do 8 indi ←Roulette chooses individuals(P);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 9 if rand () < CRt then 10 Perform adaptive crossover(indi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 11 if rand () < MRt then 12 Perform mutation(indi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 13 local best indi, local best ←Evaluate the fitness function value(P);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 14 Update the scores of crossover operators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 15 if local best > gobal best then 16 gobal best ← local best;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 17 gobal best indi ← local best indi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 18 else 19 tri1 ← tri1 + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 20 //Elite Strategy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 21 if tri1 < Thre1 and gobal best!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' = local best then 22 local best indi ← gobal best indi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 23 if local best < temp local best then 24 tri2 ← tri2 + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 25 //Local Search;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 26 if tri2 == Thre2 then 27 new indi ← Local search using Algorithm 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 28 P ← Update population by replace the worst individual by new indi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 29 tri2 ← 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 30 t ← t + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 31 temp local best ← local best;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 17 Algorithm 3: Initialization Input: population size Np, task set T Output: initial Population P0 1 Set η ← 0, PI = [];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 2 for i = 1 → Np do 3 η ← i/(Np + 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 4 indii ← Select tasks using UPF(T, η);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 5 PI ←Generate a set of tasks to be inserted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 6 while PI ̸= ∅ do 7 task ← Random select a task from PI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 8 indii ← Select a position randomly and insert task;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 9 Remove the task from PI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' remaining part generates a task set (Line 5) and uses a random approach to select genes (Line 7) and insert them to the chromosome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' After generating the initial population, the population search will be carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Fitness Evaluation The purpose of fitness evaluation is to allow the LAGA to identify the parent individuals of the change operation from the population based on individual performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Our objective function in the EEESSP problem is to maximize the profit of the detection task sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Therefore, the evaluation of individual fitness is obtained in the same way as the calculation of the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The specific calculation is shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The results of the fitness evaluation will also support the computation of the reward value for reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Individual Selection Individual selection selects individuals from the population according to a certain strat- egy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Then, offspring will generate based on the selected individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Individual selection is usually done by roulette, k-tournament, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In our algorithm, we use a roulette method to select individuals from the population for subsequent evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The equation for roulette selection of individuals is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' ¯pi = fi � i∈P fi (34) where fi denotes the fitness value of individual i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' After Selecting the individual for variation, it will perform crossover or mutation to gen- erate an offspring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Variation is vital to finding the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' A detailed description of variation is given in the subsequent section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Variation The variation consists of two population evolution operators: crossover and mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The difference between crossover and mutation is the degree of chromosome change within an individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' We restrict the population search by controlling the parameters so that the algorithm achieves efficient use of exploration and exploitation throughout the search process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' This search method allows the algorithm to find the optimal or as close to the optimal solution as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The control parameters for the probability of crossover and mutation are adapted to the problem scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Other factors, such as the length of the selected gene fragment within an individual, and the way of variation, can also affect the population evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In the following, the specific evolution operators of crossover and mutation are described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Crossover is one of the most frequent population evolution operators of genetic algorithms in the search process, attempting to explore the entire solution space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The offspring produced by using this operator will be significantly different from the parent individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' We propose an adaptive crossover operation in the LAGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' This adaptive crossover operator initializes the same score for each crossover rule when initializing the algorithm parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Afterward, weights are determined based on the crossover rules’ scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' A random choice approach is used to select a rule for crossover based on weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' This approach is extremely similar to individual selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The equation for calculating the weights is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' ˆpi = soci � i∈R soci (35) where ˆpi denotes the probability of the ith crossover rule, soci denotes the score of the ith crossover rule, and R denotes the set of crossover rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The crossover rules used in crossover operators are divided into three types, two-point crossover rule, multi-point crossover rule, and fragment flipping crossover rule, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In the following, a detailed description of the rules is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Figure 4: Two point crossover Two-point crossover rule: Two equal-length gene fragments are obtained from two dif- ferent positions of the parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Without changing their internal order, these two fragments exchange positions to generate offspring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 19 Parent 1 5 2 3 4 6 Offspring 1 3 4 7 5 2 6Figure 5: Multi-point crossover Multi-point crossover rule: Multiple genes are selected from the parent, and a new gene fragment is formed according to the relative order in the parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Then, the fragment is inserted into the remaining part at a random position to generate offspring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Fragment flipping crossover rule: Select the start and end points of a gene fragment from the parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' A new fragment is created in the order from the back to the front of the selected fragment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The new fragment is placed in the same position as the parent to generate offspring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Figure 6: Fragment flipping crossover These three crossovers affect the chromosome within individuals to different degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' How- ever, it is difficult to indicate which crossover operator is significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Each population evolu- tion is expected to have the desired effect as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Therefore, after determining the crossover used and executing the corresponding evolution, the algorithm updates the score of the used crossover operator according to the individual fitness change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The score of the updated value is determined according to the search performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' If the fitness value of the offspring is increasing compared to the fitness value of the parent, the score is in- creased by µ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' otherwise, the score is increased by µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' When a certain number of times is reached, the weights of the crossover rules will be updated according to the latest score value according to the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Compared to crossover, the mutation is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The mutation is done by double point swapping, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=', two genes are randomly selected from the chromosome of a parent and their positions are swapped to obtain an offspring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 20 Parent 1 5 2 7 3 4 6 Offspring 5 7 2 4 9 3Parent 1 5 2 7 3 4 6 Offspring 1 5 3 7 2 4 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Elite Strategy The elite strategy is designed to improve the convergence performance of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' After each generation of population search is completed, we choose to add the best individ- ual found through the search directly to the offspring population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' An elite individual can effectively improve the convergence speed, allowing the population search to find higher- quality task plans quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' When the population search reaches the threshold Thre1, the elite strategy is no longer used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Local Search Local search (LS) is a way to find the local optimum within a certain solution space, which can often play a crucial role when the search is not effective in the entire solution space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' We use a 2-opt local search operation, which is a simple and efficient method to update the neighborhood structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In the 2-opt, two genes are randomly selected from the best individual found in the population search so far, and the positions are exchanged to generate a new individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' A fitness evaluation and comparison process will determine whether to continue the local search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Local search needs to effectively balance with global search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' If too many times searches are done, it will make the algorithm solution fall into local optimization without jumping out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Therefore, when there is no further improvement in the individual fitness value, the local search stage should end and the algorithm returns to the population search stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The pseudo-code of the local search is shown in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Algorithm 4: Local Search Input: gobal best individual gobal best indi, fitness of gobal best individual gobal best Output: new individual new indi 1 while termination criterion is not met do 2 gene1, gene2 ← Random select two genes from gobal best indi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 3 new indi ← Swap two gene positions to generate a new individual;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 4 fitness ← Calculate the fitness function value (new indi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 5 if fitness > gobal best then 6 gobal best ← fitness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 7 else 8 Loop While;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' As expressed in algorithm 4, two genes are randomly selected from the chromosome of the best individual in the population search (line 2), and the positions are exchanged to obtain offspring (line 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' After that, the value of the fitness function is evaluated to determine whether to continue the local search process or to return to the population search (lines 5-8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The local search algorithm will update the best individual for the population search at the end of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Termination Conditions The LAGA ends the algorithm search after a certain search stage and outputs the optimal solution found as the final task plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In the proposed algorithm, we use the maximum number of fitness evaluations (MFE) as the algorithm termination condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The algorithm ends when the number of fitness calculations is equal to the maximum number of evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Complexity Analysis In the LAGA, the complexity of the GRU model is O(Batch ∗ |T|2 ∗ d), where d denotes the number of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The complexity of GA algorithm population search generation is O(Batch ∗ |T| ∗ |TW|), and the complexity of the local search is O(|T| ∗ |TW|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Since |TW| >> d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' So the time complexity of the LAGA is O(Batch ∗ |T| ∗ |TW|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' While batch uses population P instead, the time complexity can also be rewritten as O(|P|∗|T|∗|TW|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Experiment To verify the effectiveness of the LAGA, we design a series of experiments and use three state-of-the-art algorithms for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In addition, we also examine whether the strategies used in the algorithm can improve the search performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Experimental Setup Experimental environment: The experiments in this study are done on a desktop com- puter with Intel(R) Core(TM) I7-7700 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='6 GHz CPU, 16 GB RAM, and NVIDIA GeForce 2070Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The coding environment is Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='7 + Numpy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='3 + Pytorch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='0 (Cuda v11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Comparison algorithms: Three algorithms were used to verify the problem-solving per- formance of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' We selected a population perturbation and elimina- tion strategy based genetic algorithm (GA-PE) [17], improved adaptive large neighborhood search algorithm (ALNS-I) [38], and artificial bee colony algorithm (ABC) [39], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' GA-PE: A perturbation and elimination strategy is used in the genetic algorithm frame- work to improve the convergence speed of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' the GA-PE method has effectively solved the multi-satellite TT&C scheduling problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' ALNS-I: The improved adaptive large neighborhood search algorithm designs the corre- sponding destroy and repair method for EOSSP based on the adaptive large neighborhood search framework and can obtain high-quality observation plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' ABC: Artificial bee colony algorithm uses three types of bees: employed bees, onlooker bees, and scout bees to achieve exploration and exploitation of the search space for combi- natorial optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Experimental scenarios: Scenarios with different task scales are used to evaluate the scheduling performance of the algorithm in all aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Since there is no public test set for the EESSP, we use a random approach to generate a series of scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' To represent the scene concisely, we use the ”A-B” format to represent a scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In ”A-B”, ”A” denotes the number of tasks, and ”B” denotes the number of the scenario of task-scale ”A”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In the experiment, the tasks are sized from 100 to 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' For the convenience of differentiation, 22 we artificially divide the cases into small-scale sets (denoted as Set I), medium-scale sets (denoted as Set II), and large-scale sets (denoted as Set III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Algorithm parameter settings: The number of fitness evaluations for all algorithms is set to 5000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In LAGA, the population size Np is set to 10, the number of trajectories Ntra is set to 10, the learning rate α is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='001, the initial score of each crossover rule is set to 50, µ1 is set to 30, µ2 is set to 10, Thre1 is set to 2000, Thre2 is set to 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The parameter settings of the other comparison state-of-the-art algorithms were kept consistent with those in related studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Evaluation metrics: All algorithms run 30 times in each scenario to evaluate algorithms’ performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The optimal profit (denoted as Best), the mean profit (denoted as Mean), and the standard deviation (denoted as Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=') are set as evaluation indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Based on the data obtained, the Wilcoxon rank-sum hypothesis test (denoted as WR) is used to analyze whether the differences between the results obtained by different algorithms are significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In terms of the algorithm convergence performance, we evaluate it through convergence curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Furthermore, we analyze the effect of the strategy used in the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Results and Analysis Table 1: Scheduling Results for Set I Instance LAGA GA-PE ALNS-I ABC Best Mean Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Best Mean Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' WR Best Mean Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' WR Best Mean Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' WR 100-1 891 891.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 891 891.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 = 891 891.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 = 891 891.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 = 100-2 805 805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 805 805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 = 805 805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 = 805 805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 = 100-3 806 806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 806 806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 = 806 806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 = 806 806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 = 100-4 819 819.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 819 819.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 = 819 817.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='80 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='73 819 819.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 = 100-5 820 820.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 820 820.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 = 820 820.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 = 820 820.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 = 100-6 822 822.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 822 822.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 = 822 822.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 = 822 822.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='00 = 200-1 1550 1542.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='70 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='65 1544 1530.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='63 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='57 1539 1529.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='47 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='66 1544 1534.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='87 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='71 200-2 1666 1648.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='03 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='91 1628 1605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='33 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='61 1621 1604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='83 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='01 1624 1607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='90 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='34 200-3 1615 1608.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='27 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='49 1601 1586.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='57 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='90 300-6 2153 2134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='20 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='46 2035 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='63 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='91 2049 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='20 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='41 2030 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='77 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='55 +/-/= 0/12/6 0/12/6 0/12/6 Firstly, we verify the performance of the algorithms in Set I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' As shown in Table 1, finding the optimal solution in a 100 task-scale scenario is not difficult for all algorithms used in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Moreover, the majority of the algorithms’ search performance is stable, and only ALNS-I still has some gap between the average and optimal values of the profits obtained in the scenario 100-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Starting from scenarios with a 200 task scale, the proposed algorithm can find better optimal values than the compared algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' And the WD indicator shows that the LAGA differs significantly from the other algorithms in most of the scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The LAGA performs as well in Set II as in Set I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' As shown in Table 2, we can see that the best performance is obtained except for some scenarios where the results are not the best in terms of stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Moreover, the gap between algorithms increases significantly 23 Table 2: Scheduling Results for Set II Instance LAGA GA-PE ALNS-I ABC Best Mean Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Best Mean Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' WR Best Mean Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' WR Best Mean Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' WR 400-1 3096 3092.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='64 +/-/= 0/18/0 0/18/0 0/18/0 compared with results in Set I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In the 600 task-scale scenarios, the gap between the optimal and average profits of the LAGA and other comparison algorithms can reach about two hundred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Table 3: Scheduling Results for Set III Instance LAGA GA-PE ALNS-I ABC Best Mean Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Best Mean Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' WR Best Mean 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='20 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='13 7972 7920.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='70 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='18 +/-/= 0/18/0 0/18/0 0/18/0 After using Set II to validate algorithms, the task scale is further increased and the results in Set III are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The experiments on the algorithm’s solving ability by large- scale scenarios can well reflect the balanced performance of the algorithm’s exploration and exploitation as well as its application prospects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' It is not difficult to find that the LAGA still performs well in Set III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' This is due to the good combination of positive population exploration and local space exploitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Due to the large solution space, the local search can often be of great help in improving the solution quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' To observe the solution performance of the algorithm more intuitively, we present the results of the average performance of the 400 task-scale scenarios in the form of a bar chart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 7(a)-7(f), the average profit obtained by the proposed algorithm is 24 significantly higher than the other three algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' And the search performance of the other three algorithms is close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' From the paired statistical tests, the results obtained by LAGA are significantly different from the state-of-the-art algorithms at the level of p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' (a) Average performance of sce- nario 400-1 (b) Average performance of sce- nario 400-2 (c) Average performance of sce- nario 400-3 (d) Average performance of sce- nario 400-4 (e) Average performance of sce- nario 400-5 (f) Average performance of sce- nario 400-6 Figure 7: Average performance in different scenarios In addition to statistical analysis of algorithm differences, convergence performance is another significant evaluation criterion for search algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' We analyze convergence curves for 800 and 1000 task-scale scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 8(a)-8(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' All these four algorithms have good global search capability and can exploit solutions in the local space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The heuristic population initialization method used in the LAGA algorithm has a positive effect on finding the optimal solution, and this initialization method has outstanding performance in 1000 task-scale scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' After the search process starts, the LAGA has a fast convergence speed and can start a new search through the search strategy after the convergence encounters a bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Then, we compare the results using the algorithm with the algorithm that uses a random way to generate the initial population (denoted as LAGA/RI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' This experiment can check the effectiveness of the improved approach adopted by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='2400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='*** ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='2250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='2100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='Profit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='1950 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='1800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='1650 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='LAGA GA-PEALNS-I ABC2400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='*** ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='2250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='2100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='Profit ' metadata={'source': 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+page_content='7800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='7650 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='7500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='1000-11000-2 1000-3 1000-4 1000-5 1000-6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content='InstanceThe results for 300,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 600,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 800,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' and 1000 task-scale scenarios are shown in Figure 9(a)- 9(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' From the results, it can be seen that the use of the heuristic initialization method can effectively improve the search performance of the LAGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' This initialization method can effectively utilize the knowledge and improve the effectiveness of the algorithm in finding solutions while maintaining the diversity of populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' It is clear that as the task-scale increases, the role of the heuristic population initialization method tends to diminish and then increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' This is because the complexity of the problem depends more on the search process and solutions obtained by initialization are not decentralized throughout the solution space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Then, when the search space is large enough, knowledge again drives the population search in a good direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In summary, the above experimental results show that the LAGA performs significantly better than other state-of-the-art algorithms in solving the EESSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' From several metric aspects, the GA framework combined with an artificial neural network model and various strategies designed is effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Conclusion For the earth electromagnetic satellite scheduling problem (EESSP), we construct a mixed-integer programming model and design a learning adaptive genetic algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The LAGA effectively combines the respective advantages of evolutionary algorithms and arti- ficial neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Combined with the characteristics of population optimization, we adopt a policy gradient-based reinforcement learning training method to train the GRU model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The algorithm in this paper also uses a series of improvement strategies besides artificial neural networks to make the algorithm search more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The elite strategy al- lows the population search to have better convergence performance at the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' While focusing on the global search, we design a local search method to find the optimal local solu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' The improvement of the algorithm effectively balances the exploration and exploitation in the population search and makes it easier to find a satisfactory satellite detection plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Through extensive experiments, we verify that using adaptive learning methods to adjust the parameter configuration, which is based on the information obtained from the search, can allow the genetic algorithm to obtain better plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' In the future, we will consider other reinforcement learning methods and other ways of combining algorithms, such as using reinforcement learning methods to generate solutions or select appropriate operators for population evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Reinforcement learning can also be used to generate offspring population by selecting individuals from the parent population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' As for EESSP, some more complex situations deserve to be considered in the model design phase, such as some uncertain environmental factors or possible equipment emergencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 28 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Acknowledgements This work was supported by the National Natural Science Foundation of China (71901213, 72001212), the Special Projects in Key Fields of Universities in Guangdong (2021ZDZX1019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Jie Chun and Yanjie Song contribute equally to this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' Thanks to the editor and reviewers for their valuable 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+page_content=' Scholarpedia, 5(3), 6915.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} +page_content=' 31' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE0T4oBgHgl3EQf6gJU/content/2301.02764v1.pdf'} diff --git a/hdAzT4oBgHgl3EQfMvuF/content/tmp_files/2301.01137v1.pdf.txt b/hdAzT4oBgHgl3EQfMvuF/content/tmp_files/2301.01137v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..163632123c3c26d84310697d00c6abefcf183f38 --- /dev/null +++ b/hdAzT4oBgHgl3EQfMvuF/content/tmp_files/2301.01137v1.pdf.txt @@ -0,0 +1,532 @@ +arXiv:2301.01137v1 [math.CO] 3 Jan 2023 +On non-degenerate Berge-Tur´an problems +D´aniel Gerbner +Alfr´ed R´enyi Institute of Mathematics +gerbner.daniel@renyi.hu +Abstract +Given a hypergraph H and a graph G, we say that H is a Berge-G if there is a +bijection between the hyperedges of H and the edges of G such that each hyperedge +contains its image. We denote by exk(n, Berge-F) the largest number of hyperedges in +a k-uniform Berge-F-free graph. Let ex(n, H, F) denote the largest number of copies +of H in n-vertex F-free graphs. It is known that ex(n, Kk, F) ≤ exk(n, Berge-F) ≤ +ex(n, Kk, F)+ex(n, F), thus if χ(F) > r, then exk(n, Berge-F) = (1+o(1))ex(n, Kk, F). +We conjecture that exk(n, Berge-F) = ex(n, Kk, F) in this case. We prove this conjec- +ture in several instances, including the cases k = 3 and k = 4. We prove the general +bound exk(n, Berge-F) = ex(n, Kk, F) + O(1). +Keywords: Berge hypergraph, Tur´an +1 +Introduction +Given a hypergraph H and a graph G, we say that H is a Berge copy of G (in short: a +Berge-G) if there is a bijection between the hyperedges of H and the edges of G such that +each hyperedge contains its image. +Berge hypergraphs were introduced by Gerbner and Palmer [9] as a generalization of the +notion of hypergraph cycles due to Berge. +A closely connected area is that of generalized Tur´an problems. Given graphs H and G, +we let N (H, G) denote the number of copies of H in G. Let ex(n, H, F) := max{N (H, G) : +G is an n-vertex F-free graph}. The sytematic study of this topic was initiated by Alon +and Shikhelman [1] after several sporadic results. +The connection between Berge-Tur´an problems and generalized Tur´an problems was es- +tablished by Gerbner and Palmer [10], who showed that ex(n, Kk, F) ≤ exk(n, Berge-F) ≤ +ex(n, Kk, F) + ex(n, F). +The upper bound was strengthened by F¨uredi, Kostochka, and +Luo [3] and independently by Gerbner, Methuku and Palmer [8]. To state this result, we +need some definition. +A blue-red graph G is a graph with each edge colored blue or red. We denote by Gblue +the subgraph consisting of the blue edges and by Gred the subgraph consisting of the red +1 + +edges. We say that a blue-red graph G is F-free if G does not contain F (here we do not +care about the colors). Given an integer k ≥ 3, let g(G) := N (Kk, Gblue) + |E(Gred)|. Let +excol(n, F) := max{g(G) : G is an n-vertex F-free graph}. +Lemma 1.1 ([3],[8]). For any F we have exk(n, Berge-F) ≤ excol(n, F). +A hypergraph Tur´an problem is called degenerate if the order of magnitude of the ex- +tremal function is smaller than the largest possible, i.e. smaller than nk in our case. By +the above bounds, exk(n, Berge-F) = o(nk) if and only if ex(n, Kk, F) = o(nk), which +happens if and only if χ(F) ≤ k by a result of Alon and Shikhelman [1]. +Another re- +sult of Alon and Shikhelman [1] shows that if χ(F) = r + 1 > k, then ex(n, Kk, F) = +(1 + o(1))N (Kk, T(n, r)) = (1 + o(1)) +�r +k +� � n +r +�k. +In the non-degenerate case, for k ≥ 3 we have that ex(n, F) = O(n2) = o(ex(n, Kk, F)), +thus exk(n, Berge-F) = (1 + o(1))ex(n, Kk, F). We believe that a stronger connection also +holds. +Conjecture 1.2. If χ(F) > k and n is sufficiently large, then exk(n, Berge-F) = ex(n, Kk, F). +The above conjecture is known to hold in the case F has a color-critical edge (an edge +whose deletion decreases the chromatic number). The k-uniform expansion F +k of a graph +F is the specific k-uniform Berge copy that contains the most vertices, i.e., the k −2 vertices +added to each edge of F are distinct for different edges, and distinct from the vertices of F. +Pikhurko [19] showed that for r ≥ k, the Tur´an number of K+k +r+1 is equal to N (Kk, T(n, r)) +if n is sufficiently large. According to the survey [18] on expansions, Alon and Pikhurko +observed that Pikhurko’s proof generalizes to the case F is an (r + 1)-chromatic graph with +a color-critical edge. A simpler proof for the Berge case can be found in [7]. +In general, the above observations imply that exk(n, Berge-F) = ex(n, Kk, F) + O(n2). +This was improved to exk(n, Berge-F) = ex(n, Kk, F)+o(n2) in [4]. We further improve this +bound in our next result. +Theorem 1.3. exk(n, Berge-F) = ex(n, Kk, F) + O(1). +We show that Conjecture 1.2 holds if F contains a color-critical vertex (a vertex whose +deletion decreases the chromatic number). +Theorem 1.4. Let χ(F) > k and assume that F contains a color-critical vertex. Then for +sufficiently large n we have exk(n, Berge-F) = ex(n, Kk, F). +We show that Conjecture 1.2 holds in the 3- and 4-uniform case. Furthermore, it holds +in any uniformity if the chromatic number of F is sufficiently large. +Theorem 1.5. (i) Let χ(F) > k and k ≤ 4. Then exk(n, Berge-F) = ex(n, Kk, F) for +sufficiently large n. +(i) Let us fix k and r be sufficiently large. If χ(F) = r + 1, then exk(n, Berge-F) = +ex(n, Kk, F) for sufficiently large n. +2 + +Recall that if χ(F) > k, then the asymptotics of ex(n, Kk, F) is known, thus the asymp- +totics of exk(n, Berge-F) is known. +Even if Conjecture 1.2 is true, it only improves the +asymptotic result to an exact result in the few cases when ex(n, Kk, F) is known. Besides the +case where F has a color-critical edge, we are aware only of the following results. Let 2Kr+1 +denote two vertex-disjoint copies of Kr+1 and Br,1 denote two copies of Kr+1 sharing exactly +one vertex. Gerbner and Patk´os [12] determined ex(n, Kk, 2Kr+1) and ex(n, Kk, Br+1,1). The +first of these results was extended by Gerbner [6] to ex(n, Kk, F) in the case each component +of F either has chromatic number r + 1 and contains a color-critical edge, or has chromatic +number at most r. Gerbner [6] also determined ex(n, Kk, Qr+1) for a class of graphs Qr that +we do not define here and most values of k. +For the Berge copies of the graphs mentioned above, we can show that Conjecture 1.2 +holds. In fact, Br+1,1 and Qr each has a color-critical vertex, thus we already dealt with +them in Theorem 1.4. Let Ki + T(n − i, r) denote the graph we obtain by adding i vertices +to T(n − i, r) and joining them to every vertex. +Theorem 1.6. Let us assume that F consists of s components with chromatic number r +1, +each with a color-critical edge, and any number of components with chromatic number at +most r. Then exk(n, Berge-F) = N (Kk, Ks−1 + T(n − s + 1, r)). +To prove the above theorems, we use the following results on the structure of the extremal +graphs that are interesting on their own. Let us denote by σ(F) the smallest possible order +of a color class in a χ(F)-coloring of F. +Theorem 1.7. Let χ(F) = r + 1 > k and G be an n-vertex F-free blue-red graph with +g(G) = excol(n, F). Then the followings hold. +(i) For every vertex u of G we have that the number of blue k-cliques plus the number of +red edges containing u is at least (1 + o(1)) +�r−1 +k−1 +� +( n +r )k−1. +(ii) Let ε > 0 be sufficiently small. Then there exist an r-partition of V (G) to A1, . . . , Ar, +a constant K = K(F, ε) and a set B of at most rK(σ(F)−1) vertices such that the followings +hold. For each i we have|Ai|= (1 − o(1))n/r, each red edge is between two elements of B, +every vertex of B is adjacent to at least εn vertices in each part and to at least cn vertices in +all but one parts for some constant c = c(F). Furthermore, every vertex of Ai \B is adjacent +to at most εn vertices in Ai and all but at most ε(2rk + 1)n vertices in Aj with j ̸= i. +(iii) Let H be an n-vertex k-uniform Berge-F-free hypergraph with exk(n, Berge-F) hy- +peredges. Then every vertex of H is contained in at least (1 + o(1)) +�r−1 +k−1 +� +( n +r )k−1 hyperedges. +2 +Proofs +We will use the following stability result due to Ma and Qiu [17]. +Theorem 2.1 (Ma, Qiu [17]). Let χ(F) > k and let G be an n-vertex F-free graph that +contains ex(n, Kk, F) − o(nk) copies of Kk. Then G can be turned into T(n, r) by adding +and removing o(n2) edges. +3 + +Let us start with the proof of Theorem 1.7, that we restate here for convenience. +Theorem. Let χ(F) = r + 1 > k and G be an n-vertex F-free blue-red graph with g(G) = +excol(n, F). Then the followings hold. +(i) For every vertex u of G we have that the number of blue k-cliques plus the number of +red edges containing u is at least (1 + o(1)) +�r−1 +k−1 +� +( n +r )k−1. +(ii) Let ε > 0 be sufficiently small. Then there exist an r-partition of V (G) to A1, . . . , Ar, +a constant K = K(F, ε) and a set B of at most rK(σ(F)−1) vertices such that the followings +hold. For each i we have|Ai|= (1 − o(1))n/r, each red edge is between two elements of B, +every vertex of B is adjacent to at least εn vertices in each part and to at least cn vertices in +all but one parts for some constant c = c(F). Furthermore, every vertex of Ai \B is adjacent +to at most εn vertices in Ai and all but at most ε(2rk + 1)n vertices in Aj with j ̸= i. +(iii) Let H be an n-vertex k-uniform Berge-F-free hypergraph with exk(n, Berge-F) hy- +peredges. Then every vertex of H is contained in at least (1 + o(1)) +�r−1 +k−1 +� +( n +r )k−1 hyperedges. +We note that the analogous results for ex(n, Kk, F) can be found in [17]. Generalizations +to some other graphs in place of Kk can be found in [5] for (i) and in [6] for (ii). Our proof +follows the proofs in [5] and [6]. +Proof. Observe that G contains at least ex(n, Kk, F)−ex(n, F) blue copies of Kk, thus Gblue +can be transformed to a complete r-partite graph by adding and removing o(n2) edges by +Theorem 2.1. Note that there may be several different such complete r-partite graphs on the +vertex set V (G) that can be obtained this way, we pick one with the smallest number of edges +inside the parts and denote it by G′. It is easy to see that each part has order (1 − o(1))n/r, +otherwise the number of blue cliques is at most +�r +k +� �n +r +�k − Θ(nk). Let A1, . . . , Ar denote the +parts and let f(v) denote the number of red edges and blue k-cliques incident to v that are +removed this way. Then we have � +v∈V (G) f(v) = o(nk). Consider a set S of |V (F)| vertices +in A1 such that � +v∈S f(v) is minimal. Then by averaging � +v∈S f(v) ≤ +|S| +|V1| +� +v∈V1 f(v) = +o(nk−1). +Let us consider blue k-cliques and red edges that contain exactly one vertex s of S, and +the other vertices are in the common neighborhood of S in G. Let dG(k, S) denote the +number of such blue k-cliques and red edges. Observe that each vertex of S is in dG(k,S) +|S| +such +blue k-cliques and red edges. Clearly +dT (n,r)(k,S) +|S| += (1 + o(1)) +�r−1 +k−1 +� +( n +r )k−1. +Let x denote the number of blue k-cliques and red edges that contain u and a vertex +from S, then x = O(nk−2). Now we apply a variant of Zykov’s symmetrization [23]. If +dG(k, u) < dG(k,S) +|S| +−x, then we remove the edges incident to u from G. Then for every vertex +v that is connected to each vertex of S with a blue edge, we connect u to v with a blue edge. +For every vertex w that is connected to each vertex of S with a red edge, we connect u to +w with a red edge. This way we do not create any copy of F, as the copy should contain u, +but u could be replaced by any vertex of S that is not already in the copy, to create a copy +of F in G. We removed dG(k, u) blue k-cliqes and red edges, but added at least dG(H,S) +|S| +− x +blue k-cliqes and red edges, a contradiction. +4 + +Therefore, we have that the blue k-cliques plus the red edges containing u is at least +dG(k, S) +|S| +− x ≥ dT(n,r)(k, S) +|S| +− +� +v∈S +f(v) − x = dT(n,r)(k, S) +|S| +− o(nk−1). +This completes the proof of (i). +The proof of (iii) is similar. We pick S the same way, but instead of blue k-cliques and red +edges, we count the hyperedges containing u, let dH(k, u) denote their number. Let y denote +the number of hyperedges that contain u and a vertex from S. If dH(k, u) < dH(k,S) +|S| +−x, then +we remove the hyperedges containing u and for every hyperedge H that contains exactly one +vertex v ∈ S, we add (H \ {v}) ∪ {u} as a hyperedge. Then the same reasoning as above +completes the proof of (iii). +Let B denote the set of vertices that are adjacent to at least εn vertices in their part Ai. +Note that by the choice of G′, vertices of B are incident to at least εn vertices in each other +part. Let Bi = B ∩ Ai. +Claim 2.2. There is a K depending on ε and F such that |B|≤ K(σ(F) − 1). +The analogous claim for uncolored graphs G0 with ex(n, Kk, F) copies of Kk is in [17]. +However, the proof of that claim does not use that G0 is extremal, only that G0 contains +ex(n, Kk, F) − o(nk) copies of Kk. As this holds for G as well, the claim follows. +Consider now the set Ui of vertices v such that v ∈ Ai \ B is adjacent to less than +|Aj|−ε(2rk + 1)n vertices of some Aj. As there are o(n2) edges missing between parts, we +have that |Ui|= o(n). +Claim 2.3. For each i we have that Ui = ∅. +Proof of Claim. Let Vi = Ai \ (Bi ∩ Ui), then we have that |Vi|≥ |Ai|−εn. +Let us delete the edges from each v ∈ Ui to Ai and connect v to each vertex of each Vj, +j ̸= i with a blue edge. We claim that the resulting graph G′′ is F-free. Indeed, consider a +copy F0 of F with the smallest number of vertices in Ui. Clearly F0 contains a vertex v ∈ Ui, +as all the new edges are incident to such a vertex. Let Q be the set of vertices in F0 that +are adjacent to v in G′. They are each from ∪j̸=iVj. Their common neighborhood in Vi is +of order n +r − o(n). Therefore, at least one of the common neighbors is not in F0, thus we +can replace v with that vertex to obtain another copy of F with less vertices from ∪r +i=1Ui, a +contradiction. +We deleted at most εnk−1 blue k-cliques and red edges for each vertex v ∈ Ui, since +they each contain one of the less than ǫn edges incident to v inside Ui. We claim that we +added more than εnk−1 blue k-cliques. We consider only those blue k-cliques that contain +v, a new neighbor of v in Vj with j ̸= i, and k − 2 other vertices from other sets Vℓ. We +have at least 2rkεn choices for the neighbor and at least n/r − εn choices for the other +vertices. If ε is sufficiently small, then indeed, we obtain more than εn new blue k-cliques, +thus g(G′′) > g(G′), a contradiction unless Ui is empty. +■ +5 + +Now we show that there is a constant c = c(F) such that each vertex is adjacent to at +least cn vertices in all but one parts. Assume that v is adjacent to less than cn vertices +in A1 and in A2. Then the number of blue cliques containing v is at most +�r−2 +k−1 +� +( n +r )k−1 + +�r−2 +k−2 +� +( n +r )k−2cn + +�r−2 +k−3 +� +( n +r )k−3cn2, contradiction to (i) if c is small enough. +It is left to show that each red edge is between vertices in B. Assume that u ̸∈ B and uv +is a red edge. Let us change its color to blue. We will find more than one new blue k-clique +greedily. We can assume without loss of generality that u ∈ A1 and v is in either A1 or in +A2. Let us observe that u and v have at least cn − ε(2rk + 2)n common neighbors in Gblue +inside V3, we pick one of them. These three vertices have at least cn − 2ε(2rk + 2)n common +neighbors in Gblue inside V4, we pick one of them, and so one. We can pick k vertices if +cn−(k −2)ε(2rk + 2)n > 0, which holds if ε is small enough. Clearly we can pick more than +one blue k-clique this way, completing the proof of (ii). +■ +Theorem 1.3 is easily implied by (ii) of Theorem 1.7, since in an F-free n-vertex blue-red +graph, the number of blue k-cliques is at most ex(n, Kk, F), while the number of red edges +inside B is O(1). Theorem 1.4 is also implied by (ii) of Theorem 1.7, since a color-critical +vertex means that σ(F) = 1, thus |B|= 0, hence there are no red edges. +Let us continue with the proof of Theorem 1.5. Recall that it states that if χ(F) > k +and k ≤ 4 or if χ(F) is sufficiently large, then Conjecture 1.2 holds. +Proof of Theorem 1.5. Let χ(F) = r+1. We will use Lemma 1.1. Let G be a blue-red F-free +graph with g(G) = excol(n, F). Assume that there is a red edge uv in G and apply now (ii) +of Theorem 1.7. We obtain a partition of V (G) to A1, . . . , Ar with |Ai|= (1 + o(1))n/r such +that there are o(n) edges inside parts, and there is a set B of vertices with |B|= o(n) such +that each vertex outside B is adjacent to all but o(n) vertices in each other part. +Assume that u and v have Ω(n) common neighbors in at least k−2 of the sets A1, . . . , Ar, +say A1, . . . , Ak−2. Then at least Ω(n) of those vertices are not in B, we will use only those +vertices. We pick a common neighbor in A1 \ B, then it has Ω(n) common neighbor with u +and v in A2. Therefore, we can pick a common neighbor in A2 \ B, and so on. The resulting +cliques do not contain any vertex of B, thus by turning uv blue, we obtain multiple blue +k-cliques, thus g(G) increases, a contradiction. +We obtained that u and v have Ω(n) common neighbors in at most k − 3 of the sets Ai, +say A1, . . . , Ak−3. In the remaining r −k +3 sets Ai, they have o(n) common neighbors, thus +at least one of them, say u has at most (1 + o(1))(r − k + 3)n/2r neighbors in Ak−2, . . . , Ar. +Consider now the number of blue k-cliques containing u. There are o(nk−1) blue k-cliques +that contain u and an edge inside an Ai that is not incident to u. Therefore, we can focus +on those blue k-cliques that contain u, and the other k − 1 vertices are from different parts. +Let K be such a blue k-clique and assume that K contains i vertices from A1, . . . , Ak−3. +There are at most (1+o(1)) +�k−3 +i +� � n +r +�i ways to pick such an i-set. For the remaining k −1−i +vertices of K, we have to pick one neighbor of u from k − 1 − i of the remaining r − k + 3 +sets, and in total u has (1 + o(1))(r − k + 3)n/2r neighbors in those sets. Then the number +of (k − 1 − i)-cliques is at most ex((1 + o(1))(r − k + 3)n/2r, Kk−1−i, Kr−k+4). A theorem +6 + +of Zykov [23] states that ex(n, Ks, Kt) = N (Ks, T(n, t − 1)) = (1 + o(1)) +�t−1 +s +� � n +t−1 +�s, thus +there are at most (1 + o(1)) +�r−k+3 +k−i−1 +� � n +2r +�k−1−i ways to pick the (k − 1 − i)-clique. +We apply (i) of Theorem 1.7, thus we know that each vertex v is in at least (1 + +o(1)) +�r−1 +k−1 +� +( n +r )k−1 blue k-cliques. Therefore, +(1 + o(1)) +k−3 +� +i=0 +�k − 3 +i +� �n +r +�i �r − k + 3 +k − i − 1 +� � n +2r +�k−1−i +≥ (1 + o(1)) +�r − 1 +k − 1 +� �n +r +�k−1 +. +This holds only if +k−3 +� +i=0 +�k − 3 +i +��r − k + 3 +k − i − 1 +� �1 +2 +�k−1−i +≥ +�r − 1 +k − 1 +� +. +(1) +If k = 3, then i = 0 and +� r +k−1 +� +/4 ≥ +�r−1 +k−1 +� +, a contradiction. +If k = 4, then (1) gives +�r−1 +3 +� �1 +2 +�3 + +�r−1 +2 +� � 1 +2 +�2 ≥ +�r−1 +3 +� +. If r ≥ 6, then +�r−1 +2 +� +≤ +�r−1 +3 +� +, thus +�r−1 +3 +� � 1 +2 +�3 + +�r−1 +2 +� � 1 +2 +�2 ≤ +�r−1 +3 +� � 1 +8 + 1 +4 +� +< +�r−1 +3 +� +, a contradiction. If r = 5 or r = 4, then one +can easily obtain a contradiction as well. This completes the proof of (i). +There are several other pairs (k, r) when we could obtain a contradiction a similar way. +However, if k = r, the left hand side has a term +�k−3 +k−4 +� +/8. If k ≥ 11, then this term alone is +larger than the right hand side, thus we do not have a contradiction in general. In fact, one +can easily see that for k = r = 5 we do not obtain any contradiction. On the other hand, if +k is fixed and r grows, there is only one term on the left hand side of (1) of order rk−1, and +it is rk−1/2k−1(k − 1)!. Since the leading term on the right hand side is rk−1/(k − 1)!, we +obtain a contradiction for r large enough, proving (ii). +■ +Let us continue with the proof of Theorem 1.6 that we restate here for convenience. +Theorem. (i) Let us assume that F consists of s components with chromatic number r + 1, +each with a color-critical edge, and any number of components with chromatic number at +most r. Then exk(n, Berge-F) = N (Kk, Ks−1 + T(n − s + 1, r)). +(ii) exk(n, Berge-Br+1,1) = N (Kk, T +(n, r)). +The corresponding generalized Tur´an results are proved in [6] and we will extend the +proofs from there. We omit some details. We remark that in [6], the proof of the statement +ex(n, Kk, F) = N (Kk, Ks−1+T(n−s+1, r)) shows a bit more: if an n-vertex F-free graph G is +not a subgraph of Ks−1+T(n−s+1, r), then it contains N (Kk, Ks−1+T(n−s+1, r))−Ω(nk−1) +copies of Kk. This immediately implies for us that Gblue is a subgraph of Ks−1+T(n−s+1, r). +Changing any blue edge in Ks−1 + T(n − s + 1, r) to red destroys Θ(nk−2) copies of Kk, thus +it decreases g(G). This gives an alternative proof of (i) of Theorem 1.6. +Proof. We start with proving (i). Let G be a blue-red F-free graph with g(G) = excol(n, F). +We apply (ii) of Theorem 1.7. Assume first that there are s independent edges u1v1, . . . , usvs +inside the parts such that for each i, at least one of ui and vi are not in B. Observe that ui +7 + +and vi have Ω(n) common neighbors in each part besides the one containing them. Using +this, we can easily extend each edge to an (r + 1)-chromatic component of F, where uivi +plays the role of a color-critical edge. We can also find the other components to obtain a +copy of F in G, a contradiction. +If |B|≥ s, then we can find s distinct vertices among their neighbors not in B, resulting +in the contradiction. By similar reasoning, there are no s − |B| independent edges inside +parts but outside B. Therefore, the edges inside parts that are not incident to any vertex of +B form at most s − 1 − |B| stars plus O(1) further edges. Since the vertices outside B are +incident to o(n) edges inside parts, there are o(nk−1) k-cliques containing such a vertex. This +implies that deleting all the edges inside parts that are not incident to B, we lose o(n|V (H)|−1) +copies of H. If |B|< s − 1, then we can add a vertex to B creating Θ(n|V (H)|−1) copies of +H, a contradiction. We obtained that |B|= s − 1 and then there is no edge inside parts +but outside B. This implies that G is a subgraph of Ks−1 + T(n − s + 1, r), completing the +proof. +■ +Funding: Research supported by the National Research, Development and Innovation +Office - NKFIH under the grants SNN 129364, FK 132060, and KKP-133819. +References +[1] N. Alon, C. Shikhelman. Many T copies in H-free graphs. Journal of Combinatorial +Theory, Series B, 121, 146–172, 2016. +[2] J. I. Brown, A. Sidorenko. The inducibility of complete bipartite graphs. Journal of +Graph Theory, 18(6), 629–645, 1994. +[3] Z. F¨uredi, A. Kostochka, R. Luo. Avoiding long Berge cycles, Journal of Combinatorial +Theory, Series B 137, 55–64, 2019. +[4] D. Gerbner. Counting multiple graphs in generalized Tur´an problems. arXiv preprint +arXiv:2007.11645, 2020. +[5] D. Gerbner. Some stability and exact results in generalized Tur´an problems. arXiv +preprint arXiv:2204.04600, 2022. +[6] D. Gerbner. Some exact results for non-degenerate generalized Tur´an problems. arXiv +preprint arXiv:2209.03426, 2022. +[7] D. Gerbner. Rainbow copies of F in families of H. arXiv preprint arXiv:2211.01565, +2022. +[8] D. Gerbner, A. Methuku, C. Palmer. General lemmas for Berge-Tur´an hypergraph prob- +lems. European Journal of Combinatorics 86, Article 103082, 2020. +8 + +[9] D. Gerbner, C. Palmer, Extremal Results for Berge hypergraphs. SIAM Journal on +Discrete Mathematics, 31, 2314–2327, 2017. +[10] D. Gerbner, C. Palmer. Counting copies of a fixed subgraph in F-free graphs. European +Journal of Combinatorics 82 (2019) Article 103001. +[11] D. Gerbner, C. Palmer. Some exact results for generalized Tur´an problems. European +Journal of Combinatorics, 103, 103519, 2022. +[12] D. Gerbner, B. Patk´os. Generalized Tur´an results for intersecting cliques, arXiv preprint +arXiv:2101.08094, 2021. +[13] E. Gy˝ori, J. Pach, and M. Simonovits. On the maximal number of certain subgraphs in +Kr-free graphs. Graphs and Combinatorics, 7(1), 31–37, 1991. +[14] Doudou Hei, Xinmin Hou, Boyuan Liu, Some exact results of the generalized Tur´an +numbers for paths, arXiv preprint arXiv:2112.14895, 2021. +[15] B. Lidick´y, K. Murphy. Maximizing five-cycles in Kr-free graphs. European Journal of +Combinatorics, 97, 103367, 2021. +[16] H. Liu, O. Pikhurko, M. Sharifzadeh, and K. Staden. Stability from graph symmetri- +sation arguments with applications to inducibility. arXiv preprint arXiv:2012.10731, +2020. +[17] J. Ma, Y. Qiu, Some sharp results on the generalized Tur´an numbers. European Journal +of Combinatorics, 84, 103026, 2018. +[18] D. Mubayi, J. Verstra¨ete. A survey of Tur´an problems for expansions. Recent Trends in +Combinatorics, 117–143, 2016. +[19] O. Pikhurko. Exact computation of the hypergraph Tur´an function for expanded com- +plete 2-graphs, Journal of Combinatorial Theory, Series B, 103(2) 220–225, 2013. +[20] M. Simonovits. A method for solving extremal problems in graph theory, stability prob- +lems. Theory of Graphs, Proc. Colloq., Tihany, 1966, Academic Press, New York, 279– +319, 1968. +[21] M. Simonovits. Extremal graph problems with symmetrical extremal graphs. Additional +chromatic conditions, Discrete Math. 7, 349–376, 1974. +[22] P. Tur´an. Egy gr´afelm´eleti sz´els˝o´ert´ekfeladatr´ol. Mat. Fiz. Lapok, 48, 436–452, 1941. +[23] A. A. Zykov. On some properties of linear complexes. Matematicheskii Sbornik,66(2), +163–188, 1949. +9 + diff --git a/hdAzT4oBgHgl3EQfMvuF/content/tmp_files/load_file.txt b/hdAzT4oBgHgl3EQfMvuF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..11daa9b8c78a3fcfc0ab61d0fab6e9cef72b4c26 --- /dev/null +++ b/hdAzT4oBgHgl3EQfMvuF/content/tmp_files/load_file.txt @@ -0,0 +1,413 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf,len=412 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='01137v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='CO] 3 Jan 2023 On non-degenerate Berge-Tur´an problems D´aniel Gerbner Alfr´ed R´enyi Institute of Mathematics gerbner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='daniel@renyi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='hu Abstract Given a hypergraph H and a graph G, we say that H is a Berge-G if there is a bijection between the hyperedges of H and the edges of G such that each hyperedge contains its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' We denote by exk(n, Berge-F) the largest number of hyperedges in a k-uniform Berge-F-free graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Let ex(n, H, F) denote the largest number of copies of H in n-vertex F-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' It is known that ex(n, Kk, F) ≤ exk(n, Berge-F) ≤ ex(n, Kk, F)+ex(n, F), thus if χ(F) > r, then exk(n, Berge-F) = (1+o(1))ex(n, Kk, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' We conjecture that exk(n, Berge-F) = ex(n, Kk, F) in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' We prove this conjec- ture in several instances, including the cases k = 3 and k = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' We prove the general bound exk(n, Berge-F) = ex(n, Kk, F) + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Keywords: Berge hypergraph, Tur´an 1 Introduction Given a hypergraph H and a graph G, we say that H is a Berge copy of G (in short: a Berge-G) if there is a bijection between the hyperedges of H and the edges of G such that each hyperedge contains its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Berge hypergraphs were introduced by Gerbner and Palmer [9] as a generalization of the notion of hypergraph cycles due to Berge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' A closely connected area is that of generalized Tur´an problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Given graphs H and G, we let N (H, G) denote the number of copies of H in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Let ex(n, H, F) := max{N (H, G) : G is an n-vertex F-free graph}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' The sytematic study of this topic was initiated by Alon and Shikhelman [1] after several sporadic results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' The connection between Berge-Tur´an problems and generalized Tur´an problems was es- tablished by Gerbner and Palmer [10], who showed that ex(n, Kk, F) ≤ exk(n, Berge-F) ≤ ex(n, Kk, F) + ex(n, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' The upper bound was strengthened by F¨uredi, Kostochka, and Luo [3] and independently by Gerbner, Methuku and Palmer [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' To state this result, we need some definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' A blue-red graph G is a graph with each edge colored blue or red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' We denote by Gblue the subgraph consisting of the blue edges and by Gred the subgraph consisting of the red 1 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' We say that a blue-red graph G is F-free if G does not contain F (here we do not care about the colors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Given an integer k ≥ 3, let g(G) := N (Kk, Gblue) + |E(Gred)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Let excol(n, F) := max{g(G) : G is an n-vertex F-free graph}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='1 ([3],[8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' For any F we have exk(n, Berge-F) ≤ excol(n, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' A hypergraph Tur´an problem is called degenerate if the order of magnitude of the ex- tremal function is smaller than the largest possible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' smaller than nk in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' By the above bounds, exk(n, Berge-F) = o(nk) if and only if ex(n, Kk, F) = o(nk), which happens if and only if χ(F) ≤ k by a result of Alon and Shikhelman [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Another re- sult of Alon and Shikhelman [1] shows that if χ(F) = r + 1 > k, then ex(n, Kk, F) = (1 + o(1))N (Kk, T(n, r)) = (1 + o(1)) �r k � � n r �k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' In the non-degenerate case, for k ≥ 3 we have that ex(n, F) = O(n2) = o(ex(n, Kk, F)), thus exk(n, Berge-F) = (1 + o(1))ex(n, Kk, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' We believe that a stronger connection also holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' If χ(F) > k and n is sufficiently large, then exk(n, Berge-F) = ex(n, Kk, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' The above conjecture is known to hold in the case F has a color-critical edge (an edge whose deletion decreases the chromatic number).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' The k-uniform expansion F +k of a graph F is the specific k-uniform Berge copy that contains the most vertices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=', the k −2 vertices added to each edge of F are distinct for different edges, and distinct from the vertices of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Pikhurko [19] showed that for r ≥ k, the Tur´an number of K+k r+1 is equal to N (Kk, T(n, r)) if n is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' According to the survey [18] on expansions, Alon and Pikhurko observed that Pikhurko’s proof generalizes to the case F is an (r + 1)-chromatic graph with a color-critical edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' A simpler proof for the Berge case can be found in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' In general, the above observations imply that exk(n, Berge-F) = ex(n, Kk, F) + O(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' This was improved to exk(n, Berge-F) = ex(n, Kk, F)+o(n2) in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' We further improve this bound in our next result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' exk(n, Berge-F) = ex(n, Kk, F) + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' We show that Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='2 holds if F contains a color-critical vertex (a vertex whose deletion decreases the chromatic number).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Let χ(F) > k and assume that F contains a color-critical vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Then for sufficiently large n we have exk(n, Berge-F) = ex(n, Kk, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' We show that Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='2 holds in the 3- and 4-uniform case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Furthermore, it holds in any uniformity if the chromatic number of F is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' (i) Let χ(F) > k and k ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Then exk(n, Berge-F) = ex(n, Kk, F) for sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' (i) Let us fix k and r be sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' If χ(F) = r + 1, then exk(n, Berge-F) = ex(n, Kk, F) for sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' 2 Recall that if χ(F) > k, then the asymptotics of ex(n, Kk, F) is known, thus the asymp- totics of exk(n, Berge-F) is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Even if Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='2 is true, it only improves the asymptotic result to an exact result in the few cases when ex(n, Kk, F) is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Besides the case where F has a color-critical edge, we are aware only of the following results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Let 2Kr+1 denote two vertex-disjoint copies of Kr+1 and Br,1 denote two copies of Kr+1 sharing exactly one vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Gerbner and Patk´os [12] determined ex(n, Kk, 2Kr+1) and ex(n, Kk, Br+1,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' The first of these results was extended by Gerbner [6] to ex(n, Kk, F) in the case each component of F either has chromatic number r + 1 and contains a color-critical edge, or has chromatic number at most r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Gerbner [6] also determined ex(n, Kk, Qr+1) for a class of graphs Qr that we do not define here and most values of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' For the Berge copies of the graphs mentioned above, we can show that Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' In fact, Br+1,1 and Qr each has a color-critical vertex, thus we already dealt with them in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Let Ki + T(n − i, r) denote the graph we obtain by adding i vertices to T(n − i, r) and joining them to every vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Let us assume that F consists of s components with chromatic number r +1, each with a color-critical edge, and any number of components with chromatic number at most r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Then exk(n, Berge-F) = N (Kk, Ks−1 + T(n − s + 1, r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' To prove the above theorems, we use the following results on the structure of the extremal graphs that are interesting on their own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Let us denote by σ(F) the smallest possible order of a color class in a χ(F)-coloring of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Let χ(F) = r + 1 > k and G be an n-vertex F-free blue-red graph with g(G) = excol(n, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Then the followings hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' (i) For every vertex u of G we have that the number of blue k-cliques plus the number of red edges containing u is at least (1 + o(1)) �r−1 k−1 � ( n r )k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' (ii) Let ε > 0 be sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Then there exist an r-partition of V (G) to A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' , Ar, a constant K = K(F, ε) and a set B of at most rK(σ(F)−1) vertices such that the followings hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' For each i we have|Ai|= (1 − o(1))n/r, each red edge is between two elements of B, every vertex of B is adjacent to at least εn vertices in each part and to at least cn vertices in all but one parts for some constant c = c(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Furthermore, every vertex of Ai \\B is adjacent to at most εn vertices in Ai and all but at most ε(2rk + 1)n vertices in Aj with j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' (iii) Let H be an n-vertex k-uniform Berge-F-free hypergraph with exk(n, Berge-F) hy- peredges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Then every vertex of H is contained in at least (1 + o(1)) �r−1 k−1 � ( n r )k−1 hyperedges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' 2 Proofs We will use the following stability result due to Ma and Qiu [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='1 (Ma, Qiu [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Let χ(F) > k and let G be an n-vertex F-free graph that contains ex(n, Kk, F) − o(nk) copies of Kk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Then G can be turned into T(n, r) by adding and removing o(n2) edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' 3 Let us start with the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='7, that we restate here for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Let χ(F) = r + 1 > k and G be an n-vertex F-free blue-red graph with g(G) = excol(n, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Then the followings hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' (i) For every vertex u of G we have that the number of blue k-cliques plus the number of red edges containing u is at least (1 + o(1)) �r−1 k−1 � ( n r )k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' (ii) Let ε > 0 be sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Then there exist an r-partition of V (G) to A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' , Ar, a constant K = K(F, ε) and a set B of at most rK(σ(F)−1) vertices such that the followings hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' For each i we have|Ai|= (1 − o(1))n/r, each red edge is between two elements of B, every vertex of B is adjacent to at least εn vertices in each part and to at least cn vertices in all but one parts for some constant c = c(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Furthermore, every vertex of Ai \\B is adjacent to at most εn vertices in Ai and all but at most ε(2rk + 1)n vertices in Aj with j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' (iii) Let H be an n-vertex k-uniform Berge-F-free hypergraph with exk(n, Berge-F) hy- peredges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Then every vertex of H is contained in at least (1 + o(1)) �r−1 k−1 � ( n r )k−1 hyperedges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' We note that the analogous results for ex(n, Kk, F) can be found in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Generalizations to some other graphs in place of Kk can be found in [5] for (i) and in [6] for (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Our proof follows the proofs in [5] and [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Observe that G contains at least ex(n, Kk, F)−ex(n, F) blue copies of Kk, thus Gblue can be transformed to a complete r-partite graph by adding and removing o(n2) edges by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Note that there may be several different such complete r-partite graphs on the vertex set V (G) that can be obtained this way, we pick one with the smallest number of edges inside the parts and denote it by G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' It is easy to see that each part has order (1 − o(1))n/r, otherwise the number of blue cliques is at most �r k � �n r �k − Θ(nk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Let A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' , Ar denote the parts and let f(v) denote the number of red edges and blue k-cliques incident to v that are removed this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Then we have � v∈V (G) f(v) = o(nk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Consider a set S of |V (F)| vertices in A1 such that � v∈S f(v) is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Then by averaging � v∈S f(v) ≤ |S| |V1| � v∈V1 f(v) = o(nk−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Let us consider blue k-cliques and red edges that contain exactly one vertex s of S, and the other vertices are in the common neighborhood of S in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Let dG(k, S) denote the number of such blue k-cliques and red edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Observe that each vertex of S is in dG(k,S) |S| such blue k-cliques and red edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Clearly dT (n,r)(k,S) |S| = (1 + o(1)) �r−1 k−1 � ( n r )k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Let x denote the number of blue k-cliques and red edges that contain u and a vertex from S, then x = O(nk−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Now we apply a variant of Zykov’s symmetrization [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' If dG(k, u) < dG(k,S) |S| −x, then we remove the edges incident to u from G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Then for every vertex v that is connected to each vertex of S with a blue edge, we connect u to v with a blue edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' For every vertex w that is connected to each vertex of S with a red edge, we connect u to w with a red edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' This way we do not create any copy of F, as the copy should contain u, but u could be replaced by any vertex of S that is not already in the copy, to create a copy of F in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' We removed dG(k, u) blue k-cliqes and red edges, but added at least dG(H,S) |S| − x blue k-cliqes and red edges, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' 4 Therefore, we have that the blue k-cliques plus the red edges containing u is at least dG(k, S) |S| − x ≥ dT(n,r)(k, S) |S| − � v∈S f(v) − x = dT(n,r)(k, S) |S| − o(nk−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' This completes the proof of (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' The proof of (iii) is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' We pick S the same way, but instead of blue k-cliques and red edges, we count the hyperedges containing u, let dH(k, u) denote their number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Let y denote the number of hyperedges that contain u and a vertex from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' If dH(k, u) < dH(k,S) |S| −x, then we remove the hyperedges containing u and for every hyperedge H that contains exactly one vertex v ∈ S, we add (H \\ {v}) ∪ {u} as a hyperedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Then the same reasoning as above completes the proof of (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Let B denote the set of vertices that are adjacent to at least εn vertices in their part Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Note that by the choice of G′, vertices of B are incident to at least εn vertices in each other part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Let Bi = B ∩ Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' There is a K depending on ε and F such that |B|≤ K(σ(F) − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' The analogous claim for uncolored graphs G0 with ex(n, Kk, F) copies of Kk is in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' However, the proof of that claim does not use that G0 is extremal, only that G0 contains ex(n, Kk, F) − o(nk) copies of Kk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' As this holds for G as well, the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Consider now the set Ui of vertices v such that v ∈ Ai \\ B is adjacent to less than |Aj|−ε(2rk + 1)n vertices of some Aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' As there are o(n2) edges missing between parts, we have that |Ui|= o(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' For each i we have that Ui = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Proof of Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Let Vi = Ai \\ (Bi ∩ Ui), then we have that |Vi|≥ |Ai|−εn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Let us delete the edges from each v ∈ Ui to Ai and connect v to each vertex of each Vj, j ̸= i with a blue edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' We claim that the resulting graph G′′ is F-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Indeed, consider a copy F0 of F with the smallest number of vertices in Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Clearly F0 contains a vertex v ∈ Ui, as all the new edges are incident to such a vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Let Q be the set of vertices in F0 that are adjacent to v in G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' They are each from ∪j̸=iVj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Their common neighborhood in Vi is of order n r − o(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Therefore, at least one of the common neighbors is not in F0, thus we can replace v with that vertex to obtain another copy of F with less vertices from ∪r i=1Ui, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' We deleted at most εnk−1 blue k-cliques and red edges for each vertex v ∈ Ui, since they each contain one of the less than ǫn edges incident to v inside Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' We claim that we added more than εnk−1 blue k-cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' We consider only those blue k-cliques that contain v, a new neighbor of v in Vj with j ̸= i, and k − 2 other vertices from other sets Vℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' We have at least 2rkεn choices for the neighbor and at least n/r − εn choices for the other vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' If ε is sufficiently small, then indeed, we obtain more than εn new blue k-cliques, thus g(G′′) > g(G′), a contradiction unless Ui is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' ■ 5 Now we show that there is a constant c = c(F) such that each vertex is adjacent to at least cn vertices in all but one parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Assume that v is adjacent to less than cn vertices in A1 and in A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Then the number of blue cliques containing v is at most �r−2 k−1 � ( n r )k−1 + �r−2 k−2 � ( n r )k−2cn + �r−2 k−3 � ( n r )k−3cn2, contradiction to (i) if c is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' It is left to show that each red edge is between vertices in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Assume that u ̸∈ B and uv is a red edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Let us change its color to blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' We will find more than one new blue k-clique greedily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' We can assume without loss of generality that u ∈ A1 and v is in either A1 or in A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Let us observe that u and v have at least cn − ε(2rk + 2)n common neighbors in Gblue inside V3, we pick one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' These three vertices have at least cn − 2ε(2rk + 2)n common neighbors in Gblue inside V4, we pick one of them, and so one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' We can pick k vertices if cn−(k −2)ε(2rk + 2)n > 0, which holds if ε is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Clearly we can pick more than one blue k-clique this way, completing the proof of (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' ■ Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='3 is easily implied by (ii) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='7, since in an F-free n-vertex blue-red graph, the number of blue k-cliques is at most ex(n, Kk, F), while the number of red edges inside B is O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='4 is also implied by (ii) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='7, since a color-critical vertex means that σ(F) = 1, thus |B|= 0, hence there are no red edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Let us continue with the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Recall that it states that if χ(F) > k and k ≤ 4 or if χ(F) is sufficiently large, then Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Let χ(F) = r+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' We will use Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Let G be a blue-red F-free graph with g(G) = excol(n, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Assume that there is a red edge uv in G and apply now (ii) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' We obtain a partition of V (G) to A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' , Ar with |Ai|= (1 + o(1))n/r such that there are o(n) edges inside parts, and there is a set B of vertices with |B|= o(n) such that each vertex outside B is adjacent to all but o(n) vertices in each other part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Assume that u and v have Ω(n) common neighbors in at least k−2 of the sets A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' , Ar, say A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' , Ak−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Then at least Ω(n) of those vertices are not in B, we will use only those vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' We pick a common neighbor in A1 \\ B, then it has Ω(n) common neighbor with u and v in A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Therefore, we can pick a common neighbor in A2 \\ B, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' The resulting cliques do not contain any vertex of B, thus by turning uv blue, we obtain multiple blue k-cliques, thus g(G) increases, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' We obtained that u and v have Ω(n) common neighbors in at most k − 3 of the sets Ai, say A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' , Ak−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' In the remaining r −k +3 sets Ai, they have o(n) common neighbors, thus at least one of them, say u has at most (1 + o(1))(r − k + 3)n/2r neighbors in Ak−2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' , Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Consider now the number of blue k-cliques containing u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' There are o(nk−1) blue k-cliques that contain u and an edge inside an Ai that is not incident to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Therefore, we can focus on those blue k-cliques that contain u, and the other k − 1 vertices are from different parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Let K be such a blue k-clique and assume that K contains i vertices from A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' , Ak−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' There are at most (1+o(1)) �k−3 i � � n r �i ways to pick such an i-set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' For the remaining k −1−i vertices of K, we have to pick one neighbor of u from k − 1 − i of the remaining r − k + 3 sets, and in total u has (1 + o(1))(r − k + 3)n/2r neighbors in those sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Then the number of (k − 1 − i)-cliques is at most ex((1 + o(1))(r − k + 3)n/2r, Kk−1−i, Kr−k+4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' A theorem 6 of Zykov [23] states that ex(n, Ks, Kt) = N (Ks, T(n, t − 1)) = (1 + o(1)) �t−1 s � � n t−1 �s, thus there are at most (1 + o(1)) �r−k+3 k−i−1 � � n 2r �k−1−i ways to pick the (k − 1 − i)-clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' We apply (i) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='7, thus we know that each vertex v is in at least (1 + o(1)) �r−1 k−1 � ( n r )k−1 blue k-cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Therefore, (1 + o(1)) k−3 � i=0 �k − 3 i � �n r �i �r − k + 3 k − i − 1 � � n 2r �k−1−i ≥ (1 + o(1)) �r − 1 k − 1 � �n r �k−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' This holds only if k−3 � i=0 �k − 3 i ��r − k + 3 k − i − 1 � �1 2 �k−1−i ≥ �r − 1 k − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' (1) If k = 3, then i = 0 and � r k−1 � /4 ≥ �r−1 k−1 � , a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' If k = 4, then (1) gives �r−1 3 � �1 2 �3 + �r−1 2 � � 1 2 �2 ≥ �r−1 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' If r ≥ 6, then �r−1 2 � ≤ �r−1 3 � , thus �r−1 3 � � 1 2 �3 + �r−1 2 � � 1 2 �2 ≤ �r−1 3 � � 1 8 + 1 4 � < �r−1 3 � , a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' If r = 5 or r = 4, then one can easily obtain a contradiction as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' This completes the proof of (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' There are several other pairs (k, r) when we could obtain a contradiction a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' However, if k = r, the left hand side has a term �k−3 k−4 � /8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' If k ≥ 11, then this term alone is larger than the right hand side, thus we do not have a contradiction in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' In fact, one can easily see that for k = r = 5 we do not obtain any contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' On the other hand, if k is fixed and r grows, there is only one term on the left hand side of (1) of order rk−1, and it is rk−1/2k−1(k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='. Since the leading term on the right hand side is rk−1/(k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=', we obtain a contradiction for r large enough, proving (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' ■ Let us continue with the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='6 that we restate here for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' (i) Let us assume that F consists of s components with chromatic number r + 1, each with a color-critical edge, and any number of components with chromatic number at most r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Then exk(n, Berge-F) = N (Kk, Ks−1 + T(n − s + 1, r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' (ii) exk(n, Berge-Br+1,1) = N (Kk, T +(n, r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' The corresponding generalized Tur´an results are proved in [6] and we will extend the proofs from there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' We omit some details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' We remark that in [6], the proof of the statement ex(n, Kk, F) = N (Kk, Ks−1+T(n−s+1, r)) shows a bit more: if an n-vertex F-free graph G is not a subgraph of Ks−1+T(n−s+1, r), then it contains N (Kk, Ks−1+T(n−s+1, r))−Ω(nk−1) copies of Kk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' This immediately implies for us that Gblue is a subgraph of Ks−1+T(n−s+1, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Changing any blue edge in Ks−1 + T(n − s + 1, r) to red destroys Θ(nk−2) copies of Kk, thus it decreases g(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' This gives an alternative proof of (i) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' We start with proving (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Let G be a blue-red F-free graph with g(G) = excol(n, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' We apply (ii) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Assume first that there are s independent edges u1v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' , usvs inside the parts such that for each i, at least one of ui and vi are not in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Observe that ui 7 and vi have Ω(n) common neighbors in each part besides the one containing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Using this, we can easily extend each edge to an (r + 1)-chromatic component of F, where uivi plays the role of a color-critical edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' We can also find the other components to obtain a copy of F in G, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' If |B|≥ s, then we can find s distinct vertices among their neighbors not in B, resulting in the contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' By similar reasoning, there are no s − |B| independent edges inside parts but outside B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Therefore, the edges inside parts that are not incident to any vertex of B form at most s − 1 − |B| stars plus O(1) further edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Since the vertices outside B are incident to o(n) edges inside parts, there are o(nk−1) k-cliques containing such a vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' This implies that deleting all the edges inside parts that are not incident to B, we lose o(n|V (H)|−1) copies of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' If |B|< s − 1, then we can add a vertex to B creating Θ(n|V (H)|−1) copies of H, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' We obtained that |B|= s − 1 and then there is no edge inside parts but outside B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' This implies that G is a subgraph of Ks−1 + T(n − s + 1, r), completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' ■ Funding: Research supported by the National Research, Development and Innovation Office - NKFIH under the grants SNN 129364, FK 132060, and KKP-133819.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' References [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Alon, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Shikhelman.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Avoiding long Berge cycles, Journal of Combinatorial Theory, Series B 137, 55–64, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' [4] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Gerbner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' Counting multiple graphs in generalized Tur´an problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' arXiv preprint arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content='11645, 2020.' 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+page_content=' Matematicheskii Sbornik,66(2), 163–188, 1949.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} +page_content=' 9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAzT4oBgHgl3EQfMvuF/content/2301.01137v1.pdf'} diff --git a/hdFKT4oBgHgl3EQfuS7o/content/tmp_files/2301.11891v1.pdf.txt b/hdFKT4oBgHgl3EQfuS7o/content/tmp_files/2301.11891v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..276134bc59dc73ceb0ef7bb1f5295f18ddfc4371 --- /dev/null +++ b/hdFKT4oBgHgl3EQfuS7o/content/tmp_files/2301.11891v1.pdf.txt @@ -0,0 +1,1352 @@ + + + + +1 + +Polycraft World AI Lab (PAL) : An Extensible Platform for +Evaluating Artificial Intelligence Agents + +Stephen A. Goss1,*, Robert J. Steininger1,*, Dhruv Narayanan1, Daniel V. Olivença2, Yutong Sun2, +Peng Qiu2, Jim Amato1, Eberhard O. Voit2, Walter E. Voit1,3, Eric J. Kildebeck1,** + +1 Center for Engineering Innovation, The University of Texas at Dallas, 800 W. Campbell Road, Richardson, +Texas 75080, USA +2 Department of Biomedical Engineering, Georgia Institute of Technology and Emory University, Atlanta, +GA, United States +3 Department of Materials Science and Engineering, The University of Texas at Dallas, 800 W. Campbell +Road, Richardson, Texas 75080, USA + +Emails: +stephen@polycraftworld.com +robert.steininger@utdallas.edu +dhruv@polycraftworld.com +dolivenca3@gatech.edu +sunyutong@gatech.edu +peng.qiu@bme.gatech.edu +Jim@polycraftworld.com +eberhard.voit@bme.gatech.edu +walter.voit@utdallas.edu +eric.kildebeck@utdallas.edu + +*Co-first authors +**Corresponding author: University of Texas at Dallas, 800 W. Campbell Road, Richardson, Texas 75080, +USA + + + + + + + + + +2 + +Abstract +As artificial intelligence research advances, the platforms used to evaluate AI agents need to +adapt and grow to continue to challenge them. We present the Polycraft World AI Lab (PAL), a +task simulator with an API based on the Minecraft mod Polycraft World. Our platform is built to +allow AI agents with different architectures to easily interact with the Minecraft world, train and +be evaluated in multiple tasks. PAL enables the creation of tasks in a flexible manner as well as +having the capability to manipulate any aspect of the task during an evaluation. All actions taken +by AI agents and external actors (non-player-characters, NPCs) in the open-world environment +are logged to streamline evaluation. Here we present two custom tasks on the PAL platform, one +focused on multi-step planning and one focused on navigation, and evaluations of agents solving +them. In summary, we report a versatile and extensible AI evaluation platform with a low barrier +to entry for AI researchers to utilize. + +Introduction +Advances in artificial intelligence algorithms, in particular advances in open world learning and +lifelong learning agents, require scalable open-world testbeds for development and evaluation +(Doctor et al., 2022). Here, we present a Minecraft-based platform for simulating a wide variety +of complex tasks to evaluate the next generation of artificial intelligence agents. +Minecraft is a video game that simulates a rich 3D environment where players collect resources +and blocks to build anything from rich urban environments to virtual computing systems. Further, +Minecraft has an extensive modding community where the bounds of the base game can be +extended. We created a comprehensive, fully customizable mod to Minecraft, Polycraft World +(Smaldone et al., 2017) initially to teach materials science and polymer chemistry. The Polycraft +World mod introduces over 5,000 new materials, items, and machinery on top of base Minecraft. +In addition to increased complexity of items, this mod introduces hierarchal player permissions, +player/agent tracking and logging. Through this continued development, Polycraft World has +grown to a multipurpose platform with many applications from education to social science to +artificial intelligence. +To evaluate open-world learning, transfer learning, and lifelong learning, an ideal virtual platform +for evaluating AI agents should support a variety of tasks while still being broadly usable for +different agent types. Many platforms are limited to a task or tasks within a discrete, closed +environment (e.g., CartPole (Sutton & Barto, 2018), Arcade Learning Environment (Bellemare et +al., 2013)). Virtual 3D worlds provide added value in that they can better represent real-world +relevant tasks and complex scenarios. The Unity game engine can be a powerful platform for AI +evaluation (Juliani et al., 2018); however, there is a high development cost to create custom Unity + + + + + +3 + +based tasks. Additionally, many first-person shooter games have been adapted into AI research +platforms such as VizDoom (Doom) (Kempka et al., 2016) and DeepMind Lab (Beattie et al., 2016) +using the Quake III Arena game engine. ViZDoom and DeepMind Lab are excellent platforms to +train AI agents in a 3D visual environment, but they are essentially first-person shooters with +limited interactions with the environment often focused on movement with a single or small +number of interaction/fire commands. While more interactions can be added to these platforms, +Minecraft presents a richer out-of-the-box experience. Minecraft has been used in the past in +this capacity as well (Aluru et al., 2015; Johnson et al., 2016). Previous Minecraft platforms have +been invaluable for studying and challenging AI agents. However, they are limited by the bounds +of Minecraft itself. Tasks have limited flexibility, and agents must be adapted to function within +their system. +In this paper, we present the Polycraft World AI Lab (PAL), which retains the advantages of +Minecraft-based platforms but has the added advantage of including an API well-suited for +planning agents that creates a complex action space without requiring extensive overhead +training for movement or reinforcement learning. With this API, PAL enables more flexible tasks +that are straightforward to customize and are more approachable for various AI agents. We first +describe the platform architecture. We then explain how one would design and implement a task +on PAL. We go on to present the implementation of two custom tasks. Finally, we discuss the PAL +outputs for evaluation and platform performance metrics. + +PAL Architecture +The Polycraft World AI Lab (PAL) platform (https://github.com/PolycraftWorld/PAL) was built to +test AI agents on top of Polycraft World (Figure 1) using the popular modding API, Minecraft +Forge (Forge volunteer team, 2011). The PAL platform consists of the Minecraft Client and +Polycraft World mod connected via the Minecraft Forge API, a Web Socket API, and a +Tournament/Game manager for handling experiments with different complex tasks to test AI +agents. Step by step instructions for PAL installation can be found in the GitHub page for Linux +and Windows but any system that can use Java will be able to run PAL. Our agent API allows +interfacing with any external AI agent without specific language requirements. All that is needed +to integrate a new agent is a simple socket client script, which can be made in any language that +supports web sockets such as Java, Python, and Julia. This significantly lowers the barrier for +researchers to utilize the PAL platform as they will not need to develop or integrate with an +existing wrapper. The Tournament/Game manager is our system for facilitating evaluation of AI +agents in custom PAL task scenarios. A more detailed explanation of the PAL architecture can be +found in the Supplemental Materials, in the “1. Process Communication Standards” section. + + + + + +4 + + +Figure 1 : PAL Architecture Flow Chart +Interface - Actions +Agents interact with the world by sending commands to PAL one command per game tick. The +available commands (Table 1 and Supplemental Materials) are universal and do not change +between tasks. There are different categories of commands such as game commands (START, +RESET, GIVEUP), movement (MOVE, TP_TO), interactions (CRAFT, BREAK_BLOCK, USE_HAND, +SELECT_ITEM), and sensing (SENSE_ALL, SENSE_SCREEN). Each command has set preconditions +that must be satisfied to execute. For example, an agent cannot move forward if there is a block +in the way. Each command has an associated cost that can be used as a metric, for example, in +reinforcement learning. The cost per action is based on an estimate of the complexity of the +action. For example, Teleporting (TP_TO) 10 blocks costs 10X the cost of moving (MOVE) 1 block +and the cost of crafting increases based on the number of items used in a recipe. The reasonable +estimation of these costs allows for evaluation of agent efficiency and realism of actions in large +open-world tasks. + +Table 1: Action commands available in PAL. +Interactions +Sensing +Movement +Game commands + + + + +BREAK_BLOCK +SENSE_SCREEN +MOVE +START +PLACE +SENSE_ALL +TURN +RESET +COLLECT +SENSE_ALL_NONAV +TILT +GIVE_UP +CRAFT +SENSE_INVENTORY +TP_TO (teleport to) + +SELECT_ITEM +SENSE_LOCATIONS + + +USE_HAND +SENSE_ACTOR_ACTIONS + + + +PAL +Polycraft World +Al Agent +Web Socket APl < +(mod) +Minecraft Client +Tournament +Minecraft Forge +Tournament +Manager +API +data + + + +5 + +USE +SENSE_RECIPES + + +DELETE +SENSE_ENTITIES + + +INTERACT + + + +TRADE + + + +NOP (no operation) + + + + +For navigation, we support discrete steps by moving one block at a time or by teleporting to block +positions for agents that don’t have the capability to navigate on their own. By using the MOVE +command, an agent can move to any open adjacent square. + +Interface – Observations +Agents can sense multiple observations about the world including symbolic observations +(recipes, locations, inventory, map, actor actions, and entities) and visual observation of the +screen. There are three higher level observation commands that return specific sets of these +observations based on if the agent is using vision, is teleporting or navigating, etc. These +commands are: +• SENSE_SCREEN that returns a .PNG image file with the output of the visual screen. +• SENSE_ALL will return a JSON string with all the information about the current state of +the environment, which includes a symbolic map of the task’s terrain with block and NPC +information, inventory, NPC locations and actions, but will not include the screen +information. +• SENSE_ALL NONAV is similar to the previous command, but the symbolic map also +includes all attributes of each block in the world. + +The following commands allow the agent to access specific parts of SENSE_ALL. +• SENSE_INVENTORY will return a JSON string with the agent’s inventory contents. +• SENSE_LOCATIONS will return a JSON string with the agent’s position. +• SENSE_ACTOR_ACTIONS will return a JSON string with all the actions NPCs have +performed. +• SENSE_RECIPES will return a JSON string with all the available recipes in the CRAFT +command. +• SENSE_ENTITIES will return a JSON string with the map coordinates of all NPCs. + +Interface – Example of Command and Response +An agent can utilize the API by connecting to the default port 9000 and sending a command in +plain text. Then an agent could issue a command by sending “SELECT_ITEM + + + + + +6 + +minecraft:iron_pickaxe” to that port followed by a new line character. After the command is +processed, a text JSON response will be sent back which includes the result of the command and +task goal information: +{ + "goal": { + "goalType": "ITEM", + "goalAchieved": false, + "Distribution": "Uninformed" + }, + "command_result": { + "command": "select_item", + "argument": "minecraft:iron_pickaxe", + "result": "SUCCESS", + "message": "selected item", + "stepCost": 120 + }, + "step": 0, + "gameOver": false +} + +An example Python script can be found in the Supplemental Materials in the “2 .Example of +Command and Response” section. + +Logging +The PAL platform records logs in three different tracks which helps facilitate data analysis and +the debugging process. These logs are separated by the tournament manager, Minecraft client, +and lastly the agent. Both the tournament manager and Minecraft logs will be the same for each +evaluation; however, the agent log always records directly from the AI agent standard output, +which can be very different for each agent. These log files, produced by PAL, provide an out of +the box option for player/agent tracking and analytics. + +Task Scenario +One strength of the PAL platform is its ability to support a variety of task scenarios which can be +researcher-defined and have broad customizability. To define a PAL task scenario, one must set +a goal, define an arena and a set of available actions and interactions. A goal is what the agent +must accomplish to succeed in the task. This can be anything that a player could do in Polycraft, +whether it be obtaining/crafting an item, placing a block in a specified location, or building a + + + + + +7 + +complex system/structure in the arena. The arena is where the agent will be acting to achieve +the goal. We can customize the arena by making it larger/smaller, filling it with objects to help or +hinder the agent and adding adversarial or allied external agents. The action space of a given task +will be dependent on all the objects in the arena as well as items in the player’s inventory. Task +components are defined using the open standard file format, JSON (JavaScript Object Notation), +to allow for effortless and scalable customization of training environments in a familiar +programing format. For new tasks, Java scripts are used to generate new task definitions, which +allows quick generation of multiple seeded variations of the scenario. For tasks that resemble +existing tasks, this can be done by manually editing an existing task definition file. Below, we +present two custom task scenarios used in the DARPA SAIL-ON program (Senator, 2019) to +illustrate the types of tasks that can be created in PAL. + +Example Task Scenarios +A wide variety of tasks for AI agent evaluation can be developed utilizing the PAL platform. Here, +we present two sample task scenarios created within our PAL platform and demonstrate +evaluation of AI solutions for these tasks. The first task, called POGO, is a planning task where +agents must obtain resources and ultimately craft a pogo stick. The second task is a navigation +task named HUGA (for hunter-gatherer) where agents must find an object, pick it up, and bring +it to a target. These tasks highlight strengths in our Minecraft-based platform. In Polycraft World +rich virtual worlds can be created that, like the real world, can be challenging to navigate, are +only partially observable, and in which the optimal course of action may be unclear. + +PAL Task Scenario 1: POGO Task + +Figure 2 : POGO Task. (A) Schematic of the POGO arena. (B) First person view of the agent +within the POGO arena. (C) Recipes available to the agent. +The goal of this task is to craft a pogo stick. The agent starts in a 30 x 30 block grass field +surrounded by unbreakable bedrock walls (Figure 2A,B). The field always has 5 trees and 1 + +(B) +C) + + + +8 + +crafting table placed in the environment. Trees can be “chopped down” using the BREAK_BLOCK +command, new items can be crafted from the available recipes (Figure 2C) utilizing CRAFT +commands and the crafting table. Once the agent successfully obtains a pogo stick, the goal is +achieved, and the task ends. The task can also end if the agent gives up or if the time limit for the +task is reached. +To achieve the goal the agent must follow a series of steps including 1) chopping down trees to +collect wood, 2) crafting intermediate items including wood planks, sticks, and a tree tap, 3) +placing a tree tap on a tree, 4) collecting rubber from the tree tap, and 5) using intermediate +items and rubber to craft a pogo stick. +We developed a planner agent for this task using Planning Domain Definition Language (PDDL) +(Barrett et al., 1998; Fox & Long, 2003) and the fast forward algorithm (Hoffmann & Nebel, 2011). +The planner uses the PDDL description of the task and fast forward to get a solution which will +be used to create a sequence of actions that are necessary to achieve the task goal. It requires +knowledge of the action space and the items available in the map. The task components are +translated into to PDDL 2.1 in the form of two files: the “domain.pddl” file that sets the world +rules and the “problem.pddl” file that describes the task. The PDDL files construction is not +automated but the agent is able to adapt to differences in the map composition. Running these +files through the fast forward algorithm outputs a sequence of high-level actions that are +interpreted by the Python code. High level actions aggregate low-level actions that Minecraft can +process. As an example, a high-level action is ‘get wood block’ and the low-level actions +corresponding to it are: ’move three steps forward’, ’turn left’, ’break wood block’. For this +planner agent we use high-level actions to reduce the action space and to allow for adaptability +to changes in the task (ex: crafting table and trees changing positions in the map). This agent +does not require training. + +PAL Task Scenario 2: Hunter-Gatherer Game (HUGA) – 3D space navigation task + +Figure 3: Hunter Gatherer Task (HUGA) (A) Schematic view of the HUGA arena. (B) First person +view of the agent within the HUGA arena. (C) Aerial view of the HUGA arena. Arrows indicate a +path agent can take to complete the task + + +(A) +(B) +(C) + + + +9 + +The goal of this task is to pick up the green cross object (the MacGuffin) and place it on a Target +(the blue block). The agent starts in a four-room arena with a total size of 32 x 32 blocks (Figure +3). Each room is 14 x 14 blocks large, with inner walls and multiple walkways separating the +rooms. The agent always spawns in the bottom-left room (Room 1). The MacGuffin spawns in +the top-left room (Room 2). The top-right room is empty (Room 3). The Target spawns in the +bottom-right room (Room 4). The agent can pick up the MacGuffin by walking into it with MOVE +commands and can place the MacGuffin at the Target with the PLACE command. Once the agent +successfully places the MacGuffin at the Target, the goal is achieved and the task ends. The task +can also end if the agent gives up or if the time limit is reached. This task is designed to be similar +to other state-of-the-art 3D maze-runner tasks (e.g. VizDoom, DMLab NavMaze, Unity Obstacle +Tower Challenge). +To achieve the goal the agent must follow a series of steps (Figure 3C) including 1) walking from +Room 1 to Room 2, 2) picking up the MacGuffin, 3) walking from Room 2 to Room 4 and 4) placing +the MacGuffin at the Target. +Like the POGO task, an agent was created to solve the HUGA task. Our HUGA agent uses a Deep +Q-Network (DQN) (Mnih et al., 2013; Roderick et al., 2017) which combines deep neural networks +and Q-learning. We chose to use this architecture because the deep neural network can take +visual information as input, and the Q-learning aspect can handle the sparse reward structure in +the HUGA task. +The structure of the DQN is determined by the sizes of the state space and the action space. A +state for the visual-information-based DQN is the screenshot image of the game window at a +specific time point. The dimension of the input is (3, 84, 84), where 3 is the number of color +channels and (84, 84) is the resolution of the screenshot. The action space utilized by this agent +includes MOVE, TURN, and PLACE for the MacGuffin (green block in Figure 3B,C). The DQN is +trained to take the current state as input, and computes the expected rewards in the action +space. After the agent executes the action with the highest expected reward, the new state is fed +into the DQN to generate the next action. The process iterates until the agent completes the +game. +To complete the HUGA game, the agent needs to find the MacGuffin and bring it to the +destination (blue square in Figure 3A,C). This task can be separated into two sub-tasks. The first +sub-task is to find the MacGuffin and the second sub-task is to find the destination. We trained +two DQNs with identical architecture to complete the two sub-tasks. +More specifically, we generated 1000 instances of the HUGA task with different layouts of the +passages among rooms and various color patterns on the walls and the ground. We randomly +selected 618 instances for training, and used the remaining 382 for testing. For each training +instance of the HUGA task, we randomly initialized the location and the direction the agent is +facing (Figure 3B). We set the maximum number of actions to be 450. When the agent completed + + + + + +10 + +the sub-task within 450 actions or reaches 450 actions before completing the sub-task, the game +was reset using the next training instance of the HUGA game, and training process continued. +When implementing the agent, we check the status after each action. Before finding the +MacGuffin, the agent calls the first DQN model to decide its actions. After the MacGuffin is found, +the agent turns to the second DQN model to decide its actions. +When the agent is tested in one instance of the HUGA task, we record a success if each sub-task +is completed within 450 steps. We use the success rate as the evaluation metric for the trained +agent. We randomly selected HUGA task instances from the 382 for testing, randomly set the +location and direction the agent is facing, and tested the trained agent. We observed a success +rate of ~94% with this agent. + +PAL Extensibility + + +Figure 4 – POGO and HUGA modifications (A) Bird’s eye view of updated POGO arena (B) First +person view of updated POGO task scenario (C) Bird’s eye view of updated HUGA arena (D) First +person view of updated HUGA task scenario. + +(A) +(B) +(C) + + + +11 + +In Figure 4 we show modifications we’ve made to both the POGO and HUGA tasks that +demonstrate the extensibility of our platform. Scale and complexity of tasks can be easily +increased and decreased using our flexible task creation system based on the needs of AI +researchers and the capability of their agents. We’ve increased the complexity of the POGO and +HUGA tasks by adding more varied layouts, more complex problems, and external actors (NPCs), +which can be allied or adversarial with the AI agent. These additions increase real-world +relevance by adding scale, partial observability, and non-determinism to these task scenarios. +In the updated POGO task, the agent must collect additional resources from additional places +(added rooms in Figure 4A) and interact with trading NPCs while avoiding competitive NPCs (seen +in Figure 4B). In the updated HUGA task, the agent must navigate varied layouts with different +color schemes (Figure 4C). They must also avoid traps, obtain keys to unlock doors and obtain +advantages from helpful agents while avoiding adversarial NPCs (seen in Figure 4D). + +PAL Evaluations and Performance +Running Tournaments in PAL +A tournament manager was developed in python that manages the launching & logging of AI +Agents performing in task scenario instances in PAL. This manager can be deployed across +computing clusters during evaluations and is parameterized to accept a variety of agents and +tournaments to run. The manager runs four separate processes that communicate with each +other through sockets on the host machine: +1. PAL, the Minecraft-modded game environment. +2. AI Agent. A bash execution command is necessary to launch the agent. +3. TournamentManager, which monitors instance-ending conditions and sends the +appropriate reset command to PAL along with a JSON file describing the next instance to +load into PAL. +4. LoggingHandler that logs outputs, records all steps taken, and updates SQL databases +with this information. This process includes a tournament timeout – if no progress has +been made for more than 5*max game time (i.e., in the event any of the above processes +freeze), then the tournament ends. +We provide a copy of the tournament manager that includes processes 1,3, & 4 on the PAL GitHub +(https://github.com/PolycraftWorld/PAL). Installation instructions and execution assistance can +also be found on the PAL GitHub in the ReadME.txt file. We fully support and highly recommend +running this environment in a Debian-based UNIX operating environment (we use Ubuntu 18.04 +in our Evaluations); however, it can be successfully deployed in Windows-based environments as +well. + + + + + +12 + +AI agents are launched through a user-configurable, UNIX-compliant bash execution command +after the PAL environment has been launched and the first task instance has been loaded. +Following successful launch, the AI Agent should connect to port 9000 and send an API command +to start the tournament. +We have 547 available scenarios across 2 tasks that can be used to train and evaluate an AI agent. +Each scenario is a version of the POGO or HUGA task with additional world elements or changes +to the arena that provide different challenges to AI agents. These scenarios each have 3 +differently seeded tournament variations for a total of 1641 tournaments. Each tournament has +100 games which comes out to 164,100 different game instances. These instances are all +available on GitHub (https://github.com/PolycraftWorld/PAL) to train and evaluate on. + +PAL Platform Performance +We are limited in how many actions we can take per second by Minecraft’s internal tick rate. +Minecraft is hard coded to run at 20 ticks per second and has no setting or configuration available +to change the speed. However, this can be unlocked by using reflection to change the tick-rate +at runtime. This can be very helpful by increasing training and evaluation times by a factor of 20- +40 depending on whether the agent is using visual observations. +We have tested the limits of achievable speed (in ticks per second) in PAL and found that we can +reach speeds of 550 ticks per second on modern systems. Our tests were done on a Windows 10 +machine with a Ryzen 3900x CPU, 32GB Ram, and a GTX 1070 GPU. This number is mostly +affected by single thread processor performance. When an agent is using the screen observation +on every tick, this performance degrades significantly. The default screen output format is PNG, +which results in an average speed of 73.44 ticks per second. If we change the format of the +screen observation to JPEG we get an average speed of 300 ticks per second. We can reach +speeds closer to 550 ticks per second if an agent doesn’t request the screen after each action +(Table S1 in Supplemental Materials). + +Conclusion +Here, we present the Polycraft World AI Lab (PAL), an extensible platform for creating AI +challenges, running AI experiments, and evaluating AI agent performance. We have leveraged +the strengths of Minecraft but added additional flexibility and accessibility in the PAL system. PAL +connects with a broad range of AI agents via our socket-based command interface. Agent +evaluation includes comprehensive log files which monitor agents’ behavior throughout a task. +We present two examples of custom tasks: a planning task, where the agent is challenged to craft +a pogo stick (POGO) and a navigation task, where the agent is challenged to find an item and +bring it to a target (HUGA). These tasks were built entirely using the PAL platform. Agents were + + + + + +13 + +developed and evaluated on each task. We go on to show that we can add complexity (via added +NPCs, task problems, and domain scale) to these tasks to make them more relevant to real-world +scenarios. Finally, we discuss the PAL system performance and its ability to rapidly evaluate +agents in a tournament structure. + +Figure 5: PAL is capable of a creating a diverse set of tasks. We have created multiple arenas in +Polycraft World that have diverse goal conditions. (A) Minigames such as basketball, archery +and minigolf (B) An escape room challenge on a space station (C) Scavenger hunts on the UTD +campus (D) Virtual paintball (E) a tutorial for teaching gameplay in multiple stages. Our tasks +can have various levels of complexity. Capture the base in: (F) a simple square arena, (G) a +stadium setting, (H) a desert city. +This platform is by no means limited to the tasks presented here. We envision this platform will +be used to evaluate agents on custom open-world tasks designed by AI researchers and we have +demonstrated the capability of the platform to present a wide variety of activities for human and +AI agents (Figure 5). We believe the flexibility PAL adds to task design and agent integration are +valuable tools for fundamental AI research. +We believe that PAL will enable research on large scale challenges relevant to multi-task learning, +transfer learning, and problems encountered in open-world learning including adaptation to out- +of-distribution world states and previously unobserved environments (Boult et al., 2021; Kejriwal +& Thomas, 2021; Muhammad et al., 2021; Musliner et al., 2021). +Moving forward, this platform will be used for evaluating AI agent performance in novel AI tasks +that are increasingly complex, real-world relevant, and involve human-AI teams. + +(B) +(E) +C +(F) +X + + + +14 + + +Conflict of Interest +The authors declare that the research was conducted in the absence of any commercial or +financial relationships that could be construed as a potential conflict of interest. + +Author Contributions +S.A.G. was the lead developer for PAL, task creation, manages the project GitHub and wrote the +manuscript. R.J.S. wrote the manuscript, was involved in debugging, had several supporting roles +and bibliography research. D.N. created the tournament manager. D.V.O. developed the POGO +agent, wrote the manuscript and contributed to the project documentation. Y. S. and P. Q. +developed the HUGA agent, wrote the manuscript and contributed to the project +documentation. J.A. contributed to the project documentation and was involved in debugging. +E.K. task creation. E.K., W.V., P.Q. and E.O.V. conceived the project, reviewed the manuscript, +and provided guidance and management. All authors reviewed and edited the manuscript. + +Funding +This work was supported in part by the following grant: W911NF2020010 from the Defense +Advanced Research Projects Agency. The funding agency is not responsible for the content of this +article. + +Acknowledgements +The authors greatly appreciate the constructive feedback of colleagues in the SAIL-ON program +for their comments and help testing PAL in the early stages of the platform development. The +authors also would like to thank Willie E. Chalmers for the insightful comments and help in the +beginning of the project. + + + + + + + + +15 + +References +Aluru, K., Tellex, S., Oberlin, J., & Macglashan, J. (2015). Minecraft as an experimental world +for AI in robotics. AAAI Fall Symposium - Technical Report, FS-15-01(Makuch 2014), 5– +12. +Barrett, A., Christianson, D., Friedman, M., Kwok, C., Golden, K., Penberthy, S., Smith, D., +Sun, Y., & Weld, D. (1998). PDDL | The Planning Domain Deenition Language. +https://courses.cs.washington.edu/courses/cse473/06sp/pddl.pdf +Beattie, C., Leibo, J. 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Arxiv, 1–9. +http://arxiv.org/abs/1312.5602 +Muhammad, F., Sarathy, V., Tatiya, G., Goel, S., Gyawali, S., Guaman, M., Sinapov, J., & +Scheutz, M. (2021). A Novelty-Centric Agent Architecture for Changing Worlds. AAMAS +’21: Proceedings of the 20th International Conference on Autonomous Agents and +MultiAgent Systems, 925–933. https://doi.org/10.5555/3463952.3464062 +Musliner, D. J., Pelican, M. J. S., Mclure, M., Johnston, S., Freedman, R. G., & Knutson, C. +(2021). OpenMIND: Planning and Adapting in Domains with Novelty. Proceedings of the +Ninth +Annual +Conference +on +Advances +in +Cognitive +Systems, +1–20. +https://advancesincognitivesystems.github.io/acs2021/data/ACS-21_paper_35.pdf +Roderick, M., MacGlashan, J., & Tellex, S. (2017). Implementing the Deep Q-Network. ArXiv, +1–9. http://arxiv.org/abs/1711.07478 +Senator, T. (2019). Broad Agency Announcement: Science of Artificial Intelligence and +Learning +for +Open-world +Novelty +(SAIL-ON). +1–30. +https://research- +authority.tau.ac.il/sites/resauth.tau.ac.il/files/DARPA SAIL-ON_HR001119S0038.pdf +Smaldone, R. A., Thompson, C. M., Evans, M., & Voit, W. (2017). Teaching science through +video games. NATURE CHEMISTRY, 9, 97–102. http://go.nature.com/2hjb2l1 +Sutton, R. S., & Barto, A. G. (2018). Reinforcement learning: An introduction (The MIT Press, +Ed.). + + + + + + + + + + + +17 + +Polycraft World AI Lab (PAL) : An Extensible Platform for +Evaluating Artificial Intelligence Agents + +SUPPLEMENTARY INFORMATION + +1. Process Communication Standards +Socket communication is key in managing our training and tournament environment. PAL and +the AI Agent communicate over port 9000 (default port). The TournamentManager and PAL +communicate over port 9005. There is no direct communication between the tournament +manager and the AI Agent. +All AI Agent communications must be initiated by the AI Agent. PAL will, at no time, send a +message unprompted to the AI Agent, instead passing you key information through JSON +responses (was your command successfully executed, is the instance over due to a time-out, +etc.). In the event that a game timeout does occur, if an agent does not query PAL with a +command (like move, or sense_all), the agent will never know that the instance is in fact over +and the next instance is waiting to start. Our tournament manager will hang, and eventually, we +have a process on our end that will kill the trial and state that the agent has become non- +responsive in our logging. +AI Agent Communications are defined by our PAL API, available in the Supplements, in PAL API. +In addition to command-specific response variables that vary depending on the command sent, +every response JSON contains the following keys: +• goal: json dict that contains: +o goalType: "BLOCK_TO_LOCATION" for the HUGA Task, "POGOSTICK" for the POGO +Task +o goalAchieved: True if the goal was achieved, False otherwise. Once the goal has +been achieved, for the remainder of commands sent in that instance, this will +continue to report "True" (boolean) +o Distribution: "Uninformed" if this trial requires “System Detection” of Novelty. +"PreNovelty" or "Novelty" if this trial is “Given Detection”. (string) +• command_result: json dict that contains +o command: command sent by Agent (string) +o argument: any command arguments sent by the Agent (string) +o result: "SUCCESS" if the command was executed properly, "FAIL" if a problem +arose in command execution (string) +o message: a non-null string if the result = "FAIL" containing an error message +(string) + + + + + +18 + +o stepCost: step cost of executing a particular command (float) +• step: Step Number, 0-indexed count of commands sent by the AI Agent (integer) +• gameOver: True if the instance is over, False otherwise. (boolean) +Our LoggingHandler monitors the STDOUT and STDERR of the PAL and AI Agent process. This +process writes to separate log files all data written to STDOUT and STDERR by the PAL process +and by the Agent AI process, broken out as one file per instance. +1.1 +Communicating Game State to the AI Agent +As indicated above, the tournament manager does not directly communicate details on the state +of the trial or the state of the instance to the AI Agent. Instead, the above listed JSON response +keys convey critical information that the Agent can process after each command (step) to +understand the instance state. As an example, as soon as the goalAchieved=True is found in the +response, the Agent knows that it successfully completed the task for that particular instance. +Every JSON response has a "gameOver" key that is either True or False. The value "True" is sent +in only one json response to the first command received after of the following game-ending +criteria is met. Subsequent commands in that same game will revert to having gameOver=False, +so it is imperative that the AI Agent recognize and appropriately handle this flag as soon as it +appears. +The current instance ending conditions are: +1. Successful Completion of Task (we also include another key, "goalAchieved" in each +response that will reflect this case) +2. Game Time limit exceeded +3. Total Step Cost exceeds a step-cost maximum of 1,000,000 (an empirically determined +upper bound to prevent abuse of API commands or other counter-productive behavior +towards task completion) +4. Agent sends the GIVE_UP command +As of now, our architecture requires that the AI agent, following the successful reception of a +response JSON containing gameOver=True send one last command back to PAL to acknowledge +receipt and initiate the reset to start the next instance (doesn't matter what the command is, as +its step cost is not counted for/against your score, and its response JSON is irrelevant for your +agent as well. Most agents just pass a SENSE_ALL command and ignore the response). Ideally, +only one additional command will be processed by PAL following any of the instance end +conditions evaluating to true. +Tournament completion is not directly communicated towards the AI Agent. Instead, the process +is terminated by the parent process. + + + + + +19 + +1.2 Running Tests of your Agent in PAL +Detailed Instructions on setup and installation of the Tournament Manager environment is +available on GitHub (https://github.com/PolycraftWorld/PAL). To run an AI Agent, the +AGENT_COMMAND config variable (or command line parameter) needs to be adjusted to point +to a script that will load the AI Agent. As mentioned above, the AI agent is expected to, as part of +that startup script, try to connect to port 9000 and send an API command to begin the trial. +Though an AI agent could just be a folder of scripts as our baseline agent is, we found that many +of our partners prefer to run their AI agents in Docker containers – that works well in our current +evaluation environment, too. +1.3 Handling Updates to PAL in your Test Environment +In the event that updates or bug-fixes are made to PAL (I.e., new API commands, new novelties, +etc.), we will communicate via email/slack/teams with a detailed changelog. Please direct your +code to the appropriate branch on PAL GitHub (https://github.com/PolycraftWorld/PAL), pull the +updates, and then continue to run tests using your AI agent. + + + + + + + + +20 + +2. Example of Command and Response +An agent can use the API by connecting to the default port 9000 using the following Python script: +import socket +import json + +# connect to socket +HOST = '127.0.0.1' +PORT = 9000 +sock = socket.socket(socket.AF_INET, socket.SOCK_STREAM) +sock.connect((HOST, PORT)) + +The following script constructs a function (MC) that enables the communication with Polycraft: +def MC(command): + "function that enable the communication with minecraft" + print( command ) + sock.send(str.encode(command + '\n')) + BUFF_SIZE = 4096 # 4 KiB + data = b'' + while True: + part = sock.recv(BUFF_SIZE) + data += part + time.sleep(.001) + if len(part) < BUFF_SIZE or part[-1] == 10: + # either 0 or end of data + break + print(data) + data_dict = json.loads(data) + # print(data_dict) + return data_dict + +Now we can start the API using : +MC('START') + +And reset the task: +MC("RESET +domain +../pogo_100_PN/POGO_L00_T01_S01_X0100_U9999_V0_G00000_I0020_N0.json") + +Now we can send commands. For example: +MC('SENSE_RECIPES')) + + + + + +21 + +MC('SENSE_ALL') +MC('SELECT_ITEM minecraft:iron_pickaxe') + +After the command is processed, a text JSON response will be sent back which includes the result +of the command and task goal information. For example, the return of the last command will be: + + + + + +{ +"goal": { +"goalType": "ITEM", +"goalAchieved": false, +"Distribution": "Uninformed" +}, +"command_result": { +"command": "select_item", +"argument": "minecraft:iron_pickaxe", +"result": "SUCCESS", +"message": "selected item", +"stepCost": 120 +}, +"step": 0, +"gameOver +} + + + + + +22 + +3. PAL API +The PAL API consists of different API commands broken down into SYSTEM commands, DEV commands, +and GAME commands. For AI Agent testing using the TournamentManager, only GAME commands will +need to be used. The other commands are provided for completeness only. + +3.1 Available configuration +There are three places available for configuration settings for a tournament run. There are the +command-line arguments, the config file (PAL/PolycraftAIGym/config.py), and lastly there are +available environment variables. +3.1.1 LaunchTournament.py Command-line arguments +LaunchTournament.py Usage: +LaunchTournament.py will start a new tournament on the system. It will first start a Polycraft +client, then start a tournament manager process to initialize the first game of the tournament. +Once the game has initialized, it will start the agent. + +Format: +LaunchTournament.py -c -t -g -a + -d -x -i -m +-c How many games to run. +Default: 100 +-t Name of tournament. +Default: “POGO_L00_T01_S01_X0100_A_U9999” +-g Where games are located. +Default: “../pogo_100_PN” +-a Name of your agent. +Default: “MY_AGENT_ID” +-d Where agent is located + +Default: “” +-x Command to run agent. Text as would be entered into command-line. To escape quotes or +other special characters, use “\” + +Default: “python pogo_agent.py” +-i Seconds per game +Default: +300 +-m Max time for tournament to run in minutes +Default: 2880 + +Example: +python +LaunchTournament.py +-c +100 +-t +"POGO_L00_T01_S01_X0100_A_U9999_V0200FPS_011022" +-g +"../pogo_100_PN" +-a +"BASELINE_POGOPLAN +" +-d +"agents/pogo_stick_planner_agent/" +-x +"python +python_miner_PLANNER.py" + + + + + + +23 + +3.1.2 Environment Variables +PAL Observation Variables +SENSE_SCREEN_FORMAT – Some screen formats will compress faster than others at the expense +of quality. PNG seems to give the best quality, while JPEG typically give the best performance. +The default image type is PNG but this can be set to the following: PNG, BMP, JPEG, JPG, WBMP, +GIF. +AIGYM_REPORTING - Set this to “True” to activate AIGym style reporting. The agent will receive +a “SENSE_ALL” output after every command. +REPORT_SCREEN – If “AIGYM_REPORTING” is set to true, you can also set this to “True” to also +receive a “SENSE_SCREEN” output after every command. +PAL Ports +PAL_AGENT_PORT – Set the agent socket communication port. Default is 9000. +PAL_TM_PORT – Set the tournament manager socket communication port. Default is 9005. +PAL Speed +PAL_FPS – Set the frames per second for PAL to run. Default is 20. The maximum value here is +1000, but most systems cannot achieve that unless they have powerful single thread cores and a +powerful GPU. +Internal testing revealed some results you may be able to expect on your own machine. +Sense screen testing +Test System specs: +OS – Windows 10 Pro +CPU – Ryzen 9 3900x 12 core @4.00 GHz +RAM +– +32GB +GPU – GTX 1070 + +Table S1 – Max ticks per second. +Target ticks +per second +Tested ticks per second (with different screen output formats) + +PNG +BMP +JPG +WBMP +GIF +NONE +20 +(Default) +Min:18.79 +Max:20.07 +Mean:19.98 +Median:20.0 +Min:18.68 +Max:20.08 +Mean:19.99 +Median:20.0 +Min:19.0 +Max:20.1 +Mean:19.99 +Median:20.0 +Min:19.04 +Max:20.18 +Mean:19.99 +Median:20.0 +Min:19.0 +Max:20.06 +Mean:19.99 +Median:20.0 +Min:18.49 +Max:20.09 +Mean:19.98 +Median:20.0 +200 +Min:51.4 +Max:145.92 +Mean:78.54 +Median:74.91 +Min:95.71 +Max:174.4 +Mean:127.56 +Median:127.36 +Min:105.71 +Max:187.93 +Mean:153.39 +Median:156.33 +Min:98.24 +Max:205.07 +Mean:176.07 +Median:179.92 +Min:99.23 +Max:188.43 +Mean:123.46 +Median:114.71 +Min:169.78 +Max:209.27 +Mean:202.42 +Median:201.09 +500 +Min:42.48 +Max:93.44 +Mean:56.62 +Min:72.5 +Max:250.81 +Mean:148.03 +Min:120.09 +Max:349.22 +Mean:263.06 +Min:101.79 +Max:458.81 +Mean:305.63 +Min:90.22 +Max:199.1 +Mean:142.96 +Min:340.2 +Max:511.38 +Mean:470.79 + + + + + +24 + +Median:53.59 +Median:127.73 +Median:262.68 +Median:406.73 +Median:192.48 +Median:419.29 +800 +Min:50.3 +Max:151.18 +Mean:73.44 +Median:77.24 +Min:130.46 +Max:277.02 +Mean:227.44 +Median:130.46 +Min:121.36 +Max:409.22 +Mean:297.09 +Median:121.36 +Min:97.65 +Max:384.09 +Mean:213.5 +Median:176.36 +Min:108.61 +Max:196.33 +Mean:155.94 +Median:108.61 +Min:392.07 +Max:677.97 +Mean:549.36 +Median:539.08 +1000 +Diminishing performance past 800 + +PAL Screen Position +PAL_X – x position on screen for window to render. Default is chosen by Minecraft +PAL_Y – y position on screen for window to render. Default is chosen by Minecraft +PAL_WIDTH – Screen render width size. Default is 256 +PAL_HEIGHT – Render screen height size. Default is 256 + +3.2 GAME commands +3.2.1 Tournament Commands +These commands are sent by the AI Agent to communicate directly to the Tournament Manager +and should be used when the Agent is being officially evaluated, where appropriate. +• +GIVE_UP +o +Agent gives up task, letting tournament manager know that the next instance can be +queued for a domain reset. +o +Please note that following a GIVE_UP command, the agent will receive a gameOver=True +as part of the response JSON. The Agent is still required to send one additional command +to PAL to acknowledge receipt of the gameOver=True before the RESET domain +command is triggered by the Tournament Manager and the next instance in the trial is +loaded +• +REPORT_NOVELTY [–l lvl] [–c confidence] [–m user-message] +o +indicates that you have detected novelty with optional parameters | [-l novelty level] +o +[-c confidence interval 0f:100f] [-m user-defined message] + +3.2.2 Movement commands +These commands enable the AI Agent to move around the instance. +• +MOVE w|a|d|x +o +moves 1 meter forward (w), left (a), right (d) or back (x) (i.e. “MOVE w” moves forward 1 +block) +• +MOVE q|e|z|c +o +moves sqrt (2) distance diagonally with q,e,z,c (i.e., “MOVE q” moves diagonally forward +and leftward, relative to the player’s facing direction). Diagonal movements correspond +to relative location of letter against the WSAD keys. +• +TURN -15 +o +alters player's horizontal facing direction (yaw) in 15-degree increments (no +interpolation) +o +The parameter passed must be a multiple of 15, positive or negative. +• +SMOOTH_TILT FORWARD + + + + + +25 + +o +Sets the player’s pitch to the horizon (0 degrees) +o +While this is a “SMOOTH” command, it operates as an abrupt state transition. There are +plans to create a “TILT” command once the visual interpolation version of the command +is deployed. +• +SMOOTH_TILT DOWN +o +Sets the player’s pitch to -45 (looking directly at the ground in front of the player – ideal +for viewing the location upon which a block should be placed on the ground) +o +While this is a “SMOOTH” command, it operates as an abrupt state transition. There are +plans to create a “TILT” command once the visual interpolation version of the command +is deployed. +• +TP_TO x,y,z [distance] +o +Teleports the player to the block at x,y,z, resetting the player’s pitch and yaw to have the +player face North and set their pitch to 0 degrees (horizon), positioning the player a +default distance of 1 away from the block if no distance is passed. In other words, if +distance is Null/not given, the player’s true position would be [x,y,z-1] and executing a +MOVE W command will place the player onto the exact coordinate. +o +The optional distance parameter must be a positive integer greater than or equal to 1. +This distance will adjust the resulting position of the player to be [x,y,z-distance], facing +the coordinate desired at an offset [distance] blocks away. Running the MOVE W +command [distance] number of times will place the player on the target coordinate [x,y,z]. +o +Using distance = 2 is necessary when playing tree-taps on trees, as otherwise, the player +will be occupying the space that the tree-tap will need to occupy after being placed. +o +The distance offset must yield an allowable move_to location (i.e., it can’t be a block in +the area) or command fails. +• +TP_TO entityID +o +teleports to the location of an entity with entity_ID = entityID (i.e., TP_TO 7101 teleports +to an entity with the ID "7101") + +3.2.3 Sensing commands +• +CHECK_COST +o +returns the total stepCost incurred since the last RESET command +• +SENSE_INVENTORY +o +returns contents of player inventory in .json format +• +SENSE_LOCATIONS +o +returns sensible world environment (blocks, entities and locations) as .json +• +SENSE_RECIPES +o +Returns the list of recipes available in the experiment +• +SENSE_SCREEN +o +Returns pixels sent to the display output window, in the form of a string listing an array +of integers +• +SENSE_ALL +o +returns inventory, recipe and information on all observable blocks around the player’s +position. +• +SENSE_ALL NONAV +o +returns inventory, recipe and location information in .json +o +NONAV parameters omits information which is not needed for agents that do not +navigate the world + + + + + +26 + + +3.2.4 Interacting commands +• +SELECT_ITEM polycraft:wooden_pogo_stick +o +sets a specific item from your inventory in your hand as the active item (e.g. tool or block) +• +USE_HAND +o +to open doors with bare hand (ignores item in hand to interact) +• +BREAK_BLOCK +o +breaks block directly in front of player with selected item +o +selected item and block type yield stepCost of action +• +CRAFT 1 minecraft:log 0 0 0 +o +note that CRAFT must be followed by a "1" +o +crafts 4 Planks (uses the player’s personal 2x2 crafting grid in their inventory) +• +CRAFT 1 minecraft:planks 0 minecraft:planks 0 +o +crafts 4 Sticks (uses the player’s personal 2x2 crafting grid in their inventory) +• +CRAFT 1 minecraft:planks 0 0 minecraft:planks 0 0 0 0 0 +o +crafts 4 Sticks using a crafting table (3x3 grid) +• +CRAFT 1 minecraft:stick minecraft:stick minecraft:stick minecraft:planks minecraft:stick +minecraft:planks 0 polycraft:sack_polyisoprene_pellets 0 +o +crafts a Wooden Pogo Stick on a crafting table +• +EXTRACT_RUBBER +o +moves polycraft:sack_polyisoprene_pellets from polycraft:tree_tap to player inventory +• +PLACE_TREE_TAP +o +calls PLACE_BLOCK polycraft:tree_tap (and processes extra rules) +• +PLACE_CRAFTING_TABLE +o +calls PLACE_BLOCK minecraft:crafting_table (and processes extra rules) +• +PLACE_MACGUFFIN +o +calls PLACE_BLOCK polycraft:macguffin (and processes extra rules) + +3.2.5 Game commands +These commands should be used during developmental testing only. When testing the AI Agent +using the TournamentManager, these commands are managed by the main Python thread and +should not be called by the AI Agent. +• +START +o +no args ever used | called once to start trials +• +RESET domain ../available_tests/pogo_nonov.json +o +the base pogo experiment +• +RESET domain ../available_tests/hg_nonov.json +o +the base hunter-gatherer experiment + +3.2.6 Dev commands +Dev commands must be enabled by setting a client virtual machine argument: "-Ddev=True" Details on +setting this outside of a development environment are still being worked out, as solutions are fickle and +system dependent. Please contact us if you need these commands. +• +CHAT "Hello world." +• +CHAT /give @p minecraft:stick + + + + + +27 + +o +not used in DRY-RUN Tournaments, but active for debugging/training/development +• +TELEPORT 20 4 21 90 0 +o +not to be used in DRY-RUN Tournaments, but allows setting player location and view +direction. +o +Parameters: [x] [y] [z] [yaw] [pitch] + + + diff --git a/hdFKT4oBgHgl3EQfuS7o/content/tmp_files/load_file.txt b/hdFKT4oBgHgl3EQfuS7o/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d3fecc71da95ff38f859e96ff8de4c85ed3dc1c6 --- /dev/null +++ b/hdFKT4oBgHgl3EQfuS7o/content/tmp_files/load_file.txt @@ -0,0 +1,1008 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf,len=1007 +page_content='1 Polycraft World AI Lab (PAL) : An Extensible Platform for Evaluating Artificial Intelligence Agents Stephen A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Goss1,*, Robert J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Steininger1,*, Dhruv Narayanan1, Daniel V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Olivença2, Yutong Sun2, Peng Qiu2, Jim Amato1, Eberhard O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Voit2, Walter E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Voit1,3, Eric J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Kildebeck1,** 1 Center for Engineering Innovation, The University of Texas at Dallas, 800 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Campbell Road, Richardson, Texas 75080, USA 2 Department of Biomedical Engineering, Georgia Institute of Technology and Emory University, Atlanta, GA, United States 3 Department of Materials Science and Engineering, The University of Texas at Dallas, 800 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Campbell Road, Richardson, Texas 75080, USA Emails: stephen@polycraftworld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='com robert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='steininger@utdallas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='edu dhruv@polycraftworld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='com dolivenca3@gatech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='edu sunyutong@gatech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='edu peng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='qiu@bme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='gatech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='edu Jim@polycraftworld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='com eberhard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='voit@bme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='gatech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='edu walter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='voit@utdallas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='edu eric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='kildebeck@utdallas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='edu Co first authors **Corresponding author: University of Texas at Dallas, 800 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Campbell Road, Richardson, Texas 75080, USA 2 Abstract As artificial intelligence research advances, the platforms used to evaluate AI agents need to adapt and grow to continue to challenge them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' We present the Polycraft World AI Lab (PAL), a task simulator with an API based on the Minecraft mod Polycraft World.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Our platform is built to allow AI agents with different architectures to easily interact with the Minecraft world, train and be evaluated in multiple tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' PAL enables the creation of tasks in a flexible manner as well as having the capability to manipulate any aspect of the task during an evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' All actions taken by AI agents and external actors (non-player-characters, NPCs) in the open-world environment are logged to streamline evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Here we present two custom tasks on the PAL platform, one focused on multi-step planning and one focused on navigation, and evaluations of agents solving them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' In summary, we report a versatile and extensible AI evaluation platform with a low barrier to entry for AI researchers to utilize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Introduction Advances in artificial intelligence algorithms, in particular advances in open world learning and lifelong learning agents, require scalable open-world testbeds for development and evaluation (Doctor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Here, we present a Minecraft-based platform for simulating a wide variety of complex tasks to evaluate the next generation of artificial intelligence agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Minecraft is a video game that simulates a rich 3D environment where players collect resources and blocks to build anything from rich urban environments to virtual computing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Further, Minecraft has an extensive modding community where the bounds of the base game can be extended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' We created a comprehensive, fully customizable mod to Minecraft, Polycraft World (Smaldone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=', 2017) initially to teach materials science and polymer chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The Polycraft World mod introduces over 5,000 new materials, items, and machinery on top of base Minecraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' In addition to increased complexity of items, this mod introduces hierarchal player permissions, player/agent tracking and logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Through this continued development, Polycraft World has grown to a multipurpose platform with many applications from education to social science to artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' To evaluate open-world learning, transfer learning, and lifelong learning, an ideal virtual platform for evaluating AI agents should support a variety of tasks while still being broadly usable for different agent types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Many platforms are limited to a task or tasks within a discrete, closed environment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=', CartPole (Sutton & Barto, 2018), Arcade Learning Environment (Bellemare et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=', 2013)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Virtual 3D worlds provide added value in that they can better represent real-world relevant tasks and complex scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The Unity game engine can be a powerful platform for AI evaluation (Juliani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=', 2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' however, there is a high development cost to create custom Unity 3 based tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Additionally, many first-person shooter games have been adapted into AI research platforms such as VizDoom (Doom) (Kempka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=', 2016) and DeepMind Lab (Beattie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=', 2016) using the Quake III Arena game engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' ViZDoom and DeepMind Lab are excellent platforms to train AI agents in a 3D visual environment, but they are essentially first-person shooters with limited interactions with the environment often focused on movement with a single or small number of interaction/fire commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' While more interactions can be added to these platforms, Minecraft presents a richer out-of-the-box experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Minecraft has been used in the past in this capacity as well (Aluru et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Previous Minecraft platforms have been invaluable for studying and challenging AI agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' However, they are limited by the bounds of Minecraft itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Tasks have limited flexibility, and agents must be adapted to function within their system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' In this paper, we present the Polycraft World AI Lab (PAL), which retains the advantages of Minecraft-based platforms but has the added advantage of including an API well-suited for planning agents that creates a complex action space without requiring extensive overhead training for movement or reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' With this API, PAL enables more flexible tasks that are straightforward to customize and are more approachable for various AI agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' We first describe the platform architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' We then explain how one would design and implement a task on PAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' We go on to present the implementation of two custom tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Finally, we discuss the PAL outputs for evaluation and platform performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' PAL Architecture The Polycraft World AI Lab (PAL) platform (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='com/PolycraftWorld/PAL) was built to test AI agents on top of Polycraft World (Figure 1) using the popular modding API, Minecraft Forge (Forge volunteer team, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The PAL platform consists of the Minecraft Client and Polycraft World mod connected via the Minecraft Forge API, a Web Socket API, and a Tournament/Game manager for handling experiments with different complex tasks to test AI agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Step by step instructions for PAL installation can be found in the GitHub page for Linux and Windows but any system that can use Java will be able to run PAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Our agent API allows interfacing with any external AI agent without specific language requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' All that is needed to integrate a new agent is a simple socket client script, which can be made in any language that supports web sockets such as Java, Python, and Julia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' This significantly lowers the barrier for researchers to utilize the PAL platform as they will not need to develop or integrate with an existing wrapper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The Tournament/Game manager is our system for facilitating evaluation of AI agents in custom PAL task scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' A more detailed explanation of the PAL architecture can be found in the Supplemental Materials, in the “1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Process Communication Standards” section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' 4 Figure 1 : PAL Architecture Flow Chart Interface - Actions Agents interact with the world by sending commands to PAL one command per game tick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The available commands (Table 1 and Supplemental Materials) are universal and do not change between tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' There are different categories of commands such as game commands (START, RESET, GIVEUP), movement (MOVE, TP_TO), interactions (CRAFT, BREAK_BLOCK, USE_HAND, SELECT_ITEM), and sensing (SENSE_ALL, SENSE_SCREEN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Each command has set preconditions that must be satisfied to execute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' For example, an agent cannot move forward if there is a block in the way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Each command has an associated cost that can be used as a metric, for example, in reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The cost per action is based on an estimate of the complexity of the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' For example, Teleporting (TP_TO) 10 blocks costs 10X the cost of moving (MOVE) 1 block and the cost of crafting increases based on the number of items used in a recipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The reasonable estimation of these costs allows for evaluation of agent efficiency and realism of actions in large open-world tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Table 1: Action commands available in PAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Interactions Sensing Movement Game commands BREAK_BLOCK SENSE_SCREEN MOVE START PLACE SENSE_ALL TURN RESET COLLECT SENSE_ALL_NONAV TILT GIVE_UP CRAFT SENSE_INVENTORY TP_TO (teleport to) SELECT_ITEM SENSE_LOCATIONS USE_HAND SENSE_ACTOR_ACTIONS PAL Polycraft World Al Agent Web Socket APl < (mod) Minecraft Client Tournament Minecraft Forge Tournament Manager API data 5 USE SENSE_RECIPES DELETE SENSE_ENTITIES INTERACT TRADE NOP (no operation) For navigation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' we support discrete steps by moving one block at a time or by teleporting to block positions for agents that don’t have the capability to navigate on their own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' By using the MOVE command, an agent can move to any open adjacent square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Interface – Observations Agents can sense multiple observations about the world including symbolic observations (recipes, locations, inventory, map, actor actions, and entities) and visual observation of the screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' There are three higher level observation commands that return specific sets of these observations based on if the agent is using vision, is teleporting or navigating, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' These commands are: • SENSE_SCREEN that returns a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='PNG image file with the output of the visual screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' • SENSE_ALL will return a JSON string with all the information about the current state of the environment, which includes a symbolic map of the task’s terrain with block and NPC information, inventory, NPC locations and actions, but will not include the screen information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' • SENSE_ALL NONAV is similar to the previous command, but the symbolic map also includes all attributes of each block in the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The following commands allow the agent to access specific parts of SENSE_ALL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' • SENSE_INVENTORY will return a JSON string with the agent’s inventory contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' • SENSE_LOCATIONS will return a JSON string with the agent’s position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' • SENSE_ACTOR_ACTIONS will return a JSON string with all the actions NPCs have performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' • SENSE_RECIPES will return a JSON string with all the available recipes in the CRAFT command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' • SENSE_ENTITIES will return a JSON string with the map coordinates of all NPCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Interface – Example of Command and Response An agent can utilize the API by connecting to the default port 9000 and sending a command in plain text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Then an agent could issue a command by sending “SELECT_ITEM 6 minecraft:iron_pickaxe” to that port followed by a new line character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' After the command is processed, a text JSON response will be sent back which includes the result of the command and task goal information: { "goal": { "goalType": "ITEM", "goalAchieved": false, "Distribution": "Uninformed" }, "command_result": { "command": "select_item", "argument": "minecraft:iron_pickaxe", "result": "SUCCESS", "message": "selected item", "stepCost": 120 }, "step": 0, "gameOver": false } An example Python script can be found in the Supplemental Materials in the “2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='Example of Command and Response” section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Logging The PAL platform records logs in three different tracks which helps facilitate data analysis and the debugging process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' These logs are separated by the tournament manager, Minecraft client, and lastly the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Both the tournament manager and Minecraft logs will be the same for each evaluation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' however, the agent log always records directly from the AI agent standard output, which can be very different for each agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' These log files, produced by PAL, provide an out of the box option for player/agent tracking and analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Task Scenario One strength of the PAL platform is its ability to support a variety of task scenarios which can be researcher-defined and have broad customizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' To define a PAL task scenario, one must set a goal, define an arena and a set of available actions and interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' A goal is what the agent must accomplish to succeed in the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' This can be anything that a player could do in Polycraft, whether it be obtaining/crafting an item, placing a block in a specified location, or building a 7 complex system/structure in the arena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The arena is where the agent will be acting to achieve the goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' We can customize the arena by making it larger/smaller, filling it with objects to help or hinder the agent and adding adversarial or allied external agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The action space of a given task will be dependent on all the objects in the arena as well as items in the player’s inventory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Task components are defined using the open standard file format, JSON (JavaScript Object Notation), to allow for effortless and scalable customization of training environments in a familiar programing format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' For new tasks, Java scripts are used to generate new task definitions, which allows quick generation of multiple seeded variations of the scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' For tasks that resemble existing tasks, this can be done by manually editing an existing task definition file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Below, we present two custom task scenarios used in the DARPA SAIL-ON program (Senator, 2019) to illustrate the types of tasks that can be created in PAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Example Task Scenarios A wide variety of tasks for AI agent evaluation can be developed utilizing the PAL platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Here, we present two sample task scenarios created within our PAL platform and demonstrate evaluation of AI solutions for these tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The first task, called POGO, is a planning task where agents must obtain resources and ultimately craft a pogo stick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The second task is a navigation task named HUGA (for hunter-gatherer) where agents must find an object, pick it up, and bring it to a target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' These tasks highlight strengths in our Minecraft-based platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' In Polycraft World rich virtual worlds can be created that, like the real world, can be challenging to navigate, are only partially observable, and in which the optimal course of action may be unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' PAL Task Scenario 1: POGO Task Figure 2 : POGO Task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' (A) Schematic of the POGO arena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' (B) First person view of the agent within the POGO arena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' (C) Recipes available to the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The goal of this task is to craft a pogo stick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The agent starts in a 30 x 30 block grass field surrounded by unbreakable bedrock walls (Figure 2A,B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The field always has 5 trees and 1 (B) C) 8 crafting table placed in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Trees can be “chopped down” using the BREAK_BLOCK command, new items can be crafted from the available recipes (Figure 2C) utilizing CRAFT commands and the crafting table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Once the agent successfully obtains a pogo stick, the goal is achieved, and the task ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The task can also end if the agent gives up or if the time limit for the task is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' To achieve the goal the agent must follow a series of steps including 1) chopping down trees to collect wood, 2) crafting intermediate items including wood planks, sticks, and a tree tap, 3) placing a tree tap on a tree, 4) collecting rubber from the tree tap, and 5) using intermediate items and rubber to craft a pogo stick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' We developed a planner agent for this task using Planning Domain Definition Language (PDDL) (Barrett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=', 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Fox & Long, 2003) and the fast forward algorithm (Hoffmann & Nebel, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The planner uses the PDDL description of the task and fast forward to get a solution which will be used to create a sequence of actions that are necessary to achieve the task goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' It requires knowledge of the action space and the items available in the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The task components are translated into to PDDL 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='1 in the form of two files: the “domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='pddl” file that sets the world rules and the “problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='pddl” file that describes the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The PDDL files construction is not automated but the agent is able to adapt to differences in the map composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Running these files through the fast forward algorithm outputs a sequence of high-level actions that are interpreted by the Python code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' High level actions aggregate low-level actions that Minecraft can process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' As an example, a high-level action is ‘get wood block’ and the low-level actions corresponding to it are: ’move three steps forward’, ’turn left’, ’break wood block’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' For this planner agent we use high-level actions to reduce the action space and to allow for adaptability to changes in the task (ex: crafting table and trees changing positions in the map).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' This agent does not require training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' PAL Task Scenario 2: Hunter-Gatherer Game (HUGA) – 3D space navigation task Figure 3: Hunter Gatherer Task (HUGA) (A) Schematic view of the HUGA arena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' (B) First person view of the agent within the HUGA arena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' (C) Aerial view of the HUGA arena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Arrows indicate a path agent can take to complete the task (A) (B) (C) 9 The goal of this task is to pick up the green cross object (the MacGuffin) and place it on a Target (the blue block).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The agent starts in a four-room arena with a total size of 32 x 32 blocks (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Each room is 14 x 14 blocks large, with inner walls and multiple walkways separating the rooms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The agent always spawns in the bottom-left room (Room 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The MacGuffin spawns in the top-left room (Room 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The top-right room is empty (Room 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The Target spawns in the bottom-right room (Room 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The agent can pick up the MacGuffin by walking into it with MOVE commands and can place the MacGuffin at the Target with the PLACE command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Once the agent successfully places the MacGuffin at the Target, the goal is achieved and the task ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The task can also end if the agent gives up or if the time limit is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' This task is designed to be similar to other state-of-the-art 3D maze-runner tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' VizDoom, DMLab NavMaze, Unity Obstacle Tower Challenge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' To achieve the goal the agent must follow a series of steps (Figure 3C) including 1) walking from Room 1 to Room 2, 2) picking up the MacGuffin, 3) walking from Room 2 to Room 4 and 4) placing the MacGuffin at the Target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Like the POGO task, an agent was created to solve the HUGA task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Our HUGA agent uses a Deep Q-Network (DQN) (Mnih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Roderick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=', 2017) which combines deep neural networks and Q-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' We chose to use this architecture because the deep neural network can take visual information as input, and the Q-learning aspect can handle the sparse reward structure in the HUGA task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The structure of the DQN is determined by the sizes of the state space and the action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' A state for the visual-information-based DQN is the screenshot image of the game window at a specific time point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The dimension of the input is (3, 84, 84), where 3 is the number of color channels and (84, 84) is the resolution of the screenshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The action space utilized by this agent includes MOVE, TURN, and PLACE for the MacGuffin (green block in Figure 3B,C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The DQN is trained to take the current state as input, and computes the expected rewards in the action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' After the agent executes the action with the highest expected reward, the new state is fed into the DQN to generate the next action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The process iterates until the agent completes the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' To complete the HUGA game, the agent needs to find the MacGuffin and bring it to the destination (blue square in Figure 3A,C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' This task can be separated into two sub-tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The first sub-task is to find the MacGuffin and the second sub-task is to find the destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' We trained two DQNs with identical architecture to complete the two sub-tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' More specifically, we generated 1000 instances of the HUGA task with different layouts of the passages among rooms and various color patterns on the walls and the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' We randomly selected 618 instances for training, and used the remaining 382 for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' For each training instance of the HUGA task, we randomly initialized the location and the direction the agent is facing (Figure 3B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' We set the maximum number of actions to be 450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' When the agent completed 10 the sub-task within 450 actions or reaches 450 actions before completing the sub-task, the game was reset using the next training instance of the HUGA game, and training process continued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' When implementing the agent, we check the status after each action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Before finding the MacGuffin, the agent calls the first DQN model to decide its actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' After the MacGuffin is found, the agent turns to the second DQN model to decide its actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' When the agent is tested in one instance of the HUGA task, we record a success if each sub-task is completed within 450 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' We use the success rate as the evaluation metric for the trained agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' We randomly selected HUGA task instances from the 382 for testing, randomly set the location and direction the agent is facing, and tested the trained agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' We observed a success rate of ~94% with this agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' PAL Extensibility Figure 4 – POGO and HUGA modifications (A) Bird’s eye view of updated POGO arena (B) First person view of updated POGO task scenario (C) Bird’s eye view of updated HUGA arena (D) First person view of updated HUGA task scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' (A) (B) (C) 11 In Figure 4 we show modifications we’ve made to both the POGO and HUGA tasks that demonstrate the extensibility of our platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Scale and complexity of tasks can be easily increased and decreased using our flexible task creation system based on the needs of AI researchers and the capability of their agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' We’ve increased the complexity of the POGO and HUGA tasks by adding more varied layouts, more complex problems, and external actors (NPCs), which can be allied or adversarial with the AI agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' These additions increase real-world relevance by adding scale, partial observability, and non-determinism to these task scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' In the updated POGO task, the agent must collect additional resources from additional places (added rooms in Figure 4A) and interact with trading NPCs while avoiding competitive NPCs (seen in Figure 4B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' In the updated HUGA task, the agent must navigate varied layouts with different color schemes (Figure 4C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' They must also avoid traps, obtain keys to unlock doors and obtain advantages from helpful agents while avoiding adversarial NPCs (seen in Figure 4D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' PAL Evaluations and Performance Running Tournaments in PAL A tournament manager was developed in python that manages the launching & logging of AI Agents performing in task scenario instances in PAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' This manager can be deployed across computing clusters during evaluations and is parameterized to accept a variety of agents and tournaments to run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The manager runs four separate processes that communicate with each other through sockets on the host machine: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' PAL, the Minecraft-modded game environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' AI Agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' A bash execution command is necessary to launch the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' TournamentManager, which monitors instance-ending conditions and sends the appropriate reset command to PAL along with a JSON file describing the next instance to load into PAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' LoggingHandler that logs outputs, records all steps taken, and updates SQL databases with this information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' This process includes a tournament timeout – if no progress has been made for more than 5*max game time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=', in the event any of the above processes freeze), then the tournament ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' We provide a copy of the tournament manager that includes processes 1,3, & 4 on the PAL GitHub (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='com/PolycraftWorld/PAL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Installation instructions and execution assistance can also be found on the PAL GitHub in the ReadME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='txt file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' We fully support and highly recommend running this environment in a Debian-based UNIX operating environment (we use Ubuntu 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='04 in our Evaluations);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' however, it can be successfully deployed in Windows-based environments as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' 12 AI agents are launched through a user-configurable, UNIX-compliant bash execution command after the PAL environment has been launched and the first task instance has been loaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Following successful launch, the AI Agent should connect to port 9000 and send an API command to start the tournament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' We have 547 available scenarios across 2 tasks that can be used to train and evaluate an AI agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Each scenario is a version of the POGO or HUGA task with additional world elements or changes to the arena that provide different challenges to AI agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' These scenarios each have 3 differently seeded tournament variations for a total of 1641 tournaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Each tournament has 100 games which comes out to 164,100 different game instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' These instances are all available on GitHub (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='com/PolycraftWorld/PAL) to train and evaluate on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' PAL Platform Performance We are limited in how many actions we can take per second by Minecraft’s internal tick rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Minecraft is hard coded to run at 20 ticks per second and has no setting or configuration available to change the speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' However, this can be unlocked by using reflection to change the tick-rate at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' This can be very helpful by increasing training and evaluation times by a factor of 20- 40 depending on whether the agent is using visual observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' We have tested the limits of achievable speed (in ticks per second) in PAL and found that we can reach speeds of 550 ticks per second on modern systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Our tests were done on a Windows 10 machine with a Ryzen 3900x CPU, 32GB Ram, and a GTX 1070 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' This number is mostly affected by single thread processor performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' When an agent is using the screen observation on every tick, this performance degrades significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The default screen output format is PNG, which results in an average speed of 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='44 ticks per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' If we change the format of the screen observation to JPEG we get an average speed of 300 ticks per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' We can reach speeds closer to 550 ticks per second if an agent doesn’t request the screen after each action (Table S1 in Supplemental Materials).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Conclusion Here, we present the Polycraft World AI Lab (PAL), an extensible platform for creating AI challenges, running AI experiments, and evaluating AI agent performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' We have leveraged the strengths of Minecraft but added additional flexibility and accessibility in the PAL system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' PAL connects with a broad range of AI agents via our socket-based command interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Agent evaluation includes comprehensive log files which monitor agents’ behavior throughout a task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' We present two examples of custom tasks: a planning task, where the agent is challenged to craft a pogo stick (POGO) and a navigation task, where the agent is challenged to find an item and bring it to a target (HUGA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' These tasks were built entirely using the PAL platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Agents were 13 developed and evaluated on each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' We go on to show that we can add complexity (via added NPCs, task problems, and domain scale) to these tasks to make them more relevant to real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Finally, we discuss the PAL system performance and its ability to rapidly evaluate agents in a tournament structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Figure 5: PAL is capable of a creating a diverse set of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' We have created multiple arenas in Polycraft World that have diverse goal conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' (A) Minigames such as basketball, archery and minigolf (B) An escape room challenge on a space station (C) Scavenger hunts on the UTD campus (D) Virtual paintball (E) a tutorial for teaching gameplay in multiple stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Our tasks can have various levels of complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Capture the base in: (F) a simple square arena, (G) a stadium setting, (H) a desert city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' This platform is by no means limited to the tasks presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' We envision this platform will be used to evaluate agents on custom open-world tasks designed by AI researchers and we have demonstrated the capability of the platform to present a wide variety of activities for human and AI agents (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' We believe the flexibility PAL adds to task design and agent integration are valuable tools for fundamental AI research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' We believe that PAL will enable research on large scale challenges relevant to multi-task learning, transfer learning, and problems encountered in open-world learning including adaptation to out- of-distribution world states and previously unobserved environments (Boult et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Kejriwal & Thomas, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Muhammad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Musliner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Moving forward, this platform will be used for evaluating AI agent performance in novel AI tasks that are increasingly complex, real-world relevant, and involve human-AI teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' (B) (E) C (F) X 14 Conflict of Interest The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Author Contributions S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' was the lead developer for PAL, task creation, manages the project GitHub and wrote the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' wrote the manuscript, was involved in debugging, had several supporting roles and bibliography research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' created the tournament manager.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' developed the POGO agent, wrote the manuscript and contributed to the project documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' developed the HUGA agent, wrote the manuscript and contributed to the project documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' contributed to the project documentation and was involved in debugging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' task creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=', W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=', P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' conceived the project, reviewed the manuscript, and provided guidance and management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' All authors reviewed and edited the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Funding This work was supported in part by the following grant: W911NF2020010 from the Defense Advanced Research Projects Agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The funding agency is not responsible for the content of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Acknowledgements The authors greatly appreciate the constructive feedback of colleagues in the SAIL-ON program for their comments and help testing PAL in the early stages of the platform development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The authors also would like to thank Willie E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Chalmers for the insightful comments and help in the beginning of the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' 15 References Aluru, K.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='il/sites/resauth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='il/files/DARPA SAIL-ON_HR001119S0038.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='pdf Smaldone, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=', Thompson, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=', Evans, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=', & Voit, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Teaching science through video games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' NATURE CHEMISTRY, 9, 97–102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' http://go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='com/2hjb2l1 Sutton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=', & Barto, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Reinforcement learning: An introduction (The MIT Press, Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' 17 Polycraft World AI Lab (PAL) : An Extensible Platform for Evaluating Artificial Intelligence Agents SUPPLEMENTARY INFORMATION 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Process Communication Standards Socket communication is key in managing our training and tournament environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' PAL and the AI Agent communicate over port 9000 (default port).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The TournamentManager and PAL communicate over port 9005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' There is no direct communication between the tournament manager and the AI Agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' All AI Agent communications must be initiated by the AI Agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' PAL will, at no time, send a message unprompted to the AI Agent, instead passing you key information through JSON responses (was your command successfully executed, is the instance over due to a time-out, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' In the event that a game timeout does occur, if an agent does not query PAL with a command (like move, or sense_all), the agent will never know that the instance is in fact over and the next instance is waiting to start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Our tournament manager will hang, and eventually, we have a process on our end that will kill the trial and state that the agent has become non- responsive in our logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' AI Agent Communications are defined by our PAL API, available in the Supplements, in PAL API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' In addition to command-specific response variables that vary depending on the command sent, every response JSON contains the following keys: • goal: json dict that contains: o goalType: "BLOCK_TO_LOCATION" for the HUGA Task, "POGOSTICK" for the POGO Task o goalAchieved: True if the goal was achieved, False otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Once the goal has been achieved, for the remainder of commands sent in that instance, this will continue to report "True" (boolean) o Distribution: "Uninformed" if this trial requires “System Detection” of Novelty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' "PreNovelty" or "Novelty" if this trial is “Given Detection”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' (string) • command_result: json dict that contains o command: command sent by Agent (string) o argument: any command arguments sent by the Agent (string) o result: "SUCCESS" if the command was executed properly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' "FAIL" if a problem arose in command execution (string) o message: a non-null string if the result = "FAIL" containing an error message (string) 18 o stepCost: step cost of executing a particular command (float) • step: Step Number,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' 0-indexed count of commands sent by the AI Agent (integer) • gameOver: True if the instance is over,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' False otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' (boolean) Our LoggingHandler monitors the STDOUT and STDERR of the PAL and AI Agent process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' This process writes to separate log files all data written to STDOUT and STDERR by the PAL process and by the Agent AI process, broken out as one file per instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='1 Communicating Game State to the AI Agent As indicated above, the tournament manager does not directly communicate details on the state of the trial or the state of the instance to the AI Agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Instead, the above listed JSON response keys convey critical information that the Agent can process after each command (step) to understand the instance state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' As an example, as soon as the goalAchieved=True is found in the response, the Agent knows that it successfully completed the task for that particular instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Every JSON response has a "gameOver" key that is either True or False.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The value "True" is sent in only one json response to the first command received after of the following game-ending criteria is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Subsequent commands in that same game will revert to having gameOver=False, so it is imperative that the AI Agent recognize and appropriately handle this flag as soon as it appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The current instance ending conditions are: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Successful Completion of Task (we also include another key, "goalAchieved" in each response that will reflect this case) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Game Time limit exceeded 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Total Step Cost exceeds a step-cost maximum of 1,000,000 (an empirically determined upper bound to prevent abuse of API commands or other counter-productive behavior towards task completion) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=" Agent sends the GIVE_UP command As of now, our architecture requires that the AI agent, following the successful reception of a response JSON containing gameOver=True send one last command back to PAL to acknowledge receipt and initiate the reset to start the next instance (doesn't matter what the command is, as its step cost is not counted for/against your score, and its response JSON is irrelevant for your agent as well." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Most agents just pass a SENSE_ALL command and ignore the response).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Ideally, only one additional command will be processed by PAL following any of the instance end conditions evaluating to true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Tournament completion is not directly communicated towards the AI Agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Instead, the process is terminated by the parent process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' 19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='2 Running Tests of your Agent in PAL Detailed Instructions on setup and installation of the Tournament Manager environment is available on GitHub (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='com/PolycraftWorld/PAL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' To run an AI Agent, the AGENT_COMMAND config variable (or command line parameter) needs to be adjusted to point to a script that will load the AI Agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' As mentioned above, the AI agent is expected to, as part of that startup script, try to connect to port 9000 and send an API command to begin the trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Though an AI agent could just be a folder of scripts as our baseline agent is, we found that many of our partners prefer to run their AI agents in Docker containers – that works well in our current evaluation environment, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='3 Handling Updates to PAL in your Test Environment In the event that updates or bug-fixes are made to PAL (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=', new API commands, new novelties, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' ), we will communicate via email/slack/teams with a detailed changelog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Please direct your code to the appropriate branch on PAL GitHub (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='com/PolycraftWorld/PAL), pull the updates, and then continue to run tests using your AI agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=" Example of Command and Response An agent can use the API by connecting to the default port 9000 using the following Python script: import socket import json # connect to socket HOST = '127." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content="1' PORT = 9000 sock = socket." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='socket(socket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='AF_INET, socket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='SOCK_STREAM) sock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='connect((HOST, PORT)) The following script constructs a function (MC) that enables the communication with Polycraft: def MC(command): "function that enable the communication with minecraft" print( command ) sock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='send(str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content="encode(command + '\\n')) BUFF_SIZE = 4096 # 4 KiB data = b'' while True: part = sock." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='recv(BUFF_SIZE) data += part time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='sleep(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='001) if len(part) < BUFF_SIZE or part[-1] == 10: # either 0 or end of data break print(data) data_dict = json.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='loads(data) # print(data_dict) return data_dict Now we can start the API using : MC(\'START\') And reset the task: MC("RESET domain .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='./pogo_100_PN/POGO_L00_T01_S01_X0100_U9999_V0_G00000_I0020_N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='json") Now we can send commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=" For example: MC('SENSE_RECIPES')) 21 MC('SENSE_ALL') MC('SELECT_ITEM minecraft:iron_pickaxe') After the command is processed, a text JSON response will be sent back which includes the result of the command and task goal information." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' For example, the return of the last command will be: { "goal": { "goalType": "ITEM", "goalAchieved": false, "Distribution": "Uninformed" }, "command_result": { "command": "select_item", "argument": "minecraft:iron_pickaxe", "result": "SUCCESS", "message": "selected item", "stepCost": 120 }, "step": 0, "gameOver } 22 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' PAL API The PAL API consists of different API commands broken down into SYSTEM commands, DEV commands, and GAME commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' For AI Agent testing using the TournamentManager, only GAME commands will need to be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The other commands are provided for completeness only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='1 Available configuration There are three places available for configuration settings for a tournament run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' There are the command-line arguments, the config file (PAL/PolycraftAIGym/config.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='py), and lastly there are available environment variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='1 LaunchTournament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='py Command-line arguments LaunchTournament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='py Usage: LaunchTournament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='py will start a new tournament on the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' It will first start a Polycraft client, then start a tournament manager process to initialize the first game of the tournament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Once the game has initialized, it will start the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Format: LaunchTournament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='py -c -t -g -a -d -x -i -m -c How many games to run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Default: 100 -t Name of tournament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Default: “POGO_L00_T01_S01_X0100_A_U9999” -g Where games are located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Default: “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='./pogo_100_PN” -a Name of your agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Default: “MY_AGENT_ID” -d Where agent is located Default: “” -x Command to run agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Text as would be entered into command-line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' To escape quotes or other special characters, use “\\” Default: “python pogo_agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='py” -i Seconds per game Default: 300 -m Max time for tournament to run in minutes Default: 2880 Example: python LaunchTournament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='py c 100 t "POGO_L00_T01_S01_X0100_A_U9999_V0200FPS_011022" g ".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='./pogo_100_PN" a "BASELINE_POGOPLAN " d "agents/pogo_stick_planner_agent/" x "python python_miner_PLANNER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='py" 23 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='2 Environment Variables PAL Observation Variables SENSE_SCREEN_FORMAT – Some screen formats will compress faster than others at the expense of quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' PNG seems to give the best quality, while JPEG typically give the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The default image type is PNG but this can be set to the following: PNG, BMP, JPEG, JPG, WBMP, GIF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' AIGYM_REPORTING - Set this to “True” to activate AIGym style reporting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The agent will receive a “SENSE_ALL” output after every command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' REPORT_SCREEN – If “AIGYM_REPORTING” is set to true, you can also set this to “True” to also receive a “SENSE_SCREEN” output after every command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' PAL Ports PAL_AGENT_PORT – Set the agent socket communication port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Default is 9000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' PAL_TM_PORT – Set the tournament manager socket communication port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Default is 9005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' PAL Speed PAL_FPS – Set the frames per second for PAL to run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Default is 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The maximum value here is 1000, but most systems cannot achieve that unless they have powerful single thread cores and a powerful GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Internal testing revealed some results you may be able to expect on your own machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Sense screen testing Test System specs: OS – Windows 10 Pro CPU – Ryzen 9 3900x 12 core @4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='00 GHz RAM – 32GB GPU – GTX 1070 Table S1 – Max ticks per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Target ticks per second Tested ticks per second (with different screen output formats) PNG BMP JPG WBMP GIF NONE 20 (Default) Min:18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='79 Max:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='07 Mean:19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='98 Median:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} 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+page_content='2 Max:511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='38 Mean:470.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='79 24 Median:53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='59 Median:127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='73 Median:262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='68 Median:406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='73 Median:192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='48 Median:419.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='29 800 Min:50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='3 Max:151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='18 Mean:73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='44 Median:77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='24 Min:130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='46 Max:277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='02 Mean:227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='44 Median:130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='46 Min:121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='36 Max:409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='22 Mean:297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='09 Median:121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='36 Min:97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='65 Max:384.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='09 Mean:213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='5 Median:176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='36 Min:108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='61 Max:196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='33 Mean:155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='94 Median:108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='61 Min:392.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='07 Max:677.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='97 Mean:549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='36 Median:539.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='08 1000 Diminishing performance past 800 PAL Screen Position PAL_X – x position on screen for window to render.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Default is chosen by Minecraft PAL_Y – y position on screen for window to render.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Default is chosen by Minecraft PAL_WIDTH – Screen render width size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Default is 256 PAL_HEIGHT – Render screen height size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Default is 256 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='2 GAME commands 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='1 Tournament Commands These commands are sent by the AI Agent to communicate directly to the Tournament Manager and should be used when the Agent is being officially evaluated, where appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' • GIVE_UP o Agent gives up task, letting tournament manager know that the next instance can be queued for a domain reset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' o Please note that following a GIVE_UP command, the agent will receive a gameOver=True as part of the response JSON.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' The Agent is still required to send one additional command to PAL to acknowledge receipt of the gameOver=True before the RESET domain command is triggered by the Tournament Manager and the next instance in the trial is loaded • REPORT_NOVELTY [–l lvl] [–c confidence] [–m user-message] o indicates that you have detected novelty with optional parameters | [-l novelty level] o [-c confidence interval 0f:100f] [-m user-defined message] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='2 Movement commands These commands enable the AI Agent to move around the instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' • MOVE w|a|d|x o moves 1 meter forward (w), left (a), right (d) or back (x) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' “MOVE w” moves forward 1 block) • MOVE q|e|z|c o moves sqrt (2) distance diagonally with q,e,z,c (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=', “MOVE q” moves diagonally forward and leftward, relative to the player’s facing direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Diagonal movements correspond to relative location of letter against the WSAD keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=" • TURN -15 o alters player's horizontal facing direction (yaw) in 15-degree increments (no interpolation) o The parameter passed must be a multiple of 15, positive or negative." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' • SMOOTH_TILT FORWARD 25 o Sets the player’s pitch to the horizon (0 degrees) o While this is a “SMOOTH” command, it operates as an abrupt state transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' There are plans to create a “TILT” command once the visual interpolation version of the command is deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' • SMOOTH_TILT DOWN o Sets the player’s pitch to -45 (looking directly at the ground in front of the player – ideal for viewing the location upon which a block should be placed on the ground) o While this is a “SMOOTH” command, it operates as an abrupt state transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' There are plans to create a “TILT” command once the visual interpolation version of the command is deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' • TP_TO x,y,z [distance] o Teleports the player to the block at x,y,z, resetting the player’s pitch and yaw to have the player face North and set their pitch to 0 degrees (horizon), positioning the player a default distance of 1 away from the block if no distance is passed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' In other words, if distance is Null/not given, the player’s true position would be [x,y,z-1] and executing a MOVE W command will place the player onto the exact coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' o The optional distance parameter must be a positive integer greater than or equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' This distance will adjust the resulting position of the player to be [x,y,z-distance], facing the coordinate desired at an offset [distance] blocks away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Running the MOVE W command [distance] number of times will place the player on the target coordinate [x,y,z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' o Using distance = 2 is necessary when playing tree-taps on trees, as otherwise, the player will be occupying the space that the tree-tap will need to occupy after being placed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' o The distance offset must yield an allowable move_to location (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=', it can’t be a block in the area) or command fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' • TP_TO entityID o teleports to the location of an entity with entity_ID = entityID (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=', TP_TO 7101 teleports to an entity with the ID "7101") 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='3 Sensing commands • CHECK_COST o returns the total stepCost incurred since the last RESET command • SENSE_INVENTORY o returns contents of player inventory in .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='json format • SENSE_LOCATIONS o returns sensible world environment (blocks, entities and locations) as .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='json • SENSE_RECIPES o Returns the list of recipes available in the experiment • SENSE_SCREEN o Returns pixels sent to the display output window, in the form of a string listing an array of integers • SENSE_ALL o returns inventory, recipe and information on all observable blocks around the player’s position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' • SENSE_ALL NONAV o returns inventory, recipe and location information in .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='json o NONAV parameters omits information which is not needed for agents that do not navigate the world 26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='4 Interacting commands • SELECT_ITEM polycraft:wooden_pogo_stick o sets a specific item from your inventory in your hand as the active item (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='processes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='extra ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='rules) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='5 Game commands These commands should be used during developmental testing only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' When testing the AI Agent using the TournamentManager, these commands are managed by the main Python thread and should not be called by the AI Agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' • START o no args ever used | called once to start trials • RESET domain .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='./available_tests/pogo_nonov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='json o the base pogo experiment • RESET domain .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='./available_tests/hg_nonov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='json o the base hunter-gatherer experiment 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='6 Dev commands Dev commands must be enabled by setting a client virtual machine argument: "-Ddev=True" Details on setting this outside of a development environment are still being worked out, as solutions are fickle and system dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' Please contact us if you need these commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' • CHAT "Hello world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content='" • CHAT /give @p minecraft:stick 27 o not used in DRY-RUN Tournaments, but active for debugging/training/development • TELEPORT 20 4 21 90 0 o not to be used in DRY-RUN Tournaments, but allows setting player location and view direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} +page_content=' o Parameters: [x] [y] [z] [yaw] [pitch]' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFKT4oBgHgl3EQfuS7o/content/2301.11891v1.pdf'} diff --git a/htFLT4oBgHgl3EQfaS9u/content/tmp_files/2301.12073v1.pdf.txt b/htFLT4oBgHgl3EQfaS9u/content/tmp_files/2301.12073v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fb0f0e9e7049e47315d703feea226bd2e896a276 --- /dev/null +++ b/htFLT4oBgHgl3EQfaS9u/content/tmp_files/2301.12073v1.pdf.txt @@ -0,0 +1,1216 @@ +Towards Equitable Representation in Text-to-Image Synthesis Models with the +Cross-Cultural Understanding Benchmark (CCUB) Dataset +Zhixuan Liu 1 ∗, Youeun Shin 1, 2 ∗, Beverley-Claire Okogwu 1, Youngsik Yun 1, 2, +Lia Coleman 1, Peter Schaldenbrand 1†, Jihie Kim 2 †, Jean Oh 1 † +1 The Robotics Institute, Carnegie Mellon University +2 Department of Artificial Intelligence, Dongguk University +{zhixuan2, youeuns, bokogwu, youngsiy, liac, pschalde}@andrew.cmu.edu, +jihie.kim@dgu.edu, jeanoh@cmu.edu +Abstract +It has been shown that accurate representation in +media improves the well-being of the people who +consume it. +By contrast, inaccurate representa- +tions can negatively affect viewers and lead to +harmful perceptions of other cultures. To achieve +inclusive representation in generated images, we +propose a culturally-aware priming approach for +text-to-image synthesis using a small but cul- +turally curated dataset that we collected, known +here as Cross-Cultural Understanding Benchmark +(CCUB) Dataset, to fight the bias prevalent in gi- +ant datasets. Our proposed approach is comprised +of two fine-tuning techniques: (1) Adding visual +context via fine-tuning a pre-trained text-to-image +synthesis model, Stable Diffusion, on the CCUB +text-image pairs, and (2) Adding semantic context +via automated prompt engineering using the fine- +tuned large language model, GPT-3, trained on our +CCUB culturally-aware text data. CCUB dataset +is curated and our approach is evaluated by people +who have a personal relationship with that particu- +lar culture. Our experiments indicate that priming +using both text and image is effective in improving +the cultural relevance and decreasing the offensive- +ness of generated images while maintaining quality. +Our CCUB dataset and codes1 are publicly avail- +able. +1 +Introduction +Representation matters. +In media, +studies repeatedly +show that representation affects the well-being of its view- +ers [Shaw, 2010; Caswell et al., 2017; Elbaba, 2019]. Rep- +resentation can positively affect viewers by providing them +with role models that they identify with, but it can also neg- +atively affect viewers by creating harmful, stereotypical un- +derstandings of people and culture [Casta˜neda, 2018]. When +people are accurately represented in media, it allows peo- +ple to properly understand cultures without harmful stereo- +∗indicates equal contribution. +†indicates corresponding authors. +1https://github.com/cmubig/CCUB +Figure 1: Sample images generated for five different countries by +our proposed culturally-aware text-to-image synthesis approach; the +images in the first row show the results from the generic Stable Dif- +fusion as references. +types forming [Dixon and Linz, 2000; Mastro and Greenberg, +2000]. Despite the benefits of representation, many media +generating Artificial Intelligence (AI) models show poor rep- +resentation in their results [Ntoutsi et al., 2020]. Many of +these issues stem from their large training datasets which are +gathered by crawling the Internet without filtering supervi- +sion and contain malign stereotypes and ethnic slurs among +arXiv:2301.12073v1 [cs.CV] 28 Jan 2023 + +"A woman is painting +"A family is +"A photo of a +"Two people dressed +in a traditional style" +eating together +in traditional clothing +Generic +Stable +Diffusion +American +culture +(Ours) +Korean +culture +(Ours) +Nigerian +culture +(Ours) +Chinese +culture +(Ours) +Mexican +culture +(Ours)other problematic content [Birhane et al., 2021]. +As AI models are increasingly used to create and aid in the +production of visual content, it is important that the models +have a true understanding of culture such that it can give accu- +rate and proper representation leading to well-being rewards +for its consumers. In this paper, we aim to address such a +representation issue in image generation and introduce a new +task of culturally-aware image synthesis: generating visual +content within a cultural context that is both accurate and in- +offensive. Our overarching goal is to improve the well-being +of consumers of the AI generated images with particular at- +tention to those consumers from underrepresented groups. +Specifically, we formulate the culturally-aware text-to-image +synthesis task to take an additional input of a country name to +specify a cultural context in addition to language description. +It was found that large datasets such as the LAION- +5B [Schuhmann et al., 2021] used to train many text-to- +image synthesis models such as Stable Diffusion [Rombach +et al., 2021] are Anglo-centric and Euro-centric [Birhane et +al., 2021] as shown in the top row of Figure 1. As a con- +sequence, these powerful models may generate culturally of- +fensive images due to misrepresentation during training. Our +research question is, how can effective existing text-to-image +models be improved to become more culturally representative +and thus less offensive? It may be infeasible to vet billions of +training examples for accurate cultural content. +We hypothesize that a small dataset that is veritably repre- +sentative of a culture can be used to prime pre-trained text- +to-image models to guide the model towards more culturally +accurate content creation. To verify the hypothesis, we col- +lected a dataset of image and caption pairs for 8 cultures. +For each culture, data was collected by a few people who +are native of that culture as they are the people who prop- +erly understand it and are most affected by its misrepresenta- +tions. We call this the Cross-Cultural Understanding Bench- +mark (CCUB) dataset which comprises of 100-200 images +each with a manually written caption as shown in Figure 2. +We propose two techniques for enhancing the text-to- +image pipelines using CCUB. First, we fine-tune a text-to- +image synthesis model, Stable Diffusion, on the CCUB text- +image pairs to generate images tailored for a given cultural +context. Second, we create an automatic prompt augment- +ing approach using GPT-3 [Brown et al., 2020] fine-tuned on +CCUB to include culturally relevant details, e.g., “Two peo- +ple walking down a street” can be augmented with “using +WeChat Pay to pay a bus ticket, in Shenzhen, China.” +We evaluate our approach’s two components individually +as well as combined against the baseline of simply specifying +the culture in the text prompt. Our evaluation was performed +by native people of each country. Our survey results based on +2,244 image comparisions conducted by 72 participants from +5 countries indicate that our proposed approach is both less +offensive and more cultural relevant than simply adding the +country name as a suffix to the prompt. Our contributions are +as follows: +1. The introduction of culturally-aware text-to-image syn- +thesis as a valuable task within text-to-image synthesis; +2. The Cross-Cultural Understanding Benchmark (CCUB) +dataset consisting of 1,095 culturally representative +image-text pairs across 8 countries; and +3. Two techniques for culturally customizing a text-to- +image synthesis model. +2 +Related Work +2.1 +Cultural Datasets +Various efforts have been made to build a dataset that contains +a precise representation of each culture around the world, es- +pecially for the underrepresented groups and smaller popu- +lations, to combat the bias of benchmark datasets. MaRVL +dataset [Liu et al., 2021] created a set of cultures and lan- +guages, including Indonesian, Swahili, Tamil, Turkish, and +Mandarin Chinese, comprised of diverse cultural concepts to +mitigate existing North American or Western European bias. +While sharing the similar intuition, MaRVL was specifically +developed for the reasoning task covering common, popular +concepts only. +Dollar Street [Rojas et al., 2022] aimed to capture accurate +demographic information based on socioeconomic features, +such as everyday household items and monthly income, of 63 +countries worldwide. However, this dataset gives less diverse +scenarios. Most of the images in this dataset provide indoor +views with limited cultural features. +2.2 +Culturally Conditioned Machine Learning +Accurately representing culture with Machine Learning is an +open challenge. Many models, such as Craiyon [Dayma et +al., 2021] fail to capture certain distinguishing features relat- +ing to a country’s dominant culture [Reviriego and Merino- +G´omez, 2022]. One method to address this involves the in- +clusion of semantic understanding in a model such as the +ERNIE-ViLG 2.0. [Feng et al., 2022]. A similar approach +can be seen in Japanese Stable Diffusion [Shing and Sawada, +2022], which fine tunes the Stable Diffusion U-Net and +retrains the text encoder on the 100 million images with +Japanese caption within the LAION-5B [Schuhmann et al., +2021] dataset. +While these approaches produce better cultural representa- +tions of Japan and China, it is not easy to be used universally. +Adapting these approaches requires millions of training ex- +amples which cannot be easily met for cultures with less in- +ternet presence. Also, these datasets are so large that it is +infeasible to vet them for harmful and stereotypical informa- +tion. Our approach strives for cultural representation using a +dataset that is smaller (100-200 images) and hand selectable. +2.3 +Modifying Text-to-Image Diffusion Models +Diffusion-based text-to-image synthesis models have im- +proved incredibly over the past year in image quality and lan- +guage understanding; however, these models are still Anglo- +centric and contain gender and racial biases at least in part +due to the lack of supervision in their large text-image train- +ing datasets [Birhane et al., 2021]. One way to address this +problem while leveraging the knowledge obtained from the +large dataset is to fine-tune the latent diffusion models. [Ruiz +et al., 2022] and [Gal et al., 2022] propose textural inversion + +food & drink +clothing +artwork +dance & music +religion +architecture +people +city +nature +total +Korea +55 +9 +10 +18 +3 +23 +12 +22 +7 +159 +Japan +20 +7 +13 +15 +3 +11 +26 +13 +14 +122 +China +25 +18 +6 +12 +7 +13 +20 +23 +10 +134 +Mexico +22 +14 +23 +10 +6 +18 +19 +15 +7 +134 +Nigeria +16 +19 +12 +11 +8 +19 +12 +31 +5 +133 +Norway +24 +11 +7 +14 +5 +20 +18 +20 +13 +132 +Vietnam +27 +14 +10 +15 +7 +14 +17 +16 +10 +130 +United +States +23 +10 +15 +12 +7 +17 +22 +29 +16 +151 +Total +212 +102 +96 +107 +46 +135 +146 +169 +82 +1095 +Table 1: This table shows the scale of our CCUB dataset. The num- +ber of hand-selected images and their corresponding captions in nine +cultural categories for eight different cultures are listed. +methods that allow the latent diffusion models to generate im- +ages with specific visual concepts. However, these methods +restrict the generation to be an object or a style and cannot +generalize on the expression of abstract concepts, for exam- +ple, a culture. Inspired by the approaches of [Chambon et +al., 2022] and [Pinkney, 2022], the text-to-image diffusion +model can generate domain-specific images by fine-tuning +the U-Net of the model using a batch of data from that do- +main. +2.4 +Prompt Engineering +Originating from the field of Natural Language Process- +ing (NLP), prompt engineering, which can be conceived as +programming in natural language [Reynolds and McDonell, +2021], is to design the input text prompt to retrieve user- +desired outcomes from language models. +Inspired by the +findings in the NLP domain, researchers in computer vision +have been exploring the effects of prompt engineering. In +order to present design guidelines for better outcomes in text- +to-image generation models [Liu and Chilton, 2022], several +permutations of prompt engineering using a template were +conducted in terms of subject and style in art. To discover +some tricks and keywords to boost the quality of the out- +put image in the image generation models such as DALL-E +2 [Ramesh et al., 2022] and Midjourney [Midjourney, 2022], +various experiments through trial-and-error are ongoing on +the Internet as well. Online community [Taylor, 2022] has +come up with a prompt engineering template for artwork, +consisting of terms regarding styles, artists, vibes, and per- +spectives. +3 +Cross-Cultural Understanding Benchmark +(CCUB) Dataset +Our CCUB dataset provides text-image pairwise data across +8 cultures and is designed for text-to-image synthesis tasks. +For each culture subset, the proposed dataset contains images +among various scenarios. Moreover, the text data concisely +describes the culturally aware content of each image thus giv- +ing the language model a clear guide towards cultural aware- +ness. +Figure 2: Images and captions in our CCUB dataset for three of the +eight countries whose cultures are represented in CCUB. +3.1 +Culturally Representative Data Collection +Following the definition of culture in [Halpern, 1955] +and [Key and Comrie, 2021], nine categories are used to rep- +resent cultural elements in our dataset: food & drink, cloth- +ing, artwork, dance and music, religion, architecture, people, +city and nature. The categories are further divided into tradi- +tional and modern to reflect a characteristic of the culture that +culture changes over time. +Our CCUB image are collected based on the nine cul- +tural categories. +For collection, we recruited cultural ex- +perts who confidently know this culture well or belong to +it. Cultural experts are asked to collect 10-20 relevant im- +ages containing different objects for each cultural category. +The images were collected either from Creative Commons li- +censed images from Google searches or the collectors own +photographs. Cultural experts were also asked to select im- +ages with common or culturally representative items. +Each image in the CCUB dataset is also captioned by cul- +tural experts forming paired image-text data. Cultural experts +were asked to focus on the general and specific items in each +cultural image, rather than adding captions to subtle com- +ponents of the image. The captions accurately express cul- +tural contents in English as opposed to large datasets such as +LAION [Schuhmann et al., 2021] which are scraped from the +internet and not vetted for cultural accuracy. +3.2 +Properties +Our CCUB dataset contains culturally representative image- +text pairs in eight different cultures. As shown in Table 1, our +cultural dataset is in a minute scale considering that generic +Stable Diffusion was trained on LAION-2B-EN that includes +more than 2.3 billion text-image pairs. Our table Figure 2 +shows some selected samples of our CCUB dataset. + +Korean +cultural data +"one of the Eight +a woman dressed in +"Korean barbecue +"a Korean man +Gates in the Fortress +Hanbok is playing a +grilling meat, +and woman in +Wall of Seoul, South +traditional Korean +mushrooms and garlic" +hanbok" +Korea" +string instrument" +Mexican +cultural data +"a plate of beef +"Puebla Cholula +"some Mexican +“a woman in costume +tacos with beans +and face paint in front of +Church and +people performing +and rice on the +an altar for the Day of Popocatepetl Volcano +Voladores ritual +side" +the Dead holiday' +in Mexico' +dance" +Nigerian +cultural data +"a group of women +"chicken and fried +"a group of Nigerian +dressed in Hausa, +"Abuja Anglican +Yoruba men playing +rice in a white +Yoruba, and Igbo +church in Nigeria" +bowl in Nigeria" +the Bata drum" +traditional attires"Figure 3: Figure (a) is the overview of our two fine-tuning techniques based on our CCUB dataset. Models with cultural biases are marked by +sad faces, while culturally-aware models are marked by smiling faces. Figure (b) is the workflow for our approach toward culturally-aware +text-to-image synthesis. +4 +Culturally-aware Text-to-image Synthesis +4.1 +Fine-tuning Stable Diffusion +The Stable Diffusion [Rombach et al., 2021] pipeline gener- +ates natural images under the condition of text prompts. The +input text prompt is firstly encoded using the CLIP [Radford +et al., 2021] text encoder. A U-Net architecture model cre- +ates the output image encoding by denoising from random +noise conditioned upon the encoded text. A Variational Au- +toencoder (VAE) converts the image encoding into a high- +resolution image. +We alter Stable Diffusion to have a more accurate under- +standing of a given culture to address its known bias towards +generating Western-focused imagery. In our approach, fol- +lowing a similar approach to [Ruiz et al., 2022] and [Cham- +bon et al., 2022], the U-Net of the Stable Diffusion model is +further trained on our CCUB image-text pairs while keeping +the text encoder and autoencoder (VAE) frozen. The fine- +tuning of U-Net is equivalent to the training process of orig- +inal Latent Diffusion Models (LDMs): by minimizing the +LDM loss in several denoising time steps. The LDM loss +is given by: +LLDM := Ez∼E(x),y,ϵ∼N (0,1),t +� +∥ϵ − ϵθ (zt, t, c)∥2 +2 +� +where t is the time step; zt, the latent noise at time t; c, the +text encoding of a text prompt; ϵ, the noise sample; and ϵθ, +the noise estimating U-Net model. +Implementation detail: We set the learning rate to be +small (e.g., 1e-5) to slightly change parameters of the U-Net, +and we run 150 epochs on the culturally representative CCUB +image-text data pairs to fine-tune one Stable Diffusion. This +generally leads to convergence of the images produced by our +fine-tuned Stable Diffusion toward our culturally-aware train- +ing data. +4.2 +Prompt Augmenting +To enhance the richness of cultural representation in the text +domain, we propose an approach to automatically augment +the given text prompt with a fine-tuned large language model, +trained further based on our culturally representative text +data. Whereas current prompt augmentation in text-to-image +synthesis is processed in a fixed set of manual templates, we +benefit from the prior knowledge and generative ability of +GPT-3 [Brown et al., 2020] to describe the components of +a specific culture in a more detailed way. +First, we prepare the training dataset for fine-tuning GPT- +3 with pairs of a base prompt without culturally specific +words and its corresponding culturally-aware prompt. +We +use BLIP [Li et al., 2022], a unified vision-language under- +standing and generation model, to automatically caption our +CCUB images. The four most relevant captions are generated +per image, two based on beam search and the others by nu- +cleus sampling, respectively. We then choose the best match- +ing caption that shows the highest CLIP [Radford et al., 2021] +score. The BLIP captioning had some errors, such as cultural +misinterpretation, meaningless and repetitive words, or mis- +counted numbers. Such flaws are manually corrected before +being used as the base prompt in pairs with our CCUB text +captions. +We then fine-tune the largest GPT-3, Davinci, to translate +between the BLIP captions and the CCUB captions. During + +text +Culturally-aware +VAE +encoder +Stable Diffusion +6 +6 +‘a historical clay village in Northern Nigeri +ayounghappyNigerianwoman +U-Net +CCUB Text-image Pairs +Generic Stable Diffusion +Fine-tuned Stable Diffusion +CCUB Text Data +CCUB Dataset +Culturally-aware +Generic GPT-3 +GPT-3 +BLIP Captions +Training Data Pairs +Fine-tuned GPT-3 +(a) training time for our culturally-aware text-to-image synthesis +Culturally-aware +Augmented +Culturally-aware +Input Text prompt +Output Images +GPT-3 +Stable Diffusion +Text Prompt +(b) testing time for our culturally-aware text-to-image synthesisFigure 4: Compare between our (fine-tuned Stable Diffusion with +prompt augmenting) method with Japanese Stable Diffusion. Top +row shows the language input used to generate images. +fine-tuning, the model learns the latent concept of the specific +cultural content and text format that needs to be generated. +The base prompt is concatenated with “\n\n###\n\n” as +a separator token, and “\nEND” is appended to complete the +text as the stop token. +During inference, we sample cul- +turally augmented prompts given new input prompts using +temperature=0.7. Examples of augmented prompts are +underlined in the row of prompt augmenting in Figure 6. +5 +Experiments +We produced surveys to evaluate the effectiveness of our two +proposed techniques for culturally-aware text-to-image syn- +thesis and compare them to a baseline of simply append- +ing the culture to the prompt, e.g., “A family eating dinner +, China.” and using an existing text-to-image model. +In setting up our study, we consider a comparative structure +between images: the baseline image versus another image +from our results. The setup of a single question in our sur- +vey was as follows: Given two images, the participant selects +which image best fits three given comparative properties. The +properties analyzed were: (1) Text and Image Alignment: +Participants are given a text prompt and consider which of +the two images is more similar to the prompt; (2) Cultural +Alignment: Participants decide which of the two images is +a better representation of the country’s culture; and (3) Of- +fensiveness: Participants consider which of the two images +is more offensive to them. +The participants for the study were selected based on +whether they had a personal understanding of the culture for +which the images in the survey were generated. Participants +were recruited among university students, friends, and family +members of the authors. It was ensured that the participants +would not be able to discern the approaches used to generate +the compared images by randomizing the order of questions +and images in the survey. +6 +Results +6.1 +Baselines +Japanese Stable Diffusion: We compare our approach with +Japanese Stable Diffusion [Shing and Sawada, 2022] in Fig- +ure 4. +Both approaches achieve similar results in terms +Figure 5: Comparison between Stable Diffusion and our approach +with prompt augmentation and fine-tuning of Stable Diffusion +method. The top row shows the language input used to generate +images. +Text-Image +Alignment ↑ +Cultural- +Alignment ↑ +Offensiveness ↓ +Fine-Tuned SD +66 +67 +33 +Prompt Aug. +43 +57 +50 +Combined +57 +71 +36 +Table 2: Percentage preference between the two proposed tech- +niques (individual and combined) versus the baseline in our survey +(Section 5). +of image-text alignment and culturally relevant information +such as clothing, architecture, and food. +While both ap- +proaches are qualitatively similar, the Japanese Stable Dif- +fusion model was trained on 100 million Japanese image-text +pairs for training while ours required only 100-200. +Stable Diffusion: As a baseline, we used Stable Diffusion by +simply adding a culture name as a suffix to the text prompt. +The results in Figure 5 show that this approach fails to capture +cultural identity and instead produces many culturally biased +images. We qualitatively report the following three issues: (1) +The baseline method generates images with elements that are +still in the Western form as in Figure 5 (a), where the baseline +method generated dancers performing Western ballet dance +when the desired cultural context was Mexican, while ours is +able to generate images with more representative Mexican el- +ements. (2) The baseline method exhibits cultural misunder- +standing, as shown in Figure 5 (b), where images of Japanese +style clothing was generated when the desired cultural con- +text was Korean. (3) The baseline generates images that in- +corporate cultural stereotypes and discrimination, as in Fig- +ure 5 (c), the image generated by the baseline approach con- +tains cultural biases and discrimination, failing to represent +modern Nigerian culture. Also, in column (d), the baseline- +generated image contains offensive, stereotypical elements of +Chinese culture, especially in the facial features of the charac- +ters. All three of these issues are mitigated in our culturally- +aware text-to-image synthesis approach. +6.2 +Qualitative Results +Qualitative results from our two approaches versus a base- +line are in Figure 6. The augmented prompts are displayed at + +"Two people is +A family dressed in +"A woman is +"A photo of a building, +reading a book, in +eating together, in +traditional clothing, in +in Japan" +Japan' +Japan" +Japan" +Japanese Stable +Diffusion +s.InO"Two people dressed +"Dancers are performing +"A family is eating +"A man and a woman, +in traditional clothing, +for a crowd, in Mexico" +together, in Nigeria' +in China" +in Korea" +Stable Diffusion +(baseline) +心 +s.InO +(b) +(d) +(a)Figure 6: An ablation study of our proposed approach to culturally-aware text-to-image synthesis. The augmented prompts are underlined. + +Prompt: A photo of a modern street" +Culture Name +China +Nigeria +Korea +Mexico +United States +"A photo of a modern +"A photo of a modern +“A photo of a modern +“a photo of a modern +street, a road alongside + street, people walking +Prompt Augmenting +street, a street in the city + street, a modern street in +street in downtown +skyscrapers in Shanghai, +down Insadong street in +Detroit, in the United +of Ibadan, in Nigeria" +Monterrey, in Mexico" +in China" +Seoul, in Korea"" +States" +“Prompt, Culture Name" +Stable Diffusion +(baseline) +Prompt ++ +Finetuned Stable +Diffusion +Prompt Augmenting +Stable Diffusion +Prompt Augmenting ++ +Finetuned Stable +Diffusion +Prompt: "Musicians are practicing together" +Culture Name +China +Nigeria +Korea +Mexico +United States +"Musicians are practicing +"Musicians are practicing +"Musicians are practicing +"Musicians are practicing +together, two Chinese +Musicians are practicing together, there is a male and +together having a jam +together, musicians of +women wearing +together at an open-air +female in the front are playing +session and playing +mariachi band are +Prompt Augmenting + cheongsam are playing +drun +a traditional Korean musical +instruments including a +practicing together at +pipa, a Chinese plucked +village, in Nigeria" +instrument called a + saxophone, trumpet, and +night, in Mexico" +instrument, in China" +gayageum, in Korea" +drums, in the United +States" +"Prompt, Culture Name" +Stable Diffusion +(baseline) +"Prompt, Culture +Name" +Finetuned Stable +Diffusion +Prompt Augmenting +Stable Diffusion +Prompt Augmenting +Finetuned Stable +Diffusionthe top of each example and add culturally specific informa- +tion to the prompts such as the city of Ibadan to a Nigerian +prompt. These augmented prompts can generally help guide +Stable Diffusion to generate something more culturally rele- +vant given a text prompt without specific cultural information. +In general, faces in the fined tuned Stable Diffusion model +look more natural as seen in the musicians examples of Fig- +ure 6. Many of the Stable Diffusion examples appear older +photographs, and the fine-tuning helps give the images a more +contemporary appearance. Fine-tuning also provides Stable +Diffusion with additional image synthesis content capabili- +ties, e.g., the baseline model was unable to produce the Chi- +nese pipa instrument without the fine-tuning as seen in the +lower left of Figure 6. +Prompt augmentation had the ability to make aspects of +the given text prompt more prevalent. In the top example of +Figure 6, the augmented information added to the prompts +all contained clues to guide Stable Diffusion to create a more +modern looking image of a street in the different cultures. +6.3 +Quantitative Results +Following our experimental setup detailed in Section 5, we +present our results from our survey in Table 2. We included +five countries in our survey based on the number of evaluators +we could recruit: China, Mexico, Korea, the United States, +and Nigeria. Overall, 72 people participated in the surveys +with 2,244 image comparisons made. +Fine-Tuned Stable Diffusion The top row of Table 2 shows +our survey results for our fine-tuned Stable Diffusion model +versus the baseline. This approach greatly improved all three +metrics overall; text-image alignment and cultural alignment +were increased and offensiveness greatly decreased. +Prompt Augmentation Prompt augmentation greatly im- +proved cultural alignment over the baseline; however, the im- +ages were found to be equally offensiveness to the baseline. +Combined Our combined approach is better than the baseline +in all three metrics. It finds a middle-ground in performance +versus its individual parts for text-image alignment and of- +fensiveness; however, overall it performs best in cultural- +alignment. +7 +Discussion +In our ablation survey results shown in Table 2, we found +that prompt augmentation decreased text-image alignment. +Prompt augmentation makes alterations to the given prompt +as seen in Figure 6 which we believe contributes to the text- +image alignment performance diminishment. The augmenta- +tion is designed to add culturally relevant information to the +prompt, but this added information does not necessarily over- +lap with the content that Stable Diffusion is capable of gener- +ating. For example, in Figure 6, the middle column of the bot- +tom example adds the Korean instrument “gayageum” to the +text prompt with augmentation. The generated images failed +to accurately depict a gayageum which hurts the text-image +alignment when compared to to the baseline result which did +not include that instrument. +Combining prompt augmenting with our fine-tuned Stable +Diffusion model performed better in cultural alignment than +the two techniques separately. We hypothesize that the fine- +tuned Stable Diffusion learnt concepts that were relevant to +the information that the prompt augmentation added, there- +fore, expanding the content that Stable Diffusion can gener- +ate as in the Chinese Pipa instrument example. Further exper- +iments are needed to thoroughly explain this result. +7.1 +Limitations +Our two proposed approaches for adapting an existing text-to- +image model to be more culturally relevant were successful +versus a baseline and Japanese Stable Diffusion; however, we +believe that there is more progress to be made. In particular, +having guarantees on avoiding offensive generated images is +paramount to having trust in using these systems in the wild. +Our representation of culture is nation based in this paper. +Most nations have multiple cultures, and culture can exist out- +side of geographic borders. +The current version of CCUB dataset was adequate for +our experimental purposes. To further improve the quality +of CCUB, the number of cultural experts would need to be +increased to cross-validate the data. +In future work, we look to adapt our method to allow for +a more personal sense of culture where a user of the system +can upload their own visual and textual data according to their +own culture to tailor the text-to-image system to themselves. +8 +Conclusion +We introduce the task of culturally-aware image synthesis and +present its importance based on literature results demonstrat- +ing the importance of accurate representation in media. As +artificial intelligence improves and is used to produce more +visual content, cultural representation within this generated +media is increasingly important. +To support culturally-aware text-to-image synthesis we +collected the Cross-Cultural Understanding Benchmark +(CCUB) dataset consisting of images and captions for 8 dif- +ferent country’s cultures collected by people who are a part of +each of those cultures. We present two techniques for using +this data to alter an existing text-to-image model to be less +offensive and more culturally relevant. Our two techniques +alone and combined outperform a baseline of specifying the +culture in the text prompt. +Towards equitable representation in text-to-image synthe- +sis, we plan to continue building CCUB to include more cul- +tures in the future. +Ethical Statement +While our approach to culturally-aware text-to-image synthe- +sis shows promise in mitigating offensiveness and accurately +representing culture, it can still make mistakes and show +harmful stereotypes. Additionally, the model still carries over +other forms of bias such as gender and racial biases that the +large text-to-image models are known to have. For these rea- +sons, we would not recommend this work to be used for cases +where people could be harmed by the generated images. + +Acknowledgments +This work is in part supported by NSF IIS-2112633. Youeun +Shin, Youngsik Yun, and Jihie Kim were supported by the +MSIT (Ministry of Science, ICT), Korea, under the High- +Potential Individuals Global Training Program (RS-2022- +00155054) supervised by the IITP (Institute for Informa- +tion & Communications Technology Planning & Evaluation) +(50%). +We would like to thank Ingrid Navarro, Nariaki Kitamura, +Sindre Stoe Hobber, Huy Quyen Ngo, Yu Chen, and many +others for their help with our CCUB dataset. +References +[Birhane et al., 2021] Abeba Birhane, Vinay Uday Prabhu, +and Emmanuel Kahembwe. Multimodal datasets: misog- +yny, pornography, and malignant stereotypes. +arXiv +preprint arXiv:2110.01963, 2021. +[Brown et al., 2020] Tom B. Brown, Benjamin Mann, Nick +Ryder, Melanie Subbiah, Jared Kaplan, Prafulla Dhari- +wal, Arvind Neelakantan, Pranav Shyam, Girish Sas- +try, Amanda Askell, Sandhini Agarwal, Ariel Herbert- +Voss, Gretchen Krueger, T. J. Henighan, Rewon Child, +Aditya Ramesh, Daniel M. 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PhD thesis, University of Pennsylvania, +2010. +[Shing and Sawada, 2022] Makoto Shing and Kei Sawada. +Japanese stable diffusion. 2022. +[Taylor, 2022] Michael Taylor. Prompt engineering: From +words to art, 2022. + +Appendix +Prompt Augmenting +In this section, we describe the flexibility of our prompt engi- +neering approach with several samples of augmentation out- +puts. We demonstrate that our approach can augment a given +prompt in various ways. Some examples of food and dance in +Mexican and Chinese culture, respectively, are listed in Fig- +ure 7. The augmenting process adds semantic description in +detail as well as the cultural context of the corresponding cul- +ture. It is shown that our approach can supplement the in- +formation that is not in our training data, benefiting from the +immense knowledge of GPT-3. +Figure 7: Variety of augmented prompts for Mexican food and Chi- +nese dance. The cultural-specific words are underlined. The prompt +used for fine-tuned Stable Diffusion is a given prompt plus a cultur- +ally augmented prompt appended by its culture name. +Our fine-tuned large language model is also capable of gen- +erating relevant cultural context beyond the defined nine cul- +tural categories. As shown in Figure 8, we try two types of +new cultural elements, nature in the wild and a new year cele- +bration, out of the nine categories. Prompt augmented results +for all six cultures we address in our paper consist of explicit +cultural context, such as well-known cultural landmarks and +species originating from those cultures. +Additional Culturally-aware Text-to-image +Synthesis Results +In Figure 10, more results of our approach to achieving +culturally-aware images given a text prompt appended by a +culture name are shown. We prove the generalizability of our +model conditioned on diverse text prompts and demonstrate +that our approach is able to generate culturally aligned im- +ages within and beyond the nine categories we defined for a +culture. +Cultural Bias in Stable Diffusion 2 +In our approach, we fine-tune Stable Diffusion [Rombach et +al., 2021] version 1.4. While Stable Diffusion 2.0 has been +released recently, intrinsic cultural biases yet commonly ex- +ist. In Figure 9, we show randomly generated images based +on four prompts. We claim that Stable Diffusion 2.0: 1) gen- +erates images containing stereotypes for different cultures: in +the first and second row in Figure 9, the facial expressions for +Figure 8: Examples of augmented outputs beyond our cultural cate- +gories in the order of China, Korea, Mexico, the United States, Nige- +ria, and Japan. The cultural-specific words are underlined. +Chinese and Mexican culture are stereotypical and offensive; +2) generates discriminative items: given the prompt “A photo +of a street, in Nigeria”, Stable Diffusion 2.0 is also prone to +generate old-looking and discriminative images for Nigerian +culture; 3) contains cultural misrepresentations: in the im- +ages with the prompt “Two people in traditional clothing, in +Korea”, the traditional clothing is more in a Japanese style +instead of Korean style. +Figure 9: Random samples generated by generic Stable Diffusion +version 2.0. The first column shows the prompts. + +Givenprompt +Culturallyaugmented prompt +The father is eating a plate of fajitas while the +mother and daughter is eating red and green +A family is +salsaandchips +Food +AMexican family is eatingtacos and +eating together +guacamoletogetherattheirhome +A family is eating together at a dining table. +enjoying afeastofmolepoblano +A troupe ofPeking danceperformers decked out +in colorful costumes is performing +Dancers are +Dancers areperforming for a crowd who are +Dance +performingfor +wearing traditional Tang Dynasty costumes +a crowd +Dancers areperforming for a crowd at the +openingceremony of an ancientChinesedance +performanceGivenprompt +Culturally augmented prompt +CN +somebamboos growing amongChinese houses +KR +Korean fir trees at Jeju island +MX +two Mexican red rump tarantulas in the wild +Animalsandplantsin +the wild +US +animals and plants in the wild in the Rocky Mountains +NG +grazingherd of cows intheNigerian countryside +a Japanese sika deer standing on the snow in Japanese +JP +winter season +the square of Victoria Harbor and many bystanders are +CN +counting the last seconds of the year +KR +Koreanpeopleare celebrating theNewYear at +DongdaemunDesignPlaza +people lighting fireworks for the Mexican New Year +MX +Peopleare celebrating +celebration +the New Year +people are dancing and celebrating the New Year in Times +US +Square +NG +people celebrating the Yoruba new year +JP +people are at Shibuya Crossing in Tokyo. Japan waiting for +thefirstindicationoftheNewYear"A man and a woman, +in Mexico" ++ +Stable Diffusion 2.0 +"A family is eating +together, in China" +Stable Diffusion 2.0 +“A photo of a street, +in Nigeria" +Stable Diffusion 2.0 +"Two people in +traditional clothing: +in Korea" +Stable Diffusion 2.0Figure 10: More results of our approach to achieving culturally-aware text-to-image synthesis for six cultures. The first row is the text +prompts used, the first column is the culture condition. + +“a team of athletes +"A mascot +“"an avocado chair +“a fisherman is +"a group of student +“"a traditional +competing in a +wearing a hat"' +in a house' +fishing" + studying together" +garden" +sports game" +Generic +Stable +Diffusion +American +culture +(Ours) +Korean +culture +(Ours) +Nigerian +culture +(Ours) +LA9GR +Chinese +culture +(Ours) +Mexican +culture +(Ours) +Japanese +culture +(Ours) \ No newline at end of file diff --git a/htFLT4oBgHgl3EQfaS9u/content/tmp_files/load_file.txt b/htFLT4oBgHgl3EQfaS9u/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1c21719a56e1453a28bd80de313fbdea7d351fab --- /dev/null +++ b/htFLT4oBgHgl3EQfaS9u/content/tmp_files/load_file.txt @@ -0,0 +1,629 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf,len=628 +page_content='Towards Equitable Representation in Text-to-Image Synthesis Models with the Cross-Cultural Understanding Benchmark (CCUB) Dataset Zhixuan Liu 1 ∗, Youeun Shin 1, 2 ∗, Beverley-Claire Okogwu 1, Youngsik Yun 1, 2, Lia Coleman 1, Peter Schaldenbrand 1†, Jihie Kim 2 †, Jean Oh 1 † 1 The Robotics Institute, Carnegie Mellon University 2 Department of Artificial Intelligence, Dongguk University {zhixuan2, youeuns, bokogwu, youngsiy, liac, pschalde}@andrew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='edu, jihie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='kim@dgu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='edu, jeanoh@cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='edu Abstract It has been shown that accurate representation in media improves the well-being of the people who consume it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' By contrast, inaccurate representa- tions can negatively affect viewers and lead to harmful perceptions of other cultures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' To achieve inclusive representation in generated images, we propose a culturally-aware priming approach for text-to-image synthesis using a small but cul- turally curated dataset that we collected, known here as Cross-Cultural Understanding Benchmark (CCUB) Dataset, to fight the bias prevalent in gi- ant datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Our proposed approach is comprised of two fine-tuning techniques: (1) Adding visual context via fine-tuning a pre-trained text-to-image synthesis model, Stable Diffusion, on the CCUB text-image pairs, and (2) Adding semantic context via automated prompt engineering using the fine- tuned large language model, GPT-3, trained on our CCUB culturally-aware text data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' CCUB dataset is curated and our approach is evaluated by people who have a personal relationship with that particu- lar culture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Our experiments indicate that priming using both text and image is effective in improving the cultural relevance and decreasing the offensive- ness of generated images while maintaining quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Our CCUB dataset and codes1 are publicly avail- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' 1 Introduction Representation matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' In media, studies repeatedly show that representation affects the well-being of its view- ers [Shaw, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Caswell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Elbaba, 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Rep- resentation can positively affect viewers by providing them with role models that they identify with, but it can also neg- atively affect viewers by creating harmful, stereotypical un- derstandings of people and culture [Casta˜neda, 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' When people are accurately represented in media, it allows peo- ple to properly understand cultures without harmful stereo- ∗indicates equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' †indicates corresponding authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='com/cmubig/CCUB Figure 1: Sample images generated for five different countries by our proposed culturally-aware text-to-image synthesis approach;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' the images in the first row show the results from the generic Stable Dif- fusion as references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' types forming [Dixon and Linz, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Mastro and Greenberg, 2000].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Despite the benefits of representation, many media generating Artificial Intelligence (AI) models show poor rep- resentation in their results [Ntoutsi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Many of these issues stem from their large training datasets which are gathered by crawling the Internet without filtering supervi- sion and contain malign stereotypes and ethnic slurs among arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='12073v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='CV] 28 Jan 2023 "A woman is painting "A family is "A photo of a "Two people dressed in a traditional style" eating together in traditional clothing Generic Stable Diffusion American culture (Ours) Korean culture (Ours) Nigerian culture (Ours) Chinese culture (Ours) Mexican culture (Ours)other problematic content [Birhane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' As AI models are increasingly used to create and aid in the production of visual content, it is important that the models have a true understanding of culture such that it can give accu- rate and proper representation leading to well-being rewards for its consumers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' In this paper, we aim to address such a representation issue in image generation and introduce a new task of culturally-aware image synthesis: generating visual content within a cultural context that is both accurate and in- offensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Our overarching goal is to improve the well-being of consumers of the AI generated images with particular at- tention to those consumers from underrepresented groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Specifically, we formulate the culturally-aware text-to-image synthesis task to take an additional input of a country name to specify a cultural context in addition to language description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' It was found that large datasets such as the LAION- 5B [Schuhmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=', 2021] used to train many text-to- image synthesis models such as Stable Diffusion [Rombach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=', 2021] are Anglo-centric and Euro-centric [Birhane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=', 2021] as shown in the top row of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' As a con- sequence, these powerful models may generate culturally of- fensive images due to misrepresentation during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Our research question is, how can effective existing text-to-image models be improved to become more culturally representative and thus less offensive?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' It may be infeasible to vet billions of training examples for accurate cultural content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' We hypothesize that a small dataset that is veritably repre- sentative of a culture can be used to prime pre-trained text- to-image models to guide the model towards more culturally accurate content creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' To verify the hypothesis, we col- lected a dataset of image and caption pairs for 8 cultures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' For each culture, data was collected by a few people who are native of that culture as they are the people who prop- erly understand it and are most affected by its misrepresenta- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' We call this the Cross-Cultural Understanding Bench- mark (CCUB) dataset which comprises of 100-200 images each with a manually written caption as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' We propose two techniques for enhancing the text-to- image pipelines using CCUB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' First, we fine-tune a text-to- image synthesis model, Stable Diffusion, on the CCUB text- image pairs to generate images tailored for a given cultural context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Second, we create an automatic prompt augment- ing approach using GPT-3 [Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=', 2020] fine-tuned on CCUB to include culturally relevant details, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=', “Two peo- ple walking down a street” can be augmented with “using WeChat Pay to pay a bus ticket, in Shenzhen, China.” We evaluate our approach’s two components individually as well as combined against the baseline of simply specifying the culture in the text prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Our evaluation was performed by native people of each country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Our survey results based on 2,244 image comparisions conducted by 72 participants from 5 countries indicate that our proposed approach is both less offensive and more cultural relevant than simply adding the country name as a suffix to the prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Our contributions are as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' The introduction of culturally-aware text-to-image syn- thesis as a valuable task within text-to-image synthesis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' The Cross-Cultural Understanding Benchmark (CCUB) dataset consisting of 1,095 culturally representative image-text pairs across 8 countries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Two techniques for culturally customizing a text-to- image synthesis model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' 2 Related Work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='1 Cultural Datasets Various efforts have been made to build a dataset that contains a precise representation of each culture around the world, es- pecially for the underrepresented groups and smaller popu- lations, to combat the bias of benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' MaRVL dataset [Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=', 2021] created a set of cultures and lan- guages, including Indonesian, Swahili, Tamil, Turkish, and Mandarin Chinese, comprised of diverse cultural concepts to mitigate existing North American or Western European bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' While sharing the similar intuition, MaRVL was specifically developed for the reasoning task covering common, popular concepts only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Dollar Street [Rojas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=', 2022] aimed to capture accurate demographic information based on socioeconomic features, such as everyday household items and monthly income, of 63 countries worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' However, this dataset gives less diverse scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Most of the images in this dataset provide indoor views with limited cultural features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='2 Culturally Conditioned Machine Learning Accurately representing culture with Machine Learning is an open challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Many models, such as Craiyon [Dayma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=', 2021] fail to capture certain distinguishing features relat- ing to a country’s dominant culture [Reviriego and Merino- G´omez, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' One method to address this involves the in- clusion of semantic understanding in a model such as the ERNIE-ViLG 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' [Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' A similar approach can be seen in Japanese Stable Diffusion [Shing and Sawada, 2022], which fine tunes the Stable Diffusion U-Net and retrains the text encoder on the 100 million images with Japanese caption within the LAION-5B [Schuhmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=', 2021] dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' While these approaches produce better cultural representa- tions of Japan and China, it is not easy to be used universally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Adapting these approaches requires millions of training ex- amples which cannot be easily met for cultures with less in- ternet presence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Also, these datasets are so large that it is infeasible to vet them for harmful and stereotypical informa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Our approach strives for cultural representation using a dataset that is smaller (100-200 images) and hand selectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='3 Modifying Text-to-Image Diffusion Models Diffusion-based text-to-image synthesis models have im- proved incredibly over the past year in image quality and lan- guage understanding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' however, these models are still Anglo- centric and contain gender and racial biases at least in part due to the lack of supervision in their large text-image train- ing datasets [Birhane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' One way to address this problem while leveraging the knowledge obtained from the large dataset is to fine-tune the latent diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' [Ruiz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=', 2022] and [Gal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' 2022] propose textural inversion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='food & drink ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='clothing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='artwork ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='96 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='107 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='46 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='135 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='146 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='169 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='82 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='1095 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='Table 1: This table shows the scale of our CCUB dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' The num- ber of hand-selected images and their corresponding captions in nine cultural categories for eight different cultures are listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' methods that allow the latent diffusion models to generate im- ages with specific visual concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' However, these methods restrict the generation to be an object or a style and cannot generalize on the expression of abstract concepts, for exam- ple, a culture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Inspired by the approaches of [Chambon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=', 2022] and [Pinkney, 2022], the text-to-image diffusion model can generate domain-specific images by fine-tuning the U-Net of the model using a batch of data from that do- main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='4 Prompt Engineering Originating from the field of Natural Language Process- ing (NLP), prompt engineering, which can be conceived as programming in natural language [Reynolds and McDonell, 2021], is to design the input text prompt to retrieve user- desired outcomes from language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Inspired by the findings in the NLP domain, researchers in computer vision have been exploring the effects of prompt engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' In order to present design guidelines for better outcomes in text- to-image generation models [Liu and Chilton, 2022], several permutations of prompt engineering using a template were conducted in terms of subject and style in art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' To discover some tricks and keywords to boost the quality of the out- put image in the image generation models such as DALL-E 2 [Ramesh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=', 2022] and Midjourney [Midjourney, 2022], various experiments through trial-and-error are ongoing on the Internet as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Online community [Taylor, 2022] has come up with a prompt engineering template for artwork, consisting of terms regarding styles, artists, vibes, and per- spectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' 3 Cross-Cultural Understanding Benchmark (CCUB) Dataset Our CCUB dataset provides text-image pairwise data across 8 cultures and is designed for text-to-image synthesis tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' For each culture subset, the proposed dataset contains images among various scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Moreover, the text data concisely describes the culturally aware content of each image thus giv- ing the language model a clear guide towards cultural aware- ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Figure 2: Images and captions in our CCUB dataset for three of the eight countries whose cultures are represented in CCUB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='1 Culturally Representative Data Collection Following the definition of culture in [Halpern, 1955] and [Key and Comrie, 2021], nine categories are used to rep- resent cultural elements in our dataset: food & drink, cloth- ing, artwork, dance and music, religion, architecture, people, city and nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' The categories are further divided into tradi- tional and modern to reflect a characteristic of the culture that culture changes over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Our CCUB image are collected based on the nine cul- tural categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' For collection, we recruited cultural ex- perts who confidently know this culture well or belong to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Cultural experts are asked to collect 10-20 relevant im- ages containing different objects for each cultural category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' The images were collected either from Creative Commons li- censed images from Google searches or the collectors own photographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Cultural experts were also asked to select im- ages with common or culturally representative items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Each image in the CCUB dataset is also captioned by cul- tural experts forming paired image-text data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Cultural experts were asked to focus on the general and specific items in each cultural image, rather than adding captions to subtle com- ponents of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' The captions accurately express cul- tural contents in English as opposed to large datasets such as LAION [Schuhmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=', 2021] which are scraped from the internet and not vetted for cultural accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='2 Properties Our CCUB dataset contains culturally representative image- text pairs in eight different cultures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' As shown in Table 1, our cultural dataset is in a minute scale considering that generic Stable Diffusion was trained on LAION-2B-EN that includes more than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='3 billion text-image pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Our table Figure 2 shows some selected samples of our CCUB dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Korean cultural data "one of the Eight a woman dressed in "Korean barbecue "a Korean man Gates in the Fortress Hanbok is playing a grilling meat,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' and woman in Wall of Seoul,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' South traditional Korean mushrooms and garlic" hanbok" Korea" string instrument" Mexican cultural data "a plate of beef "Puebla Cholula "some Mexican “a woman in costume tacos with beans and face paint in front of Church and people performing and rice on the an altar for the Day of Popocatepetl Volcano Voladores ritual side" the Dead holiday\' in Mexico\' dance" Nigerian cultural data "a group of women "chicken and fried "a group of Nigerian dressed in Hausa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' "Abuja Anglican Yoruba men playing rice in a white Yoruba,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' and Igbo church in Nigeria" bowl in Nigeria" the Bata drum" traditional attires"Figure 3: Figure (a) is the overview of our two fine-tuning techniques based on our CCUB dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Models with cultural biases are marked by sad faces, while culturally-aware models are marked by smiling faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Figure (b) is the workflow for our approach toward culturally-aware text-to-image synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' 4 Culturally-aware Text-to-image Synthesis 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='1 Fine-tuning Stable Diffusion The Stable Diffusion [Rombach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=', 2021] pipeline gener- ates natural images under the condition of text prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' The input text prompt is firstly encoded using the CLIP [Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=', 2021] text encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' A U-Net architecture model cre- ates the output image encoding by denoising from random noise conditioned upon the encoded text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' A Variational Au- toencoder (VAE) converts the image encoding into a high- resolution image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' We alter Stable Diffusion to have a more accurate under- standing of a given culture to address its known bias towards generating Western-focused imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' In our approach, fol- lowing a similar approach to [Ruiz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=', 2022] and [Cham- bon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=', 2022], the U-Net of the Stable Diffusion model is further trained on our CCUB image-text pairs while keeping the text encoder and autoencoder (VAE) frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' The fine- tuning of U-Net is equivalent to the training process of orig- inal Latent Diffusion Models (LDMs): by minimizing the LDM loss in several denoising time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' The LDM loss is given by: LLDM := Ez∼E(x),y,ϵ∼N (0,1),t � ∥ϵ − ϵθ (zt, t, c)∥2 2 � where t is the time step;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' zt, the latent noise at time t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' c, the text encoding of a text prompt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' ϵ, the noise sample;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' and ϵθ, the noise estimating U-Net model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Implementation detail: We set the learning rate to be small (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=', 1e-5) to slightly change parameters of the U-Net, and we run 150 epochs on the culturally representative CCUB image-text data pairs to fine-tune one Stable Diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' This generally leads to convergence of the images produced by our fine-tuned Stable Diffusion toward our culturally-aware train- ing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='2 Prompt Augmenting To enhance the richness of cultural representation in the text domain, we propose an approach to automatically augment the given text prompt with a fine-tuned large language model, trained further based on our culturally representative text data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Whereas current prompt augmentation in text-to-image synthesis is processed in a fixed set of manual templates, we benefit from the prior knowledge and generative ability of GPT-3 [Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=', 2020] to describe the components of a specific culture in a more detailed way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' First, we prepare the training dataset for fine-tuning GPT- 3 with pairs of a base prompt without culturally specific words and its corresponding culturally-aware prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' We use BLIP [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=', 2022], a unified vision-language under- standing and generation model, to automatically caption our CCUB images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' The four most relevant captions are generated per image, two based on beam search and the others by nu- cleus sampling, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' We then choose the best match- ing caption that shows the highest CLIP [Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=', 2021] score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' The BLIP captioning had some errors, such as cultural misinterpretation, meaningless and repetitive words, or mis- counted numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Such flaws are manually corrected before being used as the base prompt in pairs with our CCUB text captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' We then fine-tune the largest GPT-3, Davinci, to translate between the BLIP captions and the CCUB captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' During ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='text ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='Culturally-aware ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='VAE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='encoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='Stable Diffusion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='‘a historical clay village in Northern Nigeri ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='ayounghappyNigerianwoman ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='U-Net ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='CCUB Text-image Pairs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='Generic Stable Diffusion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='Fine-tuned Stable Diffusion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='CCUB Text Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='CCUB Dataset ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='Culturally-aware ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='Generic GPT-3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='GPT-3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='BLIP Captions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='Training Data Pairs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='Fine-tuned GPT-3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='(a) training time for our culturally-aware text-to-image synthesis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='Culturally-aware ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='Augmented ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='Culturally-aware ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='Input Text prompt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='Output Images ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='GPT-3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='Stable Diffusion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='Text Prompt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='(b) testing time for our culturally-aware text-to-image synthesisFigure 4: Compare between our (fine-tuned Stable Diffusion with ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='prompt augmenting) method with Japanese Stable Diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Top row shows the language input used to generate images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' fine-tuning, the model learns the latent concept of the specific cultural content and text format that needs to be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' The base prompt is concatenated with “\\n\\n###\\n\\n” as a separator token, and “\\nEND” is appended to complete the text as the stop token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' During inference, we sample cul- turally augmented prompts given new input prompts using temperature=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Examples of augmented prompts are underlined in the row of prompt augmenting in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' 5 Experiments We produced surveys to evaluate the effectiveness of our two proposed techniques for culturally-aware text-to-image syn- thesis and compare them to a baseline of simply append- ing the culture to the prompt, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=', “A family eating dinner , China.” and using an existing text-to-image model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' In setting up our study, we consider a comparative structure between images: the baseline image versus another image from our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' The setup of a single question in our sur- vey was as follows: Given two images, the participant selects which image best fits three given comparative properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' The properties analyzed were: (1) Text and Image Alignment: Participants are given a text prompt and consider which of the two images is more similar to the prompt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' (2) Cultural Alignment: Participants decide which of the two images is a better representation of the country’s culture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' and (3) Of- fensiveness: Participants consider which of the two images is more offensive to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' The participants for the study were selected based on whether they had a personal understanding of the culture for which the images in the survey were generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Participants were recruited among university students, friends, and family members of the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' It was ensured that the participants would not be able to discern the approaches used to generate the compared images by randomizing the order of questions and images in the survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' 6 Results 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='1 Baselines Japanese Stable Diffusion: We compare our approach with Japanese Stable Diffusion [Shing and Sawada, 2022] in Fig- ure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Both approaches achieve similar results in terms Figure 5: Comparison between Stable Diffusion and our approach with prompt augmentation and fine-tuning of Stable Diffusion method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' The top row shows the language input used to generate images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Text-Image Alignment ↑ Cultural- Alignment ↑ Offensiveness ↓ Fine-Tuned SD 66 67 33 Prompt Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' 43 57 50 Combined 57 71 36 Table 2: Percentage preference between the two proposed tech- niques (individual and combined) versus the baseline in our survey (Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' of image-text alignment and culturally relevant information such as clothing, architecture, and food.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' While both ap- proaches are qualitatively similar, the Japanese Stable Dif- fusion model was trained on 100 million Japanese image-text pairs for training while ours required only 100-200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Stable Diffusion: As a baseline, we used Stable Diffusion by simply adding a culture name as a suffix to the text prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' The results in Figure 5 show that this approach fails to capture cultural identity and instead produces many culturally biased images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' We qualitatively report the following three issues: (1) The baseline method generates images with elements that are still in the Western form as in Figure 5 (a), where the baseline method generated dancers performing Western ballet dance when the desired cultural context was Mexican, while ours is able to generate images with more representative Mexican el- ements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' (2) The baseline method exhibits cultural misunder- standing, as shown in Figure 5 (b), where images of Japanese style clothing was generated when the desired cultural con- text was Korean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' (3) The baseline generates images that in- corporate cultural stereotypes and discrimination, as in Fig- ure 5 (c), the image generated by the baseline approach con- tains cultural biases and discrimination, failing to represent modern Nigerian culture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Also, in column (d), the baseline- generated image contains offensive, stereotypical elements of Chinese culture, especially in the facial features of the charac- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' All three of these issues are mitigated in our culturally- aware text-to-image synthesis approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='2 Qualitative Results Qualitative results from our two approaches versus a base- line are in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' The augmented prompts are displayed at "Two people is A family dressed in "A woman is "A photo of a building, reading a book, in eating together, in traditional clothing, in in Japan" Japan\' Japan" Japan" Japanese Stable Diffusion s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='InO"Two people dressed "Dancers are performing "A family is eating "A man and a woman, in traditional clothing, for a crowd, in Mexico" together, in Nigeria\' in China" in Korea" Stable Diffusion (baseline) 心 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='InO (b) (d) (a)Figure 6: An ablation study of our proposed approach to culturally-aware text-to-image synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' The augmented prompts are underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Prompt: A photo of a modern street" Culture Name China Nigeria Korea Mexico United States "A photo of a modern "A photo of a modern “A photo of a modern “a photo of a modern street,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' a road alongside street,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' people walking Prompt Augmenting street,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' a street in the city street,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' a modern street in street in downtown skyscrapers in Shanghai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' down Insadong street in Detroit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' in the United of Ibadan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' in Nigeria" Monterrey,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' in Mexico" in China" Seoul,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' in Korea"" States" “Prompt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Culture Name" Stable Diffusion (baseline) Prompt + Finetuned Stable Diffusion Prompt Augmenting Stable Diffusion Prompt Augmenting + Finetuned Stable Diffusion Prompt: "Musicians are practicing together" Culture Name China Nigeria Korea Mexico United States "Musicians are practicing "Musicians are practicing "Musicians are practicing "Musicians are practicing together,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' two Chinese Musicians are practicing together,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' there is a male and together having a jam together,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' musicians of women wearing together at an open-air female in the front are playing session and playing mariachi band are Prompt Augmenting cheongsam are playing drun a traditional Korean musical instruments including a practicing together at pipa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' a Chinese plucked village,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' in Nigeria" instrument called a saxophone,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' trumpet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' and night,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' in Mexico" instrument,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' in China" gayageum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' in Korea" drums,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' in the United States" "Prompt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Culture Name" Stable Diffusion (baseline) "Prompt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Culture Name" Finetuned Stable Diffusion Prompt Augmenting Stable Diffusion Prompt Augmenting Finetuned Stable Diffusionthe top of each example and add culturally specific informa- tion to the prompts such as the city of Ibadan to a Nigerian prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' These augmented prompts can generally help guide Stable Diffusion to generate something more culturally rele- vant given a text prompt without specific cultural information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' In general, faces in the fined tuned Stable Diffusion model look more natural as seen in the musicians examples of Fig- ure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Many of the Stable Diffusion examples appear older photographs, and the fine-tuning helps give the images a more contemporary appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Fine-tuning also provides Stable Diffusion with additional image synthesis content capabili- ties, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=', the baseline model was unable to produce the Chi- nese pipa instrument without the fine-tuning as seen in the lower left of Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Prompt augmentation had the ability to make aspects of the given text prompt more prevalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' In the top example of Figure 6, the augmented information added to the prompts all contained clues to guide Stable Diffusion to create a more modern looking image of a street in the different cultures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='3 Quantitative Results Following our experimental setup detailed in Section 5, we present our results from our survey in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' We included five countries in our survey based on the number of evaluators we could recruit: China, Mexico, Korea, the United States, and Nigeria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Overall, 72 people participated in the surveys with 2,244 image comparisons made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Fine-Tuned Stable Diffusion The top row of Table 2 shows our survey results for our fine-tuned Stable Diffusion model versus the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' This approach greatly improved all three metrics overall;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' text-image alignment and cultural alignment were increased and offensiveness greatly decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Prompt Augmentation Prompt augmentation greatly im- proved cultural alignment over the baseline;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' however, the im- ages were found to be equally offensiveness to the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Combined Our combined approach is better than the baseline in all three metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' It finds a middle-ground in performance versus its individual parts for text-image alignment and of- fensiveness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' however, overall it performs best in cultural- alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' 7 Discussion In our ablation survey results shown in Table 2, we found that prompt augmentation decreased text-image alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Prompt augmentation makes alterations to the given prompt as seen in Figure 6 which we believe contributes to the text- image alignment performance diminishment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' The augmenta- tion is designed to add culturally relevant information to the prompt, but this added information does not necessarily over- lap with the content that Stable Diffusion is capable of gener- ating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' For example, in Figure 6, the middle column of the bot- tom example adds the Korean instrument “gayageum” to the text prompt with augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' The generated images failed to accurately depict a gayageum which hurts the text-image alignment when compared to to the baseline result which did not include that instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Combining prompt augmenting with our fine-tuned Stable Diffusion model performed better in cultural alignment than the two techniques separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' We hypothesize that the fine- tuned Stable Diffusion learnt concepts that were relevant to the information that the prompt augmentation added, there- fore, expanding the content that Stable Diffusion can gener- ate as in the Chinese Pipa instrument example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Further exper- iments are needed to thoroughly explain this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='1 Limitations Our two proposed approaches for adapting an existing text-to- image model to be more culturally relevant were successful versus a baseline and Japanese Stable Diffusion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' however, we believe that there is more progress to be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' In particular, having guarantees on avoiding offensive generated images is paramount to having trust in using these systems in the wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Our representation of culture is nation based in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Most nations have multiple cultures, and culture can exist out- side of geographic borders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' The current version of CCUB dataset was adequate for our experimental purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' To further improve the quality of CCUB, the number of cultural experts would need to be increased to cross-validate the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' In future work, we look to adapt our method to allow for a more personal sense of culture where a user of the system can upload their own visual and textual data according to their own culture to tailor the text-to-image system to themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' 8 Conclusion We introduce the task of culturally-aware image synthesis and present its importance based on literature results demonstrat- ing the importance of accurate representation in media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' As artificial intelligence improves and is used to produce more visual content, cultural representation within this generated media is increasingly important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' To support culturally-aware text-to-image synthesis we collected the Cross-Cultural Understanding Benchmark (CCUB) dataset consisting of images and captions for 8 dif- ferent country’s cultures collected by people who are a part of each of those cultures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' We present two techniques for using this data to alter an existing text-to-image model to be less offensive and more culturally relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Our two techniques alone and combined outperform a baseline of specifying the culture in the text prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Towards equitable representation in text-to-image synthe- sis, we plan to continue building CCUB to include more cul- tures in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Ethical Statement While our approach to culturally-aware text-to-image synthe- sis shows promise in mitigating offensiveness and accurately representing culture, it can still make mistakes and show harmful stereotypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Additionally, the model still carries over other forms of bias such as gender and racial biases that the large text-to-image models are known to have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' For these rea- sons, we would not recommend this work to be used for cases where people could be harmed by the generated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Acknowledgments This work is in part supported by NSF IIS-2112633.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Youeun Shin, Youngsik Yun, and Jihie Kim were supported by the MSIT (Ministry of Science, ICT), Korea, under the High- Potential Individuals Global Training Program (RS-2022- 00155054) supervised by the IITP (Institute for Informa- tion & Communications Technology Planning & Evaluation) (50%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' We would like to thank Ingrid Navarro, Nariaki Kitamura, Sindre Stoe Hobber, Huy Quyen Ngo, Yu Chen, and many others for their help with our CCUB dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' References [Birhane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=', 2021] Abeba Birhane, Vinay Uday Prabhu, and Emmanuel Kahembwe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Multimodal datasets: misog- yny, 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Laion-400m: Open dataset of clip-filtered 400 million image-text pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' arXiv preprint arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='02114, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' [Shaw, 2010] Adrienne Shaw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Identity, identification, and media representation in video game play: An audience reception study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' PhD thesis, University of Pennsylvania, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' [Shing and Sawada, 2022] Makoto Shing and Kei Sawada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Japanese stable diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' [Taylor, 2022] Michael Taylor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Prompt engineering: From words to art, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Appendix Prompt Augmenting In this section, we describe the flexibility of our prompt engi- neering approach with several samples of augmentation out- puts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' We demonstrate that our approach can augment a given prompt in various ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Some examples of food and dance in Mexican and Chinese culture, respectively, are listed in Fig- ure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' The augmenting process adds semantic description in detail as well as the cultural context of the corresponding cul- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' It is shown that our approach can supplement the in- formation that is not in our training data, benefiting from the immense knowledge of GPT-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Figure 7: Variety of augmented prompts for Mexican food and Chi- nese dance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' The cultural-specific words are underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' The prompt used for fine-tuned Stable Diffusion is a given prompt plus a cultur- ally augmented prompt appended by its culture name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Our fine-tuned large language model is also capable of gen- erating relevant cultural context beyond the defined nine cul- tural categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' As shown in Figure 8, we try two types of new cultural elements, nature in the wild and a new year cele- bration, out of the nine categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Prompt augmented results for all six cultures we address in our paper consist of explicit cultural context, such as well-known cultural landmarks and species originating from those cultures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Additional Culturally-aware Text-to-image Synthesis Results In Figure 10, more results of our approach to achieving culturally-aware images given a text prompt appended by a culture name are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' We prove the generalizability of our model conditioned on diverse text prompts and demonstrate that our approach is able to generate culturally aligned im- ages within and beyond the nine categories we defined for a culture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Cultural Bias in Stable Diffusion 2 In our approach, we fine-tune Stable Diffusion [Rombach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=', 2021] version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' While Stable Diffusion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='0 has been released recently, intrinsic cultural biases yet commonly ex- ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' In Figure 9, we show randomly generated images based on four prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' We claim that Stable Diffusion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='0: 1) gen- erates images containing stereotypes for different cultures: in the first and second row in Figure 9, the facial expressions for Figure 8: Examples of augmented outputs beyond our cultural cate- gories in the order of China, Korea, Mexico, the United States, Nige- ria, and Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' The cultural-specific words are underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Chinese and Mexican culture are stereotypical and offensive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' 2) generates discriminative items: given the prompt “A photo of a street, in Nigeria”, Stable Diffusion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='0 is also prone to generate old-looking and discriminative images for Nigerian culture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' 3) contains cultural misrepresentations: in the im- ages with the prompt “Two people in traditional clothing, in Korea”, the traditional clothing is more in a Japanese style instead of Korean style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Figure 9: Random samples generated by generic Stable Diffusion version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' The first column shows the prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' Givenprompt Culturallyaugmented prompt The father is eating a plate of fajitas while the mother and daughter is eating red and green A family is salsaandchips Food AMexican family is eatingtacos and eating together guacamoletogetherattheirhome A family is eating together at a dining table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='enjoying afeastofmolepoblano ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='A troupe ofPeking danceperformers decked out ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='0 "Two people in traditional clothing: in Korea" Stable Diffusion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content='0Figure 10: More results of our approach to achieving culturally-aware text-to-image synthesis for six cultures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' The first row is the text prompts used, the first column is the culture condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} +page_content=' “a team of athletes "A mascot “"an avocado chair “a fisherman is "a group of student “"a traditional competing in a wearing a hat"\' in a house\' fishing" studying together" garden" sports game" Generic Stable Diffusion American culture (Ours) Korean culture (Ours) Nigerian culture (Ours) LA9GR Chinese culture (Ours) Mexican culture (Ours) Japanese culture (Ours)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htFLT4oBgHgl3EQfaS9u/content/2301.12073v1.pdf'} diff --git a/i9E2T4oBgHgl3EQfIAa1/vector_store/index.faiss b/i9E2T4oBgHgl3EQfIAa1/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..db1c095143daf4098a5afd13f41dcadd74f74e71 --- /dev/null +++ b/i9E2T4oBgHgl3EQfIAa1/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b4b55cf53ccc9bd699cdf820c7c462dbb2c45391c5b4fec1cb6f11b24fa40dd5 +size 1769517 diff --git a/i9E2T4oBgHgl3EQfIAa1/vector_store/index.pkl b/i9E2T4oBgHgl3EQfIAa1/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..da3f3a66e61b32dc6cee4225545833633e38312a --- /dev/null +++ b/i9E2T4oBgHgl3EQfIAa1/vector_store/index.pkl @@ -0,0 +1,3 @@ +version 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Despite this history, a human- +centered evaluation of these type systems and their usability was all but absent, with empirical evaluations +limited to testing their expressiveness in programs written by experts, i.e. the creators of the type system. +In the past few years, this has begun to change with the adoption of a version of affine types and ownership +in the popular Rust programming language. With the increase in Rust’s popularity, various studies have begun +empirically evaluating the usability of Rust’s Ownership and Lifetime rules, providing a breadth of qualitative +and quantitative information on the usability of such type systems. They found that despite Rust’s general +success in achieving its promise of safety and performance, these rules come with a steep learning curve and +have been repeatedly cited as a barrier to adopting Rust. +In this report, I provide a brief history of linear types and region-based memory management, which +directly inspired Rust’s type system. I then introduce Rust’s Ownership and Lifetime rules, and present the +state-of-the-art in academic research into their usability. I discuss both theoretical arguments and empirical +evidence for why these rules are difficult to learn and apply, and survey existing work on addressing some of +these difficulties. I also draw from broader works in the HCI and CS Education communities to recommend +future work in this area. +1 +INTRODUCTION +Despite a plethora of work on advanced type systems in the Programming Languages research +community, from Dependent types [Xi and Pfenning 1999] to Linear [Wadler 1990] and Ownership +types [Clarke et al. 2013], such type systems have rarely crossed the academic boundaries into +mainstream general-purpose programming languages. +A side-effect of this, perhaps exacerbated by a disinterest in human-centered methods in the +Programming Languages community in the past [Coblenz et al. 2018], is the lack of any notable +user evaluation of such type systems. So while their usability was repeatedly discussed, the focus +was on whether any given type system is “expressive”, i.e. can an expert write complex and useful +code that type checks in that system. This meant that, until very recently, we simply did not know +if such type systems are easy to learn, or how non-experts would learn and use them. +This has begun to change with the Rust programming language [Klabnik and Nichols 2017; +Matsakis and Klock 2014]. Rust implement a notion of “Ownership”, “Borrowing”, and “Lifetimes” as +a type system, which allows it to promise memory and thread safety at compile-time without garbage +collection. And with its emergence as an increasingly popular general-purpose programming +language, there has come a new wave of research into the usability of its Ownership model. +In this report, I aim to use this new research to better understand the usability of advanced +type systems, and to see if and how they may be adopted by mainstream software engineers. The +rest of the report is organized as follows: Sec. 2 provides a brief background in the history of +the type systems most relevant to Rust. Sec. 3 then introduces Rust’s specific implementation of +those type systems, and Sec. 4 surveys the work on evaluating and improving the usability of +Ownership in Rust. Finally, Sec. 5 combines takeaways from those works with theories from the +Human-Computer Interactions and Computer Science Education communities to discuss possible +next-steps in research on the usability of advanced type systems. +Research Exam, April 27 2022, La Jolla, CA, USA +2022. +1 +arXiv:2301.02308v1 [cs.PL] 5 Jan 2023 + +Research Exam, April 27 2022, La Jolla, CA, USA +Kasra Ferdowsi +2 +BACKGROUND +This section provides a brief overview of Linear Types, Region-Based Memory Management (RBMM) +and Ownership types, followed by a higher-level discussion of common themes among them. This +discussion is not meant to be exhaustive, as each system has a long history and would require +a separate survey paper by itself1. Instead, it is meant to provide a background for the theories +that inspired Rust, and to help better connect the usability findings for Rust to type systems more +broadly. +2.1 +Linear Types +Inspired by Girard’s Linear Logic [Jervell 1996], Linear Types were introduced by [Wadler 1990] as +a way to safely “change the world” (i.e. modify state) in functional programming languages. The +core of Linear Types are values that must be “used exactly once”, i.e. they cannot be duplicated or +implicitly discarded. Wadler pointed out that this restriction enables a number of static checks and +features that may be very useful for memory management and program correctness. +More specifically, he noted that Linear Types enable memory management of mutable values +without Garbage Collection. If a value cannot be copied or implicitly discarded and it must be used +exactly once, then we can reclaim its memory after it is used. This handles memory management, +and prevents use-after-free and double-free bugs. By prohibiting aliasing, Linear Types also solve +the problem of reasoning about mutations, both in single-threaded code (where aliased references +are a notable source of bugs), and in multi-threaded code (where aliasing can lead to race conditions). +In what will become a common theme in these type systems, Wadler also notes that strict Linear +Types are a stronger constraint than necessary, and the language he introduces is not strictly Linear. +Instead, it allows multiple “read accesses” (immutable references) to a Linear value that cannot +be used once there is a “write” access (mutable reference). I will discuss Linear Types’ relation to +Rust more in Sec. 3, but Rust also uses a looser notion than Linearity called Affine types. Rather +than being used exactly once, a value with an affine type must be used at most once, i.e. it can be +ignored [Pierce 2004]. +While I will not cover them in depth here, it is worth mentioning that Linear Types have +been implemented and extended in various works, from their implementation in the imperative +programming language Vault [Fahndrich and DeLine 2002] to their recent addition to Haskell +[Bernardy et al. 2017]. They have also been an inspiration for the rest of the type systems in this +section. Linearity is cited in both the seminal works on Region-based Memory Management [Tofte +and Talpin 1997] and Ownership Types [Clarke et al. 1998], and has affected them over time, with +[Walker and Watkins 2001] combining it with Region types, and the notion of Ownership transfer +combining linear and non-linear types [Clarke et al. 2013]. +2.2 +Region-based Memory Management +As the name suggests, Region-based Memory Management (RBMM) was an effort in static memory +management using the type system. RBMM started as an extension of Effect Type Systems [Pierce +2004] in [Tofte and Talpin 1997]. But its implementation for the Cyclone language [Grossman et al. +2002a,b; Jim et al. 2002] is the more direct influence on Rust, and so I will focus on that here. +Cyclone began as a part of the Typed Assembly Language project [Morrisett et al. 1999], but +was developed into a separate project aiming to become “a safe dialect of C” [Grossman et al. +2005]. As such it contains a number of interesting design choices and language features besides +RBMM such as tagged unions, null checks, and existential types. [Grossman et al. 2005] offers a +1Which actually exists in the case of Ownership Types [Clarke et al. 2013] +2 + +The Usability of Advanced Type Systems: Rust as a Case Study +Research Exam, April 27 2022, La Jolla, CA, USA +concise description of all these features, but here I will focus on RBMM in particular as described +in [Grossman et al. 2002a], as it is the most relevant to this report. +The key idea of RBMM is to associate a lexically-scoped part of the program with a named +“region” (a dynamically growable part of memory), and annotate the type of pointers into that +region with the region’s name. Then, the compiler can automatically deallocate the entire region +at the end of the scope, and the type-checker can guarantee that pointers into a region are not +dereferrenced outside of that region’s scope (i.e. after it is deallocated). To do this, the type system +keeps track of the set of regions that are live at each point in the program (called the “capability” at +that point), and prohibits pointer dereferencing unless that pointer’s associated region is in the +capability. +This is the core of RBMM, but to make it sufficiently expressive and guarantee soundness, there’s +a lot more subtlety involved. For example, region types can support subtyping. Since regions can be +nested, all pointers into the outer region are guaranteed to be alive during the inner one (since the +outer region is deallocated after the inner). So if a region 'a contains a smaller region 'b, pointers +annotated with 'a are a subtype of 'b. +Another detail that Cyclone developers considered was the syntactic overhead of region annota- +tions, and the need for region generics (functions with arguments and return types that are generic +over region annotations). Their solution was a combination of intraprocedural region annotation +inference, which removed the need for most explicit annotations in function bodies, and defaults +for partially-annotated function signatures. This removed a large amount of the syntactic overhead, +and made certain functions translate from C to Cyclone directly with no or minimal change. +Outside of Cyclone, RBMM has had a long history. It has been implemented for the Go program- +ming language [Davis et al. 2012], Real-Time Java [Boyapati et al. 2003b], Prolog [Makholm 2000], +GPU programming [Holk et al. 2014], and Big Data systems [Nguyen et al. 2015]. But none of these +works empirically evaluated the usability of their system on programmers, focusing instead on +benchmark performance and limiting their discussion of usability to expressiveness and syntactic +overhead. +2.3 +Ownership Types +Despite having a similar name, Ownership Types are not directly related to Rust’s notion of +Ownership. However, they share many of their goals with Rust, and are a key part of the history of +Ownership as a concept. So a discussion of relevant type systems for Rust would be incomplete +without them. +Ownership Types were developed as a part of Object-Oriented Programming (OOP) to statically +enforce a more strict notion of “encapsulation” [Clarke et al. 2013]. While the details vary greatly +between implementations, the general idea is to encode an “owning” and “owner” relationship +between objects in the type system. This places restrictions on pointer aliasing which enforce +encapsulation [Clarke et al. 1998], and enable additional checks and guarantees, such as preventing +data races and deadlocks [Boyapati et al. 2002] and dangling pointers [Boyapati et al. 2003b]. +Despite the large body of work on Ownership types, and its close relation to Java (a popular +general-purpose programming language), these types were neither widely adopted, nor evaluated +on real users. Instead, each extension, implementation or application of these types was only +evaluated by the designers of the system, who programmed real-world applications with their type +system to argue for its expressiveness [Aldrich et al. 2002; Boyapati et al. 2003a,b; Clarke et al. 2013, +1998]. +Despite having a mostly separate history, Ownership Types are conceptually very close to the +other type systems in this report. For instance, in Ownership types “a program’s heap is divided +into hierarchically nested regions, originally called ownership contexts” [Clarke et al. 2013] which +3 + +Research Exam, April 27 2022, La Jolla, CA, USA +Kasra Ferdowsi +is similar to regions in RBMM. In fact, [Boyapati et al. 2003b] combined Ownership types with +Region-based Memory Management to implement a type system for Real-Time Specification for +Java (RTSJ). This system statically guaranteed the success of runtime checks for dangling pointers, +and a lack of references to the garbage-collected heap (a requirement in RTSJ code). However, +similar to the rest of the works in this section, this system was never evaluated on real users. +2.4 +Shared Themes +So far I introduced each type system individually, but these ideas and systems are closely related, +and their development is not easily separable. So, before moving to Rust, I will first discuss the +broad trends in these works more holistically. +Linear, Region and Ownership Types are related by their attention to memory management and +safety. Each realized that type systems could be used to help programmers reason about complex +programs, prevent various errors in using aliased or freed references, and offer a provably correct +solution to memory management without the need for runtime checks or garbage collection. +They were all also quick to note and try to tackle the trade-off between the “expressiveness” of +their type systems, and their guarantees. In the paper that introduced Linear Types, [Wadler 1990] +did not enforce Linearity, but allowed combining values of Linear and non-Linear types, and (as +discussed above) loosened the definition of uniqueness to allow multiple read-only references to +Linear values. Despite touting RBMM, Cyclone [Grossman et al. 2002b] included a distinguished +garbage-collected heap region. And a few years after its introduction, early proponents of Ownership +Types were already making the case that strict uniqueness is needlessly restrictive [Clarke and +Wrigstad 2003]. +Finally, despite this attention to expressiveness, a tendency to implement their type systems +as versions or extensions of popular programming languages (ML [Tofte and Talpin 1997], C +[Grossman et al. 2002b], Java [Aldrich et al. 2002], Scala [Haller and Odersky 2010], Go [Davis +et al. 2012], Prolog [Makholm 2000], etc.), and even considering the benefit of such restrictions in +program comprehension [Aldrich et al. 2002; Clarke et al. 2013], a user-centered approach was +missing from all the works cited above. No one studied if users other than those who had invented +and implemented the type systems could easily work with the restrictions imposed by them. +3 +THE RUST PROGRAMMING LANGUAGE +Rust, though inspired by [Wadler 1990] for its notion of Ownership, and [Grossman et al. 2002b] +for its approach to lifetimes and memory management, does not directly implement any of the type +systems above. Instead, it combines them with a number of other ideas to try to guarantee memory- +and thread-safety, as well as static memory management without garbage collection. However, it +also aims to be a general-purpose systems programming language2, and so aims for “pragmatic +safety” [Evans et al. 2020] and has features that bypass its static checks for better performance or +more complex aliasing patterns. +In the rest of this section, I will first introduce Ownership as it is used in Rust, then describe +lifetimes and how they combine with Ownership, and finally describe unsafe code which bypasses +some of these checks. I have restricted my descriptions here to what’s necessary to think about +some of the usability issues I will discuss in Sec. 4, and I am ignoring many subtleties of the type +system, as well as any mention of Rust’s syntax, semantics, etc. For a good introduction to Rust in +2The term “Systems Programming Language” has caused some controversy recently [Crichton 2018]. But given its colloquial +use as a language that compiles to assembly, and offers low-level control of resources (as opposed to interpreted languages +like Python, or those that run on higher levels of abstraction such as Java), I will use that term in this report for simplicity’s +sake. +4 + +The Usability of Advanced Type Systems: Rust as a Case Study +Research Exam, April 27 2022, La Jolla, CA, USA +1 +let v = vec![1, 2]; +2 +let v2 = v; +3 +print(&v); +(a) +1 +let v = vec![1, 2]; +2 +let x = &v[0]; +3 +let v2 = v; +4 +let y = *x + 1; +(b) +1 +let mut v = vec![1, 2]; +2 +let x = &v[0]; +3 +Vec::push(&mut v, 3); +4 +let y = *x + 1; +(c) +Fig. 1. Each subfigure demonstrates a violation of the corresponding rule in Sec. 3.1. In Fig. 1a ownership of +the vector is transferred to v2 on line 2, so v cannot be borrowed for the call to print on line 3. In Fig. 1b x is +a reference to v and lives until its last use on line 4. But v only lives until the transfer of its vector to v2 on +line 3. Similarly, in Fig. 1c, the immutable borrow of v in x lives from line 2 to its last use on line 4, so v cannot +be mutably borrowed by the call to push on line 3. +general, I recommend the official Rust book [Klabnik and Nichols 2017], which is both thorough +and very readable. +3.1 +Ownership +Ownership rules in Rust are, on paper, quite simple, and various papers have attempted to +summarize them [Fulton et al. 2021; Qin et al. 2020]. Here I will use [Crichton 2020], since it is the +most simple and concise. But first, I need to introduce some terms. +In Rust, each value (a String, i32, Vec, etc.) is owned by a single variable, which is its owner. +Since working with values directly can be inconvenient, Rust also has references to values which +borrow the value from its owner. Finally, in Rust variables and references are immutable by +default. To mutate a value through one, it needs to be explicitly marked as mutable using the +mut keyword. For instance, in Fig. 1c line 1, the Vec created by the call to vec! is assigned to the +variable v, thus v owns that Vec. On line 2, x is a reference to the first element of v, and thus x +borrows v. Finally, v is marked with the mut keyword, and thus it is mutable, but x is not, and so x +can only be read, not modified or reassigned. +With ownership, references and mutability in mind, [Crichton 2020] summarizes Rust’s Owner- +ship rules like so: +(a) All values have exactly one owner. +(b) A reference to a value cannot outlive the owner. +(c) A value can have one mutable reference or many immutable references. +See Fig. 1 for a code example of what violating each of these rules may look like. Rust’s compiler +rustc contains a pass, known colloquially as the borrow-checker, which fails if it cannot statically +determine that all of these rules are being followed. +Before moving on to lifetimes, it is worth considering how these rules relate to the type systems +in Sec. 2. Rather unintuitively, Rust’s Ownership rules are not directly related to Ownership Types +discussed in Sec. 2.33. Rather, the first rule which is generally referred to as “Ownership” is most +closely tied to Linear Types: A value has a single owner at any point in the program, and while +Ownership can be transferred between variables, it cannot be implicitly duplicated. +This is not to say that Ownership Types are entirely unrelated. Mutability and borrowing are +more explicitly dealt with in Ownership Types than Linear or Region types. Certain instances of +Ownership Types restrict mutation to the owner of a value, and only permit read-access from +other objects. Similarly, the notion of borrowing appears in parts of the Ownership Type literature +with a similar function [Aldrich et al. 2002; Boyland 2001]. But Ownership Types are closely tied +3This confusion is further exacerbated by more recent papers such as [Crichton et al. 2021] which refer to Rust’s Ownership +rules as “Ownership Types”, despite the existing history of Ownership Types as a different system. +5 + +Research Exam, April 27 2022, La Jolla, CA, USA +Kasra Ferdowsi +1 +struct Foo<'small, 'large: 'small> { +2 +a: &'small str, +3 +b: &'large str, +4 +} +5 +impl<'s, 'l: 's> Foo<'s, 'l> { +6 +fn new(x: &'s str, y: &'l str) -> Self { +7 +Self { a: x, b: y } +8 +} +9 +} +10 +fn main() { +11 +let mystr = "abc"; +12 +let substr = &mystr[0..2]; +13 +let foo = Foo::new(&mystr, substr); +14 +} +Fig. 2. An example of explicit lifetime parameters in struct and fn definitions. The lifetime parameters +define the struct and fn as generic over lifetimes 'small and 'large, where 'large is a subtype of 'small. +Rust then uses these parameters to type check the use of Foo::new in the main function by comparing the +inferred lifetimes for the references passed to Foo::new with the explicit lifetime parameters. This code +compiles because substr borrows mystr and so its lifetime must be smaller than mystr, which satisfies the +subtyping requirement between 'small and 'large. +to concepts from Object-Oriented Programming (which is not Rust’s paradigm). And I have only +found a single mention of Ownership Types as an influence on Rust in the literature [Weiss et al. +2021]. +I will leave a detailed comparison to RBMM to the next section, but note how rules (a) and (b) +also allow automatic memory management: When the owner of a value goes out of scope, it is +guaranteed not to have any live references, and so the compiler can insert a call to deallocate +that value (drop the value in Rust parlance). This preserves memory-safety without the need for +garbage collection. +3.2 +Lifetimes +[Fulton et al. 2021] gives a great and concise description of lifetimes: +“A lifetime names a scope, and a lifetime annotation on a reference tells the compiler +the reference is valid only within that scope.” +Lifetime annotation here refers to the fact that Rust references are not the same as C-style “raw” +pointers. A raw pointer’s type only has the type of the value it points to. But the type of a Rust +reference is annotated with a lifetime that refers to the scope where that reference is valid, and lets +the borrow-checker keep track of which value (or other reference) it is borrowing. +Rust automatically infers all lifetimes in function bodies, and so most annotations are not visible +to the user. However, when writing functions that have references in their signatures, or data types +which store references, Rust requires users to explicitly write them as generic functions/data types +over the lifetimes of those references4. You can see examples of this in Fig. 2. +There is much more to lifetimes, how they are calculated, and their implications on expressiveness +and usability. But, as we shall see, Rust lifetimes are notoriously complex and difficult, and a full +description of these aspects is outside the scope of this report. So I will leave lifetimes here, and +end this section with a discussion of their relation to Cyclone. +As the reader may have noticed by now, lifetimes in Rust are very similar to regions and region +annotations in Cyclone, including their syntax, subtyping, and intraprocedural inference. In fact, +4There is an exception to this, which is functions whose signatures follow a particular pattern such as functions that don’t +return a reference or which take exactly one reference and return a reference. In these cases, Rust elides these lifetimes as a +syntactic convenience. Note that this is not the same as lifetime inference. Rust, similar to Cyclone, does not infer lifetimes +in function signatures. +6 + +The Usability of Advanced Type Systems: Rust as a Case Study +Research Exam, April 27 2022, La Jolla, CA, USA +many of Rust’s features were inspired by Cyclone, such automatic bounds checking, and sum types +(Rust enums and Cyclone’s tagged unions). +The main difference between the two is that Cyclone requires manually defining the syntactic +scope of regions, and using the region names (which are first-class values) to manually allocate +and initialize values inside different regions. In Rust, lifetimes are automatically determined by the +compiler, and cannot be explicitly set or used (except for generic lifetime parameters). Also, since +Rust 2018, Rust regions are not determined lexically, but are instead calculated over an intermediate +control-flow graph representation of the program [The Rust Core Team 2018]. So while the theory +behind regions and lifetimes is the same, their implementation and interaction model are different +in interesting and significant ways. +3.3 +Unsafe Rust +One of the common themes among the type systems discussed in Sec. 2 was that each found +strict adherence to its rules needlessly restrictive, and found ways to loosen it for the sake of +expressiveness. Rust is no exception to this, though its solution is rather different. +An important issue with Rust’s Ownership rules is that they are sound, but undecidable. So the +borrow-checker is incomplete, and there is plenty of safe code which follows the Ownership rules, +but the borrow-checker cannot statically verify. To alleviate this, Rust allows users to explicitly +mark functions and blocks of code as unsafe, and in these unsafe blocks, certain safety checks are +disabled5. More specifically, unsafe allows the code to: +• Dereference raw pointers +• Call unsafe functions (including C functions, compiler intrinsics, and the raw allocator) +• Implement unsafe traits +• Mutate statics +• Access fields of unions +[Rust Project Developers 2022]. This is still quite restrictive, but has serious implications. For +instance, dereferencing raw pointers can work around the Ownership rules by casting a reference +with one lifetime into a raw pointer, and dereferencing that raw pointer to borrow it again as a +new reference with a new lifetime. This allows creating multiple mutable references to a value at +the same time, which violates the third rule of Ownership. +The details of this are again notoriously complex and beyond the scope of this report, but unsafe +code is crucial to Rust’s “pragmatic safety”. It allows various performance improvements that +are too low-level for the borrow-checker to reason about, as well as interfacing with external +code, and certain aliasing patterns which the programmer can verify as safe, but do not pass the +borrow-checker. +4 +THE USABILITY OF OWNERSHIP +At the time of writing, there have been five major papers on the usability of Rust’s Ownership type +system [Coblenz et al. 2021; Crichton 2020; Fulton et al. 2021; Zeng and Crichton 2019; Zhu et al. +2022]. And, following the best practices of behavioral research [Mcgrath 1995], they use a variety of +methods to inspect a number of similar and overlapping research questions. So, rather than review +each paper individually, in this section I discuss their collective findings, introducing the papers +and their methodology as they become relevant. I also draw from research on the use of unsafe in +Rust, as well as related research on how programmers learn a new programming language. +5Some, [Zhu et al. 2022] for example, refer to unsafe code as “similar to the C Programming Language”. While this is +technically true because unsafe code can interface with arbitrary C code, unsafe code within Rust is still far more restrictive +than C. +7 + +Research Exam, April 27 2022, La Jolla, CA, USA +Kasra Ferdowsi +The high-level takeaway is that Rust’s Ownership model is indeed difficult to learn, and certain +aspects of its design remain difficult even for more experienced Rust developers. However, its +promise of safety and performance, coupled with good tooling and features for interoperability +with other languages, keep Rust popular and loved by those who succeed in adopting it. +4.1 +Is Ownership difficult to use? +“Learning Rust Ownership is like navigating a maze where the walls are made of +asbestos and frustration, and the maze has no exit, and every time you hit a dead +end you get an aneurysm and die.” +— Student participant from [Coblenz et al. 2021] +Rust is notorious for it’s “steep learning curve”, and this has been noted as a major issue in +adopting it in the industry. But interestingly, studies suggest that even experienced developers +struggle with certain aspects of Ownership in Rust. +4.1.1 +Barriers for Novice Rust Programmers. [Zeng and Crichton 2019] performed content analysis +on top posts from the /r/rust subreddit (an online community specific to Rust), and articles +and corresponding comments from Hacker News (a broader tech forum) to identify barriers to +adopting Rust. In 18 experience reports and language comparisons they inspected, they found +that “the complexity of the borrow-checker was the second most frequently mentioned complaint” +(second only to compiler version issues), where memory access patterns that were common in +other languages were disallowed by the borrow-checker, leading to frustration. +[Fulton et al. 2021] followed this work by interviewing 16 industry professionals who had +attempted to adopt Rust in their production team, and used their findings to design an online survey +which provided them with 178 more participants. They also found that Rust’s steep learning curve +was the most serious barrier to adoption. +Note that this difficulty is more than simply the difficulty of learning a new language, or indeed +learning a systems programming language without garbage collection. [Fulton et al. 2021] found +that the biggest challenge in learning Rust was specifically the borrow-checker, and the necessary +shift in programming paradigm to write code that passes the borrow-checker. And [Shrestha et al. +2020] quoted a C++ developer who said that the borrow-checker “forces a programmer to think +differently”. So it appears that Rust’s more advanced type system is the main source of its difficulty, +not just a lack of garbage collection, or more low-level programming. +Which is not to say that adding garbage collection will not ease Rust’s difficulty. [Coblenz +et al. 2021] ran a controlled study on 428 students in a sophomore-level programming course. The +students were given two weeks of lectures on Rust, and then asked to complete an assignment +which required a good understanding of Rust, its Ownership rules, and types that allow for interior +mutability. They randomly assigned students to two groups, one having to complete the assignment +using the Rust standard library data types, and one using a garbage-collected wrapper type (called +“Bronze”) which enabled a number of additional aliasing patterns to pass the borrow-checker, thus +removing the needs for more complex aliasing patterns and datatypes. +They found a significant difference in the rate of completion and the self-reported time to +completing the assignment. The students who used Bronze on average took only a third as much +time as the control group, and were approximately 2.44 times more likely to complete the assignment. +Interestingly, the time difference between the groups only appeared in the second part of the task, +which involved complex aliasing and mutability. The first part of the assignment, which focused +just on Ownership, didn’t show a significant difference between the groups. +8 + +The Usability of Advanced Type Systems: Rust as a Case Study +Research Exam, April 27 2022, La Jolla, CA, USA +4.1.2 +Barriers for Experienced Rust Programmers. Aside from the initial learning curve, studies +also suggest that aspects of Ownership remain difficult to use, even for experienced developers. +In their study of memory- and thread-safety issues in Rust, [Qin et al. 2020] inspected five Rust +systems and applications, five popular libraries, and two vulnerability databases. They found that a +common reason for blocking bugs in these codebases was a lack of “good understanding in Rust’s +lifetime rules”. This is notable since, unlike the participants in the studies above, the programmers +who worked in these codebases were presumably experienced Rust developers. +This finding is corroborated by results from the Rust community’s 2020 survey [The Rust Survey +Team 2020]. They received 8323 responses, with the largest number of participants self-reporting +their expertise as 7 out of 10. They also found that lifetimes are the most difficult topic to learn, +though unfortunately they do not report if and how this response changes according to the expertise +rating. Similarly, in their study of 100 samples of StackOverflow questions on Rust’s Ownership +rules, [Zhu et al. 2022] found that the most common cause of safety rule violations in these +questions was “complex lifetime computation”, which appeared 74 times6, 44 in intraprocedural +lifetime computation, 16 in explicit lifetime parameters, and 14 in elided ones. +From these works, it seems safe to conclude that Rust’s Ownership rules are indeed difficult +to learn. They pose a serious barrier to learning and adopting Rust, and understanding lifetimes +specifically remains a problem even for more experienced Rust developers. +4.2 +Why is Ownership difficult to use? +“I can teach the three rules [of Ownership] in a single lecture to a room of +undergrads. But the vagaries of the borrow checker still trip me up every time I +use Rust!” +— [Crichton 2020] +If we accept that Rust is indeed more difficult to learn than comparable systems programming +languages, and that this is in large part caused by its Ownership type system specifically, the next +step is to ask what about Rust’s Ownership rules is difficult to learn and apply. +4.2.1 +Change of Paradigm. One answer may be the notion of “interference” as used by [Shrestha +et al. 2020]. In that paper, they qualitatively coded 450 posts on StackOverflow for 18 different +programming languages, and interviewed 16 professional programmers, to understand how experi- +enced developers learn new programming languages, and what they struggle with in the process. +They motivated this work by borrowing the term “interference”7 from psychology and neuroscience. +The term refers to when “previous knowledge disrupts recall of newly learned information”. This +can be as simple as the difference in zero- vs. one-indexing between two languages, but it also +applies to larger differences, where programming in the new language requires a “mindshift”, or a +fundamental change in paradigms. +Learning Rust needs such a mindshift, because its Ownership rules prohibit many common +programming patterns. Consider a doubly-linked list. In most languages its implementation is +close to trivial, but it violates Rust’s rules by definition: It requires at least two mutable references +to a node, one from the previous and one from the next node. Rust has workarounds for this, +most simply datatypes with “interior mutability” that postpone checking for simultaneous mutable +access to runtime, but they are more difficult to learn and work with. So it is unsurprising that +[Shrestha et al. 2020] use Rust’s Ownership type system as an example of mindshifts, quoting a C# +developer who had to “completely rethink the problems they would have normally solved in C#”. +6The paper counts 77 violations, but I’m exlcluding 3 which were merely syntax errors. +7As well as the term “facilitation”, but that is not as relevant here. +9 + +Research Exam, April 27 2022, La Jolla, CA, USA +Kasra Ferdowsi +1 +let mut v = vec![1, 2]; +2 +let one = &mut v[0]; +3 +let two = &mut v[1]; +4 +*two += *one; +5 +(a) +1 +let mut v = vec![1, 2]; +2 +let iter = v.iter_mut(); +3 +let one = iter.next().unwrap(); +4 +let two = iter.next().unwrap(); +5 +*two += *one; +(b) +1 +let mut v = vec![1, 2]; +2 +v.insert(0, v[0]); +3 +v.get_mut(v[0]); +4 +5 +(c) +Fig. 3. The programs in Fig. 3a and Fig. 3b perform the same function, but only Fig. 3b passes the borrow- +checker. In Fig. 3c, the statements on lines 2 and 3 are nearly identical at the type-level, but only line 2 passes +the borrow-checker, presumably due to some implementation detail. +In the qualitative portion of their study, [Coblenz et al. 2021] noted a similar theme in the students’ +survey responses. For the students without the Bronze library, the second part of the assignment +required using types with interior mutability and explicit lifetime parameter declarations. Students +mentioned the difficulty of using these types, and the need for redesigning their code to use them +correctly, leading the authors of the paper to conclude that “most of the benefit of GC comes from +architectural simplification” and that “design was a significant contributor to the difference in +performance between non-Bronze and Bronze participants.” +So at least one main reason for the difficulty of learning Ownership is that it requires a change +of paradigm. A programmer who is new to Rust needs to learn entirely new patterns and ways +of structuring code at the architectural level. And their previous experience can actively interfere +with their learning, as they need to abandon common programming patterns and learn to structure +their code in new and unintuitive ways. +4.2.2 +Error Messages. [Coblenz et al. 2021] also noted that rustc’s error messages contributed to +the confusion and frustration. rustc error message not only describe the error in the code, but for +certain error patterns, suggest edits that may fix the problem. However, these edits are always local +and don’t provide any high-level design feedback which may be helpful in making the mindshift. +At best, they led the students to perform a chain of local edits that resulted in code that compiles +without them understanding why. At worst, as one student found, they could be cyclical “with +things like remove & then after removing try adding &.” This lead the authors to conclude that +Rust’s error messages do not “aid design or comprehension”. +[Zhu et al. 2022] investigated error messages more directly. They employed Cognitive Task +Analysis [Diaper 2004] to learn how experts solve 110 Rust Ownership errors they had identified in +a sample of StackOverflow questions, and compared the steps the experts took to the information +contained in the error message. They found that while for most errors the error message contained +all relevant information, for 32 errors the message failed to explain “the key steps in computing a +lifetime or a borrowing relationship”, with another 10 failing to “explain the relationship between +two lifetime annotations”, and 9 “how a safety rule works on a particular code construct”. +I will leave a broader discussion of rustc error messages to Sec. 5, but the works cited here +indicate that Rust error messages do not provide the necessary help. Programmers’ errors may be +more structural, but the error messages only suggest potentially misleading local edits. And even +for local errors, they do not always contain the necessary information to understand and fix the +error, and assume external knowledge on behalf of the programmer. But this still doesn’t explain +why an experienced Rust developer struggles with Ownership. +4.2.3 +The Curse of Incompleteness. [Crichton 2020] point out that Ownership rules are simple +and easy to learn, but statically checking for them, “like most interesting program properties”, +is undecidable. So Rust’s implementation of these rules in the borrow-checker is necessarily +10 + +The Usability of Advanced Type Systems: Rust as a Case Study +Research Exam, April 27 2022, La Jolla, CA, USA +incomplete, and a lot of the usability issues with Ownership come from this gap between the +programmer’s understanding of the rules, and the borrow-checker’s ability of verify them. +Consider the examples in Fig. 3a and Fig. 3b. Both programs perform a similar function, getting +references to two elements of a vector and incrementing one by the other. But only the second +compiles, since the borrow-checker cannot reason about indices. It conservatively assumes that +both lines 2 and 3 in Fig. 3a are mutably borrowing the entire vector, thus violating Rule (c) in +Sec. 3.1. Fig. 3b, however, uses an iterator, which the borrow-checker can reason about at the type +level8, and can successfully verify does not violate the Ownership rules. Thus Fig. 3b compiles +successfully. Note that both of these programs are “safe”, and a more advanced type system involving +dependent types could in theory statically verify the safety of Fig. 3a, but the current limitations of +the type system means that developers need to learn, not just the rules of Ownership, but how the +borrow-checker verifies them. +This issue is exacerbated by the fact that the borrow-checker’s implementation is quite complex +and sometimes very similar code may not compile for obscure reasons. The code example in Fig. 3c +has two similar calls to functions on the vector. Line 2 gets the first element of v, and inserts +it as the new first element of v. Line 3 uses the value of the first element as an index to get a +mutable reference to the second element of v. These functions have very similar types, both using +an immutable borrow of v to get an argument for a call that mutably borrows v. However, as of +Rust version 1.59.0, line 2 compiles successfully, but line 3 fails the borrow-checker9. +There are almost certainly many other large and small, obvious and subtle reasons for the +difficulty of learning and using Rust’s Ownership type system, but these three (Rust’s different +paradigm, unhelpful error messages, and the incompleteness of the borrow-checker) are the most +apparent from the works surveyed here. +4.3 +Why do developers try to use Rust anyway? +Instead of having to invoke pkg-config by hand or with Autotools macros, wran- +gling include paths for header files and library files and basically depending on +the user to ensure that the correct versions of libraries are installed, you write a +Cargo.toml file which lists the names and versions of your dependencies. [...] It +just works when you cargo build. +— [Mena-Quintero 2018]10 +The last question I will inspect here is that, if Ownership is difficult to learn and use for so many +reasons, why do developers choose to use Rust anyway? +And perhaps the first answer to that is that they don’t. Despite being the “Most Loved” program- +ming language in every StackOverflow survey since 2016 [Stack Overflow 2016, 2017, 2018, 2019, +2020, 2021], it’s user-base is small and growing slowly. In the same surveys, it appeared in the list +of Most Popular languages in 2019 at only 3.2% [Stack Overflow 2019], growing to 7.03% in the +latest survey [Stack Overflow 2021]. In comparison, the Go programming language (which is often +compared with Rust as a modern systems programming languages) was already at 8.2% in 2019, +though it only grew to 9.55% by 2021. Similarly, the TiOBE index ranks Rust at 26 [TIOBE Software +BV 2022], and the IEEE Spectrum ranks it at 17 [Cass et al. 2022], compared to 13 and 8 for Go. +8I should mention that iter_mut uses unsafe to achieve this under the hood, but since iter_mut is itself a safe function +provided by the Rust standard library, it can easily be used by novices without ever touching unsafe code. +9[Crichton 2020] speculates that the reason for this is that get_mut is defined on slices (which the Vec type implements), +while insert is implemented directly on Vec. They don’t know why this distinction matters, and it only further proves +their point. +10Quote found in [Zeng and Crichton 2019]. +11 + +Research Exam, April 27 2022, La Jolla, CA, USA +Kasra Ferdowsi +There could be many reasons for this beyond the usability of Ownership of course, and these are +not peer-reviewed sources. +But Rust’s popularity is still growing, and unsurprisingly the main reason most participants in +multiple studies cited was its promise of memory- and thread-safety [Fulton et al. 2021; Zeng and +Crichton 2019]. Unfortunately, neither of these papers go into depth about this, and only mention +that safety is the most commonly noted reason. However, other themes besides safety emerged in +these works that are far more interesting. +The first is that while safety is important, it is not enough. [Zeng and Crichton 2019] noted that +while the first and third most-noted benefits of adopting Rust were avoiding runtime errors and +data races, the second most-mentioned benefit was Rust’s build tool cargo, which avoided the many +issues of build tools for other langauges. Similarly, while [Fulton et al. 2021]’s participants cited +Rust’s safety as a benefit most frequently, they listed performance and lack of garbage collection +almost as frequently. +Another theme that came up in multiple works was Rust’s unsafe feature. Two studies which +inspected the use of unsafe in Rust code repositories found that unsafe code is common. [Evans +et al. 2020] inspected all publicly available Rust libraries on crate.io (Rust’s online library registry), +and found that explicit unsafe blocks appear in 29% of all libraries. When they filtered their results +to the most popular libraries (which accounted for 90% of downloads), this percentage increased +to 52.5%. In the 5 applications they inspected, [Qin et al. 2020] found 4990 uses of unsafe, with +a further 1581 unsafe code regions in the standard library, and concluded that unsafe code is +used “extensively”. Though they note that it is “unavoidable in many cases” and “usually for good +reasons”, including interfacing with existing libraries written in other unsafe languages such as +C, and performance improvements by a factor of 4 or 5. Interestingly, they also found cases of +the unsafe keyword being used as a warning to developers, despite the code itself being safe and +compiling without the unsafe block. Those who tried to adopt Rust also noted the many uses for +unsafe code, citing its necessity for integrating Rust into existing codebases through FFIs, accessing +hardware, and for performance reasons [Fulton et al. 2021]. So it seems that “pragmatic safety” was +an essential part of Rust’s success, as a large amount of code written in Rust would have not been +possible it if had strictly adhered to its statically-guaranteed safety rules. +It’s also interesting to note that we now have empirical evidence that, despite unsafe being +described as an “escape hatch” [Evans et al. 2020], developers can be trusted to use it responsibly, +only where necessary and while still mostly maintaining Rust’s safety guarantees. As [Evans +et al. 2020] found, most Rust codebases do not contain explicit unsafe code. And in their study of +StackOverflow questions, [Zhu et al. 2022] found that of the 110 errors in the questions, only 3 were +fixed by writing unsafe code. The rest were either fixed with simple safe code, or library functions +which used unsafe blocks internally, but exposed a safe interface (a programming pattern known +as “interior unsafe”). In larger codebases, [Qin et al. 2020] listed numerous cases of memory bugs +related to unsafe code, but they found that this has more to do with the complexity of the type +system (and lifetimes especially), and not developer negligence. And good programming patterns +such as interior unsafe, best-practices such as coding reviews, and better tools for reasoning about +lifetimes and unsafe code could alleviate many causes for these bugs. +So while Rust is not as popular as comparable languages which lack its advanced type system, +developers do like it and try to adopt it, mainly for its promise of safety. Though alongside safety, +they also value its performance, lack of a garbage collector, and great tooling. And as much as they +appreciate Rust’s safety, they still have numerous justified reasons for using unsafe code, and do so +responsibly. +12 + +The Usability of Advanced Type Systems: Rust as a Case Study +Research Exam, April 27 2022, La Jolla, CA, USA +5 +FUTURE WORK +In this section, I will briefly introduce ideas from the fields of HCI and Computer Science Education +that I believe could contribute to better understanding and improving the usability of Rust, or +indeed any other advanced type system. These ideas are eclectic, and I mean to introduce them +as a starting point for generating ideas, not as fully fleshed-out research plans. Also, as the focus +of this report is on human-centered approaches, I will not discuss techniques from Programming +Languages and Compilers research. Though, especially if combined with HCI techniques, they +could be essential to improving the usability of Rust as well. +5.1 +Better Error Messages +Perhaps the most immediately actionable takeaway from Sec. 4 is that Rust’s error messages are +an important limitation. But, paradoxically, the general community consensus is that Rust’s error +messages are more helpful than most languages [Fulton et al. 2021]. So a good next step in improving +Rust’s usability, and an important consideration when designing any language with such complex +types, is to see where Rust’s error messages succeed and how they fail. +A great starting point for this is [Becker et al. 2019]. They survey all works on compiler error +messages in the past 50 years, and provide a number of remarkable insights on the subject. For +one, they discuss empirical evidence showing that programmers, both novice and expert, do read +compiler error messages [Barik et al. 2017; Prather et al. 2017], and so improving error reporting is +indeed worthwhile. +They also compiled a list of empirically-backed guidelines for designing error messages, which +can be invaluable as a shared foundation for synthesizing various works on Rust’s usability. For +example, it could serve as the complementary theoretical background to empirical analyses (such +as in [Zhu et al. 2022]) that argue that Rust’s error messages are notably well-designed11. +It is also a good resource for understanding why Rust error messages fail, and how to improve +them. For instance, one of the important guidelines for error design is to include the relevant +context of the error directly in the message, which multiple works have argued Rust sometimes +fails to do [Blaser 2019; Dominik 2018; Zhu et al. 2022]. +Following [Coblenz et al. 2021]’s finding that Rust diagnostics lack necessary architectural +hints, the guidelines would also be invaluable in providing the human-centered design element +to programming languages or machine learning techniques that could identify such architectural +changes and present them to the programmer. +5.2 +Program Visualization Tools +An often mentioned next step for the usability of Rust’s Ownership is visualization of Rust’s +lifetimes. In their discussion of ways to address the usability issues with the incompleteness of the +borrow-checker, [Crichton 2020] recommended further research into visualizing the various static +information provided by the borrow-checker, citing the difficulty in visualizing the large amount +of information in a “succinct, non-intrusive, yet informative” manner. +Similarly, [Qin et al. 2020] suggested an IDE plugin for visualizing lock lifetimes. They found +that a notably common cause of deadlock bugs in multi-threaded code was that in Rust unlocking +a Mutex occurs implicitly when the locked value is dropped. To address this, they recommended +“plug-ins to highlight the location of Rust’s implicit unlock”, which is a specialized case of visualizing +lifetimes. +11A deeper look into this is outside the scope of this report, but my personal experience suggests that Rust follows most of +the guidelines from [Becker et al. 2019] most of the time. +13 + +Research Exam, April 27 2022, La Jolla, CA, USA +Kasra Ferdowsi +Fig. 4. A RustViz visualization. If the user hovers over the nodes and arrows on the right, it displays a pop-over +containing additional information about the event. +There are three existing but limited works attempting to visualize Ownership and lifetimes in +Rust. The first is RustViz [Gongming et al. 2020], a tool for building visualizations of Ownership and +borrowing events in Rust programs. RustViz does not generate these visualizations automatically, +but instead requires (arguably considerable) effort on the part of a “teacher” to specify them by +annotating the code being visualized, and using a Rust library to generate it. The benefit is that the +generated visualization is interactive: the “learner” can hover over parts of the image to get a brief +description of the events. See Fig. 4 for an example of RustViz’s visualizations. +The other two are related Bachelor’s Theses [Blaser 2019; Dominik 2018]. In the first [Dominik +2018], the author uses Polonius (an experimental implementation of the borrow-checking rules in +Datalog) to extract lifetime constraints from a given program, and visualize it as a directed graph. +They motivated it by arguing that rustc error messages do not always present all the necessary +information for Ownership errors, and that this visualization could address that problem. +[Blaser 2019] then extended [Dominik 2018] to improve its usability. They defined an algorithm +which filters the complete graph to a single path that contains all constraints relevant to the +particular error message, such as in Fig. 5. They also created a Visual Studio Code extension called +“Rust Life Assistant” for displaying these graphs in the IDE, and an algorithm for converting the +graph into a bullet-point list of English text explaining the cause of the error. +All these works are quite limited, and crucially none of them have been evaluated on users12. So +a great next step could be to design, implement and evaluate a tool for visualizing Ownership and +lifetimes. Such a tool could be pedagogical (similar to RustViz) or utilitarian (similar to Rust Life +Assistant), but as [Zhu et al. 2022] points out “learning Rust is a continuous process”, and I would +recommend considering existing works on education and program visualization systems to inform +the design and evaluation of any such tool. +A great survey of such systems can be found in [Sorva et al. 2013]. Besides exploring over 40 +program visualization tools, they also provide a taxonomy of visualization systems, and emphasize +that how users engage with a visualization tool is as important as the design of the tool itself. +Specifically, they cite [Hundhausen et al. 2002] who performed a meta-study of 24 algorithm +visualization (AV) tools and found that “the form of the learning exercise in which AV technology is +used is actually more important than the quality of the visualizations produced by AV technologies.” +Unfortunately most tools focus on the runtime behavior and values of programs, but they can still +help us better think about the design of a visualization system, and users’ engagement with it. +12[Gongming et al. 2020] contains a study proposal, but as of writing this report, no results of such a study have been +published. +14 + +- +fn main() +S +r1*r1 +r2|*r2 +r3|*r3 +2 +let mut s = String::from("hello"); f +3 +4 +let r1 = &s; +5 +let r2 = &s; +6 +assert!(compare_strings(r1, r2)); +7 +8 +let r3 = &mut s; +9 +clear_string(r3): +10.7The Usability of Advanced Type Systems: Rust as a Case Study +Research Exam, April 27 2022, La Jolla, CA, USA +1 +fn main() { +2 +let mut x = 4; +3 +let y = foo(&x); +4 +let z = bar(&y); +5 +let w = foobar(&z); +6 +// ... +7 +x = 5; +8 +take(w); +9 +} +10 +11 +fn foo(p: T) -> T { p } +12 +fn bar(p: T) -> T { p } +13 +fn foobar(p: T) -> T { p } +14 +fn take(p: T) { unimplemented!() } +Fig. 5. A Rust Life Assistant visualization, showing the lifetimes of the references in the code, and the +corresponding constraints. +5.3 +Grounded Theory +The works in Sec. 4 use a wide variety of research methods, including surveying existing experience +reports, semi-structured interviews, online surveys, and controlled experiments. But so far they +have left a notable gap in the methodologies, which is a deep qualitative understanding of how +Rust programmers actually write code. Questions such as what tools they use, if and how they read +error messages, how they reason about various errors, common debugging strategies, etc. are all +left out of the scope of the existing research. To fill this gap, I recommend building a Grounded +Theory of how Rust programmers write code in relation to programmers in other languages, or +using different type or memory management systems. +Grounded Theory (GT) started in the 1960s as a research method in sociology, but has since +become a standard method of qualitative research in many field as a method for developing theories +bottom-up through observation [Charmaz and Bryant 2010]. Speaking very broadly, GT as a method +is an iterative process of collecting data through interviews and observations and using open-coding +to develop a theory that is grounded in the empirical data (rather than informed by or confirming +existing theories). But the above description is too simplistic. Various versions of GT exist, each of +which make different assumptions about the nature of knowledge (positivism vs. constructivism) +and the precise steps they follow are subtly different and outside the scope of this report. +That said, GT is an established and popular method in Software Engineering research [Stol +et al. 2016], and recently [Lubin and Chasins 2021] employed Constructivist GT [Charmaz 2006] +to develop a deeper understanding of how statically-typed functional programmers write code. +Which is why I recommend building a grounded theory of “How Rust Programmers Write Code”. +Speculating on what we may learn from this is counter to GT’s philosophy. But my hope is that +such a GT will help develop the foundations that can both better motivate and help us understand +the human-centered study and design choices involved in Rust and similar advanced type systems. +ACKNOWLEDGMENTS +I would like to thank my advisors Sorin Lerner and Nadia Polikarpova for providing feedback on +this report, as well as their advice and mentorship. I would also like to thank Hila Peleg for being +my de-facto advisor in my first two years in Graduate school, and Ruanqianqian (Lisa) Huang and +Shraddha Barke for being amazing collaborators and co-authors whose research has been a major +source of motivation and inspiration in my own work. +15 + +Lifetime R2 +: &'R2 +3: let y = foo(&x); +LifetimeR26 +Z:&'R26 +Constraint +4: let z = bar(&y); +R23 may point to R2 +generated at line 3: +let y = foo(&x); +Constraint +R29 may point to R26 +generated at line 5: +LifetimeR23 +let w = foobar(&z); +y:&'R23 +3: let y = foo(&x); +LifetimeR29 +W:&'R29 +Constraint +5: let w = foobar(&z); +R26maypointtoR23 +generated at line 4: +let z = bar(&y);Research Exam, April 27 2022, La Jolla, CA, USA +Kasra Ferdowsi +REFERENCES +Jonathan Aldrich, Valentin Kostadinov, and Craig Chambers. 2002. Alias Annotations for Program Understanding. 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(2022), 13. https://doi.org/10.1145/3510003.3510164 +18 + diff --git a/kNE0T4oBgHgl3EQfYgB_/content/tmp_files/load_file.txt b/kNE0T4oBgHgl3EQfYgB_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..93bd612b33e21b222ff040a880607979ed1fa177 --- /dev/null +++ b/kNE0T4oBgHgl3EQfYgB_/content/tmp_files/load_file.txt @@ -0,0 +1,1080 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf,len=1079 +page_content='The Usability of Advanced Type Systems: Rust as a Case Study KASRA FERDOWSI, UC San Diego, USA Advanced type systems that enforce various correctness and safety guarantees—such as linear and ownership types—have a long history in the Programming Languages research community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Despite this history, a human- centered evaluation of these type systems and their usability was all but absent, with empirical evaluations limited to testing their expressiveness in programs written by experts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' the creators of the type system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' In the past few years, this has begun to change with the adoption of a version of affine types and ownership in the popular Rust programming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' With the increase in Rust’s popularity, various studies have begun empirically evaluating the usability of Rust’s Ownership and Lifetime rules, providing a breadth of qualitative and quantitative information on the usability of such type systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' They found that despite Rust’s general success in achieving its promise of safety and performance, these rules come with a steep learning curve and have been repeatedly cited as a barrier to adopting Rust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' In this report, I provide a brief history of linear types and region-based memory management, which directly inspired Rust’s type system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' I then introduce Rust’s Ownership and Lifetime rules, and present the state-of-the-art in academic research into their usability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' I discuss both theoretical arguments and empirical evidence for why these rules are difficult to learn and apply, and survey existing work on addressing some of these difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' I also draw from broader works in the HCI and CS Education communities to recommend future work in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 1 INTRODUCTION Despite a plethora of work on advanced type systems in the Programming Languages research community, from Dependent types [Xi and Pfenning 1999] to Linear [Wadler 1990] and Ownership types [Clarke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2013], such type systems have rarely crossed the academic boundaries into mainstream general-purpose programming languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' A side-effect of this, perhaps exacerbated by a disinterest in human-centered methods in the Programming Languages community in the past [Coblenz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2018], is the lack of any notable user evaluation of such type systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' So while their usability was repeatedly discussed, the focus was on whether any given type system is “expressive”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' can an expert write complex and useful code that type checks in that system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' This meant that, until very recently, we simply did not know if such type systems are easy to learn, or how non-experts would learn and use them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' This has begun to change with the Rust programming language [Klabnik and Nichols 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Matsakis and Klock 2014].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Rust implement a notion of “Ownership”, “Borrowing”, and “Lifetimes” as a type system, which allows it to promise memory and thread safety at compile-time without garbage collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' And with its emergence as an increasingly popular general-purpose programming language, there has come a new wave of research into the usability of its Ownership model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' In this report, I aim to use this new research to better understand the usability of advanced type systems, and to see if and how they may be adopted by mainstream software engineers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' The rest of the report is organized as follows: Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2 provides a brief background in the history of the type systems most relevant to Rust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 3 then introduces Rust’s specific implementation of those type systems, and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 4 surveys the work on evaluating and improving the usability of Ownership in Rust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Finally, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 5 combines takeaways from those works with theories from the Human-Computer Interactions and Computer Science Education communities to discuss possible next-steps in research on the usability of advanced type systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Research Exam, April 27 2022, La Jolla, CA, USA 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='02308v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='PL] 5 Jan 2023 Research Exam, April 27 2022, La Jolla, CA, USA Kasra Ferdowsi 2 BACKGROUND This section provides a brief overview of Linear Types, Region-Based Memory Management (RBMM) and Ownership types, followed by a higher-level discussion of common themes among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' This discussion is not meant to be exhaustive, as each system has a long history and would require a separate survey paper by itself1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Instead, it is meant to provide a background for the theories that inspired Rust, and to help better connect the usability findings for Rust to type systems more broadly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='1 Linear Types Inspired by Girard’s Linear Logic [Jervell 1996], Linear Types were introduced by [Wadler 1990] as a way to safely “change the world” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' modify state) in functional programming languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' The core of Linear Types are values that must be “used exactly once”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' they cannot be duplicated or implicitly discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Wadler pointed out that this restriction enables a number of static checks and features that may be very useful for memory management and program correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' More specifically, he noted that Linear Types enable memory management of mutable values without Garbage Collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' If a value cannot be copied or implicitly discarded and it must be used exactly once, then we can reclaim its memory after it is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' This handles memory management, and prevents use-after-free and double-free bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' By prohibiting aliasing, Linear Types also solve the problem of reasoning about mutations, both in single-threaded code (where aliased references are a notable source of bugs), and in multi-threaded code (where aliasing can lead to race conditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' In what will become a common theme in these type systems, Wadler also notes that strict Linear Types are a stronger constraint than necessary, and the language he introduces is not strictly Linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Instead, it allows multiple “read accesses” (immutable references) to a Linear value that cannot be used once there is a “write” access (mutable reference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' I will discuss Linear Types’ relation to Rust more in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 3, but Rust also uses a looser notion than Linearity called Affine types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Rather than being used exactly once, a value with an affine type must be used at most once, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' it can be ignored [Pierce 2004].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' While I will not cover them in depth here, it is worth mentioning that Linear Types have been implemented and extended in various works, from their implementation in the imperative programming language Vault [Fahndrich and DeLine 2002] to their recent addition to Haskell [Bernardy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' They have also been an inspiration for the rest of the type systems in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Linearity is cited in both the seminal works on Region-based Memory Management [Tofte and Talpin 1997] and Ownership Types [Clarke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 1998], and has affected them over time, with [Walker and Watkins 2001] combining it with Region types, and the notion of Ownership transfer combining linear and non-linear types [Clarke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2013].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='2 Region-based Memory Management As the name suggests, Region-based Memory Management (RBMM) was an effort in static memory management using the type system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' RBMM started as an extension of Effect Type Systems [Pierce 2004] in [Tofte and Talpin 1997].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' But its implementation for the Cyclone language [Grossman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2002a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Jim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2002] is the more direct influence on Rust, and so I will focus on that here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Cyclone began as a part of the Typed Assembly Language project [Morrisett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 1999], but was developed into a separate project aiming to become “a safe dialect of C” [Grossman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2005].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' As such it contains a number of interesting design choices and language features besides RBMM such as tagged unions, null checks, and existential types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' [Grossman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2005] offers a 1Which actually exists in the case of Ownership Types [Clarke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2013] 2 The Usability of Advanced Type Systems: Rust as a Case Study Research Exam, April 27 2022, La Jolla, CA, USA concise description of all these features, but here I will focus on RBMM in particular as described in [Grossman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2002a], as it is the most relevant to this report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' The key idea of RBMM is to associate a lexically-scoped part of the program with a named “region” (a dynamically growable part of memory), and annotate the type of pointers into that region with the region’s name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Then, the compiler can automatically deallocate the entire region at the end of the scope, and the type-checker can guarantee that pointers into a region are not dereferrenced outside of that region’s scope (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' after it is deallocated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' To do this, the type system keeps track of the set of regions that are live at each point in the program (called the “capability” at that point), and prohibits pointer dereferencing unless that pointer’s associated region is in the capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' This is the core of RBMM, but to make it sufficiently expressive and guarantee soundness, there’s a lot more subtlety involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' For example, region types can support subtyping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Since regions can be nested, all pointers into the outer region are guaranteed to be alive during the inner one (since the outer region is deallocated after the inner).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=" So if a region 'a contains a smaller region 'b, pointers annotated with 'a are a subtype of 'b." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Another detail that Cyclone developers considered was the syntactic overhead of region annota- tions, and the need for region generics (functions with arguments and return types that are generic over region annotations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Their solution was a combination of intraprocedural region annotation inference, which removed the need for most explicit annotations in function bodies, and defaults for partially-annotated function signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' This removed a large amount of the syntactic overhead, and made certain functions translate from C to Cyclone directly with no or minimal change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Outside of Cyclone, RBMM has had a long history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' It has been implemented for the Go program- ming language [Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2012], Real-Time Java [Boyapati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2003b], Prolog [Makholm 2000], GPU programming [Holk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2014], and Big Data systems [Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' But none of these works empirically evaluated the usability of their system on programmers, focusing instead on benchmark performance and limiting their discussion of usability to expressiveness and syntactic overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='3 Ownership Types Despite having a similar name, Ownership Types are not directly related to Rust’s notion of Ownership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' However, they share many of their goals with Rust, and are a key part of the history of Ownership as a concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' So a discussion of relevant type systems for Rust would be incomplete without them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Ownership Types were developed as a part of Object-Oriented Programming (OOP) to statically enforce a more strict notion of “encapsulation” [Clarke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2013].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' While the details vary greatly between implementations, the general idea is to encode an “owning” and “owner” relationship between objects in the type system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' This places restrictions on pointer aliasing which enforce encapsulation [Clarke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 1998], and enable additional checks and guarantees, such as preventing data races and deadlocks [Boyapati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2002] and dangling pointers [Boyapati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2003b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Despite the large body of work on Ownership types, and its close relation to Java (a popular general-purpose programming language), these types were neither widely adopted, nor evaluated on real users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Instead, each extension, implementation or application of these types was only evaluated by the designers of the system, who programmed real-world applications with their type system to argue for its expressiveness [Aldrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Boyapati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2003a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Clarke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2013, 1998].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Despite having a mostly separate history, Ownership Types are conceptually very close to the other type systems in this report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' For instance, in Ownership types “a program’s heap is divided into hierarchically nested regions, originally called ownership contexts” [Clarke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2013] which 3 Research Exam, April 27 2022, La Jolla, CA, USA Kasra Ferdowsi is similar to regions in RBMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' In fact, [Boyapati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2003b] combined Ownership types with Region-based Memory Management to implement a type system for Real-Time Specification for Java (RTSJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' This system statically guaranteed the success of runtime checks for dangling pointers, and a lack of references to the garbage-collected heap (a requirement in RTSJ code).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' However, similar to the rest of the works in this section, this system was never evaluated on real users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='4 Shared Themes So far I introduced each type system individually, but these ideas and systems are closely related, and their development is not easily separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' So, before moving to Rust, I will first discuss the broad trends in these works more holistically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Linear, Region and Ownership Types are related by their attention to memory management and safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Each realized that type systems could be used to help programmers reason about complex programs, prevent various errors in using aliased or freed references, and offer a provably correct solution to memory management without the need for runtime checks or garbage collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' They were all also quick to note and try to tackle the trade-off between the “expressiveness” of their type systems, and their guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' In the paper that introduced Linear Types, [Wadler 1990] did not enforce Linearity, but allowed combining values of Linear and non-Linear types, and (as discussed above) loosened the definition of uniqueness to allow multiple read-only references to Linear values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Despite touting RBMM, Cyclone [Grossman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2002b] included a distinguished garbage-collected heap region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' And a few years after its introduction, early proponents of Ownership Types were already making the case that strict uniqueness is needlessly restrictive [Clarke and Wrigstad 2003].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Finally, despite this attention to expressiveness, a tendency to implement their type systems as versions or extensions of popular programming languages (ML [Tofte and Talpin 1997], C [Grossman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2002b], Java [Aldrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2002], Scala [Haller and Odersky 2010], Go [Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2012], Prolog [Makholm 2000], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' ), and even considering the benefit of such restrictions in program comprehension [Aldrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Clarke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2013], a user-centered approach was missing from all the works cited above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' No one studied if users other than those who had invented and implemented the type systems could easily work with the restrictions imposed by them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 3 THE RUST PROGRAMMING LANGUAGE Rust, though inspired by [Wadler 1990] for its notion of Ownership, and [Grossman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2002b] for its approach to lifetimes and memory management, does not directly implement any of the type systems above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Instead, it combines them with a number of other ideas to try to guarantee memory- and thread-safety, as well as static memory management without garbage collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' However, it also aims to be a general-purpose systems programming language2, and so aims for “pragmatic safety” [Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2020] and has features that bypass its static checks for better performance or more complex aliasing patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' In the rest of this section, I will first introduce Ownership as it is used in Rust, then describe lifetimes and how they combine with Ownership, and finally describe unsafe code which bypasses some of these checks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' I have restricted my descriptions here to what’s necessary to think about some of the usability issues I will discuss in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 4, and I am ignoring many subtleties of the type system, as well as any mention of Rust’s syntax, semantics, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' For a good introduction to Rust in 2The term “Systems Programming Language” has caused some controversy recently [Crichton 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' But given its colloquial use as a language that compiles to assembly, and offers low-level control of resources (as opposed to interpreted languages like Python, or those that run on higher levels of abstraction such as Java), I will use that term in this report for simplicity’s sake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 4 The Usability of Advanced Type Systems: Rust as a Case Study Research Exam, April 27 2022, La Jolla, CA, USA 1 let v = vec!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' [1, 2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2 let v2 = v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 3 print(&v);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' (a) 1 let v = vec!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' [1, 2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2 let x = &v[0];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 3 let v2 = v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 4 let y = *x + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' (b) 1 let mut v = vec!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' [1, 2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2 let x = &v[0];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 3 Vec::push(&mut v, 3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 4 let y = *x + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Each subfigure demonstrates a violation of the corresponding rule in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 1a ownership of the vector is transferred to v2 on line 2, so v cannot be borrowed for the call to print on line 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 1b x is a reference to v and lives until its last use on line 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' But v only lives until the transfer of its vector to v2 on line 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Similarly, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 1c, the immutable borrow of v in x lives from line 2 to its last use on line 4, so v cannot be mutably borrowed by the call to push on line 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' general, I recommend the official Rust book [Klabnik and Nichols 2017], which is both thorough and very readable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='1 Ownership Ownership rules in Rust are, on paper, quite simple, and various papers have attempted to summarize them [Fulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Here I will use [Crichton 2020], since it is the most simple and concise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' But first, I need to introduce some terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' In Rust, each value (a String, i32, Vec, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=') is owned by a single variable, which is its owner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Since working with values directly can be inconvenient, Rust also has references to values which borrow the value from its owner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Finally, in Rust variables and references are immutable by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' To mutate a value through one, it needs to be explicitly marked as mutable using the mut keyword.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' For instance, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 1c line 1, the Vec created by the call to vec!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' is assigned to the variable v, thus v owns that Vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' On line 2, x is a reference to the first element of v, and thus x borrows v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Finally, v is marked with the mut keyword, and thus it is mutable, but x is not, and so x can only be read, not modified or reassigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' With ownership, references and mutability in mind, [Crichton 2020] summarizes Rust’s Owner- ship rules like so: (a) All values have exactly one owner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' (b) A reference to a value cannot outlive the owner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' (c) A value can have one mutable reference or many immutable references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 1 for a code example of what violating each of these rules may look like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Rust’s compiler rustc contains a pass, known colloquially as the borrow-checker, which fails if it cannot statically determine that all of these rules are being followed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Before moving on to lifetimes, it is worth considering how these rules relate to the type systems in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Rather unintuitively, Rust’s Ownership rules are not directly related to Ownership Types discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Rather, the first rule which is generally referred to as “Ownership” is most closely tied to Linear Types: A value has a single owner at any point in the program, and while Ownership can be transferred between variables, it cannot be implicitly duplicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' This is not to say that Ownership Types are entirely unrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Mutability and borrowing are more explicitly dealt with in Ownership Types than Linear or Region types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Certain instances of Ownership Types restrict mutation to the owner of a value, and only permit read-access from other objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Similarly, the notion of borrowing appears in parts of the Ownership Type literature with a similar function [Aldrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Boyland 2001].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' But Ownership Types are closely tied 3This confusion is further exacerbated by more recent papers such as [Crichton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2021] which refer to Rust’s Ownership rules as “Ownership Types”, despite the existing history of Ownership Types as a different system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 5 Research Exam, April 27 2022, La Jolla, CA, USA Kasra Ferdowsi 1 struct Foo<\'small, \'large: \'small> { 2 a: &\'small str, 3 b: &\'large str, 4 } 5 impl<\'s, \'l: \'s> Foo<\'s, \'l> { 6 fn new(x: &\'s str, y: &\'l str) -> Self { 7 Self { a: x, b: y } 8 } 9 } 10 fn main() { 11 let mystr = "abc";' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 12 let substr = &mystr[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='.2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 13 let foo = Foo::new(&mystr, substr);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 14 } Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' An example of explicit lifetime parameters in struct and fn definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=" The lifetime parameters define the struct and fn as generic over lifetimes 'small and 'large, where 'large is a subtype of 'small." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Rust then uses these parameters to type check the use of Foo::new in the main function by comparing the inferred lifetimes for the references passed to Foo::new with the explicit lifetime parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=" This code compiles because substr borrows mystr and so its lifetime must be smaller than mystr, which satisfies the subtyping requirement between 'small and 'large." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' to concepts from Object-Oriented Programming (which is not Rust’s paradigm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' And I have only found a single mention of Ownership Types as an influence on Rust in the literature [Weiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' I will leave a detailed comparison to RBMM to the next section, but note how rules (a) and (b) also allow automatic memory management: When the owner of a value goes out of scope, it is guaranteed not to have any live references, and so the compiler can insert a call to deallocate that value (drop the value in Rust parlance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' This preserves memory-safety without the need for garbage collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='2 Lifetimes [Fulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2021] gives a great and concise description of lifetimes: “A lifetime names a scope, and a lifetime annotation on a reference tells the compiler the reference is valid only within that scope.” Lifetime annotation here refers to the fact that Rust references are not the same as C-style “raw” pointers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' A raw pointer’s type only has the type of the value it points to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' But the type of a Rust reference is annotated with a lifetime that refers to the scope where that reference is valid, and lets the borrow-checker keep track of which value (or other reference) it is borrowing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Rust automatically infers all lifetimes in function bodies, and so most annotations are not visible to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' However, when writing functions that have references in their signatures, or data types which store references, Rust requires users to explicitly write them as generic functions/data types over the lifetimes of those references4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' You can see examples of this in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' There is much more to lifetimes, how they are calculated, and their implications on expressiveness and usability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' But, as we shall see, Rust lifetimes are notoriously complex and difficult, and a full description of these aspects is outside the scope of this report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' So I will leave lifetimes here, and end this section with a discussion of their relation to Cyclone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' As the reader may have noticed by now, lifetimes in Rust are very similar to regions and region annotations in Cyclone, including their syntax, subtyping, and intraprocedural inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' In fact, 4There is an exception to this, which is functions whose signatures follow a particular pattern such as functions that don’t return a reference or which take exactly one reference and return a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' In these cases, Rust elides these lifetimes as a syntactic convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Note that this is not the same as lifetime inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Rust, similar to Cyclone, does not infer lifetimes in function signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 6 The Usability of Advanced Type Systems: Rust as a Case Study Research Exam, April 27 2022, La Jolla, CA, USA many of Rust’s features were inspired by Cyclone, such automatic bounds checking, and sum types (Rust enums and Cyclone’s tagged unions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' The main difference between the two is that Cyclone requires manually defining the syntactic scope of regions, and using the region names (which are first-class values) to manually allocate and initialize values inside different regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' In Rust, lifetimes are automatically determined by the compiler, and cannot be explicitly set or used (except for generic lifetime parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Also, since Rust 2018, Rust regions are not determined lexically, but are instead calculated over an intermediate control-flow graph representation of the program [The Rust Core Team 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' So while the theory behind regions and lifetimes is the same, their implementation and interaction model are different in interesting and significant ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='3 Unsafe Rust One of the common themes among the type systems discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2 was that each found strict adherence to its rules needlessly restrictive, and found ways to loosen it for the sake of expressiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Rust is no exception to this, though its solution is rather different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' An important issue with Rust’s Ownership rules is that they are sound, but undecidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' So the borrow-checker is incomplete, and there is plenty of safe code which follows the Ownership rules, but the borrow-checker cannot statically verify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' To alleviate this, Rust allows users to explicitly mark functions and blocks of code as unsafe, and in these unsafe blocks, certain safety checks are disabled5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' More specifically, unsafe allows the code to: Dereference raw pointers Call unsafe functions (including C functions, compiler intrinsics, and the raw allocator) Implement unsafe traits Mutate statics Access fields of unions [Rust Project Developers 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' This is still quite restrictive, but has serious implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' For instance, dereferencing raw pointers can work around the Ownership rules by casting a reference with one lifetime into a raw pointer, and dereferencing that raw pointer to borrow it again as a new reference with a new lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' This allows creating multiple mutable references to a value at the same time, which violates the third rule of Ownership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' The details of this are again notoriously complex and beyond the scope of this report, but unsafe code is crucial to Rust’s “pragmatic safety”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' It allows various performance improvements that are too low-level for the borrow-checker to reason about, as well as interfacing with external code, and certain aliasing patterns which the programmer can verify as safe, but do not pass the borrow-checker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 4 THE USABILITY OF OWNERSHIP At the time of writing, there have been five major papers on the usability of Rust’s Ownership type system [Coblenz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Crichton 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Fulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Zeng and Crichton 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' And, following the best practices of behavioral research [Mcgrath 1995], they use a variety of methods to inspect a number of similar and overlapping research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' So, rather than review each paper individually, in this section I discuss their collective findings, introducing the papers and their methodology as they become relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' I also draw from research on the use of unsafe in Rust, as well as related research on how programmers learn a new programming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 5Some, [Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2022] for example, refer to unsafe code as “similar to the C Programming Language”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' While this is technically true because unsafe code can interface with arbitrary C code, unsafe code within Rust is still far more restrictive than C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 7 Research Exam, April 27 2022, La Jolla, CA, USA Kasra Ferdowsi The high-level takeaway is that Rust’s Ownership model is indeed difficult to learn, and certain aspects of its design remain difficult even for more experienced Rust developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' However, its promise of safety and performance, coupled with good tooling and features for interoperability with other languages, keep Rust popular and loved by those who succeed in adopting it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='1 Is Ownership difficult to use?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' “Learning Rust Ownership is like navigating a maze where the walls are made of asbestos and frustration, and the maze has no exit, and every time you hit a dead end you get an aneurysm and die.” — Student participant from [Coblenz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2021] Rust is notorious for it’s “steep learning curve”, and this has been noted as a major issue in adopting it in the industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' But interestingly, studies suggest that even experienced developers struggle with certain aspects of Ownership in Rust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='1 Barriers for Novice Rust Programmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' [Zeng and Crichton 2019] performed content analysis on top posts from the /r/rust subreddit (an online community specific to Rust), and articles and corresponding comments from Hacker News (a broader tech forum) to identify barriers to adopting Rust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' In 18 experience reports and language comparisons they inspected, they found that “the complexity of the borrow-checker was the second most frequently mentioned complaint” (second only to compiler version issues), where memory access patterns that were common in other languages were disallowed by the borrow-checker, leading to frustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' [Fulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2021] followed this work by interviewing 16 industry professionals who had attempted to adopt Rust in their production team, and used their findings to design an online survey which provided them with 178 more participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' They also found that Rust’s steep learning curve was the most serious barrier to adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Note that this difficulty is more than simply the difficulty of learning a new language, or indeed learning a systems programming language without garbage collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' [Fulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2021] found that the biggest challenge in learning Rust was specifically the borrow-checker, and the necessary shift in programming paradigm to write code that passes the borrow-checker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' And [Shrestha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2020] quoted a C++ developer who said that the borrow-checker “forces a programmer to think differently”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' So it appears that Rust’s more advanced type system is the main source of its difficulty, not just a lack of garbage collection, or more low-level programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Which is not to say that adding garbage collection will not ease Rust’s difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' [Coblenz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2021] ran a controlled study on 428 students in a sophomore-level programming course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' The students were given two weeks of lectures on Rust, and then asked to complete an assignment which required a good understanding of Rust, its Ownership rules, and types that allow for interior mutability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' They randomly assigned students to two groups, one having to complete the assignment using the Rust standard library data types, and one using a garbage-collected wrapper type (called “Bronze”) which enabled a number of additional aliasing patterns to pass the borrow-checker, thus removing the needs for more complex aliasing patterns and datatypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' They found a significant difference in the rate of completion and the self-reported time to completing the assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' The students who used Bronze on average took only a third as much time as the control group, and were approximately 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='44 times more likely to complete the assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Interestingly, the time difference between the groups only appeared in the second part of the task, which involved complex aliasing and mutability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' The first part of the assignment, which focused just on Ownership, didn’t show a significant difference between the groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 8 The Usability of Advanced Type Systems: Rust as a Case Study Research Exam, April 27 2022, La Jolla, CA, USA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='2 Barriers for Experienced Rust Programmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Aside from the initial learning curve, studies also suggest that aspects of Ownership remain difficult to use, even for experienced developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' In their study of memory- and thread-safety issues in Rust, [Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2020] inspected five Rust systems and applications, five popular libraries, and two vulnerability databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' They found that a common reason for blocking bugs in these codebases was a lack of “good understanding in Rust’s lifetime rules”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' This is notable since, unlike the participants in the studies above, the programmers who worked in these codebases were presumably experienced Rust developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' This finding is corroborated by results from the Rust community’s 2020 survey [The Rust Survey Team 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' They received 8323 responses, with the largest number of participants self-reporting their expertise as 7 out of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' They also found that lifetimes are the most difficult topic to learn, though unfortunately they do not report if and how this response changes according to the expertise rating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Similarly, in their study of 100 samples of StackOverflow questions on Rust’s Ownership rules, [Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2022] found that the most common cause of safety rule violations in these questions was “complex lifetime computation”, which appeared 74 times6, 44 in intraprocedural lifetime computation, 16 in explicit lifetime parameters, and 14 in elided ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' From these works, it seems safe to conclude that Rust’s Ownership rules are indeed difficult to learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' They pose a serious barrier to learning and adopting Rust, and understanding lifetimes specifically remains a problem even for more experienced Rust developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='2 Why is Ownership difficult to use?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' “I can teach the three rules [of Ownership] in a single lecture to a room of undergrads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' But the vagaries of the borrow checker still trip me up every time I use Rust!”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' — [Crichton 2020] If we accept that Rust is indeed more difficult to learn than comparable systems programming languages, and that this is in large part caused by its Ownership type system specifically, the next step is to ask what about Rust’s Ownership rules is difficult to learn and apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='1 Change of Paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' One answer may be the notion of “interference” as used by [Shrestha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' In that paper, they qualitatively coded 450 posts on StackOverflow for 18 different programming languages, and interviewed 16 professional programmers, to understand how experi- enced developers learn new programming languages, and what they struggle with in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' They motivated this work by borrowing the term “interference”7 from psychology and neuroscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' The term refers to when “previous knowledge disrupts recall of newly learned information”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' This can be as simple as the difference in zero- vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' one-indexing between two languages, but it also applies to larger differences, where programming in the new language requires a “mindshift”, or a fundamental change in paradigms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Learning Rust needs such a mindshift, because its Ownership rules prohibit many common programming patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Consider a doubly-linked list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' In most languages its implementation is close to trivial, but it violates Rust’s rules by definition: It requires at least two mutable references to a node, one from the previous and one from the next node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Rust has workarounds for this, most simply datatypes with “interior mutability” that postpone checking for simultaneous mutable access to runtime, but they are more difficult to learn and work with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' So it is unsurprising that [Shrestha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2020] use Rust’s Ownership type system as an example of mindshifts, quoting a C# developer who had to “completely rethink the problems they would have normally solved in C#”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 6The paper counts 77 violations, but I’m exlcluding 3 which were merely syntax errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 7As well as the term “facilitation”, but that is not as relevant here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 9 Research Exam, April 27 2022, La Jolla, CA, USA Kasra Ferdowsi 1 let mut v = vec!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' [1, 2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2 let one = &mut v[0];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 3 let two = &mut v[1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 4 two += *one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 5 (a) 1 let mut v = vec!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' [1, 2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2 let iter = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='iter_mut();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 3 let one = iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='next().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='unwrap();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 4 let two = iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='next().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='unwrap();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 5 two += *one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' (b) 1 let mut v = vec!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' [1, 2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='insert(0, v[0]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 3 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='get_mut(v[0]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 4 5 (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' The programs in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 3a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 3b perform the same function, but only Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 3b passes the borrow- checker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 3c, the statements on lines 2 and 3 are nearly identical at the type-level, but only line 2 passes the borrow-checker, presumably due to some implementation detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' In the qualitative portion of their study, [Coblenz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2021] noted a similar theme in the students’ survey responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' For the students without the Bronze library, the second part of the assignment required using types with interior mutability and explicit lifetime parameter declarations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Students mentioned the difficulty of using these types, and the need for redesigning their code to use them correctly, leading the authors of the paper to conclude that “most of the benefit of GC comes from architectural simplification” and that “design was a significant contributor to the difference in performance between non-Bronze and Bronze participants.” So at least one main reason for the difficulty of learning Ownership is that it requires a change of paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' A programmer who is new to Rust needs to learn entirely new patterns and ways of structuring code at the architectural level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' And their previous experience can actively interfere with their learning, as they need to abandon common programming patterns and learn to structure their code in new and unintuitive ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='2 Error Messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' [Coblenz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2021] also noted that rustc’s error messages contributed to the confusion and frustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' rustc error message not only describe the error in the code, but for certain error patterns, suggest edits that may fix the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' However, these edits are always local and don’t provide any high-level design feedback which may be helpful in making the mindshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' At best, they led the students to perform a chain of local edits that resulted in code that compiles without them understanding why.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' At worst, as one student found, they could be cyclical “with things like remove & then after removing try adding &.” This lead the authors to conclude that Rust’s error messages do not “aid design or comprehension”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' [Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2022] investigated error messages more directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' They employed Cognitive Task Analysis [Diaper 2004] to learn how experts solve 110 Rust Ownership errors they had identified in a sample of StackOverflow questions, and compared the steps the experts took to the information contained in the error message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' They found that while for most errors the error message contained all relevant information, for 32 errors the message failed to explain “the key steps in computing a lifetime or a borrowing relationship”, with another 10 failing to “explain the relationship between two lifetime annotations”, and 9 “how a safety rule works on a particular code construct”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' I will leave a broader discussion of rustc error messages to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 5, but the works cited here indicate that Rust error messages do not provide the necessary help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Programmers’ errors may be more structural, but the error messages only suggest potentially misleading local edits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' And even for local errors, they do not always contain the necessary information to understand and fix the error, and assume external knowledge on behalf of the programmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' But this still doesn’t explain why an experienced Rust developer struggles with Ownership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='3 The Curse of Incompleteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' [Crichton 2020] point out that Ownership rules are simple and easy to learn, but statically checking for them, “like most interesting program properties”, is undecidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' So Rust’s implementation of these rules in the borrow-checker is necessarily 10 The Usability of Advanced Type Systems: Rust as a Case Study Research Exam, April 27 2022, La Jolla, CA, USA incomplete, and a lot of the usability issues with Ownership come from this gap between the programmer’s understanding of the rules, and the borrow-checker’s ability of verify them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Consider the examples in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 3a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Both programs perform a similar function, getting references to two elements of a vector and incrementing one by the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' But only the second compiles, since the borrow-checker cannot reason about indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' It conservatively assumes that both lines 2 and 3 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 3a are mutably borrowing the entire vector, thus violating Rule (c) in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 3b, however, uses an iterator, which the borrow-checker can reason about at the type level8, and can successfully verify does not violate the Ownership rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Thus Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 3b compiles successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Note that both of these programs are “safe”, and a more advanced type system involving dependent types could in theory statically verify the safety of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 3a, but the current limitations of the type system means that developers need to learn, not just the rules of Ownership, but how the borrow-checker verifies them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' This issue is exacerbated by the fact that the borrow-checker’s implementation is quite complex and sometimes very similar code may not compile for obscure reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' The code example in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 3c has two similar calls to functions on the vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Line 2 gets the first element of v, and inserts it as the new first element of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Line 3 uses the value of the first element as an index to get a mutable reference to the second element of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' These functions have very similar types, both using an immutable borrow of v to get an argument for a call that mutably borrows v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' However, as of Rust version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='0, line 2 compiles successfully, but line 3 fails the borrow-checker9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' There are almost certainly many other large and small, obvious and subtle reasons for the difficulty of learning and using Rust’s Ownership type system, but these three (Rust’s different paradigm, unhelpful error messages, and the incompleteness of the borrow-checker) are the most apparent from the works surveyed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='3 Why do developers try to use Rust anyway?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Instead of having to invoke pkg-config by hand or with Autotools macros, wran- gling include paths for header files and library files and basically depending on the user to ensure that the correct versions of libraries are installed, you write a Cargo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='toml file which lists the names and versions of your dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='] It just works when you cargo build.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' — [Mena-Quintero 2018]10 The last question I will inspect here is that, if Ownership is difficult to learn and use for so many reasons, why do developers choose to use Rust anyway?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' And perhaps the first answer to that is that they don’t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Despite being the “Most Loved” program- ming language in every StackOverflow survey since 2016 [Stack Overflow 2016, 2017, 2018, 2019, 2020, 2021], it’s user-base is small and growing slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' In the same surveys, it appeared in the list of Most Popular languages in 2019 at only 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='2% [Stack Overflow 2019], growing to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='03% in the latest survey [Stack Overflow 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' In comparison, the Go programming language (which is often compared with Rust as a modern systems programming languages) was already at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='2% in 2019, though it only grew to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='55% by 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Similarly, the TiOBE index ranks Rust at 26 [TIOBE Software BV 2022], and the IEEE Spectrum ranks it at 17 [Cass et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2022], compared to 13 and 8 for Go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 8I should mention that iter_mut uses unsafe to achieve this under the hood, but since iter_mut is itself a safe function provided by the Rust standard library, it can easily be used by novices without ever touching unsafe code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 9[Crichton 2020] speculates that the reason for this is that get_mut is defined on slices (which the Vec type implements), while insert is implemented directly on Vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' They don’t know why this distinction matters, and it only further proves their point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 10Quote found in [Zeng and Crichton 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 11 Research Exam, April 27 2022, La Jolla, CA, USA Kasra Ferdowsi There could be many reasons for this beyond the usability of Ownership of course, and these are not peer-reviewed sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' But Rust’s popularity is still growing, and unsurprisingly the main reason most participants in multiple studies cited was its promise of memory- and thread-safety [Fulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Zeng and Crichton 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Unfortunately, neither of these papers go into depth about this, and only mention that safety is the most commonly noted reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' However, other themes besides safety emerged in these works that are far more interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' The first is that while safety is important, it is not enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' [Zeng and Crichton 2019] noted that while the first and third most-noted benefits of adopting Rust were avoiding runtime errors and data races, the second most-mentioned benefit was Rust’s build tool cargo, which avoided the many issues of build tools for other langauges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Similarly, while [Fulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2021]’s participants cited Rust’s safety as a benefit most frequently, they listed performance and lack of garbage collection almost as frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Another theme that came up in multiple works was Rust’s unsafe feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Two studies which inspected the use of unsafe in Rust code repositories found that unsafe code is common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' [Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2020] inspected all publicly available Rust libraries on crate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='io (Rust’s online library registry), and found that explicit unsafe blocks appear in 29% of all libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' When they filtered their results to the most popular libraries (which accounted for 90% of downloads), this percentage increased to 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' In the 5 applications they inspected, [Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2020] found 4990 uses of unsafe, with a further 1581 unsafe code regions in the standard library, and concluded that unsafe code is used “extensively”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Though they note that it is “unavoidable in many cases” and “usually for good reasons”, including interfacing with existing libraries written in other unsafe languages such as C, and performance improvements by a factor of 4 or 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Interestingly, they also found cases of the unsafe keyword being used as a warning to developers, despite the code itself being safe and compiling without the unsafe block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Those who tried to adopt Rust also noted the many uses for unsafe code, citing its necessity for integrating Rust into existing codebases through FFIs, accessing hardware, and for performance reasons [Fulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' So it seems that “pragmatic safety” was an essential part of Rust’s success, as a large amount of code written in Rust would have not been possible it if had strictly adhered to its statically-guaranteed safety rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' It’s also interesting to note that we now have empirical evidence that, despite unsafe being described as an “escape hatch” [Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2020], developers can be trusted to use it responsibly, only where necessary and while still mostly maintaining Rust’s safety guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' As [Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2020] found, most Rust codebases do not contain explicit unsafe code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' And in their study of StackOverflow questions, [Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2022] found that of the 110 errors in the questions, only 3 were fixed by writing unsafe code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' The rest were either fixed with simple safe code, or library functions which used unsafe blocks internally, but exposed a safe interface (a programming pattern known as “interior unsafe”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' In larger codebases, [Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2020] listed numerous cases of memory bugs related to unsafe code, but they found that this has more to do with the complexity of the type system (and lifetimes especially), and not developer negligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' And good programming patterns such as interior unsafe, best-practices such as coding reviews, and better tools for reasoning about lifetimes and unsafe code could alleviate many causes for these bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' So while Rust is not as popular as comparable languages which lack its advanced type system, developers do like it and try to adopt it, mainly for its promise of safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Though alongside safety, they also value its performance, lack of a garbage collector, and great tooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' And as much as they appreciate Rust’s safety, they still have numerous justified reasons for using unsafe code, and do so responsibly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 12 The Usability of Advanced Type Systems: Rust as a Case Study Research Exam, April 27 2022, La Jolla, CA, USA 5 FUTURE WORK In this section, I will briefly introduce ideas from the fields of HCI and Computer Science Education that I believe could contribute to better understanding and improving the usability of Rust, or indeed any other advanced type system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' These ideas are eclectic, and I mean to introduce them as a starting point for generating ideas, not as fully fleshed-out research plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Also, as the focus of this report is on human-centered approaches, I will not discuss techniques from Programming Languages and Compilers research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Though, especially if combined with HCI techniques, they could be essential to improving the usability of Rust as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='1 Better Error Messages Perhaps the most immediately actionable takeaway from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 4 is that Rust’s error messages are an important limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' But, paradoxically, the general community consensus is that Rust’s error messages are more helpful than most languages [Fulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' So a good next step in improving Rust’s usability, and an important consideration when designing any language with such complex types, is to see where Rust’s error messages succeed and how they fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' A great starting point for this is [Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' They survey all works on compiler error messages in the past 50 years, and provide a number of remarkable insights on the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' For one, they discuss empirical evidence showing that programmers, both novice and expert, do read compiler error messages [Barik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Prather et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2017], and so improving error reporting is indeed worthwhile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' They also compiled a list of empirically-backed guidelines for designing error messages, which can be invaluable as a shared foundation for synthesizing various works on Rust’s usability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' For example, it could serve as the complementary theoretical background to empirical analyses (such as in [Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2022]) that argue that Rust’s error messages are notably well-designed11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' It is also a good resource for understanding why Rust error messages fail, and how to improve them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' For instance, one of the important guidelines for error design is to include the relevant context of the error directly in the message, which multiple works have argued Rust sometimes fails to do [Blaser 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Dominik 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Following [Coblenz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2021]’s finding that Rust diagnostics lack necessary architectural hints, the guidelines would also be invaluable in providing the human-centered design element to programming languages or machine learning techniques that could identify such architectural changes and present them to the programmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='2 Program Visualization Tools An often mentioned next step for the usability of Rust’s Ownership is visualization of Rust’s lifetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' In their discussion of ways to address the usability issues with the incompleteness of the borrow-checker, [Crichton 2020] recommended further research into visualizing the various static information provided by the borrow-checker, citing the difficulty in visualizing the large amount of information in a “succinct, non-intrusive, yet informative” manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Similarly, [Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2020] suggested an IDE plugin for visualizing lock lifetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' They found that a notably common cause of deadlock bugs in multi-threaded code was that in Rust unlocking a Mutex occurs implicitly when the locked value is dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' To address this, they recommended “plug-ins to highlight the location of Rust’s implicit unlock”, which is a specialized case of visualizing lifetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 11A deeper look into this is outside the scope of this report, but my personal experience suggests that Rust follows most of the guidelines from [Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2019] most of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 13 Research Exam, April 27 2022, La Jolla, CA, USA Kasra Ferdowsi Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' A RustViz visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' If the user hovers over the nodes and arrows on the right, it displays a pop-over containing additional information about the event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' There are three existing but limited works attempting to visualize Ownership and lifetimes in Rust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' The first is RustViz [Gongming et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2020], a tool for building visualizations of Ownership and borrowing events in Rust programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' RustViz does not generate these visualizations automatically, but instead requires (arguably considerable) effort on the part of a “teacher” to specify them by annotating the code being visualized, and using a Rust library to generate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' The benefit is that the generated visualization is interactive: the “learner” can hover over parts of the image to get a brief description of the events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 4 for an example of RustViz’s visualizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' The other two are related Bachelor’s Theses [Blaser 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Dominik 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' In the first [Dominik 2018], the author uses Polonius (an experimental implementation of the borrow-checking rules in Datalog) to extract lifetime constraints from a given program, and visualize it as a directed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' They motivated it by arguing that rustc error messages do not always present all the necessary information for Ownership errors, and that this visualization could address that problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' [Blaser 2019] then extended [Dominik 2018] to improve its usability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' They defined an algorithm which filters the complete graph to a single path that contains all constraints relevant to the particular error message, such as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' They also created a Visual Studio Code extension called “Rust Life Assistant” for displaying these graphs in the IDE, and an algorithm for converting the graph into a bullet-point list of English text explaining the cause of the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' All these works are quite limited, and crucially none of them have been evaluated on users12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' So a great next step could be to design, implement and evaluate a tool for visualizing Ownership and lifetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Such a tool could be pedagogical (similar to RustViz) or utilitarian (similar to Rust Life Assistant), but as [Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2022] points out “learning Rust is a continuous process”, and I would recommend considering existing works on education and program visualization systems to inform the design and evaluation of any such tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' A great survey of such systems can be found in [Sorva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2013].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Besides exploring over 40 program visualization tools, they also provide a taxonomy of visualization systems, and emphasize that how users engage with a visualization tool is as important as the design of the tool itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Specifically, they cite [Hundhausen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2002] who performed a meta-study of 24 algorithm visualization (AV) tools and found that “the form of the learning exercise in which AV technology is used is actually more important than the quality of the visualizations produced by AV technologies.” Unfortunately most tools focus on the runtime behavior and values of programs, but they can still help us better think about the design of a visualization system, and users’ engagement with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 12[Gongming et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2020] contains a study proposal, but as of writing this report, no results of such a study have been published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 14 fn main() S r1*r1 r2|*r2 r3|*r3 2 let mut s = String::from("hello");' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' f 3 4 let r1 = &s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 5 let r2 = &s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 6 assert!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' (compare_strings(r1, r2));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 7 8 let r3 = &mut s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 9 clear_string(r3): 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='7The Usability of Advanced Type Systems: Rust as a Case Study Research Exam, April 27 2022, La Jolla, CA, USA 1 fn main() { 2 let mut x = 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 3 let y = foo(&x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 4 let z = bar(&y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 5 let w = foobar(&z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 6 // .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 7 x = 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 8 take(w);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 9 } 10 11 fn foo(p: T) -> T { p } 12 fn bar(p: T) -> T { p } 13 fn foobar(p: T) -> T { p } 14 fn take(p: T) { unimplemented!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' () } Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' A Rust Life Assistant visualization, showing the lifetimes of the references in the code, and the corresponding constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='3 Grounded Theory The works in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 4 use a wide variety of research methods, including surveying existing experience reports, semi-structured interviews, online surveys, and controlled experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' But so far they have left a notable gap in the methodologies, which is a deep qualitative understanding of how Rust programmers actually write code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Questions such as what tools they use, if and how they read error messages, how they reason about various errors, common debugging strategies, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' are all left out of the scope of the existing research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' To fill this gap, I recommend building a Grounded Theory of how Rust programmers write code in relation to programmers in other languages, or using different type or memory management systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Grounded Theory (GT) started in the 1960s as a research method in sociology, but has since become a standard method of qualitative research in many field as a method for developing theories bottom-up through observation [Charmaz and Bryant 2010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Speaking very broadly, GT as a method is an iterative process of collecting data through interviews and observations and using open-coding to develop a theory that is grounded in the empirical data (rather than informed by or confirming existing theories).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' But the above description is too simplistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Various versions of GT exist, each of which make different assumptions about the nature of knowledge (positivism vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' constructivism) and the precise steps they follow are subtly different and outside the scope of this report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' That said, GT is an established and popular method in Software Engineering research [Stol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2016], and recently [Lubin and Chasins 2021] employed Constructivist GT [Charmaz 2006] to develop a deeper understanding of how statically-typed functional programmers write code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Which is why I recommend building a grounded theory of “How Rust Programmers Write Code”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Speculating on what we may learn from this is counter to GT’s philosophy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' But my hope is that such a GT will help develop the foundations that can both better motivate and help us understand the human-centered study and design choices involved in Rust and similar advanced type systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' ACKNOWLEDGMENTS I would like to thank my advisors Sorin Lerner and Nadia Polikarpova for providing feedback on this report, as well as their advice and mentorship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' I would also like to thank Hila Peleg for being my de-facto advisor in my first two years in Graduate school, and Ruanqianqian (Lisa) Huang and Shraddha Barke for being amazing collaborators and co-authors whose research has been a major source of motivation and inspiration in my own work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=" 15 Lifetime R2 : &'R2 3: let y = foo(&x);" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=" LifetimeR26 Z:&'R26 Constraint 4: let z = bar(&y);" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' R23 may point to R2 generated at line 3: let y = foo(&x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Constraint R29 may point to R26 generated at line 5: LifetimeR23 let w = foobar(&z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=" y:&'R23 3: let y = foo(&x);" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=" LifetimeR29 W:&'R29 Constraint 5: let w = foobar(&z);" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' R26maypointtoR23 generated at line 4: let z = bar(&y);' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='1145/292540.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='292560 Anna Zeng and Will Crichton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Identifying barriers to adoption for Rust through online discourse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' arXiv preprint arXiv:1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='01001 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Shuofei Zhu, Ziyi Zhang, Boqin Qin, Aiping Xiong, and Linhai Song.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' Learning and Programming Challenges of Rust: A Mixed-Methods Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' (2022), 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='1145/3510003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} +page_content='3510164 18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE0T4oBgHgl3EQfYgB_/content/2301.02308v1.pdf'} diff --git a/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf b/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6d63521d81b0db60f269235d4ec99978fb01c5c6 --- /dev/null +++ b/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf @@ -0,0 +1,3 @@ +version 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Palm1,†,∗, Mark Dong1,2,†,∗, D. Andrew Golter1, Genevieve Clark1,2, Matthew +Zimmermann1, Kevin C. Chen2, Linsen Li2, Adrian Menssen2, Andrew J. Leenheer3, +Daniel Dominguez3, Gerald Gilbert4,∗, Matt Eichenfield3,5,∗, and Dirk Englund2,6∗ +1The MITRE Corporation, 202 Burlington Road, Bedford, Massachusetts 01730, USA +2Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA +3Sandia National Laboratories, P.O. Box 5800 Albuquerque, New Mexico, 87185, USA +4The MITRE Corporation, 200 Forrestal Road, Princeton, New Jersey 08540, USA +5College of Optical Sciences, University of Arizona, Tucson, Arizona 85719, USA +6Brookhaven National Laboratory, 98 Rochester St, Upton, New York 11973, USA and +†These authors contributed equally +(Dated: January 11, 2023) +A central goal in creating long-distance quantum networks and distributed quantum computing +is the development of interconnected and individually controlled qubit nodes. Atom-like emitters in +diamond have emerged as a leading system for optically networked quantum memories, motivating +the development of visible-spectrum, multi-channel photonic integrated circuit (PIC) systems for +scalable atom control. However, it has remained an open challenge to realize optical programmabil- +ity with a qubit layer that can achieve high optical detection probability over many optical channels. +Here, we address this problem by introducing a modular architecture of piezoelectrically-actuated +atom-control PICs (APICs) and artificial atoms embedded in diamond nanostructures designed for +high-efficiency free-space collection. The high-speed 4-channel APIC is based on a splitting tree +mesh with triple-phase shifter Mach-Zehnder interferometers. This design simultaneously achieves +optically broadband operation at visible wavelengths, high-fidelity switching (> 40 dB) at low +voltages, sub-µs modulation timescales (> 30 MHz), and minimal channel-to-channel crosstalk for +repeatable optical pulse carving. Via a reconfigurable free-space interconnect, we use the APIC +to address single silicon vacancy color centers in individual diamond waveguides with inverse +tapered couplers, achieving efficient single photon detection probabilities (15%) and second-order +autocorrelation measurements g(2)(0) < 0.14 for all channels. The modularity of this distributed +APIC - quantum memory system simplifies the quantum control problem, potentially enabling +further scaling to 1000s of channels. +Approved for Public Release; Distribution Unlimited. Public Release Case Number 22-4195 +© 2022 The MITRE Corporation. All rights reserved. +I. +INTRODUCTION +Solid-state artificial atoms [1], many of which have +long-lived quantum memories [2–5], can achieve photon- +mediated remote-entanglement [6, 7], and can be hetero- +geneously integrated with photonics [8, 9], are a promis- +ing platform for the construction of large-scale quan- +tum networks [10–12]. The networking of these atom- +like emitters requires an efficient and high-fidelity op- +tical interface for both reconfigurable optical address- +ing and collection of photoluminescence (PL) at visi- +ble wavelengths. The optical control layer thus presents +two challenges: i) scalable high-fidelity manipulation of +optical fields at high speeds, which necessitates high- +quality optical switches in atom-control photonic inte- +grated circuit (APIC) [13] platforms and ii) scalable high- +efficiency photon collection from remotely addressable +single emitters. While previously demonstrated visible- +wavelength APIC platforms such as thin-film lithium +∗ kpalm@mitre.org; +mdong@mitre.org; +ggilbert@mitre.org; +eichenfield@arizona.edu; englund@mit.edu +niobate [14–16], thermally-tuned silicon nitride [17–19], +and piezoelectrically-actuated silicon nitride [20–23] all +have promise for scalability, none currently combine opti- +cally broadband operation, high switching contrast (> 40 +dB) at nanosecond time scales, and low voltage oper- +ation. +On the photon collection side, efficient collec- +tion has been demonstrated using standard confocal mi- +croscopy [24–26], by leveraging photonic nanostructures +such as immersion lenses [27–29] and cavities [8, 30–33], +or single-channel fiber collection from tapered waveg- +uides [8, 34, 35]. Collection through a heterogeneously- +integrated photonic chip [9, 36, 37] at the cost of some op- +tical loss due to the diamond-chip interface has also been +reported. To date, these past works treated each side of +the optical control layer separately, but there remains an +open question of how to combine the requirements of i) +and ii) into a single scalable system. +Here we introduce an architecture for the optical con- +trol layer consisting of modular piezoelectrically-actuated +APICs and diamond microchiplets with implanted sin- +gle emitters. +In this configuration, the excitation and +collection optical paths are perpendicular, enabling the +inverse tapered diamond waveguides to take advantage +arXiv:2301.03693v1 [quant-ph] 9 Jan 2023 + +2 +Routing “Single” MZI +Input Grating +Coupler +Routing MZIs +Electrical Contacts +Switching SPSs +Switching CPSs +Edge Coupled +Outputs +ϴ +Switching “Triple” MZI +a) +b) +c) +d) +Input +Output +-ϴ +ϴ +Input +-ϴ +ɸ +-ɸ +ψ +-ψ +Output += Single MZI += Triple MZI +Input +Output +g) +500 μm +Waveguide wire +Not connected +Waveguides +f) +e) +5K +Routing “Single” MZI +Input Grating +Coupler +Routing MZIs +Electrical Contacts +Switching SPSs +Switching CPSs +Edge Coupled +Outputs +ϴ +Switching “Triple” MZI +a) +b) +c) +d) +Input +Output +-ϴ +ϴ +Input +-ϴ +ɸ +-ɸ +ψ +-ψ +Output += Single MZI += Triple MZI +Input +Output +g) +500 μm +Waveguide wire +Not connected +Waveguides +f) +e) +Routing “Single” MZI +Input Grating +Coupler +Routing MZIs +Electrical Contacts +Switching SPSs +Switching CPSs +Edge Coupled +Outputs +ϴ +Switching “Triple” MZI +a) +b) +c) +d) +Input +Output +-ϴ +ϴ +Input +-ϴ +ɸ +-ɸ +ψ +-ψ +Output += Single MZI += Triple MZI +Input +Output +g) +500 μm +Waveguide wire +Not connected +Waveguides +f) +e) +Routing MZIs +Electrical Contacts +Switching SPSs +Switching CPSs +Edge-Coupled +Outputs +d) +1 mm +a) +b) +c) +e) +f) +Excitation +Collection +Grating Coupler +Inputs +Si +C +Vacancy +g) +FIG. 1. Photonic integrated network switch architecture for local addressing of multiple quantum emitters. a) Routing “single” +MZIs to split the single input into each of the four ports and b) switching “triple” MZIs that enable fast arbitrary pulsing of light +with high extinction. The routing MZIs consist of a single cantilever phase shifter (CPS) and two 50:50 directional couplers, +while the switching MZIs consist of two CPSs, a strain-optic phase shifter (SPS), and three 50:50 couplers. c) Schematic of the +binary tree switch design. d) Microscope image of the fabricated integrated network switch with the CPSs and SPSs labeled. +Light is input through grating couplers on the left side and collected through edge-coupled outputs on the right. e) Cryostat +setup housing the quantum emitters with light from the switch projected through free-space for quantum control experiments. +f) Diamond quantum microchiplets with g) implanted Si vacancy color centers with light pulses from the chip controlling the +optical emission. The diamond nanostructure allows for high-efficiency collection of the emitter’s emission. +of free-space modal conversion for efficient collection +through the optical path parallel to the waveguides +while maintaining the ability to selectively address a +large area of distinct emitters through the perpendicu- +lar path. +We demonstrate our control scheme by first +satisfying requirement i) through our APIC switch, im- +plemented as a 4-channel binary tree mesh [13] with +visible-wavelength switching and power routing capabil- +ities. +The APIC’s switching circuit uses an optically- +broadband triple-phase shifter design that takes advan- +tage of hardware error correction [38, 39] and a stronger +strain-optic response than previous designs, enabling low +switching voltages while maintaining high-contrast (> 40 +dB) and high-speed (> 30 MHz) switching performance. +The switch shows negligible cross-talk between channels +and enables repeatable arbitrary pulse carving on all four +outputs, combined with > 1 MHz power balancing be- +tween ports. We further demonstrate requirement ii) by +applying the APIC to a local group of quantum emit- +ters by projecting the optical output channels onto ion- +implanted silicon vacancy color centers (SiVs) [28, 40] +in diamond microchiplets [9] mounted in a 5K cryostat. +Through PL excitation (PLE) and second-order autocor- +relation measurements, we demonstrate optical address- +ing with independent temporal control of four spatially +distinct color centers and achieve high (15%) collection +efficiency, single emitter linewidths of 152 MHz - 287 +MHz, and g(2)(0) of 0.06 - 0.14. The modularity of this +architecture allows for easy switching between different +sets of quantum emitters by adding different sets of dia- +mond microchiplets into the cryostat setup. Our APIC +excitation and diamond collection techniques should en- +able scalable quantum control of emitters as part of a +larger network of quantum nodes. +II. +PHOTONIC INTEGRATED SWITCH +DESIGN AND OPERATION +The schematic of our APIC-to-diamond control archi- +tecture is as follows. The APIC design consists of a “sin- +gle” routing Mach-Zender Interferometer (MZI) (Fig. 1a) +and a “triple” switching MZI (Fig. 1b) arranged in a bi- +nary tree architecture (Fig. 1c). A single cantilever phase +shifter (CPS) [22] in the routing MZIs directs the desired +amount of light to the appropriate outputs. The switch- +ing MZI uses three phase shifters: two CPSs that en- +able optically broadband and high-fidelity routing (> 40 +dB) for cross and bar ports using hardware error correc- +tion robust to fabrication imperfections [39] and a third, +strain-optic phase shifter (SPS) [21, 41], that enables a +fast phase response for on-off switching of the output + +淼Magnification: X30.0 +1.0000mman +Apo +0.28 +f=200Magnification: X30.0 +1.0000mm3 +-10 +-5 +0 +5 +10 +V (V) +0 +0.2 +0.4 +0.6 +0.8 +1 +Tnorm +-10 +-5 +0 +5 +10 +V (V) +-40 +-30 +-20 +-10 +0 +T (dB) +-20 +-10 +0 +10 +20 +V (V) +0 +0.2 +0.4 +0.6 +0.8 +1 +Tnorm +-20 +-10 +0 +10 +20 +V (V) +-30 +-25 +-20 +-15 +-10 +-5 +0 +T (dB) +50:50 coupler +50:50 coupler +Phase Shifter +50:50 coupler +50:50 coupler +Phase Shifter 1 +Phase Shifter 2 +Phase Shifter 3 +-ϴ +ϴ +ϕ +-ϕ +-ϴ +ϴ +ψ +-ψ +Tnorm +T𝜖(dB) +a) +d) +f) +g) +e) +h) +Vπ = 30.9 V +c) +b) +1 μm +1 μm +i) +j) +Vπ = 10.5 V +100 μm +100 μm +FIG. 2. Device performance and calibration. Simulated TM optical waveguide mode for the a) 400 nm waveguides and b) the +5 µm waveguides in the SPSs. c) Microscope image of a routing MZI. A voltage is anti-symmetrically applied to each side +of the phase shifter to give the maximum actuation range. d) Normalized transmission (Tnorm = T/Tmax) and e) extinction +(Tϵ = 10 × log(Tnorm)) measured from a single output with the applied voltage to the cantilever swept from -25 to 25 V. A +single phase shifter achieves 25-30 dB extinction. f) Microscope image of a switching MZI with three phase shifters. The first +two CPSs are calibrated with the SPS held at 0 V to maximize output port extinction. g) Normalized transmission Tnorm and +h) extinction Tϵ plots measured from sweeping the applied voltages of the two CPSs. The addition of the second cantilever +compared to the single MZI allows for the output extinction to exceed 40 dB. i) Normalized transmission Tnorm and j) extinction +Tϵ plots for the SPS, calibrated after the two CPSs in the switching MZI. +channel. During operation, a CPU controller programs +the two CPSs to route the light to a dump port while the +SPS is held at 0 V. We then can send an arbitrary pulse +sequence to the SPS to switch the light to the output +port without having to change the applied DC voltages +to the CPSs. +A microscope image of the APIC is shown in Fig. 1d, +with the different phase shifters and electrical contacts +labeled. +We input light into the chip with an optical +fiber array through a single grating leading to the rout- +ing MZIs, while other inputs are only used for device +calibration. We then collect the edge-coupled light from +each output with a high-NA objective, enabling imaging +of the outputs into any system for optical control exper- +iments. Figure 1e shows the optical imaging schematic +where the output channels are projected into a cryostat +to use for optical control of quantum emitters in diamond +waveguides (Fig. 1f), such as SiVs (Fig. 1g). This config- +uration enables perpendicular excitation of the diamond +waveguides, with the single photon fluorescence from the +emitters coupling to the diamond waveguide mode and +emitting vertically for collection through inverse tapered +couplers, as shown in Figure 1f. This free-space collection +allows for efficient and scalable detection due to low-loss +collection optics that are robust to misalignment when +compared with fiber coupling or PIC integration. Electri- +cal control of the integrated optical components is made +through a custom printed circuit board (PCB) with wire +bonds to the APIC. Commercial arbitrary waveform gen- +erator boards, embedded in a National Instruments PXIe +system, control the CPSs and SPSs. A single board with +22 active channels controls the CPSs, providing ± 25 V, +and two boards with four channels of arbitrary wave- +form generation each control the SPSs, providing ± 2.5 +V. High-speed amplifiers on the PCB amplify the signals +to the SPSs to ± 12.5 V. See Supplementary Sections 1 +and 2 for more details on the optical and electrical com- +ponents of the system. +Figure 2 summarizes the APIC characterization and +calibration by monitoring the transmission of each edge- +coupled optical output. For all optical tests, we use 737 +nm wavelength laser light coupled into the TM mode +of the on-chip 400 nm wide by 300 nm thick silicon ni- +tride waveguides (modal shape simulated in Fig. +2a), +which adiabatically expand to 5 µm wide in the SPS +(Fig. 2b) to increase strain-optic sensitivity [21, 41]. The +less-confined TM mode takes advantage of a higher pho- +toelastic responsivity when compared to the TE mode +[42], resulting in a lower Vπ of the phase shifter than +previously reported [21]. Our DC calibration results for +the routing MZIs (Fig. 2c) are shown in Fig. 2d-e and +for the switching MZIs (Fig. 2f) are shown in Fig. 2g- +j, highlighting the low-voltage operation of the SPS for +switching and high on-off extinction ratios. These high +extinction ratios for the triple-phase shifter are enabled +by the second CPS accounting for fabrication imperfec- +tions in the 50:50 directional couplers. For calibration +data for each of the output ports, see Supplementary +Section 3. + +1 +20 +0.8 +10 +0.6 +M +0 +> +0.4 +-10 +0.2 +-20 +0 +-20 +-10 +0 +10 +20 +V +M20 +-10 +10 +M +-20 +0 +A +-10 +-30 +-20 +-40 +-20 +-10 +0 +10 +20 +V +M +D200 μm +Magnification: 5 x200 μm +Magnification: 5 x4 +95 +100 +105 +110 +115 +120 +Time (ns) +0 +0.5 +1 +1.5 +Photodiode (V) +0 +100 +200 +300 +400 +Time (ns) +0 +0.5 +1 +1.5 +Photodiode (V) +397 +398 +399 +400 +401 +Time ( s) +-1 +0 +1 +2 +3 +Time ( s) +0 +0.5 +1 +Tnorm +200 +201 +202 +203 +204 +Time ( s) +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +4500 +Time (ns) +Tnorm (offset) +Chan 4 +Chan 3 +Chan 2 +Chan 1 +a) +b) +d) +e) +Rise time = ~20 ns +105 +106 +107 +108 +Frequency (Hz) +-6 +-5 +-4 +-3 +-2 +-1 +0 +MZI Response (dB) +ν3dB = 34 MHz +c) +FIG. 3. High-speed device pulsing qualification. a) Measured 200 ns pulse from an output port of the chip and b) inset of the +pulse showing a ∼20 ns rise time from the SPS. c) Normalized modulator response for a 3V sinusoidal signal showing the -3 dB +cutoff at ν3dB = 34 MHz. d) Pulsing scheme showing the capabilities of the binary tree for arbitrary pulsing schemes. Each +output can be pulsed at arbitrary times, lengths, shapes, and amplitudes. e) Repeated 200 ns pulses with a 50% duty cycle to +measure the consistency of our device. The standard deviation of the integrated pulse area is 6.8 × 10−4 for 1000 consecutive +pulses. +III. +PULSE CHARACTERIZATION AND +STABILITY +We tested the optical pulse carving of our switch by ap- +plying representative pulse sequences to each of the SPSs +in the switching MZIs. The “off” state of the output is de- +fined to be 0 V due to the calibration procedure, and the +full “on” state is achieved by applying the experimentally +determined cross-state voltage. Pulses of varying ampli- +tudes below the maximum are created by setting the ap- +plied voltage between these cross and bar states. Using +time-resolved measurements on a 125 MHz photodiode, +we found rise and fall times of ∼20 ns when program- +ming a 200 ns pulse (Fig. 3a,b) for all channels. The +small-signal frequency-resolved modular response (Fig. +3c) indicates a -3 dB cutoff at ν3dB = 34 MHz, allowing +for > 30 MHz optical control of each channel. The de- +vice can also be run at higher modulation speeds (> 100 +MHz) with a trade-off of lower responsivity (< −6 dB). +To explore the optical control programmability, we +tested various pulse sequences. Figure 3d shows the re- +sulting measurement of each of the outputs and shows +four different capabilities of this system: +i) Any set +of outputs can be pulsed simultaneously, ii) each pulse +width can be independently manipulated, iii) the wave- +form can be temporally amplitude modulated into differ- +ent shapes, such as square or Gaussian, and iv) the pulse +height can be independently set. With these criteria met, +our chip has the ability to create a full set of quantum ro- +tations [43]. Furthermore, we measured the consistency +of the pulsing of our device by applying repeated 200 ns +pulses with 200 ns intervals and measuring the deviations +in each pulse. We find a pulse area consistency (1σ stan- +dard deviation) of 6.8 × 10−4 for 1000 pulses, showing +robust pulse uniformity. Examples of these pulses from +the beginning, middle, and end of this pulse sequence are +shown in Fig. 3e. Lastly, we did not observe crosstalk +from either thermal, electrical, or piezo effects between +the different phase shifters (details in Supplementary Sec- +tion 4). +IV. +INDEPENDENT ADDRESSING OF +MULTIPLE SINGLE SIVS +To demonstrate the applicability of the APIC, we used +it to resonantly drive individual emitters within an en- +semble of SiVs. As shown in Fig. 4a, the APIC projects +each port perpendicularly onto separate diamond waveg- +uides in a cryostat. The diamond waveguides are fab- +ricated with inverse tapered end couplers oriented to- +wards the collection path, allowing for a high collection +efficiency of 15% (see Methods for full diamond fabri- +cation information and Suplementary Section 5 for col- +lection efficiency calculation). +The inverse tapers con- +fine the emitted PL to an NA much smaller than that of +the collection optics, allowing for scalable collection. In +the excitation path, we include a spatial light modulator +(SLM) for small spatial adjustments to each projected +beam. This allows us to independently steer each exci- +tation spot to specific SiVs in the diamond waveguides. +We note that once the SLM is initially programmed, it +is kept static over the course of the experiment, mak- +ing its slow reconfiguration time (∼100 Hz) inconsequen- +tial for the excitation experiments. We resonantly excite +each of the SiVs while collecting the phonon sideband +(PSB) emission using a 750 nm long pass filter to re- +move excess pump light. We projected this fluorescence + +5 +T = 5 K +Binary Tree +737 nm +Laser +LP +a) +b) +Normalized Counts +1 +0 +1 pixel = 16 μm x 16 μm +FIG. 4. Independent optical control of Si vacancy color centers. a) Experimental setup. 737 nm laser light is input into the +binary tree through grating couplers. The four outputs are imaged onto a diamond microchiplet with the use of an SLM to steer +the beams onto individual SiVs. The emission of the SiVs is collected, the PSB is filtered out with a 750 nm long pass filter +(LP), and imaged onto an EMCCD. b) Simultaneous PLE measurements on four different diamond waveguides. By driving +the APIC, emission from emitters in each waveguide can be independently controlled with high extinction. Each panel shows +a different iteration of outputs being driven, showing complete independence of emitter emission. +-1000 +-500 +0 +500 +1000 +Detuning (MHz) +0 +0.2 +0.4 +0.6 +0.8 +1 +PSB fluorescence (a.u.) +-1000 +-500 +0 +500 +1000 +Detuning (MHz) +0 +0.2 +0.4 +0.6 +0.8 +1 +PSB fluorescence (a.u.) +-1000 +-500 +0 +500 +1000 +Detuning (MHz) +0 +0.2 +0.4 +0.6 +0.8 +1 +PSB fluorescence (a.u.) +-1000 +-500 +0 +500 +1000 +Detuning (MHz) +0 +0.2 +0.4 +0.6 +0.8 +1 +PSB fluorescence (a.u.) +-20 +-10 +0 +10 +20 +Delay Time, (ns) +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +g(2)( ) +-20 +-10 +0 +10 +20 +Delay Time, (ns) +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +g(2)( ) +-20 +-10 +0 +10 +20 +Delay Time, (ns) +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +g(2)( ) +g(2)(0) = 0.06(9) +g(2)(0) = 0.09(14) +g(2)(0) = 0.09(10) +g(2)(0) = 0.14(11) +b) +a) +c) +d) +e) +f) +g) +h) +0 +200 +400 +600 +800 +1000 +1200 +Time (ns) +0 +0.2 +0.4 +0.6 +0.8 +1 +PSB fluorescence (a.u.) +i) +𝛤 = 160(3) MHz +𝛤 = 152(2) MHz +𝛤 = 287(6) MHz +𝛤 = 199(8) MHz +FIG. 5. Direct addressing and temporal control of single SiV emitters. a-d) PLE spectrum of single SiVs excited with the +APIC. Each vacancy is excited with light from a different APIC channel. e-h) Autocorrelation measurements of the same +single SiVs. For each emitter, g(2)(0) < 0.14, well below the 0.5 threshold to demonstrate single photon emission. i) Pulsed +fluorescence demonstrating temporal control of the emission of a single emitter. Data shown is integrated over 3 min. +onto an electron-multiplying charge-coupled device (EM- +CCD). Figure 4b shows acquisitions of 30 seconds of the +collected fluorescence normalized to the brightest point +of each image, with no further image processing. This se- +quence shows independent and simultaneous optical con- +trol of SiVs in four different diamond waveguides. Due +to variations in the local strain throughout the diamond, +the zero-phonon lines (ZPLs) have an inhomogeneous dis- +tribution that exceeds the excitation laser linewidth. To +collect SiV emission from multiple waveguides simulta- +neously, we increased the temperature of the diamond +samples to broaden the ZPL linewidths so that they are +spectrally overlapping. Thus, for these images, we likely +addressed multiple emitters in each diamond waveguide + +SLM1.2 +F0.8 +2 +90.6 +0.4 +0.2 +0 +-20 +-10 +0 +10 +20 +Delay Time, T (ns)6 +due to the high density of SiVs in our sample (> 50 emit- +ters per waveguide). +However, to show the applicability of this scheme for +controlling individual single emitters, we cooled the dia- +mond sample to a base temperature of 5 K and repeated +the excitation scheme with each channel projected onto +a spectrally resolved SiV. Figure 5a-d shows the PLE +frequency scans for SiVs in four different waveguides, +demonstrating linewidths < 290 MHz. +Second-order +correlation measurements indicate strong antibunching, +with a normalized g(2)(0) ranging from 0.06 ± 0.09 to +0.14 ± 0.11, well below the 0.5 threshold for single pho- +ton emission (Fig. +5e-h). +We find an average emitter +lifetime of 1.76(1) ns (see Supplementary Section 5), con- +sistent with other measurements on ion-implanted SiVs +[44]. With the outputs of the MZI tree projected on these +emitters simultaneously, we send pulse sequences to tem- +porally control the SiV emission. An example pulse train +is shown in Fig. 5i, where we repeatedly pulse one of the +channels (Channel 3, Fig. 5c,g) with 100 ns pulses and a +period of 250 ns and collect the fluorescence on a time- +resolved avalanche photodiode, demonstrating temporal +control of a single photon source. +V. +DISCUSSION +We introduced and demonstrated a scalable optical +control system for individual addressing of quantum +atom-like emitters. The modularity of the APICs and di- +amond microchiplets is scalable to 1000s of ports and can +be integrated with CMOS control electronics for VSLI +devices. Operating voltages can be further reduced by +allowing for a trade off of extinction and applied volt- +age, i.e. +if only 30 dB extinction is required then the +SPS can be pulsed with < 2.5 V applied signal. The dia- +mond collection architecture is also readily scalable, with +high efficiency collection of many waveguides enabled by +the modal conversion of the waveguides to a 0.26 NA +(See Supplementary Section 6). With the collection op- +tics used in this setup, this allows for the scaling to 2975 +waveguides with 3 µm spacing between waveguides in a +linear array without a loss of collection efficiency. The +losses on the chip currently limit the scalability of the +platform, with a total measured insertion loss of -19.2 +dB. This loss is dominated by a low grating coupler ef- +ficiency of 10%, which can be improved with design and +fabrication iterations (See Supplementary Section 7 for +improved grating coupler results > 40%). +Future work will use this platform for running inde- +pendent optical control schemes of quantum emitters. +Using already demonstrated strain tuning [45, 46], we en- +vision a second chip built from the same APIC platform +that allows for spectral matching of quantum emitters, a +necessary functionality for quantum computation. More +broadly, the broadband [21, 22] APIC technology can be +applied to other optically trapped atomic systems [47– +53] and will enable near-future experiments in the area +of optical quantum control. +VI. +METHODS +A. +PIC Calibration +To begin a calibration, the light from the first output +channel is focused onto a photodiode. An iris is used to +ensure that only the light from the active output is being +measured. A single laptop controls all of the equipment +in the experiment and is able to set the applied voltages +and query the measured values from the powermeter. We +first calibrate the routing MZIs. The applied voltage is +swept from -25 V to 25 V in increments of 0.1 V with +a power reading at each interval. The voltage is applied +differentially to the CPS, with +V being applied to one +cantilever and -V applied to the other, nominally giving +a θ and −θ phase shift for each path respectively. During +the calibration, all other phase shifters’ voltages are held +constant. +To find the cross and bar states, we fit an +offset sine curve to the data and take the maximum and +minimum values. +Next, we calibrate the triple-phase shifter switching +MZIs. We begin by setting the voltage of the SPS to 0 V, +and then calibrate the CPSs. Since the total extinction +of this MZI is dependent on the relationship between θ +and φ + ψ, we do a nested two dimensional sweep of +the applied voltages from -25 V to 25 V in increments +of 0.25 V. The cross and bar states are found by fitting +a two dimensional sinusoid to the data and taking the +maximum and minimum values respectively. We then set +the two CPSs to their bar state (minimum transmission) +and calibrate the SPS by sweeping the voltage from -25 +V to 25 V in increments of 0.1 V. The SPSs are also +operated differentially in a push-pull configuration. We +fit an offset sine curve to this data to find its cross and +bar states. With how we set up this calibration, the bar +state is defined to be 0 V due to the CPSs being set to +their bar state. Full calibration results are shown in the +Supporting Information Section 3. +B. +Diamond Chiplet Fabrication +For the generation of negatively charged SiV in the di- +amond, we relieved the strained surface of the diamond +plate by removing the top 7 µm using Ar/Cl2 plasma +etching followed by O2 etching. The sample was subse- +quently implanted with Si29 at 190 keV with a dose of +5×1010 ions/cm2 (Innovion Inc.). It was then annealed in +an ultra-high vacuum furnace (< 10−7 mbar) at 1200 ◦C +and cleaned in a boiling tri-acid mixture (1:1:1 nitric acid, +sulfuric acid, and perchloric acid at 345 ◦C). A 180 nm sil- +icon nitride (Si3N4) was chemical vapor deposited on the +diamond, and patterned using electron-beam lithography +and CF4 reactive-ion etching (RIE). We isotropically un- +dercut the diamond quantum microchiplet (QMC) using + +7 +an oxygen inductively coupled plasma (ICP) RIE. Lastly, +we submerged the sample in hydrofluoric acid to remove +the Si3N4 hard mask and alumina [54]. +C. +SiV Linewidth and Autocorrelation +Measurements +The diamond sample used in these experiments was +fabricated into a QMC [9] as described above. We then +broke the QMC into individual waveguides and placed +them overhanging the edge of a cleaved Si chip using +tungsten tips. We mounted the Si chip vertically in the +Montana cryostat to enable perpendicular excitation and +collection. +When measuring the individual SiV emitters, we cou- +pled a single waveguide mode at a time to a multi- +mode fiber for high-efficiency collection. We fit the emit- +ter’s PLE linewidth scans with a Voigt profile using the +Nelder-Mead simplex algorithm for the fit optimization. +For the autocorrelation measurements, we used a 50:50 +fiber splitter to send the light to two APDs in a Hanbury- +Brown-Twiss setup. During these measurements, we in- +put pulsed 532 nm light into the waveguide through the +collection objective. We input the minimum amount of +repump needed to obtain the maximum count rate, pro- +viding maximum charge state initialization. +We gated +the detectors to only collect data when the repump beam +is off. +To fit the g(2) values, we used a Lorentzian fit to the +data. +g(2)(τ) = C1 + C2 +� +1 +2Γ +τ + +� 1 +2Γ +�2 +� +(1) +where τ is the delay time between coincident counts, C1 +and C2 are the offset and scaling factors respectively, Γ is +the full-width half-max of the emitter spectrum. For the +one emitter that showed pronounced bunching behavior, +we fit the data to a three level system and added in an +overall offset and scaling factor to account for the non- +ideality of the data due to dark counts and jitter from +the APD. +g(2)(τ) = C1 + C2 +� +1 − (1 + a)e−|τ|/τ1 + ae−|τ|/τ2� +(2) +where τ is the delay time between coincident counts, C1 +and C2 are the offset and scaling factors respectively, a is +the scaling factor determining the strength of the photon +bunching, τ1 is the antibunching time constant, and τ2 +is the bunching time constant. When compared to the +standard Lorentzian fit, we obtained similar values for +g(2)(0) (0.09 vs 0.07). +The error bars reported in the +manuscript correspond to one standard deviation in the +fit parameters. +ACKNOWLEDGMENTS +Major funding for this work is provided by MITRE for +the Quantum Moonshot Program. +D.E. acknowledges +partial support from Brookhaven National Laboratory, +which is supported by the U.S. Department of Energy, +Office of Basic Energy Sciences, under Contract No. DE- +SC0012704 and the NSF RAISE TAQS program. M.E. +performed this work, in part, with funding from the Cen- +ter for Integrated Nanotechnologies, an Office of Science +User Facility operated for the U.S. Department of En- +ergy Office of Science. +M.D. and M.Z. thank MITRE +engineers L. Chan, K. Dauphinais, and S. Vergados for +their support in building mechanical and electronic com- +ponents. K.P. and M.D. thank S. Trajtenberg and Y. S. +Duan for additional experimental support and C. Li and +Y. Hu for helpful conversations and comments. +AUTHOR CONTRIBUTIONS +K.J.P. built the experimental setup and performed +the device characterization and calibration experiments. +K.J.P. and D.A.G., with assistance from G.C. and M.D., +built and performed the SiV direct excitation experi- +ments. +K.J.P. performed the data analysis. +M.Z. de- +signed the electronic control system. M.D., with assis- +tance from A.M., designed the APIC. M.E. and A.J.L., +with assistance from D.D., supervised the APIC fabri- +cation. +K.C.C. and L.L. fabricated the diamond mi- +crochiplets. K.C.C. performed the diamond waveguide +simulations. +M.D. and D.E. conceived the experiment +and device architecture. +M.D., G.G., M.E., and D.E. +supervised the project. +K.J.P. and M.D. wrote the +manuscript with input from all authors. +ADDITIONAL INFORMATION +Supplementary information is available for experimen- +tal methods related to programming and calibrating the +photonic integrated circuit and collection apparatus. +COMPETING INTERESTS +D.E. is a scientific advisor to and holds shares in QuEra +Computing. +DATA AVAILABILITY +The data that support the plots within this paper are +available from the corresponding authors upon reason- +able request. + +8 +[1] M. Atat¨ure, D. Englund, N. Vamivakas, S.-Y. Lee, and +J. Wrachtrup, Material platforms for spin-based photonic +quantum technologies, Nature Reviews Materials 3, 38 +(2018). +[2] N. Bar-Gill, L. M. Pham, A. Jarmola, D. Budker, and +R. L. Walsworth, Solid-state electronic spin coherence +time approaching one second, Nat. 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Saffman, Multi-qubit en- +tanglement and algorithms on a neutral-atom quantum +computer, Nature 604, 457 (2022). +[54] S. Mouradian, N. H. Wan, T. Schr¨oder, and D. Englund, +Rectangular photonic crystal nanobeam cavities in bulk +diamond, Appl. Phys. Lett. 111, 021103 (2017). + +Modular chip-integrated photonic control of artificial atoms in diamond +nanostructures: Supplementary Information +Kevin J. Palm1,†,∗, Mark Dong1,2,†,∗, D. Andrew Golter1, Genevieve Clark1,2, Matthew +Zimmermann1, Kevin C. Chen2, Linsen Li2, Adrian Menssen2, Andrew J. Leenheer3, +Daniel Dominguez3, Gerald Gilbert4,∗, Matt Eichenfield3,5,∗, and Dirk Englund2,6 +1The MITRE Corporation, 202 Burlington Road, Bedford, Massachusetts 01730, USA +2Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA +3Sandia National Laboratories, P.O. Box 5800 Albuquerque, New Mexico, 87185, USA +4The MITRE Corporation, 200 Forrestal Road, Princeton, New Jersey 08540, USA +5College of Optical Sciences, University of Arizona, Tucson, Arizona 85719, USA +6Brookhaven National Laboratory, 98 Rochester St, Upton, New York 11973, USA and +†These authors contributed equally +(Dated: January 11, 2023) +I. +OPTICAL EXPERIMENTAL SETUP +Figure S1 depicts the optical setup for all of the experiments demonstrated in this manuscript. Laser light was +input into the binary MZI tree through a 10-port fiber grating with a tunable laser set at 737 nm (M Squared +Lasers). We collected the edge coupled light with a 100x, 0.9 NA infinity corrected objective (Mitutoyo) and filtered +the light with a linear polarizer. We then routed the light to different characterization devices using mirrors on flip +stages for ease of switching between measurements: 1) a CCD camera for alignment, 2) a DC photodiode (Newport +818-SL) for device extinction characterization, and 3) a 125 MHz fast photodiode (New Focus 1801) for pulsed light +characterization. When performing diamond excitation experiments, we routed the light onto a spatial light modulator +(SLM) (Thorlabs Exulus) and directed the beams onto vertically-mounted diamond waveguides in a Montana cryostat. +f = 150 mm +f = 150 mm +f = 100 mm +f = 250 mm +Mirror +f = 34 mm +Mirror +Flip +Mirror +Flip Mirror +f = 500 mm +f = 500 mm +Iris +DC PD +Fast PD +737 nm +Laser +Flip BS +CCD +Iris +100x +Objective +f = 2 mm +f = 150 mm +f = 150 mm +SLM +f = 100 mm +f = 250 mm +Mirror +Cryostat +f = 34 mm +Mirror +Flip +Mirror +Flip Mirror +f = 500 mm +f = 500 mm +Iris +DC PD +Fast PD +737 nm +Laser +Flip BS +CCD +Iris +5K +Linear +Polarizer +MZI Tree +f = 2 mm +Flip +Mirror +APD +EMCCD +f = 25.4 mm +50:50 +Fiber BS +750 nm +Long Pass +APD +SLM +532 nm +Laser +AOM +f = 4 mm +600 nm +Dichroic +f = 200 mm +FIG. S1. Optical setup for binary tree experiments. Flip mirrors are used to redirect light to different optical detectors for +different types of calibration experiments. Labels: f = effective focal length, CCD = charge-coupled device, BS = beamsplitter, +SLM = spatial light modulator, AOM = acoustic optical modulator, APD = avalanche photodiode, EMCCD = electron- +multiplying charge-coupled device. +arXiv:2301.03693v1 [quant-ph] 9 Jan 2023 + +2 +c) +Waveguides +b) +a) +5 μm +5 μm +FIG. S2. Imaging edge coupled waveguides. a) Picture of the wire bonded APIC onto the PCB. Light is input into the chip +through a mounted fiber array (right) into grating couplers. The light is then collected with an objective (left) to be imaged or +routed to the cryostat. The top objective is for observing the chip and aligning the fiber array. b) Image of the edge coupled +waveguides. c) Same waveguides, but with light being evenly split through each port on the chip. +The 34 mm lens in the beam path was embedded in the side of the cryostat with a custom inset. A long working +distance objective (50x Mitutoyo) collected the emission from the diamond waveguides, whose photons then traveled +to the electron-multiplying charge-coupled device (EMCCD), or coupled one of the waveguide modes to a fiber for +autocorrelation experiments in a Hanbury-Brown-Twiss setup with two avalanche photodiodes (APD). A 532 nm +laser (Obis) was introduced in the collection path and pulsed into the diamond waveguide during autocorrelation +experiments to stabilize the SiV charge state [1]. +Figure S2a are images of the APIC coupling setup. The laser light was input through the fiber array at 18◦ (right) +and collected with the horizontal objective (left). The vertical objective was used to align the fibers and inspect the +chip. Figure S2b shows the imaged edge of the waveguides, with Fig. S2c showing the same waveguides with light +being emitted through the chip. +II. +PHOTONIC INTEGRATED CIRCUIT PACKAGING +Our photonic chip was fabricated on 200-mm Si technology with a CMOS-compatible fabrication procedure. The +binary tree was initially cleaved from the full-wafer die, and then wire-bonded to a custom printed circuit board +(PCB). The cantilever phase shifters (CPS) were electrically driven with a 32-channel voltage controller (Marvin Test +Systems GX1632e) with a voltage range of ±25 V. The CPSs were driven in a push-pull configuration for maximum +phase shift (e.g. if +10 V is applied to the top cantilever, -10 V is applied to the bottom cantilever). The strain-optic +phase shifters (SPS) were driven with two 200 MHz arbitrary waveform generator (AWG) PXIe cards (Spectrum +M4x.6622), each with four output channels with a voltage range of ± 2.5 V and 5x amplified on the PCB at a max +slew rate of 8000Vµs−1 (Texas Instruments THS3491). In the course of any experiment, the SPSs were operated in +a push-pull configuration similar to the CPSs. +III. +MZI CALIBRATION PLOTS +A labeling of the different phase shifters is shown in Figure S3. For results of a full calibration, see Fig. S4 for the +routing MZIs, Fig. S5 for the CPSs in the triple phase shifters, and Fig. S6 for the SPSs. + +an +Apo +0.28 +f=2003 +00 +10 +11 +20 30 +40 +31 +32 +33 +21 +22 +23 +41 +42 +43 +1 mm +FIG. S3. Labeling of different phase shifters for calibration. Light is input from the 6th grating coupler from the top. Output +0 is defined as the top output in this microscope image, with Output 3 as the bottom output. +-20 +-10 +0 +10 +20 +Voltage 00 (V) +0 +0.2 +0.4 +0.6 +0.8 +1 +Normalized Transmission +-20 +-10 +0 +10 +20 +Voltage 10 (V) +0 +0.2 +0.4 +0.6 +0.8 +1 +Normalized Transmission +-20 +-10 +0 +10 +20 +Voltage 11 (V) +0 +0.2 +0.4 +0.6 +0.8 +1 +Normalized Transmission +-20 +-10 +0 +10 +20 +Voltage 00 (V) +-30 +-25 +-20 +-15 +-10 +-5 +0 +Normalized Transmission (dB) +-20 +-10 +0 +10 +20 +Voltage 10 (V) +-15 +-10 +-5 +0 +Normalized Transmission (dB) +-20 +-10 +0 +10 +20 +Voltage 11 (V) +-30 +-25 +-20 +-15 +-10 +-5 +0 +Normalized Transmission (dB) +d) +e) +b) +f) +c) +a) +FIG. S4. Routing cantilever phase shifter calibration. Voltages are swept from -25 V to 25 V while all other phase shifters are +held static. Normalized transmission vs applied voltage for a) PS00, b) PS10, and c) PS11. Each phase shifter has a Vπ of ∼30 +V. Extinction measurements for d) PS00, e) PS10, and f) PS11. + +Magnification: X30.0 +1.0000mm4 +b) +d) +c) +a) +f) +h) +g) +e) +FIG. S5. +Switching cantilever phase shifter calibration. Voltages are swept from -25 V to 25 V in a nested sweep while all +other phase shifters are held static. Normalized transmission vs applied voltage for a) PS20 and PS30, b) PS21 and PS31, c) +PS22 and PS32, and d) PS23 and PS33. Extinction measurements for e) PS20 and PS30, f) PS21 and PS31, g) PS22 and PS32, +and h) PS23 and PS33. +-10 +-5 +0 +5 +10 +Voltage 40 (V) +0 +0.2 +0.4 +0.6 +0.8 +1 +Normalized Transmission +-10 +-5 +0 +5 +10 +Voltage 41 (V) +0 +0.2 +0.4 +0.6 +0.8 +1 +Normalized Transmission +-10 +-5 +0 +5 +10 +Voltage 42 (V) +0 +0.2 +0.4 +0.6 +0.8 +1 +Normalized Transmission +-10 +-5 +0 +5 +10 +Voltage 43 (V) +0 +0.2 +0.4 +0.6 +0.8 +1 +Normalized Transmission +-10 +-5 +0 +5 +10 +Voltage 40 (V) +-40 +-30 +-20 +-10 +0 +Normalized Transmission (dB) +-10 +-5 +0 +5 +10 +Voltage 41 (V) +-40 +-30 +-20 +-10 +0 +Normalized Transmission (dB) +-10 +-5 +0 +5 +10 +Voltage 42 (V) +-40 +-30 +-20 +-10 +0 +Normalized Transmission (dB) +-10 +-5 +0 +5 +10 +Voltage 43 (V) +-40 +-30 +-20 +-10 +0 +Normalized Transmission (dB) +b) +d) +c) +a) +f) +h) +g) +e) +FIG. S6. Switching strain-optic phase shifter calibration. Preceding CPSs are set to the bar state (minimum transmission) and +then the voltage of the SPS is swept from -25 V to 25 V. Normalized transmission vs applied voltage for a) PS40, b) PS41, c) +PS42, and d) PS43. Extinction measurements for e) PS40, f) PS41, g) PS42, and h) PS43. + +20 +20 +20 +20 +0.8 +0.8 +0.8 +0.8 +10 +10 +Voltage 23 (M) +10 +10 +0.6 +0.6 +0.6 +0.6 +0 +0 +0 +0 +0.4 +0.4 +0.4 +0.4 +-10 +-10 +0.2 +0.2 +0.2 +0.2 +-20 +-20 +-20 +-20 +-10 +10 +20 +-20 +-10 +10 +20 +-20 +-10 +0 +10 +20 +-20 +-10 +0 +10 +20 +0 +Voltage 30 (M) +Voltage 31 (M) +Voltage 32 (M) +Voltage 33 (M) +20 +20 +20 +20 +-10 +-10 +-10 +-1 +Voltage 20 (M) +Voltage 21 (M) +Voltage 22 (M) +10 +Voltage 23 (M) +10 +10 +10 +-20 +-20 +-2( +0 +-20 +0 +0 +0 +-10 +-10 +-10 +-10 +-3( +-30 +-30 +-30 +-20 +-20 +-20 +-20 +-40 +-4 +-40 +-40 +10 +20 +-10 +-20 +-10 +20 +-10 +0 +-20 +-10 +0 +-20 +10 +20 +10 +10 +20 +Voltage 30 (M) +Voltage 31 M) +Voltage 32 (M) +Voltage 33 (M)5 +0 +0.5 +1 +1.5 +2 +Time ( s) +0 +0.2 +0.4 +0.6 +0.8 +1 +Normalized +Transmission +0 +0.5 +1 +1.5 +2 +Time ( s) +Programmed Signal (norm) +b) +a) +Chan 4 +Chan 3 +Chan 2 +Chan 1 +Chan 3 +FIG. S7. Crosstalk investigation. a) Programmed signal to each of the four SPSs. Channel 3 was imaged onto the power meter +for measurement. The first pulse on Channel 3 is a control pulse for comparison, followed by the other three channels pulsing, +and ending with all four channels pulsing. b) Measured power of Channel 3. We do not measure any cross talk above the noise +floor between any of the channels. +IV. +CROSSTALK INVESTIGATION +To investigate whether adjacent SPSs had any effect on each other, we ran crosstalk experiments. +For these +experiments, we set a pulse series to the four SPSs and measured if signals sent to adjacent SPSs had any effect on +the measured output. An example measurement is shown in Fig. S7. In this experiment, we had an initial control +pulse, a pulse of all other channels except for the measured channel, and ending with a pulse of all four channels. We +did not measure differences between the control pulse and all four of the channels pulsing. We also did not measure +any output above the noise floor when the other three channels are pulsed and the active channel remains in the bar +state. +V. +COLLECTION EFFICIENCY CALCULATION AND LIFETIME MEASUREMENT +We used a resonant pulsing scheme to measure the collection efficiency of our system. We attached an electro- +optic modulator (EOM, iXblue) on the input of our laser source and drove it with an AWG (Tektronix 70001B) to +achieve the short excitation pulses necessary for the measurements. For all of these measurements, we drove the C +transition of the SiV (Fig. S8a). First, we excited a single emitter with a 20 ns resonant pulse and collected the +time-tagged sideband of the emission. We fit the resulting data with an exponentially damped sinusoid to extract the +Rabi frequency for our input laser power (Fig. S8b). This fit gave us an optical π-pulse of 400 ps. We then applied +this optical π-pulse repeatedly, with one π-pulse every 240 ns to ensure the emitter decayed into the ground state and +thermalized, and collected the time-tagged sideband emission. Every 400 µs we applied a 1-µs 532 nm repump pulse +to initialize the SiV to the negative charge state. Fig. S8c shows the resulting data. We fit the exponential decay of +the counts and find a lifetime of T1 = 1.76(1) ns. +To find the collection efficiency of our apparatus, we subtracted the background in Fig. S8c, sum the remaining +counts, and divided by the total number of applied π-pulses. This gave us a 0.038% chance of measuring a photon +in the sideband per pulse. A few corrections onto this value must be made to find the accurate collection efficiency. +We experimentally measured the transmission percent of the SiV emission through our optical filters by measuring +a photoluminescence spectra with 532 nm off-resonant excitation with the 750 nm longpass filter in and out of the +collection path, and obtained a sideband transmission of 15%. During this measurement a 600 nm longpass filter +was included to block the excess 532 nm pump that coupled to the waveguide mode. +In our setup, we did not +directly control the electron distribution between the two ground states split by spin-orbit coupling. This population +distribution is determined by [2]: +PLB +PUB += e−h∆gs/kBT +(S1) +where PLB and PUB are the populations in the lower and upper ground states respectively, h is Planck’s constant, + +6 +-1 +0 +1 +2 +3 +4 +5 +Time (ns) +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +Intensity (a.u.) +-2 +0 +2 +4 +6 +8 +10 +12 +14 +Time (ns) +0 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +Counts +T1 = 1.76(1) ns +Ω/2π = 1.28(3) GHz +b) +c) +406 THz +A +B +C +D +Δes = +260 GHz +Δgs = +50 GHz +a) +0.4 ns +240 ns +FIG. S8. Collection efficiency measurement. a) Electronic structure of the SiV color center. The ground and excited states +are split by spin-orbit coupling. For these experiments we drive the C transition. b) Rabi oscillations excited by a 20 ns pulse +resonant on the C transition. We fit the oscillations with an exponentially damped sinusoid and find a Rabi frequency of Ω/2π +=1.28(3) GHz, corresponding to an optical π-pulse of 400 ps. c) Pulsed excitation of a single SiV with repeated π-pulses. +Exponential fit of the decay gives a lifetime of T1 = 1.76(1) ns. Coupling efficiency is found by summing the total counts in +the decay and normalizing by applied π-pulses. +kB is the Boltzmann constant, ∆gs is the ground state splitting, and T is the temperature. At 5 K, we find that the +electron is in the lower ground state with 62% probability. We then accounted for an APD measurement efficiency +of 55% at 737 nm (Excelitas Technologies SPCM-AQRH) and the SiV quantum efficiency of 5% [3, 4]. Applying +these four corrections, we obtained a final collection efficiency of 15%. We note that this collection efficiency could +potentially be higher as the SiV was not initialized to the correct charge state with 100% fidelity. Continuous wave +measurements with 532 nm excitation indicated that the off-resonant repump put the vacancy in the correct charge +state with 60% probability, with no emission recorded when the vacancy was not in the correct charge state. We also +note that there is uncertainty in the quantum efficiency value of the emitter, as different nanostructuring schemes +cause for different modifications to this value [5]. Our design could be further improved by adding a Bragg reflector +to the back half of our diamond waveguide, which potentially increases our collection efficiency by a factor of two. +VI. +DIAMOND WAVEGUIDES NA +We used finite element simulations (COMSOL Multiphysics) to determine the far field NA of our diamond waveg- +uides for a measure of the scalability of our system, and find that our waveguides have a calculated NA = 0.26, as +shown in Fig. S9. +|E|2 +Far Field Projection +FIG. S9. Diamond waveguide far field projection simulations. Uy and Uz are the directional unit vectors and |E|2 is normalized +to 1. We find that the projected mode from our waveguides is contained in an NA = 0.26, denoted by the red circle in the plot. + +1 +0.8 +0.5 +0.6 +0 +0.4 +-0.5 +0.2 +1 +0 +-1 +-0.5 +0 +0.5 +1 +Uy7 +FIG. S10. Grating coupler optimization. The grating period and fill factor were swept around the simulated maximum value +to find the experimentally best grating coupler. The highest measured coupling is 28.3% coupling at a period of 578 nm and a +fill factor of 0.437. +VII. +IMPROVED GRATING COUPLER EFFICIENCIES +To improve the insertion loss of our device, we ran a sweep of the period and fill factors of our grating couplers +and measured the coupling efficiency. We measured the efficiency using loopback structures with the inputs and +outputs measured through a fiber array. The results of this sweep are shown in Fig. S10. The highest measured +coupling is 28.3% coupling at a period of 578 nm and a fill factor of 0.437. To improve this coupling even further, we +experimented with depositing an index matching fluid (n = 1.52) on the grating couplers, thick enough that the fiber +tips are completely submerged in the fluids when aligned for coupling. Using this technique, we are able to measure +41% grating coupler efficiency for grating couplers with a period of 621 nm and fill factor of 0.483. +[1] S. Maity, L. Shao, S. Bogdanovi´c, S. Meesala, Y.-I. Sohn, N. Sinclair, B. Pingault, M. Chalupnik, C. Chia, L. Zheng, K. Lai, +and M. Lonˇcar, Coherent acoustic control of a single silicon vacancy spin in diamond, Nat. Commun. 11, 193 (2020). +[2] K. D. Jahnke, A. Sipahigil, J. M. Binder, M. W. Doherty, M. Metsch, L. J. Rogers, N. B. Manson, M. D. Lukin, and +F. Jelezko, Electron–phonon processes of the silicon-vacancy centre in diamond, New J. Phys. 17, 043011 (2015). +[3] E. Neu, M. Agio, and C. Becher, Photophysics of single silicon vacancy centers in diamond: implications for single photon +emission, Opt. Express 20, 19956 (2012). +[4] J. Benedikter, H. Kaupp, T. H¨ummer, Y. Liang, A. Bommer, C. Becher, A. Krueger, J. M. Smith, T. W. H¨ansch, and +D. Hunger, Cavity-Enhanced Single-Photon source based on the Silicon-Vacancy center in diamond, Phys. Rev. Applied 7, +024031 (2017). +[5] J. Riedrich-M¨oller, C. Arend, C. Pauly, F. M¨ucklich, M. Fischer, S. Gsell, M. Schreck, and C. Becher, Deterministic coupling +of a single silicon-vacancy color center to a photonic crystal cavity in diamond, Nano Lett. 14, 5281 (2014). + +Grating Coupling Efficiency +0.3 +0.6 +0.25 +I Factor +0.55 +0.2 +! +0.5 +0.15 +Grating F +0.45 +0.1 +G +0.4 +0.05 +0.35 +0 +500 +520 +540 +560 +580 +600 +620 +640 +Grating Period (nm) \ No newline at end of file diff --git a/ndE2T4oBgHgl3EQfJga_/content/tmp_files/load_file.txt b/ndE2T4oBgHgl3EQfJga_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..da297b81a59de4e0cc8f853b2ae0e00cd04ea1f2 --- /dev/null +++ b/ndE2T4oBgHgl3EQfJga_/content/tmp_files/load_file.txt @@ -0,0 +1,1710 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf,len=1709 +page_content='Modular chip-integrated photonic control of artificial atoms in diamond nanostructures Kevin J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Palm1,†,∗, Mark Dong1,2,†,∗, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Andrew Golter1, Genevieve Clark1,2, Matthew Zimmermann1, Kevin C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Chen2, Linsen Li2, Adrian Menssen2, Andrew J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Leenheer3, Daniel Dominguez3, Gerald Gilbert4,∗, Matt Eichenfield3,5,∗, and Dirk Englund2,6∗ 1The MITRE Corporation, 202 Burlington Road, Bedford, Massachusetts 01730, USA 2Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 3Sandia National Laboratories, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Box 5800 Albuquerque,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' New Mexico,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 87185,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' USA 4The MITRE Corporation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 200 Forrestal Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Princeton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' New Jersey 08540,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' USA 5College of Optical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' University of Arizona,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Tucson,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Arizona 85719,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' USA 6Brookhaven National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 98 Rochester St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Upton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' New York 11973,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' USA and †These authors contributed equally (Dated: January 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 2023) A central goal in creating long-distance quantum networks and distributed quantum computing is the development of interconnected and individually controlled qubit nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Atom-like emitters in diamond have emerged as a leading system for optically networked quantum memories, motivating the development of visible-spectrum, multi-channel photonic integrated circuit (PIC) systems for scalable atom control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' However, it has remained an open challenge to realize optical programmabil- ity with a qubit layer that can achieve high optical detection probability over many optical channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Here, we address this problem by introducing a modular architecture of piezoelectrically-actuated atom-control PICs (APICs) and artificial atoms embedded in diamond nanostructures designed for high-efficiency free-space collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The high-speed 4-channel APIC is based on a splitting tree mesh with triple-phase shifter Mach-Zehnder interferometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' This design simultaneously achieves optically broadband operation at visible wavelengths, high-fidelity switching (> 40 dB) at low voltages, sub-µs modulation timescales (> 30 MHz), and minimal channel-to-channel crosstalk for repeatable optical pulse carving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Via a reconfigurable free-space interconnect, we use the APIC to address single silicon vacancy color centers in individual diamond waveguides with inverse tapered couplers, achieving efficient single photon detection probabilities (15%) and second-order autocorrelation measurements g(2)(0) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='14 for all channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The modularity of this distributed APIC - quantum memory system simplifies the quantum control problem, potentially enabling further scaling to 1000s of channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Approved for Public Release;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Distribution Unlimited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Public Release Case Number 22-4195 © 2022 The MITRE Corporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' INTRODUCTION Solid-state artificial atoms [1], many of which have long-lived quantum memories [2–5], can achieve photon- mediated remote-entanglement [6, 7], and can be hetero- geneously integrated with photonics [8, 9], are a promis- ing platform for the construction of large-scale quan- tum networks [10–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The networking of these atom- like emitters requires an efficient and high-fidelity op- tical interface for both reconfigurable optical address- ing and collection of photoluminescence (PL) at visi- ble wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The optical control layer thus presents two challenges: i) scalable high-fidelity manipulation of optical fields at high speeds, which necessitates high- quality optical switches in atom-control photonic inte- grated circuit (APIC) [13] platforms and ii) scalable high- efficiency photon collection from remotely addressable single emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' While previously demonstrated visible- wavelength APIC platforms such as thin-film lithium ∗ kpalm@mitre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='org;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' mdong@mitre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='org;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' ggilbert@mitre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='org;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' eichenfield@arizona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' englund@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='edu niobate [14–16], thermally-tuned silicon nitride [17–19], and piezoelectrically-actuated silicon nitride [20–23] all have promise for scalability, none currently combine opti- cally broadband operation, high switching contrast (> 40 dB) at nanosecond time scales, and low voltage oper- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' On the photon collection side, efficient collec- tion has been demonstrated using standard confocal mi- croscopy [24–26], by leveraging photonic nanostructures such as immersion lenses [27–29] and cavities [8, 30–33], or single-channel fiber collection from tapered waveg- uides [8, 34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Collection through a heterogeneously- integrated photonic chip [9, 36, 37] at the cost of some op- tical loss due to the diamond-chip interface has also been reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' To date, these past works treated each side of the optical control layer separately, but there remains an open question of how to combine the requirements of i) and ii) into a single scalable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Here we introduce an architecture for the optical con- trol layer consisting of modular piezoelectrically-actuated APICs and diamond microchiplets with implanted sin- gle emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' In this configuration, the excitation and collection optical paths are perpendicular, enabling the inverse tapered diamond waveguides to take advantage arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='03693v1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='[quant-ph] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='9 Jan 2023 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Waveguides ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='f) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='e) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='5K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Routing “Single” MZI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Input Grating ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Coupler ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Routing MZIs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Electrical Contacts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Switching SPSs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Switching CPSs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Edge Coupled ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Outputs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='ϴ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Switching “Triple” MZI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Output ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='ϴ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='ϴ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='ϴ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='ɸ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='ɸ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='ψ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='ψ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Output ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='= Single MZI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='= Triple MZI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Output ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='g) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='500 μm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Waveguide wire ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Not connected ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Waveguides ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='f) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='e) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Routing “Single” MZI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Input Grating ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Coupler ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Routing MZIs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Electrical Contacts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Switching SPSs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Switching CPSs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Edge Coupled ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Outputs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='ϴ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Switching “Triple” MZI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Output ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='ϴ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='ϴ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='ϴ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='ɸ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='ɸ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='ψ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='ψ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Output ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='= Single MZI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='= Triple MZI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Output ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='g) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='500 μm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Waveguide wire ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Not connected ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Waveguides ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='f) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='e) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Routing MZIs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Electrical Contacts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Switching SPSs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Switching CPSs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Edge-Coupled ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Outputs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='1 mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='e) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='f) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Excitation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Collection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Grating Coupler ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Inputs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Si ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Vacancy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='g) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Photonic integrated network switch architecture for local addressing of multiple quantum emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' a) Routing “single” MZIs to split the single input into each of the four ports and b) switching “triple” MZIs that enable fast arbitrary pulsing of light with high extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The routing MZIs consist of a single cantilever phase shifter (CPS) and two 50:50 directional couplers, while the switching MZIs consist of two CPSs, a strain-optic phase shifter (SPS), and three 50:50 couplers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' c) Schematic of the binary tree switch design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' d) Microscope image of the fabricated integrated network switch with the CPSs and SPSs labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Light is input through grating couplers on the left side and collected through edge-coupled outputs on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' e) Cryostat setup housing the quantum emitters with light from the switch projected through free-space for quantum control experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' f) Diamond quantum microchiplets with g) implanted Si vacancy color centers with light pulses from the chip controlling the optical emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The diamond nanostructure allows for high-efficiency collection of the emitter’s emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' of free-space modal conversion for efficient collection through the optical path parallel to the waveguides while maintaining the ability to selectively address a large area of distinct emitters through the perpendicu- lar path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We demonstrate our control scheme by first satisfying requirement i) through our APIC switch, im- plemented as a 4-channel binary tree mesh [13] with visible-wavelength switching and power routing capabil- ities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The APIC’s switching circuit uses an optically- broadband triple-phase shifter design that takes advan- tage of hardware error correction [38, 39] and a stronger strain-optic response than previous designs, enabling low switching voltages while maintaining high-contrast (> 40 dB) and high-speed (> 30 MHz) switching performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The switch shows negligible cross-talk between channels and enables repeatable arbitrary pulse carving on all four outputs, combined with > 1 MHz power balancing be- tween ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We further demonstrate requirement ii) by applying the APIC to a local group of quantum emit- ters by projecting the optical output channels onto ion- implanted silicon vacancy color centers (SiVs) [28, 40] in diamond microchiplets [9] mounted in a 5K cryostat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Through PL excitation (PLE) and second-order autocor- relation measurements, we demonstrate optical address- ing with independent temporal control of four spatially distinct color centers and achieve high (15%) collection efficiency, single emitter linewidths of 152 MHz - 287 MHz, and g(2)(0) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='06 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The modularity of this architecture allows for easy switching between different sets of quantum emitters by adding different sets of dia- mond microchiplets into the cryostat setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Our APIC excitation and diamond collection techniques should en- able scalable quantum control of emitters as part of a larger network of quantum nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' PHOTONIC INTEGRATED SWITCH DESIGN AND OPERATION The schematic of our APIC-to-diamond control archi- tecture is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The APIC design consists of a “sin- gle” routing Mach-Zender Interferometer (MZI) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 1a) and a “triple” switching MZI (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 1b) arranged in a bi- nary tree architecture (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' A single cantilever phase shifter (CPS) [22] in the routing MZIs directs the desired amount of light to the appropriate outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The switch- ing MZI uses three phase shifters: two CPSs that en- able optically broadband and high-fidelity routing (> 40 dB) for cross and bar ports using hardware error correc- tion robust to fabrication imperfections [39] and a third, strain-optic phase shifter (SPS) [21, 41], that enables a fast phase response for on-off switching of the output 淼Magnification: X30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='0000mman Apo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='28 f=200Magnification: X30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='0000mm3 10 5 0 5 10 V (V) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='8 1 Tnorm 10 5 0 5 10 V (V) 40 30 20 10 0 T (dB) 20 10 0 10 20 V (V) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='8 1 Tnorm 20 10 0 10 20 V (V) 30 25 20 15 10 5 0 T (dB) 50:50 coupler 50:50 coupler Phase Shifter 50:50 coupler 50:50 coupler Phase Shifter 1 Phase Shifter 2 Phase Shifter 3 ϴ ϴ ϕ ϕ ϴ ϴ ψ ψ Tnorm T𝜖(dB) a) d) f) g) e) h) Vπ = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='9 V c) b) 1 μm 1 μm i) j) Vπ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='5 V 100 μm 100 μm FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Device performance and calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Simulated TM optical waveguide mode for the a) 400 nm waveguides and b) the 5 µm waveguides in the SPSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' c) Microscope image of a routing MZI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' A voltage is anti-symmetrically applied to each side of the phase shifter to give the maximum actuation range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' d) Normalized transmission (Tnorm = T/Tmax) and e) extinction (Tϵ = 10 × log(Tnorm)) measured from a single output with the applied voltage to the cantilever swept from -25 to 25 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' A single phase shifter achieves 25-30 dB extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' f) Microscope image of a switching MZI with three phase shifters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The first two CPSs are calibrated with the SPS held at 0 V to maximize output port extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' g) Normalized transmission Tnorm and h) extinction Tϵ plots measured from sweeping the applied voltages of the two CPSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The addition of the second cantilever compared to the single MZI allows for the output extinction to exceed 40 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' i) Normalized transmission Tnorm and j) extinction Tϵ plots for the SPS, calibrated after the two CPSs in the switching MZI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' During operation, a CPU controller programs the two CPSs to route the light to a dump port while the SPS is held at 0 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We then can send an arbitrary pulse sequence to the SPS to switch the light to the output port without having to change the applied DC voltages to the CPSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' A microscope image of the APIC is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 1d, with the different phase shifters and electrical contacts labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We input light into the chip with an optical fiber array through a single grating leading to the rout- ing MZIs, while other inputs are only used for device calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We then collect the edge-coupled light from each output with a high-NA objective, enabling imaging of the outputs into any system for optical control exper- iments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Figure 1e shows the optical imaging schematic where the output channels are projected into a cryostat to use for optical control of quantum emitters in diamond waveguides (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 1f), such as SiVs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 1g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' This config- uration enables perpendicular excitation of the diamond waveguides, with the single photon fluorescence from the emitters coupling to the diamond waveguide mode and emitting vertically for collection through inverse tapered couplers, as shown in Figure 1f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' This free-space collection allows for efficient and scalable detection due to low-loss collection optics that are robust to misalignment when compared with fiber coupling or PIC integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Electri- cal control of the integrated optical components is made through a custom printed circuit board (PCB) with wire bonds to the APIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Commercial arbitrary waveform gen- erator boards, embedded in a National Instruments PXIe system, control the CPSs and SPSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' A single board with 22 active channels controls the CPSs, providing ± 25 V, and two boards with four channels of arbitrary wave- form generation each control the SPSs, providing ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='5 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' High-speed amplifiers on the PCB amplify the signals to the SPSs to ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='5 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' See Supplementary Sections 1 and 2 for more details on the optical and electrical com- ponents of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Figure 2 summarizes the APIC characterization and calibration by monitoring the transmission of each edge- coupled optical output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' For all optical tests, we use 737 nm wavelength laser light coupled into the TM mode of the on-chip 400 nm wide by 300 nm thick silicon ni- tride waveguides (modal shape simulated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 2a), which adiabatically expand to 5 µm wide in the SPS (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 2b) to increase strain-optic sensitivity [21, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The less-confined TM mode takes advantage of a higher pho- toelastic responsivity when compared to the TE mode [42], resulting in a lower Vπ of the phase shifter than previously reported [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Our DC calibration results for the routing MZIs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 2c) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 2d-e and for the switching MZIs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 2f) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 2g- j, highlighting the low-voltage operation of the SPS for switching and high on-off extinction ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' These high extinction ratios for the triple-phase shifter are enabled by the second CPS accounting for fabrication imperfec- tions in the 50:50 directional couplers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' For calibration data for each of the output ports, see Supplementary Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 1 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='6 M 0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='4 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 20 0 20 10 0 10 20 V M20 10 10 M 20 0 A 10 30 20 40 20 10 0 10 20 V M D200 μm Magnification: 5 x200 μm Magnification: 5 x4 95 100 105 110 115 120 Time (ns) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='5 Photodiode (V) 0 100 200 300 400 Time (ns) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='5 Photodiode (V) 397 398 399 400 401 Time ( s) 1 0 1 2 3 Time ( s) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='5 1 Tnorm 200 201 202 203 204 Time ( s) 500 1000 1500 2000 2500 3000 3500 4000 4500 Time (ns) Tnorm (offset) Chan 4 Chan 3 Chan 2 Chan 1 a) b) d) e) Rise time = ~20 ns 105 106 107 108 Frequency (Hz) 6 5 4 3 2 1 0 MZI Response (dB) ν3dB = 34 MHz c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' High-speed device pulsing qualification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' a) Measured 200 ns pulse from an output port of the chip and b) inset of the pulse showing a ∼20 ns rise time from the SPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' c) Normalized modulator response for a 3V sinusoidal signal showing the -3 dB cutoff at ν3dB = 34 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' d) Pulsing scheme showing the capabilities of the binary tree for arbitrary pulsing schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Each output can be pulsed at arbitrary times, lengths, shapes, and amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' e) Repeated 200 ns pulses with a 50% duty cycle to measure the consistency of our device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The standard deviation of the integrated pulse area is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='8 × 10−4 for 1000 consecutive pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' PULSE CHARACTERIZATION AND STABILITY We tested the optical pulse carving of our switch by ap- plying representative pulse sequences to each of the SPSs in the switching MZIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The “off” state of the output is de- fined to be 0 V due to the calibration procedure, and the full “on” state is achieved by applying the experimentally determined cross-state voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Pulses of varying ampli- tudes below the maximum are created by setting the ap- plied voltage between these cross and bar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Using time-resolved measurements on a 125 MHz photodiode, we found rise and fall times of ∼20 ns when program- ming a 200 ns pulse (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 3a,b) for all channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The small-signal frequency-resolved modular response (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 3c) indicates a -3 dB cutoff at ν3dB = 34 MHz, allowing for > 30 MHz optical control of each channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The de- vice can also be run at higher modulation speeds (> 100 MHz) with a trade-off of lower responsivity (< −6 dB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' To explore the optical control programmability, we tested various pulse sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Figure 3d shows the re- sulting measurement of each of the outputs and shows four different capabilities of this system: i) Any set of outputs can be pulsed simultaneously, ii) each pulse width can be independently manipulated, iii) the wave- form can be temporally amplitude modulated into differ- ent shapes, such as square or Gaussian, and iv) the pulse height can be independently set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' With these criteria met, our chip has the ability to create a full set of quantum ro- tations [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Furthermore, we measured the consistency of the pulsing of our device by applying repeated 200 ns pulses with 200 ns intervals and measuring the deviations in each pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We find a pulse area consistency (1σ stan- dard deviation) of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='8 × 10−4 for 1000 pulses, showing robust pulse uniformity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Examples of these pulses from the beginning, middle, and end of this pulse sequence are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 3e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Lastly, we did not observe crosstalk from either thermal, electrical, or piezo effects between the different phase shifters (details in Supplementary Sec- tion 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' INDEPENDENT ADDRESSING OF MULTIPLE SINGLE SIVS To demonstrate the applicability of the APIC, we used it to resonantly drive individual emitters within an en- semble of SiVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 4a, the APIC projects each port perpendicularly onto separate diamond waveg- uides in a cryostat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The diamond waveguides are fab- ricated with inverse tapered end couplers oriented to- wards the collection path, allowing for a high collection efficiency of 15% (see Methods for full diamond fabri- cation information and Suplementary Section 5 for col- lection efficiency calculation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The inverse tapers con- fine the emitted PL to an NA much smaller than that of the collection optics, allowing for scalable collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' In the excitation path, we include a spatial light modulator (SLM) for small spatial adjustments to each projected beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' This allows us to independently steer each exci- tation spot to specific SiVs in the diamond waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We note that once the SLM is initially programmed, it is kept static over the course of the experiment, mak- ing its slow reconfiguration time (∼100 Hz) inconsequen- tial for the excitation experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We resonantly excite each of the SiVs while collecting the phonon sideband (PSB) emission using a 750 nm long pass filter to re- move excess pump light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We projected this fluorescence 5 T = 5 K Binary Tree 737 nm Laser LP a) b) Normalized Counts 1 0 1 pixel = 16 μm x 16 μm FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Independent optical control of Si vacancy color centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' a) Experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 737 nm laser light is input into the binary tree through grating couplers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The four outputs are imaged onto a diamond microchiplet with the use of an SLM to steer the beams onto individual SiVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The emission of the SiVs is collected, the PSB is filtered out with a 750 nm long pass filter (LP), and imaged onto an EMCCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' b) Simultaneous PLE measurements on four different diamond waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' By driving the APIC, emission from emitters in each waveguide can be independently controlled with high extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Each panel shows a different iteration of outputs being driven, showing complete independence of emitter emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 1000 500 0 500 1000 Detuning (MHz) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='8 1 PSB fluorescence (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=') 1000 500 0 500 1000 Detuning (MHz) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='8 1 PSB fluorescence (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=') 1000 500 0 500 1000 Detuning (MHz) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='8 1 PSB fluorescence (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=') 1000 500 0 500 1000 Detuning (MHz) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='8 1 PSB fluorescence (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=') 20 10 0 10 20 Delay Time, (ns) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 g(2)( ) 20 10 0 10 20 Delay Time, (ns) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 g(2)( ) 20 10 0 10 20 Delay Time, (ns) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 g(2)( ) g(2)(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='06(9) g(2)(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='09(14) g(2)(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='09(10) g(2)(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='14(11) b) a) c) d) e) f) g) h) 0 200 400 600 800 1000 1200 Time (ns) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='8 1 PSB fluorescence (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=') i) 𝛤 = 160(3) MHz 𝛤 = 152(2) MHz 𝛤 = 287(6) MHz 𝛤 = 199(8) MHz FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Direct addressing and temporal control of single SiV emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' a-d) PLE spectrum of single SiVs excited with the APIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Each vacancy is excited with light from a different APIC channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' e-h) Autocorrelation measurements of the same single SiVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' For each emitter, g(2)(0) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='14, well below the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='5 threshold to demonstrate single photon emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' i) Pulsed fluorescence demonstrating temporal control of the emission of a single emitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Data shown is integrated over 3 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' onto an electron-multiplying charge-coupled device (EM- CCD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Figure 4b shows acquisitions of 30 seconds of the collected fluorescence normalized to the brightest point of each image, with no further image processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' This se- quence shows independent and simultaneous optical con- trol of SiVs in four different diamond waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Due to variations in the local strain throughout the diamond, the zero-phonon lines (ZPLs) have an inhomogeneous dis- tribution that exceeds the excitation laser linewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' To collect SiV emission from multiple waveguides simulta- neously, we increased the temperature of the diamond samples to broaden the ZPL linewidths so that they are spectrally overlapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Thus, for these images, we likely addressed multiple emitters in each diamond waveguide SLM1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='8 2 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 0 20 10 0 10 20 Delay Time, T (ns)6 due to the high density of SiVs in our sample (> 50 emit- ters per waveguide).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' However, to show the applicability of this scheme for controlling individual single emitters, we cooled the dia- mond sample to a base temperature of 5 K and repeated the excitation scheme with each channel projected onto a spectrally resolved SiV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Figure 5a-d shows the PLE frequency scans for SiVs in four different waveguides, demonstrating linewidths < 290 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Second-order correlation measurements indicate strong antibunching, with a normalized g(2)(0) ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='09 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='11, well below the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='5 threshold for single pho- ton emission (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 5e-h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We find an average emitter lifetime of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='76(1) ns (see Supplementary Section 5), con- sistent with other measurements on ion-implanted SiVs [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' With the outputs of the MZI tree projected on these emitters simultaneously, we send pulse sequences to tem- porally control the SiV emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' An example pulse train is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 5i, where we repeatedly pulse one of the channels (Channel 3, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 5c,g) with 100 ns pulses and a period of 250 ns and collect the fluorescence on a time- resolved avalanche photodiode, demonstrating temporal control of a single photon source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' DISCUSSION We introduced and demonstrated a scalable optical control system for individual addressing of quantum atom-like emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The modularity of the APICs and di- amond microchiplets is scalable to 1000s of ports and can be integrated with CMOS control electronics for VSLI devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Operating voltages can be further reduced by allowing for a trade off of extinction and applied volt- age, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' if only 30 dB extinction is required then the SPS can be pulsed with < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='5 V applied signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The dia- mond collection architecture is also readily scalable, with high efficiency collection of many waveguides enabled by the modal conversion of the waveguides to a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='26 NA (See Supplementary Section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' With the collection op- tics used in this setup, this allows for the scaling to 2975 waveguides with 3 µm spacing between waveguides in a linear array without a loss of collection efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The losses on the chip currently limit the scalability of the platform, with a total measured insertion loss of -19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' This loss is dominated by a low grating coupler ef- ficiency of 10%, which can be improved with design and fabrication iterations (See Supplementary Section 7 for improved grating coupler results > 40%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Future work will use this platform for running inde- pendent optical control schemes of quantum emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Using already demonstrated strain tuning [45, 46], we en- vision a second chip built from the same APIC platform that allows for spectral matching of quantum emitters, a necessary functionality for quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' More broadly, the broadband [21, 22] APIC technology can be applied to other optically trapped atomic systems [47– 53] and will enable near-future experiments in the area of optical quantum control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' METHODS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' PIC Calibration To begin a calibration, the light from the first output channel is focused onto a photodiode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' An iris is used to ensure that only the light from the active output is being measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' A single laptop controls all of the equipment in the experiment and is able to set the applied voltages and query the measured values from the powermeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We first calibrate the routing MZIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The applied voltage is swept from -25 V to 25 V in increments of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='1 V with a power reading at each interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The voltage is applied differentially to the CPS, with +V being applied to one cantilever and -V applied to the other, nominally giving a θ and −θ phase shift for each path respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' During the calibration, all other phase shifters’ voltages are held constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' To find the cross and bar states, we fit an offset sine curve to the data and take the maximum and minimum values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Next, we calibrate the triple-phase shifter switching MZIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We begin by setting the voltage of the SPS to 0 V, and then calibrate the CPSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Since the total extinction of this MZI is dependent on the relationship between θ and φ + ψ, we do a nested two dimensional sweep of the applied voltages from -25 V to 25 V in increments of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='25 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The cross and bar states are found by fitting a two dimensional sinusoid to the data and taking the maximum and minimum values respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We then set the two CPSs to their bar state (minimum transmission) and calibrate the SPS by sweeping the voltage from -25 V to 25 V in increments of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='1 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The SPSs are also operated differentially in a push-pull configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We fit an offset sine curve to this data to find its cross and bar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' With how we set up this calibration, the bar state is defined to be 0 V due to the CPSs being set to their bar state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Full calibration results are shown in the Supporting Information Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Diamond Chiplet Fabrication For the generation of negatively charged SiV in the di- amond, we relieved the strained surface of the diamond plate by removing the top 7 µm using Ar/Cl2 plasma etching followed by O2 etching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The sample was subse- quently implanted with Si29 at 190 keV with a dose of 5×1010 ions/cm2 (Innovion Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' It was then annealed in an ultra-high vacuum furnace (< 10−7 mbar) at 1200 ◦C and cleaned in a boiling tri-acid mixture (1:1:1 nitric acid, sulfuric acid, and perchloric acid at 345 ◦C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' A 180 nm sil- icon nitride (Si3N4) was chemical vapor deposited on the diamond, and patterned using electron-beam lithography and CF4 reactive-ion etching (RIE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We isotropically un- dercut the diamond quantum microchiplet (QMC) using 7 an oxygen inductively coupled plasma (ICP) RIE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Lastly, we submerged the sample in hydrofluoric acid to remove the Si3N4 hard mask and alumina [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' SiV Linewidth and Autocorrelation Measurements The diamond sample used in these experiments was fabricated into a QMC [9] as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We then broke the QMC into individual waveguides and placed them overhanging the edge of a cleaved Si chip using tungsten tips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We mounted the Si chip vertically in the Montana cryostat to enable perpendicular excitation and collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' When measuring the individual SiV emitters, we cou- pled a single waveguide mode at a time to a multi- mode fiber for high-efficiency collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We fit the emit- ter’s PLE linewidth scans with a Voigt profile using the Nelder-Mead simplex algorithm for the fit optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' For the autocorrelation measurements, we used a 50:50 fiber splitter to send the light to two APDs in a Hanbury- Brown-Twiss setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' During these measurements, we in- put pulsed 532 nm light into the waveguide through the collection objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We input the minimum amount of repump needed to obtain the maximum count rate, pro- viding maximum charge state initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We gated the detectors to only collect data when the repump beam is off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' To fit the g(2) values, we used a Lorentzian fit to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' g(2)(τ) = C1 + C2 � 1 2Γ τ + � 1 2Γ �2 � (1) where τ is the delay time between coincident counts, C1 and C2 are the offset and scaling factors respectively, Γ is the full-width half-max of the emitter spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' For the one emitter that showed pronounced bunching behavior, we fit the data to a three level system and added in an overall offset and scaling factor to account for the non- ideality of the data due to dark counts and jitter from the APD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' g(2)(τ) = C1 + C2 � 1 − (1 + a)e−|τ|/τ1 + ae−|τ|/τ2� (2) where τ is the delay time between coincident counts, C1 and C2 are the offset and scaling factors respectively, a is the scaling factor determining the strength of the photon bunching, τ1 is the antibunching time constant, and τ2 is the bunching time constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' When compared to the standard Lorentzian fit, we obtained similar values for g(2)(0) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='09 vs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='07).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The error bars reported in the manuscript correspond to one standard deviation in the fit parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' ACKNOWLEDGMENTS Major funding for this work is provided by MITRE for the Quantum Moonshot Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' acknowledges partial support from Brookhaven National Laboratory, which is supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Department of Energy, Office of Basic Energy Sciences, under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' DE- SC0012704 and the NSF RAISE TAQS program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' performed this work, in part, with funding from the Cen- ter for Integrated Nanotechnologies, an Office of Science User Facility operated for the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Department of En- ergy Office of Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' thank MITRE engineers L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Chan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Dauphinais, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Vergados for their support in building mechanical and electronic com- ponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' thank S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Trajtenberg and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Duan for additional experimental support and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Li and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Hu for helpful conversations and comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' AUTHOR CONTRIBUTIONS K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' built the experimental setup and performed the device characterization and calibration experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='A.' metadata={'source': 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+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' performed the data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' de- signed the electronic control system.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' performed the diamond waveguide simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' conceived the experiment and device architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=', and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' supervised the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' wrote the manuscript with input from all authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' ADDITIONAL INFORMATION Supplementary information is available for experimen- tal methods related to programming and calibrating the photonic integrated circuit and collection apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' COMPETING INTERESTS D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' is a scientific advisor to and holds shares in QuEra Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' DATA AVAILABILITY The data that support the plots within this paper are available from the corresponding authors upon reason- able request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 8 [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Atat¨ure, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Englund, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Vamivakas, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Lee, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Wrachtrup, Material platforms for spin-based photonic quantum technologies, Nature Reviews Materials 3, 38 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' [2] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Bar-Gill, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Pham, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Jarmola, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Budker, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Walsworth, Solid-state electronic spin coherence time approaching one second, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 4, 1743 (2013).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Wan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Schr¨oder, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Englund, Rectangular photonic crystal nanobeam cavities in bulk diamond, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 111, 021103 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Modular chip-integrated photonic control of artificial atoms in diamond nanostructures: Supplementary Information Kevin J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Palm1,†,∗, Mark Dong1,2,†,∗, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Andrew Golter1, Genevieve Clark1,2, Matthew Zimmermann1, Kevin C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Chen2, Linsen Li2, Adrian Menssen2, Andrew J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Leenheer3, Daniel Dominguez3, Gerald Gilbert4,∗, Matt Eichenfield3,5,∗, and Dirk Englund2,6 1The MITRE Corporation, 202 Burlington Road, Bedford, Massachusetts 01730, USA 2Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 3Sandia National Laboratories, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Box 5800 Albuquerque, New Mexico, 87185, USA 4The MITRE Corporation, 200 Forrestal Road, Princeton, New Jersey 08540, USA 5College of Optical Sciences, University of Arizona, Tucson, Arizona 85719, USA 6Brookhaven National Laboratory, 98 Rochester St, Upton, New York 11973, USA and †These authors contributed equally (Dated: January 11, 2023) I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' OPTICAL EXPERIMENTAL SETUP Figure S1 depicts the optical setup for all of the experiments demonstrated in this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Laser light was input into the binary MZI tree through a 10-port fiber grating with a tunable laser set at 737 nm (M Squared Lasers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We collected the edge coupled light with a 100x, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='9 NA infinity corrected objective (Mitutoyo) and filtered the light with a linear polarizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We then routed the light to different characterization devices using mirrors on flip stages for ease of switching between measurements: 1) a CCD camera for alignment, 2) a DC photodiode (Newport 818-SL) for device extinction characterization, and 3) a 125 MHz fast photodiode (New Focus 1801) for pulsed light characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' When performing diamond excitation experiments, we routed the light onto a spatial light modulator (SLM) (Thorlabs Exulus) and directed the beams onto vertically-mounted diamond waveguides in a Montana cryostat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' f = 150 mm f = 150 mm f = 100 mm f = 250 mm Mirror f = 34 mm Mirror Flip Mirror Flip Mirror f = 500 mm f = 500 mm Iris DC PD Fast PD 737 nm Laser Flip BS CCD Iris 100x Objective f = 2 mm f = 150 mm f = 150 mm SLM f = 100 mm f = 250 mm Mirror Cryostat f = 34 mm Mirror Flip Mirror Flip Mirror f = 500 mm f = 500 mm Iris DC PD Fast PD 737 nm Laser Flip BS CCD Iris 5K Linear Polarizer MZI Tree f = 2 mm Flip Mirror APD EMCCD f = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='4 mm 50:50 Fiber BS 750 nm Long Pass APD SLM 532 nm Laser AOM f = 4 mm 600 nm Dichroic f = 200 mm FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Optical setup for binary tree experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Flip mirrors are used to redirect light to different optical detectors for different types of calibration experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Labels: f = effective focal length, CCD = charge-coupled device, BS = beamsplitter, SLM = spatial light modulator, AOM = acoustic optical modulator, APD = avalanche photodiode, EMCCD = electron- multiplying charge-coupled device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='03693v1 [quant-ph] 9 Jan 2023 2 c) Waveguides b) a) 5 μm 5 μm FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Imaging edge coupled waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' a) Picture of the wire bonded APIC onto the PCB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Light is input into the chip through a mounted fiber array (right) into grating couplers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The light is then collected with an objective (left) to be imaged or routed to the cryostat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The top objective is for observing the chip and aligning the fiber array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' b) Image of the edge coupled waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' c) Same waveguides, but with light being evenly split through each port on the chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The 34 mm lens in the beam path was embedded in the side of the cryostat with a custom inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' A long working distance objective (50x Mitutoyo) collected the emission from the diamond waveguides, whose photons then traveled to the electron-multiplying charge-coupled device (EMCCD), or coupled one of the waveguide modes to a fiber for autocorrelation experiments in a Hanbury-Brown-Twiss setup with two avalanche photodiodes (APD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' A 532 nm laser (Obis) was introduced in the collection path and pulsed into the diamond waveguide during autocorrelation experiments to stabilize the SiV charge state [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Figure S2a are images of the APIC coupling setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The laser light was input through the fiber array at 18◦ (right) and collected with the horizontal objective (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The vertical objective was used to align the fibers and inspect the chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Figure S2b shows the imaged edge of the waveguides, with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' S2c showing the same waveguides with light being emitted through the chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' PHOTONIC INTEGRATED CIRCUIT PACKAGING Our photonic chip was fabricated on 200-mm Si technology with a CMOS-compatible fabrication procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The binary tree was initially cleaved from the full-wafer die, and then wire-bonded to a custom printed circuit board (PCB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The cantilever phase shifters (CPS) were electrically driven with a 32-channel voltage controller (Marvin Test Systems GX1632e) with a voltage range of ±25 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The CPSs were driven in a push-pull configuration for maximum phase shift (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' if +10 V is applied to the top cantilever, -10 V is applied to the bottom cantilever).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The strain-optic phase shifters (SPS) were driven with two 200 MHz arbitrary waveform generator (AWG) PXIe cards (Spectrum M4x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='6622), each with four output channels with a voltage range of ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='5 V and 5x amplified on the PCB at a max slew rate of 8000Vµs−1 (Texas Instruments THS3491).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' In the course of any experiment, the SPSs were operated in a push-pull configuration similar to the CPSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' MZI CALIBRATION PLOTS A labeling of the different phase shifters is shown in Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' For results of a full calibration, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' S4 for the routing MZIs, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' S5 for the CPSs in the triple phase shifters, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' S6 for the SPSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' an Apo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='28 f=2003 00 10 11 20 30 40 31 32 33 21 22 23 41 42 43 1 mm FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Labeling of different phase shifters for calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Light is input from the 6th grating coupler from the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Output 0 is defined as the top output in this microscope image, with Output 3 as the bottom output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 20 10 0 10 20 Voltage 00 (V) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='8 1 Normalized Transmission 20 10 0 10 20 Voltage 10 (V) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='8 1 Normalized Transmission 20 10 0 10 20 Voltage 11 (V) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='8 1 Normalized Transmission 20 10 0 10 20 Voltage 00 (V) 30 25 20 15 10 5 0 Normalized Transmission (dB) 20 10 0 10 20 Voltage 10 (V) 15 10 5 0 Normalized Transmission (dB) 20 10 0 10 20 Voltage 11 (V) 30 25 20 15 10 5 0 Normalized Transmission (dB) d) e) b) f) c) a) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Routing cantilever phase shifter calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Voltages are swept from -25 V to 25 V while all other phase shifters are held static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Normalized transmission vs applied voltage for a) PS00, b) PS10, and c) PS11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Each phase shifter has a Vπ of ∼30 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Extinction measurements for d) PS00, e) PS10, and f) PS11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Magnification: X30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='0000mm4 b) d) c) a) f) h) g) e) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Switching cantilever phase shifter calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Voltages are swept from -25 V to 25 V in a nested sweep while all other phase shifters are held static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Normalized transmission vs applied voltage for a) PS20 and PS30, b) PS21 and PS31, c) PS22 and PS32, and d) PS23 and PS33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Extinction measurements for e) PS20 and PS30, f) PS21 and PS31, g) PS22 and PS32, and h) PS23 and PS33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 10 5 0 5 10 Voltage 40 (V) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='8 1 Normalized Transmission 10 5 0 5 10 Voltage 41 (V) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='8 1 Normalized Transmission 10 5 0 5 10 Voltage 42 (V) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='8 1 Normalized Transmission 10 5 0 5 10 Voltage 43 (V) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='8 1 Normalized Transmission 10 5 0 5 10 Voltage 40 (V) 40 30 20 10 0 Normalized Transmission (dB) 10 5 0 5 10 Voltage 41 (V) 40 30 20 10 0 Normalized Transmission (dB) 10 5 0 5 10 Voltage 42 (V) 40 30 20 10 0 Normalized Transmission (dB) 10 5 0 5 10 Voltage 43 (V) 40 30 20 10 0 Normalized Transmission (dB) b) d) c) a) f) h) g) e) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Switching strain-optic phase shifter calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Preceding CPSs are set to the bar state (minimum transmission) and then the voltage of the SPS is swept from -25 V to 25 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Normalized transmission vs applied voltage for a) PS40, b) PS41, c) PS42, and d) PS43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Extinction measurements for e) PS40, f) PS41, g) PS42, and h) PS43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 20 20 20 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='8 10 10 Voltage 23 (M) 10 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='6 0 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='4 10 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 20 20 20 20 10 10 20 20 10 10 20 20 10 0 10 20 20 10 0 10 20 0 Voltage 30 (M) Voltage 31 (M) Voltage 32 (M) Voltage 33 (M) 20 20 20 20 10 10 10 1 Voltage 20 (M) Voltage 21 (M) Voltage 22 (M) 10 Voltage 23 (M) 10 10 10 20 20 2( 0 20 0 0 0 10 10 10 10 3( 30 30 30 20 20 20 20 40 4 40 40 10 20 10 20 10 20 10 0 20 10 0 20 10 20 10 10 20 Voltage 30 (M) Voltage 31 M) Voltage 32 (M) Voltage 33 (M)5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='5 2 Time ( s) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='8 1 Normalized Transmission 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='5 2 Time ( s) Programmed Signal (norm) b) a) Chan 4 Chan 3 Chan 2 Chan 1 Chan 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Crosstalk investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' a) Programmed signal to each of the four SPSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Channel 3 was imaged onto the power meter for measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The first pulse on Channel 3 is a control pulse for comparison, followed by the other three channels pulsing, and ending with all four channels pulsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' b) Measured power of Channel 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We do not measure any cross talk above the noise floor between any of the channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' CROSSTALK INVESTIGATION To investigate whether adjacent SPSs had any effect on each other, we ran crosstalk experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' For these experiments, we set a pulse series to the four SPSs and measured if signals sent to adjacent SPSs had any effect on the measured output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' An example measurement is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' In this experiment, we had an initial control pulse, a pulse of all other channels except for the measured channel, and ending with a pulse of all four channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We did not measure differences between the control pulse and all four of the channels pulsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We also did not measure any output above the noise floor when the other three channels are pulsed and the active channel remains in the bar state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' COLLECTION EFFICIENCY CALCULATION AND LIFETIME MEASUREMENT We used a resonant pulsing scheme to measure the collection efficiency of our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We attached an electro- optic modulator (EOM, iXblue) on the input of our laser source and drove it with an AWG (Tektronix 70001B) to achieve the short excitation pulses necessary for the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' For all of these measurements, we drove the C transition of the SiV (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' S8a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' First, we excited a single emitter with a 20 ns resonant pulse and collected the time-tagged sideband of the emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We fit the resulting data with an exponentially damped sinusoid to extract the Rabi frequency for our input laser power (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' S8b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' This fit gave us an optical π-pulse of 400 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We then applied this optical π-pulse repeatedly, with one π-pulse every 240 ns to ensure the emitter decayed into the ground state and thermalized, and collected the time-tagged sideband emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Every 400 µs we applied a 1-µs 532 nm repump pulse to initialize the SiV to the negative charge state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' S8c shows the resulting data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We fit the exponential decay of the counts and find a lifetime of T1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='76(1) ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' To find the collection efficiency of our apparatus, we subtracted the background in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' S8c, sum the remaining counts, and divided by the total number of applied π-pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' This gave us a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='038% chance of measuring a photon in the sideband per pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' A few corrections onto this value must be made to find the accurate collection efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We experimentally measured the transmission percent of the SiV emission through our optical filters by measuring a photoluminescence spectra with 532 nm off-resonant excitation with the 750 nm longpass filter in and out of the collection path, and obtained a sideband transmission of 15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' During this measurement a 600 nm longpass filter was included to block the excess 532 nm pump that coupled to the waveguide mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' In our setup, we did not directly control the electron distribution between the two ground states split by spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' This population distribution is determined by [2]: PLB PUB = e−h∆gs/kBT (S1) where PLB and PUB are the populations in the lower and upper ground states respectively, h is Planck’s constant, 6 1 0 1 2 3 4 5 Time (ns) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='4 Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=') 2 0 2 4 6 8 10 12 14 Time (ns) 0 1000 2000 3000 4000 5000 6000 7000 8000 Counts T1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='76(1) ns Ω/2π = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='28(3) GHz b) c) 406 THz A B C D Δes = 260 GHz Δgs = 50 GHz a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='4 ns 240 ns FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Collection efficiency measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' a) Electronic structure of the SiV color center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The ground and excited states are split by spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' For these experiments we drive the C transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' b) Rabi oscillations excited by a 20 ns pulse resonant on the C transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We fit the oscillations with an exponentially damped sinusoid and find a Rabi frequency of Ω/2π =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='28(3) GHz, corresponding to an optical π-pulse of 400 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' c) Pulsed excitation of a single SiV with repeated π-pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Exponential fit of the decay gives a lifetime of T1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='76(1) ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Coupling efficiency is found by summing the total counts in the decay and normalizing by applied π-pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' kB is the Boltzmann constant, ∆gs is the ground state splitting, and T is the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' At 5 K, we find that the electron is in the lower ground state with 62% probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We then accounted for an APD measurement efficiency of 55% at 737 nm (Excelitas Technologies SPCM-AQRH) and the SiV quantum efficiency of 5% [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Applying these four corrections, we obtained a final collection efficiency of 15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We note that this collection efficiency could potentially be higher as the SiV was not initialized to the correct charge state with 100% fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Continuous wave measurements with 532 nm excitation indicated that the off-resonant repump put the vacancy in the correct charge state with 60% probability, with no emission recorded when the vacancy was not in the correct charge state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We also note that there is uncertainty in the quantum efficiency value of the emitter, as different nanostructuring schemes cause for different modifications to this value [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Our design could be further improved by adding a Bragg reflector to the back half of our diamond waveguide, which potentially increases our collection efficiency by a factor of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' DIAMOND WAVEGUIDES NA We used finite element simulations (COMSOL Multiphysics) to determine the far field NA of our diamond waveg- uides for a measure of the scalability of our system, and find that our waveguides have a calculated NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='26, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' |E|2 Far Field Projection FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Diamond waveguide far field projection simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Uy and Uz are the directional unit vectors and |E|2 is normalized to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We find that the projected mode from our waveguides is contained in an NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='26, denoted by the red circle in the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 1 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='5 1 Uy7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Grating coupler optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The grating period and fill factor were swept around the simulated maximum value to find the experimentally best grating coupler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The highest measured coupling is 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='3% coupling at a period of 578 nm and a fill factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='437.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' IMPROVED GRATING COUPLER EFFICIENCIES To improve the insertion loss of our device, we ran a sweep of the period and fill factors of our grating couplers and measured the coupling efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' We measured the efficiency using loopback structures with the inputs and outputs measured through a fiber array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The results of this sweep are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' The highest measured coupling is 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='3% coupling at a period of 578 nm and a fill factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='437.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' To improve this coupling even further, we experimented with depositing an index matching fluid (n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='52) on the grating couplers, thick enough that the fiber tips are completely submerged in the fluids when aligned for coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Using this technique, we are able to measure 41% grating coupler efficiency for grating couplers with a period of 621 nm and fill factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='483.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Maity, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Shao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Bogdanovi´c, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Meesala, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Sohn, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Sinclair, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Pingault, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Chalupnik, C.' metadata={'source': 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Schreck, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Becher, Deterministic coupling of a single silicon-vacancy color center to a photonic crystal cavity in diamond, Nano Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 14, 5281 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' Grating Coupling Efficiency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='25 I Factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='2 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='15 Grating F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE2T4oBgHgl3EQfJga_/content/2301.03693v1.pdf'} 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a/otFPT4oBgHgl3EQfLDSW/content/tmp_files/2301.13021v1.pdf.txt b/otFPT4oBgHgl3EQfLDSW/content/tmp_files/2301.13021v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ad4f829281b4634fc824435a294dbd21f348142c --- /dev/null +++ b/otFPT4oBgHgl3EQfLDSW/content/tmp_files/2301.13021v1.pdf.txt @@ -0,0 +1,2246 @@ +ROBUST DPG FORTIN OPERATORS +THOMAS FÜHRER AND NORBERT HEUER +Abstract. At the fully discrete setting, stability of the discontinuous Petrov–Galerkin (DPG) +method with optimal test functions requires local test spaces that ensure the existence of Fortin +operators. We construct such operators for H1 and H(div) on simplices in any space dimension and +arbitrary polynomial degree. The resulting test spaces are smaller than previously analyzed cases. +For parameter-dependent norms, we achieve uniform boundedness by the inclusion of exponential +layers. As an example, we consider a canonical DPG setting for reaction-dominated diffusion. Our +test spaces guarantee uniform stability and quasi-optimal convergence of the scheme. We present +numerical experiments that illustrate the loss of stability and error control by the residual for +small diffusion coefficient when using standard polynomial test spaces, whereas we observe uniform +stability and error control with our construction. +1. Introduction +Fortin operators are a critical tool for the stability analysis of mixed finite element schemes, cf. [2]. +The discontinuous Petrov–Galerkin (DPG) method with optimal test functions, on the other hand, +is a framework that aims at automatic inf-sup stability. In practice, optimal test functions have to +be approximated and the question of existence of Fortin operators re-appears. In this case, local +(element-wise) operators are sufficient. A first answer was given in [12], with subsequent studies +in [5, 15, 9, 11]. In the case of singularly-perturbed problems, uniform discrete stability, or robust- +ness of the method, requires the existence of uniformly bounded Fortin operators. This has been +an open problem. In this paper, we present local Fortin operators for H1 and H(div), on sim- +plices in arbitrary dimension and arbitrary polynomial degree. In contrast to previous results, our +constructions are explicit (not needed in applications) and require fewer degrees of freedom. More +importantly, we include parameter-dependent exponential layers that guarantee uniform bounded- +ness of our operators for parameter-dependent norms (the H(div)-case is restricted to two and +three space dimensions). We illustrate their application to a DPG method for a reaction-dominated +diffusion problem, leading to robustness, i.e., uniform stability, error control, and convergence. In +this case, we consider the energy-norm induced by the problem. We have not analyzed the case +of balanced norms as proposed in [14]. This and possible extensions to other singularly-perturbed +problems like advection-dominated diffusion are left to future research. +Let us shortly discuss the abstract setting of the DPG method: Consider the variational formu- +lation +u ∈ U : +b(u, v) = L(v) +∀v ∈ V, +where U, V are Hilbert spaces with norms ∥·∥U, ∥·∥V , b(·, ·) is a bounded bilinear form and induces a +boundedly invertible operator B : U → V ′, u �→ b(u, ·). Choosing finite dimensional spaces Uh ⊂ U, +Vh ⊂ V , the fully discrete DPG method reads: +uh ∈ Uh : +b(uh, v) = L(v) +∀v ∈ Θh(Uh), +(1) +Date: January 31, 2023. +2010 Mathematics Subject Classification. 65N30, 65N12 . +Key words and phrases. DPG method, Fortin operators, singularly perturbed problems, reaction-diffusion. +Acknowledgment. This work was supported by ANID through FONDECYT projects 1210391 and 1230013. +1 +arXiv:2301.13021v1 [math.NA] 30 Jan 2023 + +where Θh : U → Vh is defined through ((·, ·)V being the inner product on V ) +(Θhu, vh)V = b(u, vh) +∀vh ∈ Vh. +Well-posedness of discrete DPG is ensured if there exists a Fortin operator ΠF : V → Vh such that +∥ΠFv∥V ≤ CΠF∥v∥V , +b(uh, v − ΠFv) = 0 +∀uh ∈ Uh, v ∈ V, +(2) +see, [12]. Furthermore, the existence of a Fortin operator also implies quasi-optimality, +∥u − uh∥U ≤ CCΠF min +wh∈Uh ∥u − wh∥U +with C = Cb/cb where Cb and cb are the boundedness and inf-sup constants of b(·, ·), respectively. +It also plays an important role in the a posteriori error control, see [4, Theorem 2.1], +∥u − uh∥U ≂ ∥Buh − L∥V ′ + osc(L), +where osc(L) = sup0̸=v∈V L(v − ΠFv)/∥v∥V . +One of the main motivations that had driven the development of the DPG method was to derive +robust numerical schemes for singularly perturbed problems, see, e.g., [8, 14]. All these problems +have in common that they naturally lead to parameter dependent trial and test norms. +For a +concrete example consider the test space H1(T) (here T as an element of the mesh T ) equipped +with the norm +∥v∥T,α = ∥v∥T + α∥∇v∥T +for some fixed α > 0. +In [12, 5, 9] Fortin operators are constructed (resp. +their existence is shown). +Let us write +ΠF : H1(T) → P k(T) (a polynomial space). Besides some conditions they satisfy the bounded- +ness estimates +∥ΠFv∥T ≲ ∥v∥T + hT ∥∇v∥T , +∥∇ΠFv∥T ≲ ∥∇v∥T . +Combining the latter two estimates yields +∥ΠFv∥T,α ≲ max{1, α−1hT }∥v∥T,α. +If hT ≲ α it is clear that ∥ΠFv∥T,α ≲ ∥v∥T,α uniformly. However, when α ≲ hT — a case often +encountered with singularly perturbed problems — then we get ∥ΠFv∥T,α ≲ α−1hT ∥v∥T,α. This +means that, particularly on coarse meshes, the Fortin operator is not uniformly bounded. By our +prior considerations, this means that quasi-optimality as well as a posteriori error control is spoiled. +One of the main objectives of this work is to define discrete spaces Vh and construct corresponding +Fortin operators ΠF : V → Vh with boundedness constant CΠF ≂ 1 for small parameters (α ≲ hT +in the previous example). We do this by first revisiting the construction of Fortin operators for the +spaces H1(T) and H(div; T) in the case hT ≲ α. Contrary to prior works we construct our Fortin +operators in an explicit manner. This allows to precisely write down a basis for the discrete test +spaces yielding — compared with the operators from [12, 5, 9] — smaller dimensions. The novel +idea of definition also requires a different analysis which we present in detail. Our constructions are +valid for any polynomial degree (this statement will be made precise below) and arbitrary dimension +(except for some operators from Section 4). However, main advantage is that the definition and +construction can be extended to the case α ≲ hT . Specifically, we use modified face bubble functions +ηα,F instead of polynomial face bubble functions ηF . They are defined in such a way that their +volume norm ∥ηα,F ∥T resp. ∥∇ηα,F ∥T scales differently than ∥ηF ∥T resp. ∥∇ηF ∥T depending on +the ratio α/hT , though ηα,F |∂T = ηF |∂T . The analysis requires some additional tools and steps. We +also consider low order polynomial cases which allow for even smaller dimensions in the test space. +As mentioned above, Fortin operators for the DPG method have been constructed in various +works: The first one was [12]. Other articles that analyze the existence of Fortin operators for +second-order PDEs include [5, 15, 9]. The latter references consider all arbitrary fixed polynomial +2 + +orders. For low order methods with smaller test space dimensions we refer to [6, 7]. For a fourth- +order PDE model problem we have shown existence of Fortin operators in [11]. +The remainder of this work is organized as follows: In Section 2 we introduce some notation and +define basis functions as well as novel face bubble functions. Section 3 and Section 4 discuss the +construction of Fortin operators for H1 and H(div), respectively. Section 5 concludes this article +with a short description of a DPG method for a singularly perturbed reaction-diffusion problem and +numerical examples. +2. Preliminaries +The notation a ≲ b (a ≳ b) for a, b > 0 means that there exists C > 0 such that a ≤ C b +(C a ≥ b). We write a ≂ b for a, b > 0 if a ≲ b ≲ a. The generic constant C is independent of +involved functions, the diameter of elements, and parameters like α and ε, where present. +2.1. Mesh and spaces. Let T denote a shape-regular simplicial mesh of a Lipschitz domain Ω +with diam(Ω) ≤ 1. Throughout, T ∈ T is some fixed element, �T is the reference element given as +the convex hull of the origin and the n coordinate axis vectors. E.g., for n = 2, 3 it reads +�T = +� +conv{(0, 0)⊤, (1, 0)⊤, (0, 1)⊤}, +n = 2, +conv{(0, 0, 0)⊤, (1, 0, 0)⊤, (0, 1, 0)⊤, (0, 0, 1)⊤}, +n = 3. +Here, we understand conv as the interior of the convex hull of a set. +We adopt the standard notation for Lebesgue and Sobolev spaces, L2(ω), L2(ω) := L2(ω; Rn), +H1(ω) = +� +v ∈ L2(ω) : ∇v ∈ L2(ω) +� +, +H(div; ω) = +� +τ ∈ L2(ω) : div τ ∈ L2(ω) +� +for a Lipschitz domain ω ⊂ Rn. +With nω we denote the normal vector on the boundary of ω +pointing from ω to its complement Rn \ω. Recall that traces of H1(ω) elements are well defined (in +the sense of trace operators) and the canonic trace space is H1/2(∂ω). Normal traces of H(div; ω) +elements are well defined (in a duality sense) and the canonic trace space is H−1/2(∂ω). We simply +write τ · nω for the normal trace of τ ∈ H(div; ω). +We denote by ∥ · ∥ω the canonical L2(ω) norm induced by the L2(ω) inner product (·, ·)ω for a +Lipschitz domain ω ⊂ Rn. The volume measure of ω is given by |ω|. The same notation for norm +and inner product is used for L2(ω; Rn). The surface measure of γ ⊆ ∂ω is denoted by |γ| and +∥v∥γ is the L2(γ) norm induced by the inner product ⟨·, ·⟩γ. We also use the same notation for the +duality between H−1/2(∂ω) and H1/2(∂ω), +⟨φ, v⟩∂ω +for φ ∈ H−1/2(∂ω), v ∈ H1/2(∂ω). +Recall the following relation between traces of H1(ω) and H(div; ω), +⟨τ · nω , v⟩∂ω = (div τ , v)ω + (τ , ∇v)ω +(3) +for all τ ∈ H(div; ω), v ∈ H1(ω). +Obviously, for sufficiently regular functions this is just the +integration by parts formula. +Let Πq +T : L2(T) → P q(T) denote the L2(T) orthogonal projection on P q(T), the space of polyno- +mials on T of degree less than or equal to q ∈ N0. For vector-valued polynomials (each component +is a polynomial of degree less than or equal to q) we use the symbol P q(T). Recall the first-order +approximation property +∥v − Πq +T v∥T ≤ ∥v − Π0 +T v∥T ≲ hT ∥∇v∥T +for all v ∈ H1(T). +An important tool is the following (multiplicative version) of the trace inequality. It can be de- +rived from [3, Theorem 1.6.6] with a scaling argument (using the reference element �T) and the +approximation property of Π0 +T . +3 + +Lemma 1. For any v ∈ H1(T) we have +∥v − Π0 +T v∥∂T ≲ ∥v − Π0 +T v∥1/2 +T ∥∇v∥1/2 +T +≲ h1/2 +T ∥∇v∥T +(4) +with hidden constants only depending on the shape of T. +We denote by VT the set of the n + 1 vertices of T, FT is the set of n + 1 faces of T and +VF denotes the set of n vertices of F. For z ∈ VT let Fz ∈ FT be the face opposite to z, i.e., +Fz = conv +� +VT \ {z} +� +. Similarly, for F ∈ FT let zF ∈ VT be the vertex opposite to F. For F ∈ FT +let P q(FT ) ⊂ L2(∂T) denote face-wise polynomials of degree less than or equal to q ∈ N0 and +P q +c (FT ) := P q(FT ) ∩ C0(∂T). Note that nT is face-wise constant. For the fixed element T ∈ T and +any F ∈ FT we abbreviate nF = nT |F . For a vertex z ∈ VT we denote by Ez the set of n edges +that share the same vertex z, i.e., for each E ∈ Ez there is a z′ ∈ VT \ {z} with E = conv{z, z′}. +To each E = conv{z, z′} ∈ Ez we associate the (tangential) vector tE = z′ − z (the orientation does +not matter for our analysis nor for implementation). +Furthermore, hT = diam(T), hT ≂ diam(F) for all F ∈ FT with hidden constants only depending +on the shape of T. Some other relations that we frequently use without further notice are |T| ≂ hn +T , +|F| ≂ hn−1 +T +, |∂T| ≂ |F|, |T| ≂ |F|hT for any F ∈ FT . +In the following subsections we define special functions that will be used for the construction of +the Fortin operators. For ease of reading and reference they are listed, together with their relevant +properties, in Table 1 below. +2.2. Low-order basis functions. The functions ηz ∈ P 1(T) are canonical basis functions with +ηz(z′) = δz,z′ for z, z′ ∈ VT and δ·,· denoting the Kronecker-δ, +ηF = +� +z∈VF +ηz ∈ P n(T) +are the face bubble functions, and ηT = � +z∈VT ηz ∈ P n+1(T) is the element bubble function. +Clearly, span{ηz : z ∈ VT } = P 1(T). Alternatively, we may use the following basis for P 1(T): +First, we abbreviate dF = ηzF . Then, for F ∈ FT we define +νF = +� +z∈VF +ηz − (n − 1)dF . +For space P 0(FT ) we use the characteristic functions χF |F ′ = δF,F ′ for F ′ ∈ FT as basis functions. +Lemma 2. We have +⟨νF , χF ′⟩∂T = |F|δF,F ′ +∀F, F ′ ∈ FT +(5) +and +span +� +νF : F ∈ FT +� += P 1(T), +span +� +νF |∂T : F ∈ FT +� += P 1 +c (FT ). +Proof. Note that (5) implies that νF |∂T , F ∈ FT are linearly independent. This implies the last +two assertions because dim(P 1(T)) = n + 1 = dim(P 1 +c (FT )). It only remains to prove (5): Let +F ∈ FT . From its definition we see that νF |F = 1, thus, ⟨νF , χF ⟩∂T = |F|. Let F ′ ∈ FT \ {F}. +Using +� +F ′ ηz dx = |F ′|n−1 for z ∈ VF ′ one verifies that +⟨νF , χF ′⟩∂T = +� +z∈VF ∩VF ′ +� +F ′ ηz dx − (n − 1) +� +F ′ dF dx += (n − 1)|F ′|n−1 − (n − 1)|F ′|n−1 = 0, +finishing the proof. +□ +4 + +Let RT 0(T) = +� +ψ ∈ L2(T) : ψ = α + βx, α ∈ Rn, β ∈ R +� +denote the lowest-order Raviart– +Thomas space where x: T → Rn, z �→ z. Let ψF ∈ RT 0(T) denote the canonical Raviart–Thomas +basis function with +ψF · nT |F ′ = δF,F ′ +∀F, F ′ ∈ FT +and ∥ψF ∥T ≂ |T|1/2. One verifies the explicit representation ψF (z) = +|F| +n|T|(z − zF ). Note that by +Lemma 2 we have that +⟨ψF · nT , νF ′⟩∂T = |F|δF,F ′ +∀F, F ′ ∈ FT . +We also use Bernardi–Raugel elements, see [1], +ηF := ηF nF , +F ∈ FT +for which we get +⟨ηF · nT , νF ′⟩∂T = ⟨ηF · nT , 1⟩F δF,F ′ ≂ |F|δF,F ′ +∀F, F ′ ∈ FT . +We also define edge based functions: Fix a vertex z∗ ∈ VT and set E∗ = Ez∗. Clearly, tE (E ∈ E∗) +are linearly independent and span Rn. Let σE, (E ∈ E∗) denote a basis of P 0(T) such that (for any +z ∈ T) +σE(z) · tE′ = δE,E′ +∀E, E′ ∈ E∗. +We have that |σE(z)| ≂ 1 with constants only depending on the shape of T. With these preparations +we define edge functions by ηE = � +z∈VT ∩E ηz and tangential edge functions by +ηE := ηEtE +∀E ∈ E∗. +These functions play the role of element bubble functions in H(div; T) as can be seen from the next +result. +Lemma 3. We have ηE ∈ P 2(T) and +ηE · nT |∂T = 0, +(σE , ηE′)T = (σE , ηE)T δE,E′ ≂ |T|δE,E′ +∀E, E′ ∈ E∗. +Proof. Clearly, ηE ∈ P 2(T), thus, ηE ∈ P 2(T). Note that ηE|∂T is supported on n − 1 faces, +say Fj, j = 1, . . . , n − 1. +For these faces we also have tE ⊆ Fj yielding tE · nFj = 0. +We +conclude ηE · nT |∂T = 0. The final assertion follows from the definition of σE and ηE and scaling +arguments. +□ +2.3. Higher-order basis functions. For F ∈ FT let �χF,j ∈ P p(T), j = 1, . . . , dim(P p(F)) be +such that �χF,j|F is a basis of P p(F) with ∥�χF,j∥∞ = ∥�χF,j|F ∥∞ ≂ 1 and define +ηF,j = ηF �χF,j, +j = 1, . . . , dim P p(F), F ∈ FT . +Let χF,j ∈ P p(FT ), j = 1, . . . , dim(P p(F)), be such that χF,j|F ′ = 0 for F ′ ∈ FT \ {F} and +⟨χF,j , ηF,k⟩F = δj,k|F|, +j, k = 1, . . . , dim(P p(F)). +Let �χT,j ∈ P p(T), j = 1, . . . , dim(P p(T)), denote a basis of P p(T) with ∥�χT,j∥∞ ≂ 1 and define +ηT,j = ηT �χT,j, +j = 1, . . . , dim(P p(T)). +Furthermore, let χT,j, j = 1, . . . , dim(P p(T)) be such that +(χT,j , ηT,k)T = |T|δj,k, +j, k = 1, . . . , dim(P p(T)). +By scaling arguments one verifies that ∥χF,j∥∞ ≂ 1 and ∥χT,j∥∞ ≂ 1. +5 + +Let �P p(T) denote the orthogonal complement of P p +b (T) = +� +v ∈ P p(T) : v|∂T = 0 +� +in P p(T). +Let �ν∂T,j, j = 1, . . . , dim( �P p+1(T)), denote a basis of �P p+1(T) with ∥�ν∂T,j∥∞ ≂ 1. Furthermore, let +ν∂T,j ∈ �P p+1(T), j = 1, . . . , dim( �P p+1(T)), denote a basis with +⟨ν∂T,j , �ν∂T,k⟩∂T = |∂T|δj,k +j, k = 1, . . . , dim( �P p+1(T)). +One verifies that ∥ν∂T,j∥∞ ≂ 1 by scaling arguments. Define ψ∂T = � +F∈FT ψF and note that +ψ∂T · nT |∂T = 1. Define +ψ∂T,j = ψ∂T �ν∂T,j +j = 1, . . . , dim( �P p+1(T)). +Thus, by our previous considerations, ⟨ψ∂T,j · nT , ν∂T,k⟩∂T = |∂T|δj,k, j, k = 1, . . . , dim( �P p+1(T)). +We also define higher order edge functions: For E ∈ E∗, j = 1, . . . , dim(P p(T)), define +ηE,j = ηE �χT,j. +For E ∈ E∗ let χE,j ∈ P p(T), j = 1, . . . , dim(P p(T)), denote a basis with +(χE,j , �χT,kηE)T = |T|δj,k +j, k = 1, . . . , dim(P p(T)). +One verifies that ∥χE,j∥∞ ≂ 1. Furthermore, define +σE,j = σEχE,j +j = 1, . . . , dim(P p(T)), E ∈ E∗. +The proof of the next result follows the arguments given in Lemma 3 together with the aforegoing +definitions and is thus omitted. +Lemma 4. We have that ηE,j · nT |∂T = 0, ηE,j ∈ P p+2(T), and +(σE,j , ηE′,k)T = δE,E′δj,k|T| +for all E, E′ ∈ E∗, j, k = 1, . . . , dim(P p(T)). +□ +2.4. Modified face bubble functions. We introduce modified face bubble functions. Before we +come to their definition and analysis we state the following result: +Lemma 5. Let R > 0. Consider R ≥ κ > 0 and the function φκ : [0, 1] → [0, 1], t �→ e−t/κ. Then, +∥φκ∥L2(0,1) ≂ κ1/2, +∥φ′ +κ∥L2(0,1) ≂ κ−1/2, +φκ(0) = 1. +The hidden constants only depend on R. +Proof. The results follow from straightforward calculations. +□ +Recall that dF = ηzF . We can interpret dF as a relative distance function that is 0 when restricted +to F and 1 when evaluated at the vertex opposite to F. Considering φ := φα/hT , i.e., +φ(t) = e−hT t/α, +define for F ∈ FT the modified face bubble function by +ηα,F := (φ ◦ dF )ηF . +(6) +Some basic properties of this modified function are given in the next result. A visualization of ηα,F +is presented in Figure 1. +Lemma 6. Suppose that 0 < α ≲ hT . For any F ∈ FT we have that +∥ηα,F ∥T ≲ |T|1/2 +� α +hT +�1/2 +, +∥∇ηα,F ∥T ≲ h−1 +T |T|1/2 +� α +hT +�−1/2 +, +ηα,F |∂T = ηF |∂T . +6 + +Figure 1. Visualization of the face bubble functions ηF and ηα,F on the reference +element �T and face F = (0, 1) × {0}. +Proof. The identity ηα,F |∂T = ηF |∂T follows since φ(0) = 1 and ηF |∂T\F ′ = 0 for F ′ ∈ FT \ {F}. +We show the details for n = 2. For n ≥ 3 we may argue similarly. Let AT : �T → T denote the +affine element mapping. W.l.o.g. let F ∈ FT be the face such AT : �T → T maps F to the edge +(0, 1) × {0}. Then, dF ◦ AT (�x, �y) = �y for (�x, �y) ∈ �T. Moreover, +ηα,F ◦ AT (�x, �y) = �ηα(�x, �y) := e−�yhT /α(1 − �x − �y)�x. +The remaining assertions follow by standard calculations, e.g., +∥ηα,F ∥2 +T = 2|T|∥�ηα∥2 +�T ≲ α +hT +|T|. +□ +With the same logic as for the definition of ηF,j we define +ηα,F,j = ηα,F �χF,j, +j = 1, . . . , dim(P p(F)), F ∈ FT . +The same proof as for Lemma 6 shows +Lemma 7. Suppose that α ≲ hT . Then, for any F, j the function ηα,F,j satisfies the assertions of +Lemma 6 (replacing ηα,F by ηα,F,j and ηF by ηF,j). +Define the modified Bernardi–Raugel elements by +ηα,F := ηα,F nF = (φ ◦ dF )ηF . +Some important properties of ηα,F follow directly from its definition and are summarized in the +next result. Its proof follows the same ideas as the proof of Lemma 6. +7 + +"f +na,F C=1 +0.3, +0.3, +0.2, +0.2, +0.1J +0.1 +0. +0 +0 +0 +0.5 +0.5 +0 +0.5 +1 +0 +0.5 +x +y +x +y +na,F α=10-1 +"a,F (=10-2 +0.3, +0.3, +0.2, +0.2, +0.1, +0.1 J +0. +0, +0 +0 +0.5 +0.5 +0 +0.5 +1 +0 +0.5 +y +X +yLemma 8. For F ∈ FT we have ηα,F · nT |∂T = ηF · nT |∂T and if α ≲ hT , then, +∥ηα,F ∥T ≲ |T|1/2 +� α +hT +�1/2 +, +∥ div ηα,F ∥T ≲ |T|1/2h−1 +T +� α +hT +�−1/2 +. +We also need higher order variants: Set ηα,∂T = � +F∈FT ηα,F and +ηα,∂T,j = ηα,∂T �ν∂T,j +j = 1, . . . , dim(P p+1 +c +(FT )). +Let χ∂T,j ∈ P p+1 +c +(FT ), j = 1, . . . , dim(P p+1 +c +(FT )) denote a basis of P p+1 +c +(FT ) with +⟨ηα,∂T,j · nT , χ∂T,k⟩∂T = |∂T|δj,k, +j, k = 1, . . . , dim(P p+1 +c +(∂T)). +The proof of the next result follows the proof of Lemma 8. +Lemma 9. The boundedness estimates of Lemma 8 hold with ηα,F replaced by ηα,∂T,j. +□ +To close this section and to have a better overview we summarize the most important basis +functions used in the remainder of this work in Table 1. +3. Fortin operator in H1(T) +We consider a fixed parameter α > 0 and space H1(T) equipped with the (squared) norm +∥v∥2 +T,α := ∥v∥2 +T + α2∥∇v∥2 +T . +The idea of this section is to construct Fortin operators, say Π∇ +F : H1(T) → V ∇ +h (with V ∇ +h ⊂ H1(T) +being some finite-dimensional subspace) such that, for a fixed p ∈ N0 and for all v ∈ H1(T), +∥Π∇ +F v∥T,α ≤ CF∥v∥T,α, +(7a) +⟨σ , v − Π∇ +F v⟩∂T = 0 +∀σ ∈ P p(FT ), +(7b) +(u, v − Π∇ +F v)T = 0 +∀u ∈ P p(T) +(7c) +with CF > 0 independent of α, hT (but possibly dependent on p). Note that (7b)–(7c) imply +(σ , ∇(1 − Π∇ +F )v)T = 0 +∀σ ∈ P p(T). +(7d) +This can be seen from integration by parts: Take σ ∈ P p(T), v ∈ H1(T). Then, div σ ∈ P p−1(T) +and +(σ , ∇Π∇ +F v)T = −(div σ , Π∇ +F v)T + ⟨σ · nT , Π∇ +F v⟩∂T += −(div σ , v)T + ⟨σ · nT , v⟩∂T = (σ , ∇v)T . +From the last identities we also see that the weaker condition +(u, v − Π∇ +F v)T = 0 +∀u ∈ P p−1(T) +(7c’) +would be sufficient to conclude (7d). However, depending on the problem, condition (7c) is needed, +e.g., in the presence of reaction terms as in the DPG method in Section 5. We stress that our Fortin +operator can be easily modified to satisfy (7c’) only. +8 + +function +space +basis +definition +property +low order +χF +P 0(FT ) +yes +χF |F ′ = δF,F ′ +– +νF +P 1(T) +yes +� +z∈VF ηz − (n − 1)dF +⟨νF , χF ′⟩∂T = |F|δF,F ′ +ψF +RT 0(T) +yes +ψF · nT |∂T = χF +⟨ψF · nT , νF ′⟩∂T = |F|δF,F ′ +ηF +P p(T) +no +ηF nF +⟨ηF · nT , νF ′⟩∂T = ⟨ηF , 1⟩F δF,F ′ +σE +P 0(T) +yes +– +σE · tE′ = δE,E′ +ηE +P 2(T) +no +ηEtE +(σE , ηE′)T = (σE , ηE)T δE,E′ +higher order +χT,j, �χT,j +P p(T) +yes +– +– +ηT,j +P p+n+1(T) +no +ηT �χT,j +(ηT,j , χT,k)T = |T|δj,k +�χF,j +P p(T) +yes +– +– +χF,j +P p(FT ) +yes +– +– +ηF,j +P p+n(T) +no +ηF �χF,j +⟨ηF,j , χF ′,k⟩∂T = δj,kδF,F ′|F| +ν∂T,j, �ν∂T,j +�P p+1(T) +yes +– +⟨ν∂T,j , �ν∂T,k⟩∂T = |∂T|δj,k +ψ∂T,j +P p+2(T) +no +�ν∂T,j +� +F∈FT ψF +⟨ψ∂T,j · nT , ν∂T,k⟩∂T = δj,k|∂T| +χE,j +P p(T) +yes +– +(χE,j , �χT,kηE)T = δj,k|T| +σE,j +P p(T) +yes +σEχE,j +– +ηE,j +P p+2(T) +no +ηE �χT,j +(σE,j , ηE′,k)T = δj,kδE,E′|T| +modified +φ +– +– +t �→ e−hT /αt +– +ηα,F +– +– +(φ ◦ dF )ηF +ηα,F |∂T = ηF |∂T +ηα,F,j +– +– +ηα,F �χF,j +ηα,F,j|∂T = ηF,j|∂T +ηα,F +– +– +ηα,F nF +ηα,F · nT |∂T = ηF · nT |∂T +χ∂T,j +P p+1 +c +(FT ) +yes +– +– +ηα,∂T,j +– +– +ηα,F �ν∂T,j +⟨ηα,F,j · nT , χ∂T,k⟩∂T = |∂T|δj,k +Table 1. Overview of special functions together with some of their main properties. +Here, ηz is the canonical Lagrange basis function of P 1(T), ηE, ηF , ηT are edge, +face, and element bubble functions, respectively, and dF = ηzF . In column basis we +indicate whether the respective family of functions generates the indicated space. +For a detailed description we refer to Sections 2.2–2.4. +3.1. Constructions for moderate parameter. Define the space +�V ∇ +hp = P 0(T) + span +� +ηF,j : j = 1, . . . , dim(P p(F)), F ∈ FT +� +and operator �Π∇ +F,hp : H1(T) → �V ∇ +hp for v ∈ H1(T) by +�Π∇ +F,hpv = Π0 +T v + +� +F∈FT +dim(P p(F)) +� +j=1 +⟨χF,j , (1 − Π0 +T )v⟩F +⟨χF,j , ηF,j⟩F +ηF,j. +The following result collects its main properties. +Lemma 10. Operator Π∇ +F = �Π∇ +F,hp is idempotent on P 0(T), satisfies property (7b) and +∥�Π∇ +F,hpv∥T ≲ ∥v∥T + hT ∥∇v∥T , +∥∇�Π∇ +F,hpv∥T ≲ ∥∇v∥T , +∥(1 − �Π∇ +F,hp)v∥T ≲ hT ∥∇v∥T +9 + +for all v ∈ H1(T). +Proof. Idempotency can be seen from the definition, since v ∈ P 0(T) implies that Π0 +T v = v, thus, +(1 − Π0 +T )v = 0. +Let v ∈ H1(T). To see (7b) we employ the orthogonality property ⟨ηF,j , χF ′,k⟩∂T = δF,F ′δj,k|F| +to get +⟨χF ′,k , v − �Π∇ +F v⟩∂T = ⟨χF ′,k , v − Π0 +T v⟩∂T − +� +F∈FT +dim(P p(F)) +� +j=1 +⟨χF,j , (1 − Π0 +T )v⟩F +⟨χF,j , ηF,j⟩F +⟨χF ′,k , ηF,j⟩∂T += ⟨χF ′,k , v − Π0 +T v⟩∂T − ⟨χF ′,k , (1 − Π0 +T )v⟩∂T = 0 +Since F ′ and k were arbitrary, condition (7b) follows. +The boundedness follows from the triangle inequality, the Cauchy–Schwarz inequality and bound- +edness of Π0 +T , i.e., +∥�Π∇ +F,hpv∥T ≤ ∥v∥T + +� +F∈FT +dim(P p(T)) +� +j=1 +∥χF,j∥F ∥v − Π0 +T v∥F +⟨χF,j , ηF,j⟩F +∥ηF,j∥T . +Note that ⟨χF,j , ηF,j⟩F = |F|, ∥ηF,j∥T ≂ |T|1/2, ∥χF,j∥F ≂ |F|1/2 which follows by standard scaling +arguments and the properties of the basis functions discussed in Section 2. Applying the trace +inequality (4) we see that +∥χF,j∥F ∥v − Π0 +T v∥F +⟨χF,j , ηF,j⟩F +∥ηF,j∥T ≂ |F|−1/2|T|1/2∥v − Π0 +T v∥F ≲ h1/2 +T h1/2 +T ∥∇v∥T . +Thus, we conclude that ∥�Π∇ +F,hpv∥T ≲ ∥v∥T + hT ∥∇v∥T . Then, with similar arguments but using +the inverse estimate ∥∇ηF,j∥T ≲ h−1 +T |T|1/2, we see that (∇Π0 +T v = 0) +∥∇�Π∇ +F,hpv∥T ≤ +� +F∈FT +dim(P p(F)) +� +j=1 +∥χF,j∥F ∥v − Π0 +T v∥F +⟨χF,j , ηF,j⟩F +∥∇ηF,j∥T ≲ ∥∇v∥T . +Finally, the approximation property is derived by using the idempotency, and the established bound- +edness estimates, i.e., +∥(1 − �Π∇ +F,hp)v∥T = ∥(1 − �Π∇ +F,hp)(v − Π0 +T v)∥T ≲ ∥v − Π0 +T v∥T + hT ∥∇(v − Π0 +T v)∥T ≲ hT ∥∇v∥T . +This concludes the proof. +□ +To obtain an operator that also satisfies property (7c) we consider slight modifications by adding +a correction term based on element bubbles. Define the space +V ∇ +hp = �V ∇ +hp + span +� +ηT,j : j = 1, . . . , dim(P p(T)) +� +and the operator Π∇ +F,hp : H1(T) → V ∇ +hp for all v ∈ H1(T) by +Π∇ +F,hpv = �Π∇ +F,hpv + +dim(P p(T)) +� +j=1 +(χT,j , (1 − �Π∇ +F,hp)v)T +(χT,j , ηT,j)T +ηT,j. +Theorem 11. Operator Π∇ +F = Π∇ +F,hp is idempotent on P 0(T), satisfies (7b)–(7c) and +∥Π∇ +F,hpv∥T ≲ ∥v∥T + hT ∥∇v∥T , +∥∇Π∇ +F,hpv∥T ≲ ∥∇v∥T , +∥(1 − Π∇ +F,hp)v∥T ≲ hT ∥∇v∥T +for all v ∈ H1(T). +10 + +Proof. The idempotency on P 0(T) follows from the idempotency of �Π∇ +F,hp (Lemma 10). +State- +ment (7b) follows also from Lemma 10 since the element bubbles ηT,j vanish on the boundary +and, therefore, Π∇ +F,hpv|∂T = �Π∇ +F,hpv|∂T . To see (7c) a simple calculation using the orthogonality +(χT,j , ηT,k)T = |T|δj,k yields +(χT,k , (1 − Π∇ +F )v)T = (χT,k , (1 − �Π∇ +F,hp)v)T − +dim(P p(T)) +� +j=1 +(χT,k , (1 − �Π∇ +F,hp)v)T +(χT,j , ηT,j)T +(χT,k , ηT,j)T += (χT,k , (1 − �Π∇ +F,hp)v)T − (χT,k , (1 − �Π∇ +F,hp)v)T = 0. +It remains to prove the boundedness estimates which follow — besides standard arguments — from +the boundedness estimates of �Π∇,j +F +(see Lemma 10). First, using the triangle inequality, Cauchy– +Schwarz inequality and scaling arguments we estimate +∥Π∇ +F,hpv∥T ≲ ∥�Π∇ +F,hpv∥T + +dim(P p(T)) +� +j=1 +|T|−1/2∥v − �Π∇ +F,hpv∥T ∥ηT,j∥T +≲ ∥�Π∇ +F,hpv∥T + ∥v − �Π∇ +F,hpv∥T ≲ ∥v∥T + hT ∥∇v∥T . +The gradient contribution is estimated by employing the inverse estimate ∥∇ηT,j∥T ≲ h−1 +T ∥ηT,j∥T +together with Lemma 10 to give +∥∇Π∇ +F,hpv∥T ≲ ∥∇�Π∇ +F,hpv∥T + h−1 +T ∥v − �Π∇ +F,hpv∥T ≲≲ ∥∇v∥T . +The final assertion ∥(1 − Π∇ +F,hp)v∥T ≲ hT ∥∇v∥T follows as in Lemma 10. +□ +Corollary 12. Suppose that hT ≲ α. From Lemma 10 and Theorem 11 it follows that Π∇ +F = �Π∇ +F,hp +and Π∇ +F = Π∇ +F,hp satisfy (7a). Particularly, Π∇ +F = Π∇ +F,hp has properties (7). +Remark 13. In general, discrete test spaces are chosen such that a Fortin operator exists, which +not necessarily implies approximation results of the form +min +vh∈Vh ∥v − vh∥T + ∥∇(v − vh)∥T ≲ hT |v|2,T +for v ∈ H2(T). +Here, | · |2,T denotes the H2(T) seminorm. For certain supercloseness results in the DPG method +the latter approximation property is needed, see [10]. To ensure this property one can simply require +P 1(T) ⊂ Vh. +3.2. Constructions for small parameter. In this section we focus on the case α ≲ hT . Let +Π∇ +F = �Π∇ +F,hp or Π∇ +F = Π∇ +F,hp. By Lemma 10 resp. Theorem 11 we have the boundedness +∥Π∇ +F v∥T ≲ ∥v∥T + hT ∥∇v∥T ≲ max{1, hT α−1} (∥v∥T + α∥∇v∥T ) . +(8) +We conclude that ∥Π∇ +F v∥T,α ≲ max{1, hT α−1}∥v∥T,α. This tells us that Π∇ +F is only conditionally +uniformly bounded. +Particularly, for small parameters α and coarse meshes huge boundedness +constants are expected so that robustness of the numerical methods is likely to be lost. This can +actually be observed in our numerical experiments presented in Section 5. Let us remark that the +operators constructed in [12, 5, 9] also satisfy (8) and are not suited for small parameters, α ≲ hT . +To overcome this problem we construct an operator based on the modified face bubble functions +ηα,F,j instead of ηF,j. The construction of the novel Fortin operators follows the definition of �Π∇ +F,hp +and Π∇ +F,hp replacing ηF,j with the modified face bubble functions ηα,F,j. First, set +�V ∇ +hp,α := P 0(T) + span{ηα,F,j : j = 1, . . . , dim(P p(F)), F ∈ FT }, +11 + +and define �Π∇ +F,hp,α : H1(T) → �V ∇ +hp,α for all v ∈ H1(T) by +�Π∇ +F,hp,αv := Π0 +T v + +� +F∈FT +dim(P p(F)) +� +j=1 +⟨χF,j , (1 − Π0 +T )v⟩F +⟨χF,j , ηα,F,j⟩F +ηα,F,j. +Its main properties are given in +Lemma 14. Operator Π∇ +F = �Π∇ +F,hp,α satisfies (7b) and is idempotent on P 0(T). If α ≲ hT , then +∥�Π∇ +F,hp,αv∥T,α ≲ ∥v∥T,α, +∥v − �Π∇ +F,hp,αv∥T ≲ hT ∥∇v∥T +for all v ∈ H1(T). +Proof. The idempotency on P 0(T) can be seen directly from the definition of the operator. Noting +that ηα,F,j|∂T = ηF,j|∂T for any F, j the proof of Fortin property (7b) follows as in Lemma 10. +It remains to prove the boundedness estimates. Suppose that α ≲ hT . First, using the triangle +inequality and the Cauchy–Schwarz inequality together with the properties of the modified bubble +(Lemma 7), ∥χF,j∥F ≂ |F|1/2, ⟨χF,j , ηα,F,j⟩F = |F| we infer +∥�Π∇ +F,hp,αv∥T ≤ ∥v∥T + +� +F∈FT +dim(P p(F)) +� +j=1 +∥χF,j∥F ∥v − Π0 +T v∥F +⟨χF,j , ηα,F,j⟩ +∥ηα,F,j∥T +≲ ∥v∥T + +� +F∈FT +∥v − Π0 +T v∥F |F|−1/2|T|1/2α1/2h−1/2 +T +≂ ∥v∥T + +� +F∈FT +∥v − Π0 +T v∥F α1/2. +Then, with the multiplicative trace inequality (4) and Young’s inequality we further get +∥v − Π0 +T v∥F α1/2 ≲ ∥v − Π0 +T v∥1/2 +T α1/2∥∇v∥1/2 +T +≲ ∥v − Π0 +T v∥T + α∥∇v∥T . +Putting all the estimates together we infer that +∥�Π∇ +F,hp,αv∥T ≲ ∥v∥T + +� +F∈FT +∥v − Π0 +T v∥F α1/2 ≲ ∥v∥T + α∥∇v∥T . +We are left with the gradient contribution of the norm. With similar arguments as before we get +α∥∇�Π∇ +F,hp,αv∥T ≤ α +� +F∈FT +dim(P p(F)) +� +j=1 +∥χF,j∥F ∥v − Π0 +T v∥F +⟨χF,j , ηα,F,j⟩ +∥∇ηα,F,j∥T +≲ α +� +F∈FT +α−1/2∥v − Π0 +T v∥F +≲ α1/2∥v − Π0 +T v∥1/2 +T ∥∇v∥1/2 +T +≲ ∥v∥T + α∥∇v∥T . +Combining all the estimates from above we conclude the boundedness of the Fortin operator. +Finally, to see the approximation property note that �Π∇ +F,hp,α1 = 1, thus, +∥(1 − �Π∇ +F,hp,α)v∥T = ∥(1 − �Π∇ +F,hp,α)(v − Π0 +T v)∥T ≲ ∥v − Π0 +T v∥T + α∥∇v∥T ≲ hT ∥∇v∥T . +Note that we have used that α ≲ hT . +□ +To ensure property (7c) we follow the idea of the definition of Π∇ +F,hp by adding correction terms +based on element bubble functions. Contrary to the face bubble functions, the element bubble +functions do not need to be modified. Set +V ∇ +hp,α = �V ∇ +hp,α + span +� +ηT,j : j = 1, . . . , dim(P p(T)) +� +12 + +and define Π∇ +F,hp,α : H1(T) → V ∇ +hp,α for all v ∈ H1(T) by +Π∇ +F,hp,αv := �Π∇ +F,hp,αv + +dim(P p(T)) +� +j=1 +(χT,j , (1 − �Π∇ +F,hp,α)v)T +(χT,j , ηT,j)T +ηT,j. +The following theorem is one of our main results. +Theorem 15. Suppose that α ≲ hT . Then, Π∇ +F = Π∇ +F,hp,α is idempotent on P 0(T) and satisfies (7). +Proof. Idempotency follows from the definition. Verifying (7b)–(7c) follows as in Theorem 11. For +the boundedness estimate we also use the arguments displayed in the proof of Theorem 11. Note +that with Lemma 14 and ∥∇ηT,j∥T ≲ h−1 +T |T|1/2 we see that, for v ∈ H1(T), +∥Π∇ +F,hp,αv∥T,α ≲ ∥�Π∇ +F,hp,αv∥T,α + +dim(P p(T)) +� +j=1 +(χT,j , v − �Π∇ +F,hp,αv)T +(χT,j , ηT,j)T +(∥ηT,j∥T + α∥∇ηT,j∥T ) +≲ ∥v∥T,α + |T|−1/2∥v − �Π∇ +F,hp,αv∥T |T|1/2 + |T|−1/2∥v − �Π∇ +F,hp,αv∥T αh−1 +T |T|1/2 +≲ ∥v∥T,α + ∥v − �Π∇ +F,hp,αv∥T . +In the last step we used α ≲ hT . Boundedness of �Π∇ +F,hp,α finishes the proof. +□ +3.3. Alternative operator for lowest-order spaces and moderate parameter. In this section +we construct a Fortin operator Π∇ +F : H1(T) → V ∇,0 +h +such that (7) is satisfied for the lowest-order +case p = 0. In [6] it is shown that for a low-order DPG method for the Poisson problem, test space +V ∇ +h += P 1(T) for the scalar test functions is sufficient to guarantee well-posedness. The authors +of [6] did use different techniques. Here, we complement their results by the construction of a Fortin +operator. Define the spaces +�V ∇,0 +h += P 1(T), +V ∇,0 +h += �V ∇,0 +h ++ span{ηT } +and operators �Π∇,0 +F +: H1(T) → �V ∇,0 +h +, Π∇,0 +F +: H1(T) → V ∇,0 +h +for all v ∈ H1(T) by +�Π∇,0 +F v = Π0 +T v + +� +F∈FT +⟨1, (1 − Π0 +T )v⟩F +⟨1, νF ⟩F +νF , +Π∇,0 +F v = �Π∇,0 +F v + (1, (1 − �Π∇,0 +F )v)T +(1, ηT )T +ηT . +The analysis of these operators can be done as in Section 3.1. The details are left to the reader. +Theorem 16. Let p = 0. +Operator Π∇ +F = �Π∇,0 +F +resp. +Π∇ +F = Π∇,0 +F +is idempotent on P 0(T), +satisfies (7b) and +∥Π∇ +F v∥T ≲ ∥v∥T + hT ∥∇v∥T , +∥∇Π∇ +F v∥T ≲ ∥∇v∥T +for all v ∈ H1(T). Operator Π∇ +F = Π∇,0 +F +additionally satisfies (7c). +□ +3.4. Comparison with existing Fortin operators. As mentioned in the introduction, several +works have already dealt with the construction of Fortin operators on a simplex that satisfy (7), +see, e.g., [12, 5, 9]. These works all have in common that they construct resp. prove the existence +of a Fortin operator Π∇ +F : H1(T) → P p+n(T) which satisfy (7a)-(7b) (for α ≳ hT ) and +(u, (1 − Π∇ +F )v)T = 0 +∀u ∈ P p−1(T), v ∈ H1(T). +The latter condition is not the same as (7c). In order to satisfy (7c) one needs to increase the +polynomial degree by one, i.e., P p+1+n(T) instead of P p+n(T). +13 + +Comparing the dimension of P p+1+n(T) and the space V ∇ +hp, we get +dim(P p+1+n(T)) = +�n +j=1(p + 1 + n + j) +n! +, +dim(V ∇ +hp) = 1 + (n + 1) +�n−1 +j=1 (p + j) +(n − 1)! ++ +�n +j=1(p + j) +n! +. +For the lowest-order case p = 0 we thus find +dim(P n+1(T)) = +� +10 +n = 2, +35 +n = 3, +and +dim(V ∇ +h0) = +� +5 +n = 2, +6 +n = 3. +For operator Π∇,0 +F +: H1(T) → V ∇,0 +h +we even have a reduction to +dim(V ∇,0 +h +) = n + 1 + 1 = +� +4 +n = 2, +5 +n = 3. +In conclusion, our test spaces are systematically smaller than previously used ones, and guarantee +robustness contrary to the previous cases. +4. Fortin operator in H(div; T) +We consider a fixed parameter α > 0 and space H(div; T) equipped with the (squared) norm +∥τ∥2 +T,α = ∥τ∥2 +T + α2∥ div τ∥2 +T +for τ ∈ H(div; T). +The motivation for this section is the construction of Fortin operators, say Πdiv +F : H(div; T) → Vh +(where Vh ⊆ H(div; T) is some discrete space) such that, for all τ ∈ H(div; T), +∥Πdiv +F τ∥T,α ≤ CF∥τ∥T,α, +(9a) +⟨(τ − Πdiv +F τ) · nT , u⟩∂T = 0 +∀u ∈ P p+1 +c +(FT ), +(9b) +(σ , τ − Πdiv +F τ)T = 0 +∀σ ∈ P p(T) +(9c) +with CF > 0 independent of α, hT (but possibly dependent on p ∈ N0). The latter two identities +also imply +(u, div(τ − Πdiv +F τ))T = 0 +∀u ∈ P p+1(T), +(9d) +which can be seen from integration by parts: Let u ∈ P p+1(T) be given, then +(u, div Πdiv +F τ)T = −(∇u, Πdiv +F τ)T + ⟨Πdiv +F τ · nT , u|∂T ⟩∂T += −(∇u, τ)T + ⟨τ · nT , u|∂T ⟩∂T = (u, div τ)T +∀τ ∈ H(div; T). +4.1. Construction for moderate parameter. Define +�V div +hp = span +� +ψ∂T,j : j = 1, . . . , dim( �P p+1(T)) +� +⊂ P p+2(T) +and operator �Πdiv +F,hp : H(div; T) → �V div +hp +by +�Πdiv +F,hpτ = +dim(P p+1 +c +(∂T)) +� +j=1 +⟨τ · nT , ν∂T,j⟩∂T +⟨ψ∂T,j · nT , ν∂T,j⟩∂T +ψ∂T,j. +We collect its main properties. +Lemma 17. Operator Πdiv +F += �Πdiv +F,hp satisfies (9b) and +∥�Πdiv +F,hpτ∥T ≲ ∥τ∥T + hT ∥ div τ∥T +∀τ ∈ H(div; T). +14 + +Proof. To see (9b) a simple computation yields +⟨(1 − �Πdiv +F,hp)τ · nT , ν∂T,k⟩∂T = ⟨τ · nT , ν∂T,k⟩∂T − +� +j +⟨τ · nT , ν∂T,j⟩∂T +⟨ψ∂T,j · nT , ν∂T,j⟩∂T +⟨ψ∂T,j · nT , ν∂T,k⟩∂T += ⟨τ · nT , ν∂T,k⟩∂T − ⟨τ · nT , ν∂T,k⟩∂T = 0. +Resolving the duality term, the Cauchy–Schwarz inequality and properties of basis functions give +|⟨τ · nT , ν∂T,j⟩∂T | = |(τ , ∇ν∂T,j)T + (div τ , ν∂T,j)T | +≲ ∥τ∥T h−1 +T |T|1/2 + ∥ div τ∥T |T|1/2. +With the triangle inequality we conclude that +∥�Πdiv +F,hpτ∥T ≤ +� +j +|⟨τ · nT , ν∂T,j⟩∂T | +⟨ψ∂T,j · nT , ν∂T,j⟩∂T +∥ψ∂T,j∥T +≲ |∂T|−1(∥τ∥T h−1 +T |T|1/2 + ∥ div τ∥T |T|1/2)|T|1/2 ≲ ∥τ∥T + hT ∥ div τ∥T +which finishes the proof. +□ +For the definition of operators that ensure (9c) we add a correction term based on the functions +ηE,j. We set +V div +hp = �V div +hp + span +� +ηE,j : j = 1, . . . , dim P p(T), E ∈ E∗ +� +and define Πdiv +F,hp : H(div; T) → V div +hp +for all τ ∈ H(div; T) by +Πdiv +F,hpτ = �Πdiv +F,hpτ + +� +E∈E∗ +dim(P p(T)) +� +j=1 +(σE,j , (1 − �Πdiv +F,hp)τ)T +(σE,j , ηE,j)T +ηE,j. +Theorem 18. Operator Πdiv +F += Πdiv +F,hp satisfies (9b)–(9c) and +∥Πdiv +F,hpτ∥T ≲ ∥τ∥T + hT ∥ div τ∥T +∀τ ∈ H(div; T). +Furthermore, div ◦Πdiv +F,hp = Πp+1 +T +◦ div. If hT ≲ α, then Πdiv +F += Πdiv +F,hp satisfies (9a). +Proof. Since ηE,j · nT |∂T = 0, we get Πdiv +F,hpτ · nT |∂T = �Πdiv +F,hpτ · nT |∂T , thus, condition (9b) follows +from Lemma 17. To see (9c) we compute for any E′, k +(σE′,k , (1 − Πdiv +F,hp)τ)T = (σE′,k , (1 − �Πdiv +F,hp)τ)T − +� +E∈E∗ +� +j +(σE,j , (1 − �Πdiv +F,hp)τ)T +(σE,j , ηE,j)T +(σE′,k , ηE,j)T += (σE′,k , (1 − �Πdiv +F,hp)τ)T − (σE′,k , (1 − �Πdiv +F,hp)τ)T = 0. +Noting that properties of the basis functions and boundedness by Lemma 17 give +|(σE,j , (1 − �Πdiv +F,hp)τ)T | +(σE,j , ηE,j)T +∥ηE,j∥T ≲ ∥(1 − �Πdiv +F,hp)τ∥T ≲ ∥τ∥T + hT ∥ div τ∥T , +we see that +∥Πdiv +F,hpτ∥T ≤ ∥�Πdiv +F,hpτ∥T + +� +E∈E∗ +� +j +|(σE,j , (1 − �Πdiv +F,hp)τ)T | +(σE,j , ηE,j)T +∥ηE,j∥T ≲ ∥τ∥T + hT ∥ div τ∥T . +The commutativity property follows from (9d) and the fact that V div +hp +⊆ P p+2(T) and, therefore, +div Πdiv +F,hpτ ∈ P p+1(T). +15 + +For the final assertion note that ∥ div Πdiv +F,hpτ∥T = ∥Πp+1 +T +div τ∥T ≤ ∥ div τ∥T by the commuta- +tivity property. Together with boundedness in the L2(T) norm established above, this finishes the +proof. +□ +Remark 19. In general, discrete test spaces are chosen such that a Fortin operator exists, which +not necessarily implies approximation results of the form +min +τ h∈Vh ∥τ − τ h∥T + ∥ div(τ − τ h)∥T ≲ hT ∥∇τ∥T + ∥(1 − Π0 +T ) div τ∥T +for τ ∈ H1(T). +For certain supercloseness results in the DPG method the latter approximation property is required, +see [10]. To ensure this property one can simply require RT 0(T) ⊂ Vh. +4.2. Construction for small parameter. For a small parameter, i.e., α ≲ hT , we build a Fortin +operator quite a bit different to the ones presented in the previous section. The proof of boundedness +requires as in the scalar case the multiplicative version of the trace inequality, but also the following +Helmholtz decomposition together with elliptic regularity. +Lemma 20. Let τ ∈ H(div; T). There exist r ∈ H1 +0(T), q ∈ H(curl ; T) (for n = 2 we have +q ∈ H1(T)) such that τ = ∇r + curl q and +∥curl q∥2 +T + ∥∇r∥2 +T = ∥τ∥2 +T , +∥D2r∥T ≲ ∥ div τ∥T +with constants independent of T. +Proof. Define r ∈ H1 +0(T) as the weak solution of ∆r = div τ, r|∂T = 0. Elliptic regularity implies +(note that div τ ∈ L2(T) and T is convex) that r ∈ H2(T). Moreover, [13, Theorem 3.1.1.2] implies +that +� +|β|=2 +∥Dβr∥T ≤ C∥∆r∥T +with a constant independent of T. Finally, div(τ − ∇r) = 0 by construction implies that τ − ∇r = +curl q for some q ∈ H(curl ; T) for n = 3 resp. q ∈ H1(T) for n = 2. +□ +The definition and analysis of our Fortin operator is based on the Helmholtz decomposition +τ = ∇r + curl q. For the curl q contribution we use the Fortin operator defined in the previous +section. For ∇r we consider the following definitions and analysis: Define the space +�V div,aux +hp,α += P 0(T) + span +� +ηα,F,j : j = 1, . . . , dim(P p+1 +c +(∂T)) +� +and operator �Πdiv,aux +F,hp +: H(div; T) → �V div,aux +hp,α +for all τ ∈ H(div; T) by +�Πdiv,aux +F,hp,α τ = Π0 +T τ + +dim(P p+1 +c +(FT )) +� +j=1 +⟨(1 − Π0 +T )τ · nT , χ∂T,j⟩∂T +⟨ηα,∂T,j · nT , χ∂T,j⟩∂T +ηα,∂T,j. +Lemma 21. Operator Πdiv +F += �Πdiv,aux +F,hp,α satisfies (9b). If α ≲ hT , then +∥�Πdiv,aux +F,hp,α ∇r∥T,α ≲ ∥∇r∥T,α +for all r ∈ H1 +0(T) ∩ H2(T). +Proof. The verification of (9b) follows similarly as in Lemma 17. We leave the details to the reader +and focus on details for the proof of boundedness. Let r ∈ H1 +0(T) ∩ H2(T). Note that elliptic +16 + +regularity yields ∥D2r∥T ≲ ∥ div ∇r∥T with constant independent of T. +Using standard norm +estimates, the multiplicative version of the trace inequality (see Lemma 1) and Lemma 9, we get +|⟨(1 − Π0 +T )∇r · nT , χ∂T,j⟩∂T | +⟨ηα,∂T,j · nT , χ∂T,j⟩∂T +∥ηα,∂T,j∥T ≲ ∥(1 − Π0 +T )∇r∥∂T |∂T|−1/2|T|1/2(α/hT )1/2 +≲ ∥(1 − Π0 +T )∇r∥1/2 +T α1/2∥D2r∥1/2 +T +≲ ∥(1 − Π0 +T )∇r∥T + α∥∆r∥T ≤ ∥∇r∥T + α∥∆r∥T . +By summing over all indices and bounding ∥Π0 +T ∇r∥T ≤ ∥∇r∥T we find that +∥�Πdiv,aux +F,hp,α ∇r∥T ≲ ∥Π0 +T ∇r∥T + ∥∇r∥T + α∥∆r∥T ≲ ∥∇r∥T + α∥ div ∇r∥T . +The same argumentation also proves +α|⟨(1 − Π0 +T )∇r · nT , χ∂T,j⟩∂T | +⟨ηα,∂T,j · nT , χ∂T,j⟩∂T +∥ div ηα,∂T,j∥T ≲ α∥(1 − Π0 +T )∇r∥∂T |∂T|−1/2|T|1/2h−1 +T (α/hT )−1/2 +≲ ∥(1 − Π0 +T )∇r∥1/2 +T α1/2∥D2r∥1/2 +T +≲ ∥(1 − Π0 +T )∇r∥T + α∥ div ∇r∥T +≤ ∥∇r∥T + α∥ div ∇r∥T . +Summing over all indices and using div Π0 +T ∇r = 0 we conclude that +α∥ div �Πdiv,aux +F,hp,α ∇r∥T ≲ ∥∇r∥T + α∥ div ∇r∥T . +Combining all estimates finishes the proof. +□ +For the definition of operators that ensure (9c) we add — as before — a correction term based +on the functions ηE,j. We stress that these edge functions do not need to be modified. Set +V div +hp,α = V div +hp + �V div,aux +hp +and define Πdiv +F,hp : H(div; T) → V div +hp +for all τ = ∇r + curl q ∈ H(div; T) by +Πdiv +F,hp,ατ = Πdiv +F,hpcurl q + �Πdiv,aux +F,hp,α ∇r + +� +E∈E∗ +dim(P p(T)) +� +j=1 +(σE,j , (1 − �Πdiv,aux +F,hp,α )∇r)T +(σE,j , ηE,j)T +ηE,j. +The following is one of our main results. +Theorem 22. Operator Πdiv +F += Πdiv +F,hp,α satisfies (9b)–(9c). If α ≲ hT , then +∥Πdiv +F,hp,ατ∥T,α ≲ ∥τ∥T,α +∀τ ∈ H(div; T). +Proof. The verification of (9b)–(9c) follows from the arguments already seen in Theorem 18 and +Lemma 21. +It only remains to prove boundedness. By the triangle inequality we get +∥Πdiv +F,hp,ατ∥T,α ≤ ∥Πdiv +F,hp,αcurl q∥T,α + ∥Πdiv +F,hp,α∇r∥T,α +for τ = ∇r + curl q ∈ H(div; T) where r, q are defined as in Lemma 20. +Note that Πdiv +F,hp,αcurl q = Πdiv +F,hpcurl q and applying Theorem 18 and Lemma 20 we see that +∥Πdiv +F,hpcurl q∥T ≲ ∥curl q∥T + hT ∥ div curl q∥T = ∥curl q∥T ≤ ∥τ∥T +as well as div Πdiv +F,hpcurl q = Πp+1 +T +div curl q = 0. We conclude that +∥Πdiv +F,hp,αcurl q∥T,α ≲ ∥τ∥T ≤ ∥τ∥T,α. +17 + +It remains to estimate ∥Πdiv +F,hp,α∇r∥T,α. Using the triangle inequality, the Cauchy–Schwarz inequal- +ity, estimates for basis functions and Lemma 21, we get +∥Πdiv +F,hp,α∇r∥T ≤ ∥�Πdiv,aux +F,hp,α ∇r∥T + +� +E∈E∗ +� +j +|(σE,j , (1 − �Πdiv,aux +F,hp,α )∇r)T | +(σE,j , ηE,j)T +∥ηE,j∥T +≲ ∥�Πdiv,aux +F,hp,α ∇r∥T + |T|1/2∥(1 − �Πdiv,aux +F,hp,α )∇r∥T |T|−1|T|1/2 ≲ ∥∇r∥T,α. +For the divergence part in the norm, recall that ηE,j ∈ P p+2(T). Therefore, div ηE,j ∈ P p+1(T) +and (9d) implies +(div ηE,j , div(1 − Πdiv +F,hp,α)∇r)T = 0, +j = 1, . . . , dim(P p(T)), E ∈ E∗. +We conclude that +∥ div(1 − Πdiv +F,hp,α)∇r∥2 +T = (div(1 − Πdiv +F,hp,α)∇r , div(1 − Πdiv +F,hp,α)∇r)T += (div(1 − �Πdiv,aux +F,hp,α )∇r , div(1 − Πdiv +F,hp,α)∇r)T +≤ ∥ div(1 − �Πdiv,aux +F,hp,α )∇r∥T ∥ div(1 − Πdiv +F,hp,α)∇r∥T . +It follows that ∥ div(1 − Πdiv +F,hp,α)∇r∥T ≤ ∥ div(1 − �Πdiv,aux +F,hp,α )∇r∥T and with the triangle inequality +and Lemma 21 we get that +α∥ div Πdiv +F,hp,α∇r∥T ≤ α∥ div ∇r∥T + α∥ div(1 − �Πdiv,aux +F,hp,α )∇r∥T ≲ ∥∇r∥T,α +which finishes the proof together with ∥∇r∥T,α ≤ ∥τ∥T,α (Lemma 20). +□ +4.3. Alternative operator for lowest order and moderate parameter. First, consider the +spaces +�V div,1 +h += RT 0(T), +�V div,2 +h += P 0(T) + span +� +ηF : F ∈ FT +� +and operators �Πdiv,j +F +: H(div; T) → �V div,j +h +, j = 1, 2, for all τ ∈ H(div; T) by +�Πdiv,1 +F +τ = +� +F∈FT +⟨τ · nT , νF ⟩∂T +⟨ψF · nT , νF ⟩∂T +ψF , +(10a) +�Πdiv,2 +F +τ = Π0 +T τ + +� +F∈FT +⟨(1 − Π0 +T )τ · nT , νF ⟩∂T +⟨ηF · nT , νF ⟩∂T +ηF . +(10b) +One verifies that �Πdiv,1 +F +is a projector whereas �Πdiv,2 +F +is idempotent on P 0(T). Defining the spaces +V div,j +h += �V div,j +h ++ span +� +ηE : E ∈ E∗ +� +, +j = 1, 2, +we introduce operators Πdiv,j +F +: V → V div,j +h +, j = 1, 2, for all τ ∈ H(div; T) by +Πdiv,j +F +τ = �Πdiv,j +F +τ + +� +E∈E∗ +(σE , (1 − �Πdiv,j +F +)τ)T +(σE , ηE)T +ηE. +Theorem 23. Operator Πdiv +F +∈ +� +Πdiv,j +F +: j = 1, 2 +� +satisfies (9b)–(9c) and +∥Πdiv +F τ∥T ≲ ∥τ∥T + hT ∥ div τ∥T , +∥ div Πdiv +F τ∥T ≲ ∥ div τ∥T +∀τ ∈ H(div; T). +Moreover, div ◦Πdiv,1 +F += Π1 +T ◦div. Furthermore, operator Πdiv,1 +F +is idempotent on RT 0(T) and Πdiv,2 +F +is idempotent on P 0(T). +18 + +Proof. Idempotency of the operators follows from their definitions. Let Πdiv +F +∈ +� +Πdiv,j +F +: j = 1, 2 +� +and �Πdiv +F +∈ +��Πdiv,j +F +: j = 1, 2 +� +. +First, we check condition (9b). +It holds for �Πdiv +F +as can be +seen with the same arguments as in Lemma 17. +Since ηE · nT |∂T = 0 by Lemma 3 we have +Πdiv +F τ · nT |∂T = �Πdiv +F τ · nT |∂T . We conclude that (9b) follows. +Second, condition (9c) can be seen as follows, +(σE′ , Πdiv +F τ)T = (σE′ , �Πdiv +F τ)T + +� +E∈E∗ +(σE , (1 − �Πdiv +F )τ)T +(σE , ηE)T +(σE′ , ηE)T += (σE′ , �Πdiv +F τ)T + (σE′ , (1 − �Πdiv +F )τ)T = (σE′ , τ)T +∀E′ ∈ E∗. +Next we prove boundedness. Estimate ∥�Πdiv,1 +F +τ∥T ≲ ∥τ∥T + hT ∥ div τ∥T follows as in Lemma 17. +For the second operator we stress that +|⟨(1 − Π0 +T )τ · nT , νF ⟩T | = |(div τ , νF )T + ((1 − Π0 +T )τ , ∇νF )T | = |(div τ , νF )T | ≲ ∥ div τ∥T |T|1/2. +The second identity follows since ∇νF ∈ P 0(T). This allows us to estimate +∥�Πdiv,2 +F +τ∥T ≲ ∥Π0 +T τ∥T + +� +F∈FT +|F|−1|T|∥ div τ∥T ≲ ∥τ∥T + hT ∥ div τ∥T . +Then, by the triangle inequality, the Cauchy–Schwarz inequality, norm estimates of basis functions +and the previously established boundedness estimates, we see that +∥Πdiv +F τ∥T ≤ ∥�Πdiv +F τ∥T + +� +E∈E∗ +|(σE , (1 − �Πdiv +F )τ)T | +(σE , ηE)T +∥ηE∥T +≲ ∥�Πdiv +F τ∥T + |T|1/2∥(1 − �Πdiv +F )τ∥T |T|−1|T|1/2 +≲ ∥�Πdiv +F τ∥T + ∥(1 − �Πdiv +F )τ∥T ≲ ∥τ∥T + hT ∥ div τ∥T . +Furthermore, the same arguments and div Π0 +T τ = 0, ∥ div ηF ∥T ≂ h−1 +T |T|1/2 show that +∥ div �Πdiv,2 +F +τ∥T ≲ ∥ div τ∥T . +Note that Πdiv,1 +F +τ ∈ P 2(T), thus, div Πdiv,1 +F +τ ∈ P 1(T). Consequently, (9d) implies the commu- +tativity property of Πdiv,1 +F +. Clearly, this also yields ∥ div Πdiv,1 +F +τ∥T ≤ ∥ div τ∥T . It thus remains +to prove that ∥ div Πdiv,2 +F +τ∥T ≲ ∥ div τ∥T . +To do so we argue as in the proof of Theorem 22 +to derive ∥ div(1 − Πdiv,2 +F +)τ∥T ≤ ∥ div(1 − �Πdiv,2 +F +)τ∥T . Together with the triangle inequality and +∥ div �Πdiv,2 +F +τ∥T ≲ ∥ div τ∥T we get that +∥ div Πdiv,2 +F +τ∥T ≤ ∥ div τ∥T + ∥ div(1 − �Πdiv,2 +F +)τ∥T ≲ ∥ div τ∥T . +This finishes the proof. +□ +4.4. Alternative operator for lowest order and small parameter. In this section we construct +a simpler Fortin operator for α ≲ hT and the lowest-order case (p = 0 in (9)), based on Πdiv,2 +F +from +the previous section. Define the spaces +�V div +h,α = P 0(T) + span +� +ηα,F : F ∈ FT +� +, +V div +h,α = �V div +h,α + span +� +ηE : E ∈ E∗ +� +19 + +and operators �Πdiv +F,α : H(div; T) → �V div +h,α , Πdiv +F,α : H(div; T) → V div +h,α for all τ ∈ H(div; T) by +�Πdiv +F,ατ = Π0 +T τ + +� +F∈FT +⟨(1 − Π0 +T )τ · nT , νF ⟩∂T +⟨ηα,F · nT , νF ⟩∂T +ηα,F , +Πdiv +F,ατ = �Πdiv +F,ατ + +� +E∈E∗ +(σE , (1 − �Πdiv +F,α)τ)T +(σE , ηE)T +ηE. +Theorem 24. Operator Πdiv +F += Πdiv +F,α satisfies (9b)–(9c) for p = 0. If α ≲ hT then +∥Πdiv +F,ατ∥T,α ≲ ∥τ∥T,α +∀τ ∈ H(div; T). +Proof. The verification of (9b)–(9c) follows as in Theorem 18. +It remains to prove boundedness. First, we show boundedness of �Πdiv +F,α. Let τ = ∇r + curl q ∈ +H(div; T) with r, q as in Lemma 20 be given. Using ∇νF ∈ P 0(T), div curl q = 0 we obtain +�Πdiv +F,αcurl q = Π0 +T curl q + +� +F∈FT +(div(1 − Π0 +T )curl q , νF )T + ((1 − Π0 +T )curl q , ∇νF )T +⟨ηα,F · nT , νF ⟩∂T +ηα,F += Π0 +T curl q. +Then, ∥�Πdiv +F,αcurl q∥T,α = ∥Π0 +T curl q∥T ≤ ∥curl q∥T ≤ ∥τ∥T,α. +With the multiplicative trace +inequality (Lemma 1), properties of the basis functions, Lemma 8 and Lemma 20 we get +|⟨(1 − Π0 +T )∇r · nT , νF ⟩∂T | +⟨ηα,F · nT , νF ⟩∂T +∥ηα,F ∥T ≲ |F|−1|∂T|1/2∥(1 − Π0 +T )∇r∥∂T ∥ηα,F ∥T +≲ |∂T|−1/2∥(1 − Π0 +T )∇r∥1/2 +T ∥D2r∥1/2 +T α1/2h−1/2 +T +|T|1/2 +≲ ∥∇r∥1/2 +T α1/2∥ div τ∥1/2 +T +≲ ∥τ∥T + α∥ div τ∥T . +The last estimates yield ∥�Πdiv +F,α∇r∥T ≲ ∥τ∥T,α. For the divergence contribution the same arguments +prove +α|⟨(1 − Π0 +T )∇r · nT , νF ⟩∂T | +⟨ηα,F · nT , νF ⟩∂T +∥ div ηα,F ∥T ≲ |F|−1|∂T|1/2∥(1 − Π0 +T )∇r∥∂T h−1 +T |T|1/2α1/2h1/2 +T +≲ ∥∇r∥1/2 +T α1/2∥ div τ∥1/2 +T +≲ ∥τ∥T + α∥ div τ∥T . +We conclude that α∥ div �Πdiv +F,α∇r∥T ≲ ∥τ∥T,α. Putting all estimates together we have shown that +∥�Πdiv +F,ατ∥T,α ≤ ∥�Πdiv +F,αcurl q∥T,α + ∥�Πdiv +F,α∇r∥T,α ≲ ∥τ∥T,α. +Arguing as in the proof of Theorem 23 we find that +∥Πdiv +F,ατ∥T ≲ ∥�Πdiv +F,ατ∥T + ∥(1 − �Πdiv +F,α)τ∥T +giving us ∥Πdiv +F,ατ∥T ≲ ∥τ∥T,α. Finally, arguing as in the proof of Theorem 22 we find that +∥ div(1 − Πdiv +F,α)τ∥T ≤ ∥ div(1 − �Πdiv +F,α)τ∥T . +Together with the boundedness of �Πdiv +F,α we conclude that α∥ div Πdiv +F,ατ∥T ≲ ∥τ∥T,α which finishes +the proof. +□ +20 + +4.5. Comparison with existing Fortin operators. Fortin operators that satisfy (9) are con- +structed in [12, 5, 9]. In [12, Lemma 3.3], it is shown that there exists a Fortin operator, map- +ping into the discrete test space P p+2(T) and in [9], the authors impose the minimal condition +RT p+1(T) ⊂ V div +h +to ensure the existence of a Fortin operator satisfying (9). Here, RT p+1(T) = +P p+1(T) + xP p+1(T) denotes the Raviart–Thomas space. We stress that all these mentioned oper- +ators are uniformly bounded only if hT ≲ α. +Computing dimensions we get +dim(P p+2(T)) = n +�n +j=1(j + p + 2) +n! +, +dim(RT p+1(T)) = n +�n +j=1(j + p) +n! ++ (n + 1) +�n−1 +j=1 (j + p + 1) +(n − 1)! +. +To compute the dimension of V div +hp +we note that dim(P p+1 +c +(FT )) = dim(P p+1(T)) − dim(P p+1 +b +(T)) +where we recall that P p +b (T) denotes the space of element bubbles. This yields +dim(V div +hp ) = dim(P p+1 +c +(FT )) + dim(P p(T)) += +�n +j=1(j + p + 1) +n! +− +�n +j=1(j + p − n) +n! ++ n +�n +j=1(j + p) +n! +. +For the lowest-order case p = 0 we thus get +dim(P 2(T)) = +� +12 +n = 2, +30 +n = 3, +dim(RT 1(T)) = +� +8 +n = 2, +15 +n = 3, +and +dim(V div +h0 ) = +� +5 +n = 2, +7 +n = 3. +As in Section 3.4 we conclude that our test spaces are systematically smaller than previously used +ones, and guarantee robustness, contrary to the previous cases. +5. Numerical experiment +In this section we consider the reaction-diffusion problem +−ε2∆u + u = f +in Ω, +u|∂Ω = 0. +First, we give a brief overview of a DPG method for the latter problem and, then, discuss results +of our numerical experiment. +5.1. DPG method for reaction-diffusion problem. We introduce the trace operators +tr∇ : H1(Ω) → (H(div; T ))′, +trdiv : H(div; Ω) → (H1(T ))′, +defined for u ∈ H1(Ω), σ ∈ H(div; Ω) by +⟨tr∇ u, τ⟩∂T = (u, divT τ)Ω + (∇u, τ)Ω +∀τ ∈ H(div; T ) = +� +T∈T +H(div; T), +⟨trdiv σ , v⟩∂T = (σ , ∇T v)Ω + (div σ , v)Ω +∀v ∈ H1(T ) = +� +T∈T +H1(T). +21 + +Here, divT : H(div; T ) → L2(Ω) is given by divT τ|T = div(τ|T ) for T ∈ T , τ = (τ T )T∈T ∈ +H(div; T ) and ∇T : H1(T ) → L2(Ω) is defined similarly. The trace spaces H1/2 +00 (∂T ) = tr∇(H1 +0(Ω)), +H−1/2(∂T ) = trdiv(H(div; Ω)) are closed with respect to the canonical norms (see [5]) +∥�u∥1/2,ε = inf +� +∥u∥Ω,ε : u ∈ H1(Ω), tr∇ u = �u +� +, +∥�σ∥−1/2,ε = inf +� +∥σ∥Ω,ε : σ ∈ H(div; Ω), trdiv σ = �σ +� +. +Introducing the spaces +U = L2(Ω) × L2(Ω) × H1/2 +00 (∂T ) × H−1/2(∂T ), +V = H1(T ) × H(div; T ), +where U is equipped with the canonical product norm and V with the (squared) norm +∥(v, τ)∥2 +V = ∥v∥2 +T ,ε + ∥τ∥2 +T ,ε := +� +T∈T +∥v∥2 +T,ε + ∥τ∥2 +T,ε +we obtain the ultraweak formulation of the reaction-diffusion problem by defining σ = ε∇u and +element-wise integration by parts. This yields +u = (u, σ, �u, �σ) ∈ U : +b(u, v) = L(v) +∀v = (v, τ) ∈ V, +(11) +where for u ∈ U, v ∈ V and given f ∈ L2(Ω) we define +b(u, v) = (u, ε divT τ + v)Ω + (σ , ε∇T v + τ)Ω − ε⟨�u, τ⟩∂T − ε⟨�σ , v⟩∂T , +L(v) = (f , v)Ω. +The following result contains well-posedness of the ultraweak formulation (11). It can be derived +by following the abstract theory presented in [5] together with our discussions on the fully-discrete +scheme (1) from the introduction. +Proposition 25. The ultraweak formulation (11) admits a unique solution u ∈ U with ∥u∥U ≲ +∥f∥Ω. Let Uh ⊂ U, Vh ⊂ V denote finite-dimensional subspaces and suppose that there exists a +Fortin operator ΠF : V → Vh satisfying (2). Then, with uh ∈ U denoting the solution of (1), +∥u − uh∥U ≲ min +v∈Uh ∥u − vh∥U. +The hidden constants are independent of ε and the mesh-size. +5.2. Results for reaction-diffusion problem. In this section we consider the manufactured so- +lution +u(x, y) = v(x)v(y) +for (x, y) ∈ Ω = (0, 1)2, +where +v(x) = 1 − (1 − e−1/( +√ +2ε))e−(1−x)/( +√ +2ε) + e−x/( +√ +2ε) +1 − e−2/( +√ +2ε) +. +One verifies that u is the solution of +−ε2∆u + u = f, +u|∂Ω = 0 +with f(x, y) = 1 +2(v(x) + v(y)). +We use the DPG method from Section 5.1 with test spaces +Vh,pol = +� +T∈T +Vh,pol(T), +Vh,ε = +� +T∈T +Vh,ε(T) +22 + +101 +102 +103 +104 +105 +106 +10−3 +10−2 +10−1 +100 +degrees of freedom +Vh,pol, ε = 10−3 +est +∥u − uh∥ +∥σ − σh∥ +101 +102 +103 +104 +105 +106 +10−3 +10−2 +10−1 +100 +degrees of freedom +Vh,ε, ε = 10−3 +est +∥u − uh∥ +∥σ − σh∥ +101 +102 +103 +104 +105 +106 +10−3 +10−2 +10−1 +degrees of freedom +Vh,pol, ε = 10−4 +est +∥u − uh∥ +∥σ − σh∥ +101 +102 +103 +104 +105 +106 +10−3 +10−2 +10−1 +degrees of freedom +Vh,ε, ε = 10−4 +est +∥u − uh∥ +∥σ − σh∥ +Figure 2. Errors in the field variables compared with DPG estimator for ε = 10−3 +and ε = 10−4. The left resp. right column shows the results using test space Vh,pol +resp. Vh,ε. +where +Vh,pol(T) = P 3(T) × P 2(T), +Vh,ε(T) = +� +V ∇ +h0 × V div,2 +h +, +ε > hT , +V ∇ +h0,ε × V div +h,ε , +ε ≤ hT . +The trial space is +Uh = P 0(T ) × P 0(T ) × tr∇(P 1(T ) ∩ H1 +0(Ω)) × trdiv(RT 0(T )), +where RT 0(T ) = +� +τ ∈ L2(Ω) : τ|T ∈ RT 0(T), T ∈ T +� +∩ H(div; Ω). We stress that by our +constructions from Sections 3 and 4, Vh = Vh,ε is a test space that allows for a uniformly bounded +Fortin operator (2). On the other hand, test space Vh = Vh,pol allows for a Fortin operator whose +norm depends on ε, cf. [12]. +23 + +10−6 +10−5 +10−4 +10−3 +10−2 +10−1 +100 +10−1 +100 +101 +102 +ε +Vh,pol +Vh,ε +Figure 3. Ratio ρ = ∥u − uh∥/ est for the two different test spaces Vh,pol and Vh,ε +on a fixed mesh. The black dotted line corresponds to O(ε−1/2). +We also define the DPG error estimator by +est = +sup +0̸=vh∈Vh +b(uh, vh) − L(vh) +∥vh∥V +. +Clearly, this estimator depends on the choice of the test space. In [4] it is shown that est is, up to +an oscillation term, equivalent to the error +∥u − uh∥U ≂ est +osc(f) = est + +sup +0̸=v=(v,τ)∈V +L(v − ΠFv) +∥v∥V +, +provided there exists a uniformly bounded Fortin operator ΠF : V → Vh. +Figure 2 shows the errors of the field variables for ε ∈ {10−3, 10−4} and estimator est. We observe +differences when using Vh = Vh,pol or Vh = Vh,ε as test space: For coarse meshes and Vh = Vh,pol the +estimator est underestimates the errors in the field variables. This effect is more severe for smaller +parameters. This can also be seen in Figure 3. There, we fix a mesh with four elements and only +vary ε. We plot the index ρ = ∥u − uh∥/ est. When using Vh = Vh,pol we observe that ρ = O(ε−1/2) +for ε → 0 whereas ρ = O(1) when using Vh = Vh,ε. We conclude that the DPG method with +Vh = Vh,pol is not robust, whereas with the new test space Vh = Vh,ε it is. +5.3. Discrete stability. Let Ω = �T, T = { �T}. In this section we want to study stability of the +method by investigating the norm equivalence constants λmin, λmax in +λmin∥uh∥2 +U ≤ b(uh, Θhuh) ≤ λmax∥uh∥2 +U +∀uh ∈ Uh. +(12) +Here, Uh is defined as in the previous section. We use two different test spaces, Vh,ε (defined in the +previous section) and +�Vh = V ∇ +h0 × V div,2 +h0 +. +24 + +10−5 +10−4 +10−3 +10−2 +10−1 +10−4 +10−3 +10−2 +10−1 +100 +ε +λmax, �Vh +λmin, �Vh +λmax, Vh,ε +λmin, Vh,ε +Figure 4. Constants λmax and λmin from (12) for the test spaces �Vh and Vh,ε (see +Section 5.3). The black dotted line corresponds to O(ε−1). +The difficulty in checking the norm equivalence is the implementation of the trace norms. Note that +due to inclusion of boundary conditions and the fact that all nodes of T are on the boundary, we +do not have to consider ∥�uh∥1/2,ε. To calculate ∥�σh∥−1/2,ε we generate a submesh �T of T such that +all elements that have a boundary face have diameter less than or equal to ε/2. This, heuristically, +resolves possible boundary layers. To evaluate ∥�σh∥−1/2,ε we approximate the PDE +τ ∈ H(div; Ω): +− ε2∇ div τ + τ = 0, +τ · nΩ|∂Ω = �σh +by a standard FEM on �T using lowest-order Raviart–Thomas elements. +The H(div; Ω) norm +∥τ h∥Ω,ε of the approximation τ h ∈ RT 0(T ) is taken as ∥�σh∥−1/2,ε. +In view of norm equivalence (12) we stress that λmax ≲ 1 independent of ε and test space Vh +since the bilinear form is uniformly bounded. However, the lower bound is directly related to the +stability of the DPG method, i.e., λmin depends on the discrete inf-sup constant which for the +DPG method is related to the norm of Fortin operators as we already discussed in the introduction. +Figure 4 visualizes λmin and λmax for Vh = Vh,ε and Vh = �Vh. We observe that λmax is uniformly +bounded for both test spaces (for small ε we can not even distinguish them in the plot), whereas +λmin deteriorates for Vh = �Vh and is essentially constant for Vh = Vh,ε. This illustrates that the +DPG method with test space Vh,ε is uniformly stable, in contrast to the canonical method with test +space �Vh. +References +[1] C. Bernardi and G. Raugel. Analysis of some finite elements for the Stokes problem. Math. Comp., 44(169):71–79, +1985. +[2] D. Boffi, F. Brezzi, and M. Fortin. Mixed finite element methods and applications, volume 44 of Springer Series +in Computational Mathematics. Springer, Heidelberg, 2013. +25 + +[3] S. C. Brenner and L. R. Scott. The mathematical theory of finite element methods, volume 15 of Texts in Applied +Mathematics. Springer, New York, third edition, 2008. +[4] C. Carstensen, L. Demkowicz, and J. Gopalakrishnan. A posteriori error control for DPG methods. SIAM J. +Numer. Anal., 52(3):1335–1353, 2014. +[5] C. Carstensen, L. Demkowicz, and J. Gopalakrishnan. Breaking spaces and forms for the DPG method and +applications including Maxwell equations. Comput. Math. Appl., 72(3):494–522, 2016. +[6] C. Carstensen, D. Gallistl, F. Hellwig, and L. Weggler. Low-order dPG-FEM for an elliptic PDE. Comput. Math. +Appl., 68(11):1503–1512, 2014. +[7] C. Carstensen and F. Hellwig. Low-order discontinuous Petrov-Galerkin finite element methods for linear elas- +ticity. SIAM J. Numer. Anal., 54(6):3388–3410, 2016. +[8] L. Demkowicz and N. Heuer. Robust DPG method for convection-dominated diffusion problems. SIAM J. Numer. +Anal., 51(5):2514–2537, 2013. +[9] L. Demkowicz and P. Zanotti. Construction of DPG Fortin operators revisited. Comput. Math. Appl., +80(11):2261–2271, 2020. +[10] T. Führer. Superconvergent DPG methods for second-order elliptic problems. Comput. Methods Appl. Math., +19(3):483–502, 2019. +[11] T. Führer and N. Heuer. Fully discrete DPG methods for the Kirchhoff-Love plate bending model. Comput. +Methods Appl. Mech. Engrg., 343:550–571, 2019. +[12] J. Gopalakrishnan and W. Qiu. An analysis of the practical DPG method. Math. Comp., 83(286):537–552, 2014. +[13] P. Grisvard. Elliptic problems in nonsmooth domains, volume 24 of Monographs and Studies in Mathematics. +Pitman (Advanced Publishing Program), Boston, MA, 1985. +[14] N. Heuer and M. Karkulik. A robust DPG method for singularly perturbed reaction-diffusion problems. SIAM +J. Numer. Anal., 55(3):1218–1242, 2017. +[15] S. Nagaraj, S. Petrides, and L. F. Demkowicz. Construction of DPG Fortin operators for second order problems. +Comput. Math. Appl., 74(8):1964–1980, 2017. +Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Santiago, Chile +Email address: {tofuhrer,nheuer}@mat.uc.cl +26 + diff --git a/otFPT4oBgHgl3EQfLDSW/content/tmp_files/load_file.txt b/otFPT4oBgHgl3EQfLDSW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ec2b4b6a26955eb0ba704d2512bb7958881c56b9 --- /dev/null +++ b/otFPT4oBgHgl3EQfLDSW/content/tmp_files/load_file.txt @@ -0,0 +1,991 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf,len=990 +page_content='ROBUST DPG FORTIN OPERATORS THOMAS FÜHRER AND NORBERT HEUER Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' At the fully discrete setting, stability of the discontinuous Petrov–Galerkin (DPG) method with optimal test functions requires local test spaces that ensure the existence of Fortin operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We construct such operators for H1 and H(div) on simplices in any space dimension and arbitrary polynomial degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The resulting test spaces are smaller than previously analyzed cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' For parameter-dependent norms, we achieve uniform boundedness by the inclusion of exponential layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' As an example, we consider a canonical DPG setting for reaction-dominated diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Our test spaces guarantee uniform stability and quasi-optimal convergence of the scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We present numerical experiments that illustrate the loss of stability and error control by the residual for small diffusion coefficient when using standard polynomial test spaces, whereas we observe uniform stability and error control with our construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Introduction Fortin operators are a critical tool for the stability analysis of mixed finite element schemes, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The discontinuous Petrov–Galerkin (DPG) method with optimal test functions, on the other hand, is a framework that aims at automatic inf-sup stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' In practice, optimal test functions have to be approximated and the question of existence of Fortin operators re-appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' In this case, local (element-wise) operators are sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' A first answer was given in [12], with subsequent studies in [5, 15, 9, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' In the case of singularly-perturbed problems, uniform discrete stability, or robust- ness of the method, requires the existence of uniformly bounded Fortin operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' This has been an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' In this paper, we present local Fortin operators for H1 and H(div), on sim- plices in arbitrary dimension and arbitrary polynomial degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' In contrast to previous results, our constructions are explicit (not needed in applications) and require fewer degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' More importantly, we include parameter-dependent exponential layers that guarantee uniform bounded- ness of our operators for parameter-dependent norms (the H(div)-case is restricted to two and three space dimensions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We illustrate their application to a DPG method for a reaction-dominated diffusion problem, leading to robustness, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=', uniform stability, error control, and convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' In this case, we consider the energy-norm induced by the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We have not analyzed the case of balanced norms as proposed in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' This and possible extensions to other singularly-perturbed problems like advection-dominated diffusion are left to future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Let us shortly discuss the abstract setting of the DPG method: Consider the variational formu- lation u ∈ U : b(u, v) = L(v) ∀v ∈ V, where U, V are Hilbert spaces with norms ∥·∥U, ∥·∥V , b(·, ·) is a bounded bilinear form and induces a boundedly invertible operator B : U → V ′, u �→ b(u, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Choosing finite dimensional spaces Uh ⊂ U, Vh ⊂ V , the fully discrete DPG method reads: uh ∈ Uh : b(uh, v) = L(v) ∀v ∈ Θh(Uh), (1) Date: January 31, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 65N30, 65N12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' DPG method, Fortin operators, singularly perturbed problems, reaction-diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Acknowledgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' This work was supported by ANID through FONDECYT projects 1210391 and 1230013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='13021v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='NA] 30 Jan 2023 where Θh : U → Vh is defined through ((·, ·)V being the inner product on V ) (Θhu, vh)V = b(u, vh) ∀vh ∈ Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Well-posedness of discrete DPG is ensured if there exists a Fortin operator ΠF : V → Vh such that ∥ΠFv∥V ≤ CΠF∥v∥V , b(uh, v − ΠFv) = 0 ∀uh ∈ Uh, v ∈ V, (2) see, [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Furthermore, the existence of a Fortin operator also implies quasi-optimality, ∥u − uh∥U ≤ CCΠF min wh∈Uh ∥u − wh∥U with C = Cb/cb where Cb and cb are the boundedness and inf-sup constants of b(·, ·), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' It also plays an important role in the a posteriori error control, see [4, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='1], ∥u − uh∥U ≂ ∥Buh − L∥V ′ + osc(L), where osc(L) = sup0̸=v∈V L(v − ΠFv)/∥v∥V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' One of the main motivations that had driven the development of the DPG method was to derive robust numerical schemes for singularly perturbed problems, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=', [8, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' All these problems have in common that they naturally lead to parameter dependent trial and test norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' For a concrete example consider the test space H1(T) (here T as an element of the mesh T ) equipped with the norm ∥v∥T,α = ∥v∥T + α∥∇v∥T for some fixed α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' In [12, 5, 9] Fortin operators are constructed (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' their existence is shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Let us write ΠF : H1(T) → P k(T) (a polynomial space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Besides some conditions they satisfy the bounded- ness estimates ∥ΠFv∥T ≲ ∥v∥T + hT ∥∇v∥T , ∥∇ΠFv∥T ≲ ∥∇v∥T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Combining the latter two estimates yields ∥ΠFv∥T,α ≲ max{1, α−1hT }∥v∥T,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' If hT ≲ α it is clear that ∥ΠFv∥T,α ≲ ∥v∥T,α uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' However, when α ≲ hT — a case often encountered with singularly perturbed problems — then we get ∥ΠFv∥T,α ≲ α−1hT ∥v∥T,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' This means that, particularly on coarse meshes, the Fortin operator is not uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' By our prior considerations, this means that quasi-optimality as well as a posteriori error control is spoiled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' One of the main objectives of this work is to define discrete spaces Vh and construct corresponding Fortin operators ΠF : V → Vh with boundedness constant CΠF ≂ 1 for small parameters (α ≲ hT in the previous example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We do this by first revisiting the construction of Fortin operators for the spaces H1(T) and H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T) in the case hT ≲ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Contrary to prior works we construct our Fortin operators in an explicit manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' This allows to precisely write down a basis for the discrete test spaces yielding — compared with the operators from [12, 5, 9] — smaller dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The novel idea of definition also requires a different analysis which we present in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Our constructions are valid for any polynomial degree (this statement will be made precise below) and arbitrary dimension (except for some operators from Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' However, main advantage is that the definition and construction can be extended to the case α ≲ hT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Specifically, we use modified face bubble functions ηα,F instead of polynomial face bubble functions ηF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' They are defined in such a way that their volume norm ∥ηα,F ∥T resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' ∥∇ηα,F ∥T scales differently than ∥ηF ∥T resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' ∥∇ηF ∥T depending on the ratio α/hT , though ηα,F |∂T = ηF |∂T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The analysis requires some additional tools and steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We also consider low order polynomial cases which allow for even smaller dimensions in the test space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' As mentioned above, Fortin operators for the DPG method have been constructed in various works: The first one was [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Other articles that analyze the existence of Fortin operators for second-order PDEs include [5, 15, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The latter references consider all arbitrary fixed polynomial 2 orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' For low order methods with smaller test space dimensions we refer to [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' For a fourth- order PDE model problem we have shown existence of Fortin operators in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The remainder of this work is organized as follows: In Section 2 we introduce some notation and define basis functions as well as novel face bubble functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Section 3 and Section 4 discuss the construction of Fortin operators for H1 and H(div), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Section 5 concludes this article with a short description of a DPG method for a singularly perturbed reaction-diffusion problem and numerical examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Preliminaries The notation a ≲ b (a ≳ b) for a, b > 0 means that there exists C > 0 such that a ≤ C b (C a ≥ b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We write a ≂ b for a, b > 0 if a ≲ b ≲ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The generic constant C is independent of involved functions, the diameter of elements, and parameters like α and ε, where present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Mesh and spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Let T denote a shape-regular simplicial mesh of a Lipschitz domain Ω with diam(Ω) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Throughout, T ∈ T is some fixed element, �T is the reference element given as the convex hull of the origin and the n coordinate axis vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=', for n = 2, 3 it reads �T = � conv{(0, 0)⊤, (1, 0)⊤, (0, 1)⊤}, n = 2, conv{(0, 0, 0)⊤, (1, 0, 0)⊤, (0, 1, 0)⊤, (0, 0, 1)⊤}, n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Here, we understand conv as the interior of the convex hull of a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We adopt the standard notation for Lebesgue and Sobolev spaces, L2(ω), L2(ω) := L2(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Rn), H1(ω) = � v ∈ L2(ω) : ∇v ∈ L2(ω) � , H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' ω) = � τ ∈ L2(ω) : div τ ∈ L2(ω) � for a Lipschitz domain ω ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' With nω we denote the normal vector on the boundary of ω pointing from ω to its complement Rn \\ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Recall that traces of H1(ω) elements are well defined (in the sense of trace operators) and the canonic trace space is H1/2(∂ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Normal traces of H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' ω) elements are well defined (in a duality sense) and the canonic trace space is H−1/2(∂ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We simply write τ · nω for the normal trace of τ ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We denote by ∥ · ∥ω the canonical L2(ω) norm induced by the L2(ω) inner product (·, ·)ω for a Lipschitz domain ω ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The volume measure of ω is given by |ω|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The same notation for norm and inner product is used for L2(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The surface measure of γ ⊆ ∂ω is denoted by |γ| and ∥v∥γ is the L2(γ) norm induced by the inner product ⟨·, ·⟩γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We also use the same notation for the duality between H−1/2(∂ω) and H1/2(∂ω), ⟨φ, v⟩∂ω for φ ∈ H−1/2(∂ω), v ∈ H1/2(∂ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Recall the following relation between traces of H1(ω) and H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' ω), ⟨τ · nω , v⟩∂ω = (div τ , v)ω + (τ , ∇v)ω (3) for all τ ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' ω), v ∈ H1(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Obviously, for sufficiently regular functions this is just the integration by parts formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Let Πq T : L2(T) → P q(T) denote the L2(T) orthogonal projection on P q(T), the space of polyno- mials on T of degree less than or equal to q ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' For vector-valued polynomials (each component is a polynomial of degree less than or equal to q) we use the symbol P q(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Recall the first-order approximation property ∥v − Πq T v∥T ≤ ∥v − Π0 T v∥T ≲ hT ∥∇v∥T for all v ∈ H1(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' An important tool is the following (multiplicative version) of the trace inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' It can be de- rived from [3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='6] with a scaling argument (using the reference element �T) and the approximation property of Π0 T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 3 Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' For any v ∈ H1(T) we have ∥v − Π0 T v∥∂T ≲ ∥v − Π0 T v∥1/2 T ∥∇v∥1/2 T ≲ h1/2 T ∥∇v∥T (4) with hidden constants only depending on the shape of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We denote by VT the set of the n + 1 vertices of T, FT is the set of n + 1 faces of T and VF denotes the set of n vertices of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' For z ∈ VT let Fz ∈ FT be the face opposite to z, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=', Fz = conv � VT \\ {z} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Similarly, for F ∈ FT let zF ∈ VT be the vertex opposite to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' For F ∈ FT let P q(FT ) ⊂ L2(∂T) denote face-wise polynomials of degree less than or equal to q ∈ N0 and P q c (FT ) := P q(FT ) ∩ C0(∂T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Note that nT is face-wise constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' For the fixed element T ∈ T and any F ∈ FT we abbreviate nF = nT |F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' For a vertex z ∈ VT we denote by Ez the set of n edges that share the same vertex z, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=', for each E ∈ Ez there is a z′ ∈ VT \\ {z} with E = conv{z, z′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' To each E = conv{z, z′} ∈ Ez we associate the (tangential) vector tE = z′ − z (the orientation does not matter for our analysis nor for implementation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Furthermore, hT = diam(T), hT ≂ diam(F) for all F ∈ FT with hidden constants only depending on the shape of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Some other relations that we frequently use without further notice are |T| ≂ hn T , |F| ≂ hn−1 T , |∂T| ≂ |F|, |T| ≂ |F|hT for any F ∈ FT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' In the following subsections we define special functions that will be used for the construction of the Fortin operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' For ease of reading and reference they are listed, together with their relevant properties, in Table 1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Low-order basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The functions ηz ∈ P 1(T) are canonical basis functions with ηz(z′) = δz,z′ for z, z′ ∈ VT and δ·,· denoting the Kronecker-δ, ηF = � z∈VF ηz ∈ P n(T) are the face bubble functions, and ηT = � z∈VT ηz ∈ P n+1(T) is the element bubble function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Clearly, span{ηz : z ∈ VT } = P 1(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Alternatively, we may use the following basis for P 1(T): First, we abbreviate dF = ηzF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Then, for F ∈ FT we define νF = � z∈VF ηz − (n − 1)dF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' For space P 0(FT ) we use the characteristic functions χF |F ′ = δF,F ′ for F ′ ∈ FT as basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We have ⟨νF , χF ′⟩∂T = |F|δF,F ′ ∀F, F ′ ∈ FT (5) and span � νF : F ∈ FT � = P 1(T), span � νF |∂T : F ∈ FT � = P 1 c (FT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Note that (5) implies that νF |∂T , F ∈ FT are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' This implies the last two assertions because dim(P 1(T)) = n + 1 = dim(P 1 c (FT )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' It only remains to prove (5): Let F ∈ FT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' From its definition we see that νF |F = 1, thus, ⟨νF , χF ⟩∂T = |F|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Let F ′ ∈ FT \\ {F}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Using � F ′ ηz dx = |F ′|n−1 for z ∈ VF ′ one verifies that ⟨νF , χF ′⟩∂T = � z∈VF ∩VF ′ � F ′ ηz dx − (n − 1) � F ′ dF dx = (n − 1)|F ′|n−1 − (n − 1)|F ′|n−1 = 0, finishing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' □ 4 Let RT 0(T) = � ψ ∈ L2(T) : ψ = α + βx, α ∈ Rn, β ∈ R � denote the lowest-order Raviart– Thomas space where x: T → Rn, z �→ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Let ψF ∈ RT 0(T) denote the canonical Raviart–Thomas basis function with ψF · nT |F ′ = δF,F ′ ∀F, F ′ ∈ FT and ∥ψF ∥T ≂ |T|1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' One verifies the explicit representation ψF (z) = |F| n|T|(z − zF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Note that by Lemma 2 we have that ⟨ψF · nT , νF ′⟩∂T = |F|δF,F ′ ∀F, F ′ ∈ FT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We also use Bernardi–Raugel elements, see [1], ηF := ηF nF , F ∈ FT for which we get ⟨ηF · nT , νF ′⟩∂T = ⟨ηF · nT , 1⟩F δF,F ′ ≂ |F|δF,F ′ ∀F, F ′ ∈ FT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We also define edge based functions: Fix a vertex z∗ ∈ VT and set E∗ = Ez∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Clearly, tE (E ∈ E∗) are linearly independent and span Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Let σE, (E ∈ E∗) denote a basis of P 0(T) such that (for any z ∈ T) σE(z) · tE′ = δE,E′ ∀E, E′ ∈ E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We have that |σE(z)| ≂ 1 with constants only depending on the shape of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' With these preparations we define edge functions by ηE = � z∈VT ∩E ηz and tangential edge functions by ηE := ηEtE ∀E ∈ E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' These functions play the role of element bubble functions in H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T) as can be seen from the next result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We have ηE ∈ P 2(T) and ηE · nT |∂T = 0, (σE , ηE′)T = (σE , ηE)T δE,E′ ≂ |T|δE,E′ ∀E, E′ ∈ E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Clearly, ηE ∈ P 2(T), thus, ηE ∈ P 2(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Note that ηE|∂T is supported on n − 1 faces, say Fj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' For these faces we also have tE ⊆ Fj yielding tE · nFj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We conclude ηE · nT |∂T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The final assertion follows from the definition of σE and ηE and scaling arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Higher-order basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' For F ∈ FT let �χF,j ∈ P p(T), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , dim(P p(F)) be such that �χF,j|F is a basis of P p(F) with ∥�χF,j∥∞ = ∥�χF,j|F ∥∞ ≂ 1 and define ηF,j = ηF �χF,j, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , dim P p(F), F ∈ FT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Let χF,j ∈ P p(FT ), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , dim(P p(F)), be such that χF,j|F ′ = 0 for F ′ ∈ FT \\ {F} and ⟨χF,j , ηF,k⟩F = δj,k|F|, j, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , dim(P p(F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Let �χT,j ∈ P p(T), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , dim(P p(T)), denote a basis of P p(T) with ∥�χT,j∥∞ ≂ 1 and define ηT,j = ηT �χT,j, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , dim(P p(T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Furthermore, let χT,j, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , dim(P p(T)) be such that (χT,j , ηT,k)T = |T|δj,k, j, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , dim(P p(T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' By scaling arguments one verifies that ∥χF,j∥∞ ≂ 1 and ∥χT,j∥∞ ≂ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 5 Let �P p(T) denote the orthogonal complement of P p b (T) = � v ∈ P p(T) : v|∂T = 0 � in P p(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Let �ν∂T,j, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , dim( �P p+1(T)), denote a basis of �P p+1(T) with ∥�ν∂T,j∥∞ ≂ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Furthermore, let ν∂T,j ∈ �P p+1(T), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , dim( �P p+1(T)), denote a basis with ⟨ν∂T,j , �ν∂T,k⟩∂T = |∂T|δj,k j, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , dim( �P p+1(T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' One verifies that ∥ν∂T,j∥∞ ≂ 1 by scaling arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Define ψ∂T = � F∈FT ψF and note that ψ∂T · nT |∂T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Define ψ∂T,j = ψ∂T �ν∂T,j j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , dim( �P p+1(T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Thus, by our previous considerations, ⟨ψ∂T,j · nT , ν∂T,k⟩∂T = |∂T|δj,k, j, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , dim( �P p+1(T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We also define higher order edge functions: For E ∈ E∗, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , dim(P p(T)), define ηE,j = ηE �χT,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' For E ∈ E∗ let χE,j ∈ P p(T), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , dim(P p(T)), denote a basis with (χE,j , �χT,kηE)T = |T|δj,k j, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , dim(P p(T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' One verifies that ∥χE,j∥∞ ≂ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Furthermore, define σE,j = σEχE,j j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , dim(P p(T)), E ∈ E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The proof of the next result follows the arguments given in Lemma 3 together with the aforegoing definitions and is thus omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We have that ηE,j · nT |∂T = 0, ηE,j ∈ P p+2(T), and (σE,j , ηE′,k)T = δE,E′δj,k|T| for all E, E′ ∈ E∗, j, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , dim(P p(T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Modified face bubble functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We introduce modified face bubble functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Before we come to their definition and analysis we state the following result: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Let R > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Consider R ≥ κ > 0 and the function φκ : [0, 1] → [0, 1], t �→ e−t/κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Then, ∥φκ∥L2(0,1) ≂ κ1/2, ∥φ′ κ∥L2(0,1) ≂ κ−1/2, φκ(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The hidden constants only depend on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The results follow from straightforward calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' □ Recall that dF = ηzF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We can interpret dF as a relative distance function that is 0 when restricted to F and 1 when evaluated at the vertex opposite to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Considering φ := φα/hT , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=', φ(t) = e−hT t/α, define for F ∈ FT the modified face bubble function by ηα,F := (φ ◦ dF )ηF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' (6) Some basic properties of this modified function are given in the next result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' A visualization of ηα,F is presented in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Suppose that 0 < α ≲ hT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' For any F ∈ FT we have that ∥ηα,F ∥T ≲ |T|1/2 � α hT �1/2 , ∥∇ηα,F ∥T ≲ h−1 T |T|1/2 � α hT �−1/2 , ηα,F |∂T = ηF |∂T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 6 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Visualization of the face bubble functions ηF and ηα,F on the reference element �T and face F = (0, 1) × {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The identity ηα,F |∂T = ηF |∂T follows since φ(0) = 1 and ηF |∂T\\F ′ = 0 for F ′ ∈ FT \\ {F}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We show the details for n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' For n ≥ 3 we may argue similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Let AT : �T → T denote the affine element mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' let F ∈ FT be the face such AT : �T → T maps F to the edge (0, 1) × {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Then, dF ◦ AT (�x, �y) = �y for (�x, �y) ∈ �T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Moreover, ηα,F ◦ AT (�x, �y) = �ηα(�x, �y) := e−�yhT /α(1 − �x − �y)�x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The remaining assertions follow by standard calculations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=', ∥ηα,F ∥2 T = 2|T|∥�ηα∥2 �T ≲ α hT |T|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' □ With the same logic as for the definition of ηF,j we define ηα,F,j = ηα,F �χF,j, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , dim(P p(F)), F ∈ FT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The same proof as for Lemma 6 shows Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Suppose that α ≲ hT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Then, for any F, j the function ηα,F,j satisfies the assertions of Lemma 6 (replacing ηα,F by ηα,F,j and ηF by ηF,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Define the modified Bernardi–Raugel elements by ηα,F := ηα,F nF = (φ ◦ dF )ηF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Some important properties of ηα,F follow directly from its definition and are summarized in the next result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Its proof follows the same ideas as the proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 7 "f na,F C=1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='1J 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='5 x y x y na,F α=10-1 "a,F (=10-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='1 J 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 0, 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='5 y X yLemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' For F ∈ FT we have ηα,F · nT |∂T = ηF · nT |∂T and if α ≲ hT , then, ∥ηα,F ∥T ≲ |T|1/2 � α hT �1/2 , ∥ div ηα,F ∥T ≲ |T|1/2h−1 T � α hT �−1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We also need higher order variants: Set ηα,∂T = � F∈FT ηα,F and ηα,∂T,j = ηα,∂T �ν∂T,j j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , dim(P p+1 c (FT )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Let χ∂T,j ∈ P p+1 c (FT ), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , dim(P p+1 c (FT )) denote a basis of P p+1 c (FT ) with ⟨ηα,∂T,j · nT , χ∂T,k⟩∂T = |∂T|δj,k, j, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , dim(P p+1 c (∂T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The proof of the next result follows the proof of Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The boundedness estimates of Lemma 8 hold with ηα,F replaced by ηα,∂T,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' □ To close this section and to have a better overview we summarize the most important basis functions used in the remainder of this work in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Fortin operator in H1(T) We consider a fixed parameter α > 0 and space H1(T) equipped with the (squared) norm ∥v∥2 T,α := ∥v∥2 T + α2∥∇v∥2 T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The idea of this section is to construct Fortin operators, say Π∇ F : H1(T) → V ∇ h (with V ∇ h ⊂ H1(T) being some finite-dimensional subspace) such that, for a fixed p ∈ N0 and for all v ∈ H1(T), ∥Π∇ F v∥T,α ≤ CF∥v∥T,α, (7a) ⟨σ , v − Π∇ F v⟩∂T = 0 ∀σ ∈ P p(FT ), (7b) (u, v − Π∇ F v)T = 0 ∀u ∈ P p(T) (7c) with CF > 0 independent of α, hT (but possibly dependent on p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Note that (7b)–(7c) imply (σ , ∇(1 − Π∇ F )v)T = 0 ∀σ ∈ P p(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' (7d) This can be seen from integration by parts: Take σ ∈ P p(T), v ∈ H1(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Then, div σ ∈ P p−1(T) and (σ , ∇Π∇ F v)T = −(div σ , Π∇ F v)T + ⟨σ · nT , Π∇ F v⟩∂T = −(div σ , v)T + ⟨σ · nT , v⟩∂T = (σ , ∇v)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' From the last identities we also see that the weaker condition (u, v − Π∇ F v)T = 0 ∀u ∈ P p−1(T) (7c’) would be sufficient to conclude (7d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' However, depending on the problem, condition (7c) is needed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=', in the presence of reaction terms as in the DPG method in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We stress that our Fortin operator can be easily modified to satisfy (7c’) only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 8 function space basis definition property low order χF P 0(FT ) yes χF |F ′ = δF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='F ′ – νF P 1(T) yes � z∈VF ηz − (n − 1)dF ⟨νF ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' χF ′⟩∂T = |F|δF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='F ′ ψF RT 0(T) yes ψF · nT |∂T = χF ⟨ψF · nT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' νF ′⟩∂T = |F|δF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='F ′ ηF P p(T) no ηF nF ⟨ηF · nT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' νF ′⟩∂T = ⟨ηF ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 1⟩F δF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='F ′ σE P 0(T) yes – σE · tE′ = δE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='E′ ηE P 2(T) no ηEtE (σE ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' ηE′)T = (σE ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' ηE)T δE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='E′ higher order χT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' �χT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='j P p(T) yes – – ηT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='j P p+n+1(T) no ηT �χT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='j (ηT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' χT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='k)T = |T|δj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='k �χF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='j P p(T) yes – – χF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='j P p(FT ) yes – – ηF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='j P p+n(T) no ηF �χF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='j ⟨ηF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' χF ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='k⟩∂T = δj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='kδF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='F ′|F| ν∂T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' �ν∂T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='j �P p+1(T) yes – ⟨ν∂T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' �ν∂T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='k⟩∂T = |∂T|δj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='k ψ∂T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='j P p+2(T) no �ν∂T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='j � F∈FT ψF ⟨ψ∂T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='j · nT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' ν∂T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='k⟩∂T = δj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='k|∂T| χE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='j P p(T) yes – (χE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' �χT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='kηE)T = δj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='k|T| σE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='j P p(T) yes σEχE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='j – ηE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='j P p+2(T) no ηE �χT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='j (σE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' ηE′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='k)T = δj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='kδE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='E′|T| modified φ – – t �→ e−hT /αt – ηα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='F – – (φ ◦ dF )ηF ηα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='F |∂T = ηF |∂T ηα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='j – – ηα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='F �χF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='j ηα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='j|∂T = ηF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='j|∂T ηα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='F – – ηα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='F nF ηα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='F · nT |∂T = ηF · nT |∂T χ∂T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='j P p+1 c (FT ) yes – – ηα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='∂T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='j – – ηα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='F �ν∂T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='j ⟨ηα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='j · nT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' χ∂T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='k⟩∂T = |∂T|δj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='k Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Overview of special functions together with some of their main properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Here, ηz is the canonical Lagrange basis function of P 1(T), ηE, ηF , ηT are edge, face, and element bubble functions, respectively, and dF = ηzF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' In column basis we indicate whether the respective family of functions generates the indicated space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' For a detailed description we refer to Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='2–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Constructions for moderate parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Define the space �V ∇ hp = P 0(T) + span � ηF,j : j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , dim(P p(F)), F ∈ FT � and operator �Π∇ F,hp : H1(T) → �V ∇ hp for v ∈ H1(T) by �Π∇ F,hpv = Π0 T v + � F∈FT dim(P p(F)) � j=1 ⟨χF,j , (1 − Π0 T )v⟩F ⟨χF,j , ηF,j⟩F ηF,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The following result collects its main properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Operator Π∇ F = �Π∇ F,hp is idempotent on P 0(T), satisfies property (7b) and ∥�Π∇ F,hpv∥T ≲ ∥v∥T + hT ∥∇v∥T , ∥∇�Π∇ F,hpv∥T ≲ ∥∇v∥T , ∥(1 − �Π∇ F,hp)v∥T ≲ hT ∥∇v∥T 9 for all v ∈ H1(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Idempotency can be seen from the definition, since v ∈ P 0(T) implies that Π0 T v = v, thus, (1 − Π0 T )v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Let v ∈ H1(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' To see (7b) we employ the orthogonality property ⟨ηF,j , χF ′,k⟩∂T = δF,F ′δj,k|F| to get ⟨χF ′,k , v − �Π∇ F v⟩∂T = ⟨χF ′,k , v − Π0 T v⟩∂T − � F∈FT dim(P p(F)) � j=1 ⟨χF,j , (1 − Π0 T )v⟩F ⟨χF,j , ηF,j⟩F ⟨χF ′,k , ηF,j⟩∂T = ⟨χF ′,k , v − Π0 T v⟩∂T − ⟨χF ′,k , (1 − Π0 T )v⟩∂T = 0 Since F ′ and k were arbitrary, condition (7b) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The boundedness follows from the triangle inequality, the Cauchy–Schwarz inequality and bound- edness of Π0 T , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=', ∥�Π∇ F,hpv∥T ≤ ∥v∥T + � F∈FT dim(P p(T)) � j=1 ∥χF,j∥F ∥v − Π0 T v∥F ⟨χF,j , ηF,j⟩F ∥ηF,j∥T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Note that ⟨χF,j , ηF,j⟩F = |F|, ∥ηF,j∥T ≂ |T|1/2, ∥χF,j∥F ≂ |F|1/2 which follows by standard scaling arguments and the properties of the basis functions discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Applying the trace inequality (4) we see that ∥χF,j∥F ∥v − Π0 T v∥F ⟨χF,j , ηF,j⟩F ∥ηF,j∥T ≂ |F|−1/2|T|1/2∥v − Π0 T v∥F ≲ h1/2 T h1/2 T ∥∇v∥T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Thus, we conclude that ∥�Π∇ F,hpv∥T ≲ ∥v∥T + hT ∥∇v∥T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Then, with similar arguments but using the inverse estimate ∥∇ηF,j∥T ≲ h−1 T |T|1/2, we see that (∇Π0 T v = 0) ∥∇�Π∇ F,hpv∥T ≤ � F∈FT dim(P p(F)) � j=1 ∥χF,j∥F ∥v − Π0 T v∥F ⟨χF,j , ηF,j⟩F ∥∇ηF,j∥T ≲ ∥∇v∥T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Finally, the approximation property is derived by using the idempotency, and the established bound- edness estimates, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=', ∥(1 − �Π∇ F,hp)v∥T = ∥(1 − �Π∇ F,hp)(v − Π0 T v)∥T ≲ ∥v − Π0 T v∥T + hT ∥∇(v − Π0 T v)∥T ≲ hT ∥∇v∥T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' □ To obtain an operator that also satisfies property (7c) we consider slight modifications by adding a correction term based on element bubbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Define the space V ∇ hp = �V ∇ hp + span � ηT,j : j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , dim(P p(T)) � and the operator Π∇ F,hp : H1(T) → V ∇ hp for all v ∈ H1(T) by Π∇ F,hpv = �Π∇ F,hpv + dim(P p(T)) � j=1 (χT,j , (1 − �Π∇ F,hp)v)T (χT,j , ηT,j)T ηT,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Operator Π∇ F = Π∇ F,hp is idempotent on P 0(T), satisfies (7b)–(7c) and ∥Π∇ F,hpv∥T ≲ ∥v∥T + hT ∥∇v∥T , ∥∇Π∇ F,hpv∥T ≲ ∥∇v∥T , ∥(1 − Π∇ F,hp)v∥T ≲ hT ∥∇v∥T for all v ∈ H1(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 10 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The idempotency on P 0(T) follows from the idempotency of �Π∇ F,hp (Lemma 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' State- ment (7b) follows also from Lemma 10 since the element bubbles ηT,j vanish on the boundary and, therefore, Π∇ F,hpv|∂T = �Π∇ F,hpv|∂T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' To see (7c) a simple calculation using the orthogonality (χT,j , ηT,k)T = |T|δj,k yields (χT,k , (1 − Π∇ F )v)T = (χT,k , (1 − �Π∇ F,hp)v)T − dim(P p(T)) � j=1 (χT,k , (1 − �Π∇ F,hp)v)T (χT,j , ηT,j)T (χT,k , ηT,j)T = (χT,k , (1 − �Π∇ F,hp)v)T − (χT,k , (1 − �Π∇ F,hp)v)T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' It remains to prove the boundedness estimates which follow — besides standard arguments — from the boundedness estimates of �Π∇,j F (see Lemma 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' First, using the triangle inequality, Cauchy– Schwarz inequality and scaling arguments we estimate ∥Π∇ F,hpv∥T ≲ ∥�Π∇ F,hpv∥T + dim(P p(T)) � j=1 |T|−1/2∥v − �Π∇ F,hpv∥T ∥ηT,j∥T ≲ ∥�Π∇ F,hpv∥T + ∥v − �Π∇ F,hpv∥T ≲ ∥v∥T + hT ∥∇v∥T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The gradient contribution is estimated by employing the inverse estimate ∥∇ηT,j∥T ≲ h−1 T ∥ηT,j∥T together with Lemma 10 to give ∥∇Π∇ F,hpv∥T ≲ ∥∇�Π∇ F,hpv∥T + h−1 T ∥v − �Π∇ F,hpv∥T ≲≲ ∥∇v∥T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The final assertion ∥(1 − Π∇ F,hp)v∥T ≲ hT ∥∇v∥T follows as in Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' □ Corollary 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Suppose that hT ≲ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' From Lemma 10 and Theorem 11 it follows that Π∇ F = �Π∇ F,hp and Π∇ F = Π∇ F,hp satisfy (7a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Particularly, Π∇ F = Π∇ F,hp has properties (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Remark 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' In general, discrete test spaces are chosen such that a Fortin operator exists, which not necessarily implies approximation results of the form min vh∈Vh ∥v − vh∥T + ∥∇(v − vh)∥T ≲ hT |v|2,T for v ∈ H2(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Here, | · |2,T denotes the H2(T) seminorm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' For certain supercloseness results in the DPG method the latter approximation property is needed, see [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' To ensure this property one can simply require P 1(T) ⊂ Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Constructions for small parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' In this section we focus on the case α ≲ hT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Let Π∇ F = �Π∇ F,hp or Π∇ F = Π∇ F,hp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' By Lemma 10 resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Theorem 11 we have the boundedness ∥Π∇ F v∥T ≲ ∥v∥T + hT ∥∇v∥T ≲ max{1, hT α−1} (∥v∥T + α∥∇v∥T ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' (8) We conclude that ∥Π∇ F v∥T,α ≲ max{1, hT α−1}∥v∥T,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' This tells us that Π∇ F is only conditionally uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Particularly, for small parameters α and coarse meshes huge boundedness constants are expected so that robustness of the numerical methods is likely to be lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' This can actually be observed in our numerical experiments presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Let us remark that the operators constructed in [12, 5, 9] also satisfy (8) and are not suited for small parameters, α ≲ hT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' To overcome this problem we construct an operator based on the modified face bubble functions ηα,F,j instead of ηF,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The construction of the novel Fortin operators follows the definition of �Π∇ F,hp and Π∇ F,hp replacing ηF,j with the modified face bubble functions ηα,F,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' First, set �V ∇ hp,α := P 0(T) + span{ηα,F,j : j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , dim(P p(F)), F ∈ FT }, 11 and define �Π∇ F,hp,α : H1(T) → �V ∇ hp,α for all v ∈ H1(T) by �Π∇ F,hp,αv := Π0 T v + � F∈FT dim(P p(F)) � j=1 ⟨χF,j , (1 − Π0 T )v⟩F ⟨χF,j , ηα,F,j⟩F ηα,F,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Its main properties are given in Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Operator Π∇ F = �Π∇ F,hp,α satisfies (7b) and is idempotent on P 0(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' If α ≲ hT , then ∥�Π∇ F,hp,αv∥T,α ≲ ∥v∥T,α, ∥v − �Π∇ F,hp,αv∥T ≲ hT ∥∇v∥T for all v ∈ H1(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The idempotency on P 0(T) can be seen directly from the definition of the operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Noting that ηα,F,j|∂T = ηF,j|∂T for any F, j the proof of Fortin property (7b) follows as in Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' It remains to prove the boundedness estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Suppose that α ≲ hT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' First, using the triangle inequality and the Cauchy–Schwarz inequality together with the properties of the modified bubble (Lemma 7), ∥χF,j∥F ≂ |F|1/2, ⟨χF,j , ηα,F,j⟩F = |F| we infer ∥�Π∇ F,hp,αv∥T ≤ ∥v∥T + � F∈FT dim(P p(F)) � j=1 ∥χF,j∥F ∥v − Π0 T v∥F ⟨χF,j , ηα,F,j⟩ ∥ηα,F,j∥T ≲ ∥v∥T + � F∈FT ∥v − Π0 T v∥F |F|−1/2|T|1/2α1/2h−1/2 T ≂ ∥v∥T + � F∈FT ∥v − Π0 T v∥F α1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Then, with the multiplicative trace inequality (4) and Young’s inequality we further get ∥v − Π0 T v∥F α1/2 ≲ ∥v − Π0 T v∥1/2 T α1/2∥∇v∥1/2 T ≲ ∥v − Π0 T v∥T + α∥∇v∥T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Putting all the estimates together we infer that ∥�Π∇ F,hp,αv∥T ≲ ∥v∥T + � F∈FT ∥v − Π0 T v∥F α1/2 ≲ ∥v∥T + α∥∇v∥T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We are left with the gradient contribution of the norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' With similar arguments as before we get α∥∇�Π∇ F,hp,αv∥T ≤ α � F∈FT dim(P p(F)) � j=1 ∥χF,j∥F ∥v − Π0 T v∥F ⟨χF,j , ηα,F,j⟩ ∥∇ηα,F,j∥T ≲ α � F∈FT α−1/2∥v − Π0 T v∥F ≲ α1/2∥v − Π0 T v∥1/2 T ∥∇v∥1/2 T ≲ ∥v∥T + α∥∇v∥T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Combining all the estimates from above we conclude the boundedness of the Fortin operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Finally, to see the approximation property note that �Π∇ F,hp,α1 = 1, thus, ∥(1 − �Π∇ F,hp,α)v∥T = ∥(1 − �Π∇ F,hp,α)(v − Π0 T v)∥T ≲ ∥v − Π0 T v∥T + α∥∇v∥T ≲ hT ∥∇v∥T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Note that we have used that α ≲ hT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' □ To ensure property (7c) we follow the idea of the definition of Π∇ F,hp by adding correction terms based on element bubble functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Contrary to the face bubble functions, the element bubble functions do not need to be modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Set V ∇ hp,α = �V ∇ hp,α + span � ηT,j : j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , dim(P p(T)) � 12 and define Π∇ F,hp,α : H1(T) → V ∇ hp,α for all v ∈ H1(T) by Π∇ F,hp,αv := �Π∇ F,hp,αv + dim(P p(T)) � j=1 (χT,j , (1 − �Π∇ F,hp,α)v)T (χT,j , ηT,j)T ηT,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The following theorem is one of our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Theorem 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Suppose that α ≲ hT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Then, Π∇ F = Π∇ F,hp,α is idempotent on P 0(T) and satisfies (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Idempotency follows from the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Verifying (7b)–(7c) follows as in Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' For the boundedness estimate we also use the arguments displayed in the proof of Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Note that with Lemma 14 and ∥∇ηT,j∥T ≲ h−1 T |T|1/2 we see that, for v ∈ H1(T), ∥Π∇ F,hp,αv∥T,α ≲ ∥�Π∇ F,hp,αv∥T,α + dim(P p(T)) � j=1 (χT,j , v − �Π∇ F,hp,αv)T (χT,j , ηT,j)T (∥ηT,j∥T + α∥∇ηT,j∥T ) ≲ ∥v∥T,α + |T|−1/2∥v − �Π∇ F,hp,αv∥T |T|1/2 + |T|−1/2∥v − �Π∇ F,hp,αv∥T αh−1 T |T|1/2 ≲ ∥v∥T,α + ∥v − �Π∇ F,hp,αv∥T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' In the last step we used α ≲ hT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Boundedness of �Π∇ F,hp,α finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Alternative operator for lowest-order spaces and moderate parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' In this section we construct a Fortin operator Π∇ F : H1(T) → V ∇,0 h such that (7) is satisfied for the lowest-order case p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' In [6] it is shown that for a low-order DPG method for the Poisson problem, test space V ∇ h = P 1(T) for the scalar test functions is sufficient to guarantee well-posedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The authors of [6] did use different techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Here, we complement their results by the construction of a Fortin operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Define the spaces �V ∇,0 h = P 1(T), V ∇,0 h = �V ∇,0 h + span{ηT } and operators �Π∇,0 F : H1(T) → �V ∇,0 h , Π∇,0 F : H1(T) → V ∇,0 h for all v ∈ H1(T) by �Π∇,0 F v = Π0 T v + � F∈FT ⟨1, (1 − Π0 T )v⟩F ⟨1, νF ⟩F νF , Π∇,0 F v = �Π∇,0 F v + (1, (1 − �Π∇,0 F )v)T (1, ηT )T ηT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The analysis of these operators can be done as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The details are left to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Theorem 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Let p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Operator Π∇ F = �Π∇,0 F resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Π∇ F = Π∇,0 F is idempotent on P 0(T), satisfies (7b) and ∥Π∇ F v∥T ≲ ∥v∥T + hT ∥∇v∥T , ∥∇Π∇ F v∥T ≲ ∥∇v∥T for all v ∈ H1(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Operator Π∇ F = Π∇,0 F additionally satisfies (7c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Comparison with existing Fortin operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' As mentioned in the introduction, several works have already dealt with the construction of Fortin operators on a simplex that satisfy (7), see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=', [12, 5, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' These works all have in common that they construct resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' prove the existence of a Fortin operator Π∇ F : H1(T) → P p+n(T) which satisfy (7a)-(7b) (for α ≳ hT ) and (u, (1 − Π∇ F )v)T = 0 ∀u ∈ P p−1(T), v ∈ H1(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The latter condition is not the same as (7c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' In order to satisfy (7c) one needs to increase the polynomial degree by one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=', P p+1+n(T) instead of P p+n(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 13 Comparing the dimension of P p+1+n(T) and the space V ∇ hp, we get dim(P p+1+n(T)) = �n j=1(p + 1 + n + j) n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , dim(V ∇ hp) = 1 + (n + 1) �n−1 j=1 (p + j) (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' + �n j=1(p + j) n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' For the lowest-order case p = 0 we thus find dim(P n+1(T)) = � 10 n = 2, 35 n = 3, and dim(V ∇ h0) = � 5 n = 2, 6 n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' For operator Π∇,0 F : H1(T) → V ∇,0 h we even have a reduction to dim(V ∇,0 h ) = n + 1 + 1 = � 4 n = 2, 5 n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' In conclusion, our test spaces are systematically smaller than previously used ones, and guarantee robustness contrary to the previous cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Fortin operator in H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T) We consider a fixed parameter α > 0 and space H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T) equipped with the (squared) norm ∥τ∥2 T,α = ∥τ∥2 T + α2∥ div τ∥2 T for τ ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The motivation for this section is the construction of Fortin operators, say Πdiv F : H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T) → Vh (where Vh ⊆ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T) is some discrete space) such that, for all τ ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T), ∥Πdiv F τ∥T,α ≤ CF∥τ∥T,α, (9a) ⟨(τ − Πdiv F τ) · nT , u⟩∂T = 0 ∀u ∈ P p+1 c (FT ), (9b) (σ , τ − Πdiv F τ)T = 0 ∀σ ∈ P p(T) (9c) with CF > 0 independent of α, hT (but possibly dependent on p ∈ N0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The latter two identities also imply (u, div(τ − Πdiv F τ))T = 0 ∀u ∈ P p+1(T), (9d) which can be seen from integration by parts: Let u ∈ P p+1(T) be given, then (u, div Πdiv F τ)T = −(∇u, Πdiv F τ)T + ⟨Πdiv F τ · nT , u|∂T ⟩∂T = −(∇u, τ)T + ⟨τ · nT , u|∂T ⟩∂T = (u, div τ)T ∀τ ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Construction for moderate parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Define �V div hp = span � ψ∂T,j : j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , dim( �P p+1(T)) � ⊂ P p+2(T) and operator �Πdiv F,hp : H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T) → �V div hp by �Πdiv F,hpτ = dim(P p+1 c (∂T)) � j=1 ⟨τ · nT , ν∂T,j⟩∂T ⟨ψ∂T,j · nT , ν∂T,j⟩∂T ψ∂T,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We collect its main properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Operator Πdiv F = �Πdiv F,hp satisfies (9b) and ∥�Πdiv F,hpτ∥T ≲ ∥τ∥T + hT ∥ div τ∥T ∀τ ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 14 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' To see (9b) a simple computation yields ⟨(1 − �Πdiv F,hp)τ · nT , ν∂T,k⟩∂T = ⟨τ · nT , ν∂T,k⟩∂T − � j ⟨τ · nT , ν∂T,j⟩∂T ⟨ψ∂T,j · nT , ν∂T,j⟩∂T ⟨ψ∂T,j · nT , ν∂T,k⟩∂T = ⟨τ · nT , ν∂T,k⟩∂T − ⟨τ · nT , ν∂T,k⟩∂T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Resolving the duality term, the Cauchy–Schwarz inequality and properties of basis functions give |⟨τ · nT , ν∂T,j⟩∂T | = |(τ , ∇ν∂T,j)T + (div τ , ν∂T,j)T | ≲ ∥τ∥T h−1 T |T|1/2 + ∥ div τ∥T |T|1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' With the triangle inequality we conclude that ∥�Πdiv F,hpτ∥T ≤ � j |⟨τ · nT , ν∂T,j⟩∂T | ⟨ψ∂T,j · nT , ν∂T,j⟩∂T ∥ψ∂T,j∥T ≲ |∂T|−1(∥τ∥T h−1 T |T|1/2 + ∥ div τ∥T |T|1/2)|T|1/2 ≲ ∥τ∥T + hT ∥ div τ∥T which finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' □ For the definition of operators that ensure (9c) we add a correction term based on the functions ηE,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We set V div hp = �V div hp + span � ηE,j : j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , dim P p(T), E ∈ E∗ � and define Πdiv F,hp : H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T) → V div hp for all τ ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T) by Πdiv F,hpτ = �Πdiv F,hpτ + � E∈E∗ dim(P p(T)) � j=1 (σE,j , (1 − �Πdiv F,hp)τ)T (σE,j , ηE,j)T ηE,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Theorem 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Operator Πdiv F = Πdiv F,hp satisfies (9b)–(9c) and ∥Πdiv F,hpτ∥T ≲ ∥τ∥T + hT ∥ div τ∥T ∀τ ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Furthermore, div ◦Πdiv F,hp = Πp+1 T div.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' If hT ≲ α, then Πdiv F = Πdiv F,hp satisfies (9a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Since ηE,j · nT |∂T = 0, we get Πdiv F,hpτ · nT |∂T = �Πdiv F,hpτ · nT |∂T , thus, condition (9b) follows from Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' To see (9c) we compute for any E′, k (σE′,k , (1 − Πdiv F,hp)τ)T = (σE′,k , (1 − �Πdiv F,hp)τ)T − � E∈E∗ � j (σE,j , (1 − �Πdiv F,hp)τ)T (σE,j , ηE,j)T (σE′,k , ηE,j)T = (σE′,k , (1 − �Πdiv F,hp)τ)T − (σE′,k , (1 − �Πdiv F,hp)τ)T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Noting that properties of the basis functions and boundedness by Lemma 17 give |(σE,j , (1 − �Πdiv F,hp)τ)T | (σE,j , ηE,j)T ∥ηE,j∥T ≲ ∥(1 − �Πdiv F,hp)τ∥T ≲ ∥τ∥T + hT ∥ div τ∥T , we see that ∥Πdiv F,hpτ∥T ≤ ∥�Πdiv F,hpτ∥T + � E∈E∗ � j |(σE,j , (1 − �Πdiv F,hp)τ)T | (σE,j , ηE,j)T ∥ηE,j∥T ≲ ∥τ∥T + hT ∥ div τ∥T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The commutativity property follows from (9d) and the fact that V div hp ⊆ P p+2(T) and, therefore, div Πdiv F,hpτ ∈ P p+1(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 15 For the final assertion note that ∥ div Πdiv F,hpτ∥T = ∥Πp+1 T div τ∥T ≤ ∥ div τ∥T by the commuta- tivity property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Together with boundedness in the L2(T) norm established above, this finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' □ Remark 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' In general, discrete test spaces are chosen such that a Fortin operator exists, which not necessarily implies approximation results of the form min τ h∈Vh ∥τ − τ h∥T + ∥ div(τ − τ h)∥T ≲ hT ∥∇τ∥T + ∥(1 − Π0 T ) div τ∥T for τ ∈ H1(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' For certain supercloseness results in the DPG method the latter approximation property is required, see [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' To ensure this property one can simply require RT 0(T) ⊂ Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Construction for small parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' For a small parameter, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=', α ≲ hT , we build a Fortin operator quite a bit different to the ones presented in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The proof of boundedness requires as in the scalar case the multiplicative version of the trace inequality, but also the following Helmholtz decomposition together with elliptic regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Let τ ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' There exist r ∈ H1 0(T), q ∈ H(curl ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T) (for n = 2 we have q ∈ H1(T)) such that τ = ∇r + curl q and ∥curl q∥2 T + ∥∇r∥2 T = ∥τ∥2 T , ∥D2r∥T ≲ ∥ div τ∥T with constants independent of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Define r ∈ H1 0(T) as the weak solution of ∆r = div τ, r|∂T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Elliptic regularity implies (note that div τ ∈ L2(T) and T is convex) that r ∈ H2(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Moreover, [13, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='2] implies that � |β|=2 ∥Dβr∥T ≤ C∥∆r∥T with a constant independent of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Finally, div(τ − ∇r) = 0 by construction implies that τ − ∇r = curl q for some q ∈ H(curl ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T) for n = 3 resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' q ∈ H1(T) for n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' □ The definition and analysis of our Fortin operator is based on the Helmholtz decomposition τ = ∇r + curl q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' For the curl q contribution we use the Fortin operator defined in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' For ∇r we consider the following definitions and analysis: Define the space �V div,aux hp,α = P 0(T) + span � ηα,F,j : j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , dim(P p+1 c (∂T)) � and operator �Πdiv,aux F,hp : H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T) → �V div,aux hp,α for all τ ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T) by �Πdiv,aux F,hp,α τ = Π0 T τ + dim(P p+1 c (FT )) � j=1 ⟨(1 − Π0 T )τ · nT , χ∂T,j⟩∂T ⟨ηα,∂T,j · nT , χ∂T,j⟩∂T ηα,∂T,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Operator Πdiv F = �Πdiv,aux F,hp,α satisfies (9b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' If α ≲ hT , then ∥�Πdiv,aux F,hp,α ∇r∥T,α ≲ ∥∇r∥T,α for all r ∈ H1 0(T) ∩ H2(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The verification of (9b) follows similarly as in Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We leave the details to the reader and focus on details for the proof of boundedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Let r ∈ H1 0(T) ∩ H2(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Note that elliptic 16 regularity yields ∥D2r∥T ≲ ∥ div ∇r∥T with constant independent of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Using standard norm estimates, the multiplicative version of the trace inequality (see Lemma 1) and Lemma 9, we get |⟨(1 − Π0 T )∇r · nT , χ∂T,j⟩∂T | ⟨ηα,∂T,j · nT , χ∂T,j⟩∂T ∥ηα,∂T,j∥T ≲ ∥(1 − Π0 T )∇r∥∂T |∂T|−1/2|T|1/2(α/hT )1/2 ≲ ∥(1 − Π0 T )∇r∥1/2 T α1/2∥D2r∥1/2 T ≲ ∥(1 − Π0 T )∇r∥T + α∥∆r∥T ≤ ∥∇r∥T + α∥∆r∥T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' By summing over all indices and bounding ∥Π0 T ∇r∥T ≤ ∥∇r∥T we find that ∥�Πdiv,aux F,hp,α ∇r∥T ≲ ∥Π0 T ∇r∥T + ∥∇r∥T + α∥∆r∥T ≲ ∥∇r∥T + α∥ div ∇r∥T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The same argumentation also proves α|⟨(1 − Π0 T )∇r · nT , χ∂T,j⟩∂T | ⟨ηα,∂T,j · nT , χ∂T,j⟩∂T ∥ div ηα,∂T,j∥T ≲ α∥(1 − Π0 T )∇r∥∂T |∂T|−1/2|T|1/2h−1 T (α/hT )−1/2 ≲ ∥(1 − Π0 T )∇r∥1/2 T α1/2∥D2r∥1/2 T ≲ ∥(1 − Π0 T )∇r∥T + α∥ div ∇r∥T ≤ ∥∇r∥T + α∥ div ∇r∥T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Summing over all indices and using div Π0 T ∇r = 0 we conclude that α∥ div �Πdiv,aux F,hp,α ∇r∥T ≲ ∥∇r∥T + α∥ div ∇r∥T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Combining all estimates finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' □ For the definition of operators that ensure (9c) we add — as before — a correction term based on the functions ηE,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We stress that these edge functions do not need to be modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Set V div hp,α = V div hp + �V div,aux hp and define Πdiv F,hp : H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T) → V div hp for all τ = ∇r + curl q ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T) by Πdiv F,hp,ατ = Πdiv F,hpcurl q + �Πdiv,aux F,hp,α ∇r + � E∈E∗ dim(P p(T)) � j=1 (σE,j , (1 − �Πdiv,aux F,hp,α )∇r)T (σE,j , ηE,j)T ηE,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The following is one of our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Theorem 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Operator Πdiv F = Πdiv F,hp,α satisfies (9b)–(9c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' If α ≲ hT , then ∥Πdiv F,hp,ατ∥T,α ≲ ∥τ∥T,α ∀τ ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The verification of (9b)–(9c) follows from the arguments already seen in Theorem 18 and Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' It only remains to prove boundedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' By the triangle inequality we get ∥Πdiv F,hp,ατ∥T,α ≤ ∥Πdiv F,hp,αcurl q∥T,α + ∥Πdiv F,hp,α∇r∥T,α for τ = ∇r + curl q ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T) where r, q are defined as in Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Note that Πdiv F,hp,αcurl q = Πdiv F,hpcurl q and applying Theorem 18 and Lemma 20 we see that ∥Πdiv F,hpcurl q∥T ≲ ∥curl q∥T + hT ∥ div curl q∥T = ∥curl q∥T ≤ ∥τ∥T as well as div Πdiv F,hpcurl q = Πp+1 T div curl q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We conclude that ∥Πdiv F,hp,αcurl q∥T,α ≲ ∥τ∥T ≤ ∥τ∥T,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 17 It remains to estimate ∥Πdiv F,hp,α∇r∥T,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Using the triangle inequality, the Cauchy–Schwarz inequal- ity, estimates for basis functions and Lemma 21, we get ∥Πdiv F,hp,α∇r∥T ≤ ∥�Πdiv,aux F,hp,α ∇r∥T + � E∈E∗ � j |(σE,j , (1 − �Πdiv,aux F,hp,α )∇r)T | (σE,j , ηE,j)T ∥ηE,j∥T ≲ ∥�Πdiv,aux F,hp,α ∇r∥T + |T|1/2∥(1 − �Πdiv,aux F,hp,α )∇r∥T |T|−1|T|1/2 ≲ ∥∇r∥T,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' For the divergence part in the norm, recall that ηE,j ∈ P p+2(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Therefore, div ηE,j ∈ P p+1(T) and (9d) implies (div ηE,j , div(1 − Πdiv F,hp,α)∇r)T = 0, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , dim(P p(T)), E ∈ E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We conclude that ∥ div(1 − Πdiv F,hp,α)∇r∥2 T = (div(1 − Πdiv F,hp,α)∇r , div(1 − Πdiv F,hp,α)∇r)T = (div(1 − �Πdiv,aux F,hp,α )∇r , div(1 − Πdiv F,hp,α)∇r)T ≤ ∥ div(1 − �Πdiv,aux F,hp,α )∇r∥T ∥ div(1 − Πdiv F,hp,α)∇r∥T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' It follows that ∥ div(1 − Πdiv F,hp,α)∇r∥T ≤ ∥ div(1 − �Πdiv,aux F,hp,α )∇r∥T and with the triangle inequality and Lemma 21 we get that α∥ div Πdiv F,hp,α∇r∥T ≤ α∥ div ∇r∥T + α∥ div(1 − �Πdiv,aux F,hp,α )∇r∥T ≲ ∥∇r∥T,α which finishes the proof together with ∥∇r∥T,α ≤ ∥τ∥T,α (Lemma 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Alternative operator for lowest order and moderate parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' First, consider the spaces �V div,1 h = RT 0(T), �V div,2 h = P 0(T) + span � ηF : F ∈ FT � and operators �Πdiv,j F : H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T) → �V div,j h , j = 1, 2, for all τ ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T) by �Πdiv,1 F τ = � F∈FT ⟨τ · nT , νF ⟩∂T ⟨ψF · nT , νF ⟩∂T ψF , (10a) �Πdiv,2 F τ = Π0 T τ + � F∈FT ⟨(1 − Π0 T )τ · nT , νF ⟩∂T ⟨ηF · nT , νF ⟩∂T ηF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' (10b) One verifies that �Πdiv,1 F is a projector whereas �Πdiv,2 F is idempotent on P 0(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Defining the spaces V div,j h = �V div,j h + span � ηE : E ∈ E∗ � , j = 1, 2, we introduce operators Πdiv,j F : V → V div,j h , j = 1, 2, for all τ ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T) by Πdiv,j F τ = �Πdiv,j F τ + � E∈E∗ (σE , (1 − �Πdiv,j F )τ)T (σE , ηE)T ηE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Theorem 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Operator Πdiv F ∈ � Πdiv,j F : j = 1, 2 � satisfies (9b)–(9c) and ∥Πdiv F τ∥T ≲ ∥τ∥T + hT ∥ div τ∥T , ∥ div Πdiv F τ∥T ≲ ∥ div τ∥T ∀τ ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Moreover, div ◦Πdiv,1 F = Π1 T ◦div.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Furthermore, operator Πdiv,1 F is idempotent on RT 0(T) and Πdiv,2 F is idempotent on P 0(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 18 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Idempotency of the operators follows from their definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Let Πdiv F ∈ � Πdiv,j F : j = 1, 2 � and �Πdiv F ∈ ��Πdiv,j F : j = 1, 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' First, we check condition (9b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' It holds for �Πdiv F as can be seen with the same arguments as in Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Since ηE · nT |∂T = 0 by Lemma 3 we have Πdiv F τ · nT |∂T = �Πdiv F τ · nT |∂T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We conclude that (9b) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Second, condition (9c) can be seen as follows, (σE′ , Πdiv F τ)T = (σE′ , �Πdiv F τ)T + � E∈E∗ (σE , (1 − �Πdiv F )τ)T (σE , ηE)T (σE′ , ηE)T = (σE′ , �Πdiv F τ)T + (σE′ , (1 − �Πdiv F )τ)T = (σE′ , τ)T ∀E′ ∈ E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Next we prove boundedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Estimate ∥�Πdiv,1 F τ∥T ≲ ∥τ∥T + hT ∥ div τ∥T follows as in Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' For the second operator we stress that |⟨(1 − Π0 T )τ · nT , νF ⟩T | = |(div τ , νF )T + ((1 − Π0 T )τ , ∇νF )T | = |(div τ , νF )T | ≲ ∥ div τ∥T |T|1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The second identity follows since ∇νF ∈ P 0(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' This allows us to estimate ∥�Πdiv,2 F τ∥T ≲ ∥Π0 T τ∥T + � F∈FT |F|−1|T|∥ div τ∥T ≲ ∥τ∥T + hT ∥ div τ∥T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Then, by the triangle inequality, the Cauchy–Schwarz inequality, norm estimates of basis functions and the previously established boundedness estimates, we see that ∥Πdiv F τ∥T ≤ ∥�Πdiv F τ∥T + � E∈E∗ |(σE , (1 − �Πdiv F )τ)T | (σE , ηE)T ∥ηE∥T ≲ ∥�Πdiv F τ∥T + |T|1/2∥(1 − �Πdiv F )τ∥T |T|−1|T|1/2 ≲ ∥�Πdiv F τ∥T + ∥(1 − �Πdiv F )τ∥T ≲ ∥τ∥T + hT ∥ div τ∥T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Furthermore, the same arguments and div Π0 T τ = 0, ∥ div ηF ∥T ≂ h−1 T |T|1/2 show that ∥ div �Πdiv,2 F τ∥T ≲ ∥ div τ∥T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Note that Πdiv,1 F τ ∈ P 2(T), thus, div Πdiv,1 F τ ∈ P 1(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Consequently, (9d) implies the commu- tativity property of Πdiv,1 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Clearly, this also yields ∥ div Πdiv,1 F τ∥T ≤ ∥ div τ∥T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' It thus remains to prove that ∥ div Πdiv,2 F τ∥T ≲ ∥ div τ∥T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' To do so we argue as in the proof of Theorem 22 to derive ∥ div(1 − Πdiv,2 F )τ∥T ≤ ∥ div(1 − �Πdiv,2 F )τ∥T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Together with the triangle inequality and ∥ div �Πdiv,2 F τ∥T ≲ ∥ div τ∥T we get that ∥ div Πdiv,2 F τ∥T ≤ ∥ div τ∥T + ∥ div(1 − �Πdiv,2 F )τ∥T ≲ ∥ div τ∥T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Alternative operator for lowest order and small parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' In this section we construct a simpler Fortin operator for α ≲ hT and the lowest-order case (p = 0 in (9)), based on Πdiv,2 F from the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Define the spaces �V div h,α = P 0(T) + span � ηα,F : F ∈ FT � , V div h,α = �V div h,α + span � ηE : E ∈ E∗ � 19 and operators �Πdiv F,α : H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T) → �V div h,α , Πdiv F,α : H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T) → V div h,α for all τ ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T) by �Πdiv F,ατ = Π0 T τ + � F∈FT ⟨(1 − Π0 T )τ · nT , νF ⟩∂T ⟨ηα,F · nT , νF ⟩∂T ηα,F , Πdiv F,ατ = �Πdiv F,ατ + � E∈E∗ (σE , (1 − �Πdiv F,α)τ)T (σE , ηE)T ηE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Theorem 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Operator Πdiv F = Πdiv F,α satisfies (9b)–(9c) for p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' If α ≲ hT then ∥Πdiv F,ατ∥T,α ≲ ∥τ∥T,α ∀τ ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The verification of (9b)–(9c) follows as in Theorem 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' It remains to prove boundedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' First, we show boundedness of �Πdiv F,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Let τ = ∇r + curl q ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T) with r, q as in Lemma 20 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Using ∇νF ∈ P 0(T), div curl q = 0 we obtain �Πdiv F,αcurl q = Π0 T curl q + � F∈FT (div(1 − Π0 T )curl q , νF )T + ((1 − Π0 T )curl q , ∇νF )T ⟨ηα,F · nT , νF ⟩∂T ηα,F = Π0 T curl q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Then, ∥�Πdiv F,αcurl q∥T,α = ∥Π0 T curl q∥T ≤ ∥curl q∥T ≤ ∥τ∥T,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' With the multiplicative trace inequality (Lemma 1), properties of the basis functions, Lemma 8 and Lemma 20 we get |⟨(1 − Π0 T )∇r · nT , νF ⟩∂T | ⟨ηα,F · nT , νF ⟩∂T ∥ηα,F ∥T ≲ |F|−1|∂T|1/2∥(1 − Π0 T )∇r∥∂T ∥ηα,F ∥T ≲ |∂T|−1/2∥(1 − Π0 T )∇r∥1/2 T ∥D2r∥1/2 T α1/2h−1/2 T |T|1/2 ≲ ∥∇r∥1/2 T α1/2∥ div τ∥1/2 T ≲ ∥τ∥T + α∥ div τ∥T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The last estimates yield ∥�Πdiv F,α∇r∥T ≲ ∥τ∥T,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' For the divergence contribution the same arguments prove α|⟨(1 − Π0 T )∇r · nT , νF ⟩∂T | ⟨ηα,F · nT , νF ⟩∂T ∥ div ηα,F ∥T ≲ |F|−1|∂T|1/2∥(1 − Π0 T )∇r∥∂T h−1 T |T|1/2α1/2h1/2 T ≲ ∥∇r∥1/2 T α1/2∥ div τ∥1/2 T ≲ ∥τ∥T + α∥ div τ∥T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We conclude that α∥ div �Πdiv F,α∇r∥T ≲ ∥τ∥T,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Putting all estimates together we have shown that ∥�Πdiv F,ατ∥T,α ≤ ∥�Πdiv F,αcurl q∥T,α + ∥�Πdiv F,α∇r∥T,α ≲ ∥τ∥T,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Arguing as in the proof of Theorem 23 we find that ∥Πdiv F,ατ∥T ≲ ∥�Πdiv F,ατ∥T + ∥(1 − �Πdiv F,α)τ∥T giving us ∥Πdiv F,ατ∥T ≲ ∥τ∥T,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Finally, arguing as in the proof of Theorem 22 we find that ∥ div(1 − Πdiv F,α)τ∥T ≤ ∥ div(1 − �Πdiv F,α)τ∥T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Together with the boundedness of �Πdiv F,α we conclude that α∥ div Πdiv F,ατ∥T ≲ ∥τ∥T,α which finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' □ 20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Comparison with existing Fortin operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Fortin operators that satisfy (9) are con- structed in [12, 5, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' In [12, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='3], it is shown that there exists a Fortin operator, map- ping into the discrete test space P p+2(T) and in [9], the authors impose the minimal condition RT p+1(T) ⊂ V div h to ensure the existence of a Fortin operator satisfying (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Here, RT p+1(T) = P p+1(T) + xP p+1(T) denotes the Raviart–Thomas space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We stress that all these mentioned oper- ators are uniformly bounded only if hT ≲ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Computing dimensions we get dim(P p+2(T)) = n �n j=1(j + p + 2) n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' , dim(RT p+1(T)) = n �n j=1(j + p) n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' + (n + 1) �n−1 j=1 (j + p + 1) (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' To compute the dimension of V div hp we note that dim(P p+1 c (FT )) = dim(P p+1(T)) − dim(P p+1 b (T)) where we recall that P p b (T) denotes the space of element bubbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' This yields dim(V div hp ) = dim(P p+1 c (FT )) + dim(P p(T)) = �n j=1(j + p + 1) n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' − �n j=1(j + p − n) n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' + n �n j=1(j + p) n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' For the lowest-order case p = 0 we thus get dim(P 2(T)) = � 12 n = 2, 30 n = 3, dim(RT 1(T)) = � 8 n = 2, 15 n = 3, and dim(V div h0 ) = � 5 n = 2, 7 n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' As in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='4 we conclude that our test spaces are systematically smaller than previously used ones, and guarantee robustness, contrary to the previous cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Numerical experiment In this section we consider the reaction-diffusion problem −ε2∆u + u = f in Ω, u|∂Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' First, we give a brief overview of a DPG method for the latter problem and, then, discuss results of our numerical experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' DPG method for reaction-diffusion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We introduce the trace operators tr∇ : H1(Ω) → (H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T ))′, trdiv : H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Ω) → (H1(T ))′, defined for u ∈ H1(Ω), σ ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Ω) by ⟨tr∇ u, τ⟩∂T = (u, divT τ)Ω + (∇u, τ)Ω ∀τ ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T ) = � T∈T H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T), ⟨trdiv σ , v⟩∂T = (σ , ∇T v)Ω + (div σ , v)Ω ∀v ∈ H1(T ) = � T∈T H1(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 21 Here, divT : H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T ) → L2(Ω) is given by divT τ|T = div(τ|T ) for T ∈ T , τ = (τ T )T∈T ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T ) and ∇T : H1(T ) → L2(Ω) is defined similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The trace spaces H1/2 00 (∂T ) = tr∇(H1 0(Ω)), H−1/2(∂T ) = trdiv(H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Ω)) are closed with respect to the canonical norms (see [5]) ∥�u∥1/2,ε = inf � ∥u∥Ω,ε : u ∈ H1(Ω), tr∇ u = �u � , ∥�σ∥−1/2,ε = inf � ∥σ∥Ω,ε : σ ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Ω), trdiv σ = �σ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Introducing the spaces U = L2(Ω) × L2(Ω) × H1/2 00 (∂T ) × H−1/2(∂T ), V = H1(T ) × H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' T ), where U is equipped with the canonical product norm and V with the (squared) norm ∥(v, τ)∥2 V = ∥v∥2 T ,ε + ∥τ∥2 T ,ε := � T∈T ∥v∥2 T,ε + ∥τ∥2 T,ε we obtain the ultraweak formulation of the reaction-diffusion problem by defining σ = ε∇u and element-wise integration by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' This yields u = (u, σ, �u, �σ) ∈ U : b(u, v) = L(v) ∀v = (v, τ) ∈ V, (11) where for u ∈ U, v ∈ V and given f ∈ L2(Ω) we define b(u, v) = (u, ε divT τ + v)Ω + (σ , ε∇T v + τ)Ω − ε⟨�u, τ⟩∂T − ε⟨�σ , v⟩∂T , L(v) = (f , v)Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The following result contains well-posedness of the ultraweak formulation (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' It can be derived by following the abstract theory presented in [5] together with our discussions on the fully-discrete scheme (1) from the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Proposition 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The ultraweak formulation (11) admits a unique solution u ∈ U with ∥u∥U ≲ ∥f∥Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Let Uh ⊂ U, Vh ⊂ V denote finite-dimensional subspaces and suppose that there exists a Fortin operator ΠF : V → Vh satisfying (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Then, with uh ∈ U denoting the solution of (1), ∥u − uh∥U ≲ min v∈Uh ∥u − vh∥U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The hidden constants are independent of ε and the mesh-size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Results for reaction-diffusion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' In this section we consider the manufactured so- lution u(x, y) = v(x)v(y) for (x, y) ∈ Ω = (0, 1)2, where v(x) = 1 − (1 − e−1/( √ 2ε))e−(1−x)/( √ 2ε) + e−x/( √ 2ε) 1 − e−2/( √ 2ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' One verifies that u is the solution of −ε2∆u + u = f, u|∂Ω = 0 with f(x, y) = 1 2(v(x) + v(y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We use the DPG method from Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='1 with test spaces Vh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='pol = � T∈T Vh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='pol(T),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Vh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='ε = � T∈T Vh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='ε(T) 22 101 102 103 104 105 106 10−3 10−2 10−1 100 degrees of freedom Vh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='pol,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' ε = 10−3 est ∥u − uh∥ ∥σ − σh∥ 101 102 103 104 105 106 10−3 10−2 10−1 100 degrees of freedom Vh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' ε = 10−3 est ∥u − uh∥ ∥σ − σh∥ 101 102 103 104 105 106 10−3 10−2 10−1 degrees of freedom Vh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='pol,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' ε = 10−4 est ∥u − uh∥ ∥σ − σh∥ 101 102 103 104 105 106 10−3 10−2 10−1 degrees of freedom Vh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' ε = 10−4 est ∥u − uh∥ ∥σ − σh∥ Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Errors in the field variables compared with DPG estimator for ε = 10−3 and ε = 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The left resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' right column shows the results using test space Vh,pol resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Vh,ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' where Vh,pol(T) = P 3(T) × P 2(T), Vh,ε(T) = � V ∇ h0 × V div,2 h , ε > hT , V ∇ h0,ε × V div h,ε , ε ≤ hT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The trial space is Uh = P 0(T ) × P 0(T ) × tr∇(P 1(T ) ∩ H1 0(Ω)) × trdiv(RT 0(T )), where RT 0(T ) = � τ ∈ L2(Ω) : τ|T ∈ RT 0(T), T ∈ T � ∩ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We stress that by our constructions from Sections 3 and 4, Vh = Vh,ε is a test space that allows for a uniformly bounded Fortin operator (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' On the other hand, test space Vh = Vh,pol allows for a Fortin operator whose norm depends on ε, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 23 10−6 10−5 10−4 10−3 10−2 10−1 100 10−1 100 101 102 ε Vh,pol Vh,ε Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Ratio ρ = ∥u − uh∥/ est for the two different test spaces Vh,pol and Vh,ε on a fixed mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The black dotted line corresponds to O(ε−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We also define the DPG error estimator by est = sup 0̸=vh∈Vh b(uh, vh) − L(vh) ∥vh∥V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Clearly, this estimator depends on the choice of the test space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' In [4] it is shown that est is, up to an oscillation term, equivalent to the error ∥u − uh∥U ≂ est +osc(f) = est + sup 0̸=v=(v,τ)∈V L(v − ΠFv) ∥v∥V , provided there exists a uniformly bounded Fortin operator ΠF : V → Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Figure 2 shows the errors of the field variables for ε ∈ {10−3, 10−4} and estimator est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We observe differences when using Vh = Vh,pol or Vh = Vh,ε as test space: For coarse meshes and Vh = Vh,pol the estimator est underestimates the errors in the field variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' This effect is more severe for smaller parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' This can also be seen in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' There, we fix a mesh with four elements and only vary ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We plot the index ρ = ∥u − uh∥/ est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' When using Vh = Vh,pol we observe that ρ = O(ε−1/2) for ε → 0 whereas ρ = O(1) when using Vh = Vh,ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We conclude that the DPG method with Vh = Vh,pol is not robust, whereas with the new test space Vh = Vh,ε it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Discrete stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Let Ω = �T, T = { �T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' In this section we want to study stability of the method by investigating the norm equivalence constants λmin, λmax in λmin∥uh∥2 U ≤ b(uh, Θhuh) ≤ λmax∥uh∥2 U ∀uh ∈ Uh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' (12) Here, Uh is defined as in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We use two different test spaces, Vh,ε (defined in the previous section) and �Vh = V ∇ h0 × V div,2 h0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' 24 10−5 10−4 10−3 10−2 10−1 10−4 10−3 10−2 10−1 100 ε λmax, �Vh λmin, �Vh λmax, Vh,ε λmin, Vh,ε Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Constants λmax and λmin from (12) for the test spaces �Vh and Vh,ε (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The black dotted line corresponds to O(ε−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The difficulty in checking the norm equivalence is the implementation of the trace norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Note that due to inclusion of boundary conditions and the fact that all nodes of T are on the boundary, we do not have to consider ∥�uh∥1/2,ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' To calculate ∥�σh∥−1/2,ε we generate a submesh �T of T such that all elements that have a boundary face have diameter less than or equal to ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' This, heuristically, resolves possible boundary layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' To evaluate ∥�σh∥−1/2,ε we approximate the PDE τ ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Ω): − ε2∇ div τ + τ = 0, τ · nΩ|∂Ω = �σh by a standard FEM on �T using lowest-order Raviart–Thomas elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' The H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Ω) norm ∥τ h∥Ω,ε of the approximation τ h ∈ RT 0(T ) is taken as ∥�σh∥−1/2,ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' In view of norm equivalence (12) we stress that λmax ≲ 1 independent of ε and test space Vh since the bilinear form is uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' However, the lower bound is directly related to the stability of the DPG method, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=', λmin depends on the discrete inf-sup constant which for the DPG method is related to the norm of Fortin operators as we already discussed in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Figure 4 visualizes λmin and λmax for Vh = Vh,ε and Vh = �Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' We observe that λmax is uniformly bounded for both test spaces (for small ε we can not even distinguish them in the plot), whereas λmin deteriorates for Vh = �Vh and is essentially constant for Vh = Vh,ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' This illustrates that the DPG method with test space Vh,ε is uniformly stable, in contrast to the canonical method with test space �Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' References [1] C.' metadata={'source': 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J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Gopalakrishnan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Breaking spaces and forms for the DPG method and applications including Maxwell equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=', 72(3):494–522, 2016.' metadata={'source': 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Construction of DPG Fortin operators for second order problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=', 74(8):1964–1980, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content=' Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Santiago, Chile Email address: {tofuhrer,nheuer}@mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='uc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} +page_content='cl 26' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFPT4oBgHgl3EQfLDSW/content/2301.13021v1.pdf'} diff --git a/pNFQT4oBgHgl3EQfrzZ8/content/tmp_files/2301.13385v1.pdf.txt b/pNFQT4oBgHgl3EQfrzZ8/content/tmp_files/2301.13385v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2403b74bbcb2dbf5945b5da6ca399f2728cce3dc --- /dev/null +++ b/pNFQT4oBgHgl3EQfrzZ8/content/tmp_files/2301.13385v1.pdf.txt @@ -0,0 +1,456 @@ +Fisheye traffic data set of point center markers +Chung-I Huang +1, Wei-Yu Chen +1, Wei Jan Ko +2, Jih-Sheng Chang +1 + Chen-Kai Sun +1 , Hui Hung Yu +1, Fang-Pang Lin +1 + +1National Center for High-performance Computing +2National Yang Ming Chiao Tung University + + Abstract +This study presents an open data- +market +platform +and +a +dataset +containing 160,000 markers and 18,000 +images. We hope that this dataset will +bring more new data value and +applications In this paper, we introduce +the format and usage of the dataset, and +we show a demonstration of deep +learning vehicle detection trained by this +dataset. + +I. +Introduction +This study proposes an open +application and challenging new traffic +dataset: +point-centered +annotated +fisheye traffic dataset. This dataset +contains fisheye traffic images of four +important intersections in Hsinchu City. +Traffic images of four important +intersections in Hsinchu City, with the +following features: +(1) the use of point-centered +annotation. +(2) Three types of objects: large +cars, +small +cars, +and +motorcycles. +(3) About 18,000 images and +160,000 markers are included. +In addition to providing + +In addition to providing the dataset, this +study also develops AI models, and the +model experiments prove that the +dataset can provide traffic information. +This dataset can provide traffic-related +artificial +intelligence +results +and +applications. +The +dataset +will +be +released and validated. In addition to the +release and validation of the dataset, +this study also proposes an open +platform and model for data release. + +II. +Proposed Method +This study proposes an open +platform and mode of data release for +other needy data providers. The hope is +to provide new data to bring new + +applications and to revitalize the data. +We hope to not only provide new data to +bring new applications, but also to +revitalize +data +energy +to +create +opportunities. +Technologies such as intelligent +transportation and even autonomous +driving rely heavily on a large amount of +real-world data in order to develop, test, +and validate AI models. Some of the +traffic datasets are based on computer +vision for autonomous driving, including +[1-4]. +Although most of the typical object +detection methods are marked using the +pull frame method, whether it is RCNN +[5], FastRCNN [6], or Yolo [7], the cost of +marking, the range of objects marked, +and other conditions need to be weighed. +The current method of annotating +by marking the center also has good +results. CornerNet [8] detects two +boundary box corners as key points, +while ExtremeNet [9] detects the top, +left, bottom, and right-most points, as +well as the center point of all objects. +CenterNet [10] simply extracts a center +point for each object without grouping +or post-processing. +The Fisheye traffic data set contains +point center markers that indicate the +location of vehicles or other objects in +traffic data. These markers are typically +derived from cameras, sensors, or other +imaging systems, and they provide a +detailed picture of the traffic behavior in +a given area. The data set also includes +information about the speed, direction, +and other characteristics of the traffic +flow. This information can be used to +better understand how traffic moves in a +given area and to make predictions +about future traffic patterns. Recent +advances in vehicle detection and +tracking algorithms have enabled them +to be used in a range of fields such as +autonomous +driving, +traffic +flow +estimation and traffic control. +However, these models are mainly +developed for rectilinear-lens cameras +and are not optimized for fisheye images, +which +suffer +from +strong +radial +distortion, especially in the periphery. +Owing to their ability to provide dynamic +viewing angles and reduce blind spots, +fisheye +cameras +are +increasingly +replacing +traditional +surveillance +cameras in applications such as airports +and hotels. Additionally, the field of +fisheye vehicle detection and tracking is +relatively new and in need of a +comprehensive dataset for validating +performance. + The +point-centered +annotated + +fisheye traffic dataset is a newly +proposed, open-access dataset that +presents +a +unique +challenge +to +researchers in the field of traffic +monitoring. This dataset includes fisheye +images of four intersections, each with +annotated points of interest. These +images provide a valuable resource for +the development of cutting-edge traffic +monitoring and analysis technologies. + +III. +Experiment + In +this +study, +we +propose +a +comprehensive approach to traffic data +analysis and applications by releasing a +dataset and developing AI models. The +effectiveness of the dataset and models +is +demonstrated +through +model +experiments, highlighting the potential +for AI applications in traffic. To facilitate +the use of this data, we also propose an +open platform and model for data +release. The release of a dataset and AI +models for traffic data analysis allows +researchers and practitioners to analyze +traffic data and develop AI-based +solutions for traffic-related problems. +The validation of the dataset and models +through experiments helps to ensure the +reliability and effectiveness of the data +and models for traffic data analysis. + This marker dataset is a useful tool +for training and evaluating object +detection models, as it includes images +of three types of traffic vehicles. This +dataset is helpful for developing AI +applications because it allows for +accurate +testing +of +AI +model +performance. It contains traffic data +from four intersections in Hsinchu City, +including approximately 18,000 images +and around 160,000 point annotations. +(Fig.1) + +● +Use the point center labeling +method, +● +Only objects within the ROI +range are marked +● +The content of the label is +divided into: large car, small +car, locomotive + +Each +compressed +file +includes +a +collection +of +png +images, +json +annotation files, and bound annotation +images. For example: + +● +01_roi.txt => mark the roi +range +● +frame_00000.png => original +image +● +frame_00000.json => +annotation file +● +simulation/frame_00000_out.p + +ng => Binding annotated +image +The marked frame_00000.png, red dots +are locomotives, green dots are small cars, +blue dots are large cars, the ROI range is +near the intersection area within the zebra +crossing, please refer to the roi.txt of each +intersection for details + +Fig.1 fisheye camera from in Hsinchu City + +Each line represents a ROI line expressed +as [[x1, y1], [x2, y2]], where (x1, y1) and +(x2, y2) represent the two endpoints of +the ROI line. (Fig.2) +Fig.2 ROI line Information + +Fig.3 Label data format + +The annotation file is described in +JSON format. The 'image' field contains +the width and height and the file name. +The 'points' field contains a description +of the center annotation of each point. +Each number has an ID, and the x,y +values +are +the +coordinates. +The +'category_id' field indicates the type of +transportation, where 0 is a motorcycle, +1 is a small car, and 2 is a large car. (Fig.3) + +In +this +study, +a +comprehensive +approach was taken to analyze the fisheye +traffic dataset. The VGG19 pre-training +model was utilized, along with the Adam +Optimizer, Sigmoid classifier, and Loss +function to compute errors. The prediction +results were then obtained through visual +processing. This methodology allowed for +a thorough and accurate analysis of the +dataset, leading to reliable and valuable +insights (Fig.4). + +CH1 2022-02-24 08:10:23 +車 +py busuzbu"1ine':[[1732.4,436.8],[1796.6,582.4]] +"iine": +[[1796.6,582.4],[1826.2,708.3]] +"1ine":[[955.1,76.5],[1238.9,96.2]] +"1ine":[[417.1,407.2],[525.7,269]] +"1ine":[[525.7,269],[654,170.3]] +"line": +[[387.5,925.4],[1021.7,1083.4]] +"1ine":[[1021.7,1083.4],[1705.3,1142.6]]2 +"image":( +3 +"file name": +"frame01545.png", +4 +"width":1920, +5 +"height":1920 +6 +8 +9 +"id":264312, +10 +"x":1619, +11 +y:336, +12 +"category_id':o +13 +14 +15 +"id"264462, +16 +x:661, +17 +"y:460, +18 +"category id: +19 +1 +20 +21 +"id":264463, +22 +x":1296, +23 +"y:89. +24 +"category id":1 +25 +26 +Fig.4 Fisheye data detection results +(https://youtu.be/sjUQ-Ayxxtk) + +IV. +Data acquisition +The dataset is maintained by the +National Web Center SciDM platform +(http://dx.doi.org/10.30193/SciDM.DB_Fis +heye_Camera_Dataset/Dataset ). iService +account is required for data applicants, +which is free of charge. The data owner will +receive the application, review it, and notify +the user that it is available for downloading, +or to add additional information to the +application. +V. +Conclusion : +This study presents a detailed description +of the fisheye traffic dataset, including its +central +point +and +the +models +and +applications used to analyze it. Looking +towards the future, we hope to continue +expanding upon this dataset by releasing +higher quality and larger datasets with +additional features such as tracking IDs, +more road segments, and longer time +periods. +This +will +provide +a +more +comprehensive understanding of traffic +patterns and behaviors, and enable the +development of more effective AI-based +solutions for traffic-related issues. +[1] G. J. Brostow, J. Fauqueur, and R. +Cipolla, “Semantic object classes in video: +A high-definition ground truth database,” +Pattern Recognition Letters, vol. 30, no. +2, pp. 88–97, 2009. +[2] G. Pandey, J. R. McBride, and R. M. +Eustice, “Ford campus vision and LIDAR +data set,” The International Journal of +Robotics Research, vol. 30, no. 13, pp. +1543–1552, 2011. +[3] D. Pfeiffer, S. Gehrig, and N. +Schneider, “Exploiting the power of +stereo confidences,” in Proceedings of +the IEEE Conference on Computer Vision +and Pattern Recognition, 2013, pp. 297– +304. +[4] J.-L. Blanco-Claraco, F.-A . Moreno- +Duen˜as, and J. Gonza lez-Jime nez, “The +M alaga urban dataset: High-rate stereo +and LiDAR in a realistic urban scenario,” +The International Journal of Robotics +Research, vol. 33, no. 2, pp. 207–214, +2014. +[5] R. Girshick, J. Donahue, T. Darrell, and +J. Malik. Rich. “feature hierarchies for +accurate object detection and semantic +segmentation.” In CVPR, 2014. + +車 +不[6] J. R. Uijlings, K. E. Van De Sande, T. +Gevers, and A. W. Smeulders. “Selective +search for object recognition.” IJCV, +2013. +[7] R. Girshick. “Fast r-cnn.” In ICCV, 2015. +[8] X. Zhou, J. Zhuo, and P. Kr ahenb uhl. +“Bottom-up +object +detection +by +grouping extreme and center points.” In +CVPR, 2019. +[9] ZHOU, Xingyi; WANG, Dequan; +KRÄHENBÜHL, +Philipp. +“Objects +as +points.” In arXiv:1904.07850, 2019. +[10] Kaiwen Duan, Song Bai, Lingxi Xie, +Honggang Qi, Qingming Huang, Qi +Tian;”Centernet: Keypoint triplets for +object detection.” In ICCV, 2019 + diff --git a/pNFQT4oBgHgl3EQfrzZ8/content/tmp_files/load_file.txt b/pNFQT4oBgHgl3EQfrzZ8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e33690c00358bf17bd3ee2682dd31df2cbcb1296 --- /dev/null +++ b/pNFQT4oBgHgl3EQfrzZ8/content/tmp_files/load_file.txt @@ -0,0 +1,170 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf,len=169 +page_content='Fisheye traffic data set of point center markers Chung-I Huang 1, Wei-Yu Chen 1, Wei Jan Ko 2, Jih-Sheng Chang 1 Chen-Kai Sun 1 , Hui Hung Yu 1, Fang-Pang Lin 1 1National Center for High-performance Computing 2National Yang Ming Chiao Tung University Abstract This study presents an open data- market platform and a dataset containing 160,000 markers and 18,000 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' We hope that this dataset will bring more new data value and applications In this paper, we introduce the format and usage of the dataset, and we show a demonstration of deep learning vehicle detection trained by this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' Introduction This study proposes an open application and challenging new traffic dataset: point-centered annotated fisheye traffic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' This dataset contains fisheye traffic images of four important intersections in Hsinchu City.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' Traffic images of four important intersections in Hsinchu City, with the following features: (1) the use of point-centered annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' (2) Three types of objects: large cars, small cars, and motorcycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' (3) About 18,000 images and 160,000 markers are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' In addition to providing In addition to providing the dataset, this study also develops AI models, and the model experiments prove that the dataset can provide traffic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' This dataset can provide traffic-related artificial intelligence results and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' The dataset will be released and validated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' In addition to the release and validation of the dataset, this study also proposes an open platform and model for data release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' Proposed Method This study proposes an open platform and mode of data release for other needy data providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' The hope is to provide new data to bring new applications and to revitalize the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' We hope to not only provide new data to bring new applications, but also to revitalize data energy to create opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' Technologies such as intelligent transportation and even autonomous driving rely heavily on a large amount of real-world data in order to develop, test, and validate AI models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' Some of the traffic datasets are based on computer vision for autonomous driving, including [1-4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' Although most of the typical object detection methods are marked using the pull frame method, whether it is RCNN [5], FastRCNN [6], or Yolo [7], the cost of marking, the range of objects marked, and other conditions need to be weighed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' The current method of annotating by marking the center also has good results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' CornerNet [8] detects two boundary box corners as key points, while ExtremeNet [9] detects the top, left, bottom, and right-most points, as well as the center point of all objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' CenterNet [10] simply extracts a center point for each object without grouping or post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' The Fisheye traffic data set contains point center markers that indicate the location of vehicles or other objects in traffic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' These markers are typically derived from cameras, sensors, or other imaging systems, and they provide a detailed picture of the traffic behavior in a given area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' The data set also includes information about the speed, direction, and other characteristics of the traffic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' This information can be used to better understand how traffic moves in a given area and to make predictions about future traffic patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' Recent advances in vehicle detection and tracking algorithms have enabled them to be used in a range of fields such as autonomous driving, traffic flow estimation and traffic control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' However, these models are mainly developed for rectilinear-lens cameras and are not optimized for fisheye images, which suffer from strong radial distortion, especially in the periphery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' Owing to their ability to provide dynamic viewing angles and reduce blind spots, fisheye cameras are increasingly replacing traditional surveillance cameras in applications such as airports and hotels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' Additionally, the field of fisheye vehicle detection and tracking is relatively new and in need of a comprehensive dataset for validating performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' The point-centered annotated fisheye traffic dataset is a newly proposed, open-access dataset that presents a unique challenge to researchers in the field of traffic monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' This dataset includes fisheye images of four intersections, each with annotated points of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' These images provide a valuable resource for the development of cutting-edge traffic monitoring and analysis technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' Experiment In this study, we propose a comprehensive approach to traffic data analysis and applications by releasing a dataset and developing AI models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' The effectiveness of the dataset and models is demonstrated through model experiments, highlighting the potential for AI applications in traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' To facilitate the use of this data, we also propose an open platform and model for data release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' The release of a dataset and AI models for traffic data analysis allows researchers and practitioners to analyze traffic data and develop AI-based solutions for traffic-related problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' The validation of the dataset and models through experiments helps to ensure the reliability and effectiveness of the data and models for traffic data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' This marker dataset is a useful tool for training and evaluating object detection models, as it includes images of three types of traffic vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' This dataset is helpful for developing AI applications because it allows for accurate testing of AI model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' It contains traffic data from four intersections in Hsinchu City, including approximately 18,000 images and around 160,000 point annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content='1) Use the point center labeling method, Only objects within the ROI range are marked The content of the label is divided into: large car, small car, locomotive Each compressed file includes a collection of png images, json annotation files, and bound annotation images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' For example: 01_roi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content='txt => mark the roi range frame_00000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content='png => original image frame_00000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content='json => annotation file simulation/frame_00000_out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content='p ng => Binding annotated image The marked frame_00000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content='png, red dots are locomotives, green dots are small cars, blue dots are large cars, the ROI range is near the intersection area within the zebra crossing, please refer to the roi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content='txt of each intersection for details Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content='1 fisheye camera from in Hsinchu City Each line represents a ROI line expressed as [[x1, y1], [x2, y2]], where (x1, y1) and (x2, y2) represent the two endpoints of the ROI line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content='2) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content='2 ROI line Information Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content='3 Label data format The annotation file is described in JSON format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=" The 'image' field contains the width and height and the file name." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=" The 'points' field contains a description of the center annotation of each point." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' Each number has an ID, and the x,y values are the coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=" The 'category_id' field indicates the type of transportation, where 0 is a motorcycle, 1 is a small car, and 2 is a large car." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content='3) In this study, a comprehensive approach was taken to analyze the fisheye traffic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' The VGG19 pre-training model was utilized, along with the Adam Optimizer, Sigmoid classifier, and Loss function to compute errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' The prediction results were then obtained through visual processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' This methodology allowed for a thorough and accurate analysis of the dataset, leading to reliable and valuable insights (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content='4).' metadata={'source': 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National Web Center SciDM platform (http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content='30193/SciDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content='DB_Fis heye_Camera_Dataset/Dataset ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' iService account is required for data applicants, which is free of charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' The data owner will receive the application, review it, and notify the user that it is available for downloading, or to add additional information to the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' Conclusion : This study presents a detailed description of the fisheye traffic dataset, including its central point and the models and applications used to analyze it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' Looking towards the future, we hope to continue expanding upon this dataset by releasing higher quality and larger datasets with additional features such as tracking IDs, more road segments, and longer time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' This will provide a more comprehensive understanding of traffic patterns and behaviors, and enable the development of more effective AI-based solutions for traffic-related issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' Brostow, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} +page_content=' Fauqueur, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFQT4oBgHgl3EQfrzZ8/content/2301.13385v1.pdf'} 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sha256:be58efbf3a4b075c76c57ade1d025b99c24f372bd41b6ce9bcebd220d8f948d9 +size 231060 diff --git a/s9E2T4oBgHgl3EQffgcE/content/tmp_files/2301.03926v1.pdf.txt b/s9E2T4oBgHgl3EQffgcE/content/tmp_files/2301.03926v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e16ab4eaeec1e4d3e7799d6091824be13a5c4a58 --- /dev/null +++ b/s9E2T4oBgHgl3EQffgcE/content/tmp_files/2301.03926v1.pdf.txt @@ -0,0 +1,865 @@ +1 +Video Surveillance System Incorporating Expert +Decision-making Process: A Case Study on +Detecting Calving Signs in Cattle +Ryosuke Hyodo∗, Susumu Saito∗†, Teppei Nakano∗†, Makoto Akabane∗, Ryoichi Kasuga‡, Tetsuji Ogawa∗ +∗ Waseda University, Tokyo, Japan +† Intelligent Framework Lab, Tokyo, Japan +‡ Farmers Support, Kagoshima, Japan +Abstract—Through a user study in the field of livestock +farming, we verify the effectiveness of an XAI framework for +video surveillance systems. The systems can be made inter- +pretable by incorporating experts’ decision-making processes. +AI systems are becoming increasingly common in real-world +applications, especially in fields related to human decision- +making, and its interpretability is necessary. However, there are +still relatively few standard methods for assessing and addressing +the interpretability of machine learning-based systems in real- +world applications. In this study, we examine the framework of +a video surveillance AI system that presents the reasoning behind +predictions by incorporating experts’ decision-making processes +with rich domain knowledge of the notification target. While +general black-box AI systems can only present final probability +values, the proposed framework can present information relevant +to experts’ decisions, which is expected to be more helpful for +their decision-making. In our case study, we designed a system +for detecting signs of calving in cattle based on the proposed +framework and evaluated the system through a user study (N=6) +with people involved in livestock farming. A comparison with the +black-box AI system revealed that many participants referred to +the presented reasons for the prediction results, and five out +of six participants selected the proposed system as the system +they would like to use in the future. It became clear that we +need to design a user interface that considers the reasons for the +prediction results. +Index Terms-XAI, machine learning, user study, precision +livestock farming +I. INTRODUCTION +In recent years, artificial intelligence (AI) and machine +learning (ML) have been used in areas related to human +decision-making, including the medical field [1]. When devel- +oping AI systems, an end-to-end approach is generally con- +sidered to be an indispensable technique due to its simplicity. +However, it has been pointed out that this approach is black- +box in its internal behavior and is not a sufficient system for +supporting human decision-making [2]. Meanwhile, there has +been growing interest in the interpretability of AI systems +as explainable AI (XAI) [3], [4]. Representative technologies +of XAI include visualization techniques such as CAM [5] +and Grad-CAM [6], and post-hoc explanations such as local +interpretable model agnostic explanations (LIME) [7] and +Shapley additive explanations (SHAP) [8]. However, many +developments in AI and ML tend to suffer from a lack +of usability and practical interpretability for real decision- +makers [3], [4], [9]. +In recent years, the human-computer interaction (HCI) +community has recognized the importance of human-centered +evaluation, which incorporates user evaluation into the inter- +pretability of AI systems. There has been increasing research +on the user evaluation of these explanatory techniques using +experimental data sets [10], [11], [12]. However, when imple- +menting AI systems for highly specialized tasks in industry, the +aforementioned generic explanatory techniques and the find- +ings on general user evaluations using experimental datasets do +not lead to interpretability for experts. Thus, system design and +empirical experiments in highly specialized tasks are essential; +however, there is still a lack of research on XAI frameworks +for industrial use and their validation through user studies [13]. +This study examines the framework of a video surveillance +system that can provide relevant explanations for expert judg- +ments by incorporating their decision-making processes into +the neural networks. The experts watch a video and determine +whether there are any abnormalities after considering vari- +ous attributes based on their domain knowledge. We design +a neural network to incorporate this process into anomaly +detection. Specifically, we extract different features related to +an anomaly based on domain knowledge, and each stream +with extracted features as input determine whether the features +are anomalous. Here, explicitly extracting the features that +experts use to make judgments enables them to be presented +as interpretable information for the experts +The above framework is applied to video-based detection for +calving signs in cattle thorough a user study with livestock +farmers (Fig. 1). Since calf deaths during calving can be +very costly for farmers [14], accurately detecting the signs +of calving is important for livestock management. Although +contact-type sensors, which are attached directly to cattle, +are widely used, a detection system using a camera, which +is a non-contact-type sensor, is valuable in terms of man- +agement efficiency and animal welfare. First, to apply the +above framework, we reviewed the literature on animal studies +and conducted farmer interviews to collect domain knowledge +about calving signs. Using our findings, we designed a system +that explicitly extracts statistical information on the posture, +rotation, and movement of cattle relevant to calving and then +identifies the signs from these features. We designed the +user interface of the notification screen using the information +provided by the feature extractor. In the user interface, the fre- +arXiv:2301.03926v1 [cs.HC] 10 Jan 2023 + +2 +quency of posture, amount of rotation, amount of movement, +and the pattern of movement (bird’s eye view) of cattle are +presented to farmers. Compared with the user interface of the +end-to-end system, which only presents the probability values +of calving signs, the presentation of the internal state of the +system is expected to be more interpretable for farmers. The +user interface of the end-to-end system and the proposed sys- +tem were evaluated by people who were involved in livestock +farming (N=6). We anticipate that our findings will be useful +in developing human-centered explainable AI-based systems +that effectively incorporate experts’ knowledge. +The rest of the paper is organized as follows. In Section II, +we briefly explain the framework which enables us to under- +stand the reasoning behind predictions. Section III describes +the case study on a calving detection system in livestock +farming based on our framework. In Section IV, we discuss the +findings from the case study and future work of the proposed +framework. We conclude in Section V with a brief summary. +II. FRAMEWORK +The proposed framework incorporates the users’ decision- +making process into a prediction network and allows for +interpretation on the basis of their domain knowledge. A +typical machine leaning-based system uses an end-to-end +approach with experts’ annotations to model the notification +target. In contrast, the proposed framework uses a human- +centric approach and models the notification target through +user interviews. This section describes the following four +phases of the design procedure for the proposed framework: +• Interview with an expert +• Designing streams to extract the information that charac- +terizes what is being monitored +• Designing detectors by integrating information from mul- +tiple streams +• Designing the notification interface +In the first phase, we interview experts on the notification +target. In interviews, we ask them to verbalize their decision- +making processes when detecting the target, and determine +the features they focus on in daily operations. Then we divide +those features into more detailed features that can be judged +even by non-experts. +In the next phase, we extract more detailed features from the +ones that experts focus on when they detect the notification +target, and then we develop a stream network based on the +extracted features. Dividing the features into more detailed +features that can be judged even by non-experts, enables the +addition of crowd-sourced annotations. This is important in +terms of making the surveillance system sustainable, as it +allows us to increase the amount of training data without +having to rely on experts in the domain. +In the third phase, we develop a network that integrates +multiple stream networks on the basis of different attribute +features. The users are notified based on the posterior proba- +bility of the network. Here, even if the detection of each stream +network is not necessarily effective, if the predictions of each +stream are complementary, the detection can be improved. +Finally, we design the user interface that is presented to the +users. In addition to the final posterior probability of the no- +tification target, multiple features extracted from each stream +network can be presented as the reasons for the prediction +results. These features are designed with consideration for the +responses in the user interviews, so they provide information +necessary for users’ decision-making processes. +III. CASE STUDY +In this section, we verify the effectiveness of the proposed +framework for detecting signs of calving in cattle through a +user study with livestock farmers. First, we describe the system +design based on our framework introduced in Section II, and +then we describe the procedure for the user experiments and +the results and findings. +A. System Design +In this section, we describe the design of our calving detec- +tion system based on the proposed framework. We describe the +domain knowledge of calving signs and the network structure +that explicitly extracts features relevant to calving. Finally, we +explain the user interface that presents the farmers with the +reasons for the prediction results. +1) Calving Signs Observable from Video: This section +presents an overview of the domain knowledge about calving +signs. We interviewed farmers and reviewed the animal sci- +ence literature to identify cattle behaviors related to calving. +Changes in postures and behaviors related to calving have +been extensively investigated in animal science. This part +corresponds to the implementation of the first phase of the +proposed framework. +The following are typical posture-based calving signs that +can be observed from images: +• Switching between standing and lying postures: About +two to six hours before calving, the number of posture +changes (e.g., switching between standing and lying) +become more frequent [15], [16], [17] and the time spent +lying increases two hours before calving [17]. +• Tail raising: About four to six hours before calving, tail +raising becomes more frequent [17] and the position of +the tail before calving is elevated [18], [19]. +The following action-based calving signs were observed in +the video: +• Increase in the number of rotations and turns: Char- +acteristic walking patterns (e.g., rotations and turns) can +be observed four hours before calving and become more +frequent three hours before calving [20]. +• Increase in aimless walking time: The duration of +walking on the calving day increases [16], [17]. Aimless +walking time apparently increases about 140 minutes +before calving [19]. +2) System Architecture: We designed a multi-stream net- +work [21] that extracts statistical information on posture, +rotation, and movement based on calving signs identified in +prior studies and integrates the three streams depending on +the situation. This part corresponds to the implementation + +3 +Fig. 1. User study of livestock farmers. +of the second and third phases of the proposed framework. +Specifically, each stream identifies calving signs for each 30- +minute input video based on features as follows: +• Posture-based feature: The appearance of a cattle stand- +ing, lying, and raising its tail are captured for each video +frame using ResNet-50 [22] and then accumulated into +the relevant frequencies using temporal pooling tech- +niques. +• Rotation-based feature: Information on body direction +is extracted from each video frame using ResNet-50 and +accumulated into a statistic on the cattle’s rotation by +measuring the changes in the body direction using the +M-measure [23], [24]. +• Movement-based feature: The region of the cattle’s +body is detected in each video frame using YOLOv3 [25] +and differences in locations across frames are accumu- +lated into a statistic on the cattle’s movement. +The calving-relevant features are designed to be extracted from +information that can be judged by non-experts, such as posture, +neck and tail positions, and positional coordinates. This makes +it possible to collect data using crowdsourcing, and re-training +the feature extraction mechanism becomes easier. +3) User Interface of Notification Screen: In this section, +we describe the design of the user interface of the proposed +system. This part corresponds to the implementation of the +fourth phase of the proposed framework. In addition to the +final posterior probability of calving signs, the proposed frame- +work provides the following information related to calving +signs for each frame: +• Posterior probability of cattle’s posture, 1) standing cattle +with tail raised, 2) standing cattle without tail raised, 3) +lying cattle, and 4) can’t tell +• Heatmaps of cattle’s body direction +• Position coordinates of cattle +Displaying these data in an easy-to-understand representation +will help farmers to estimate when cattle start calving. +Fig. 2 shows the user interface designed using the infor- +mation described above. In addition to the monitoring video +and the posterior probability values of the system in this +scene, the interface also displays information on the frequency +of posture, amount of rotation, amount of movement, and +trajectory of the cattle as seen from directly above. The upper +right bar graph shows the posterior probability of calving signs +which the system output using the 30-minute video frames. +Four graphs are presented, each with posterior probabilities +and their mean values based on the statistics on posture, +rotation, and movement. Users can check which information +the system considers to be a calving sign. The statistics on +posture, rotation, and movement are presented at the bottom +of the screen. The circle graph on the left shows the frequency +of these 30-minute postures calculated from the posterior +probability of the posture classification obtained by the feature +extractor. This graph can be interpreted as the reasons for the +prediction results of the posture-based stream. The comparison +with the normal state in the amount of rotation (shown in +the center) is calculated from the estimated heatmaps of the +cattle’s body direction obtained from the feature extractor. It +is possible to quantify how much the cattle turn during these +30 minutes compared with their default state, which can then +be interpreted as reasons for the prediction results. In the +prototype user interface, we visualized the time-series changes +in the angle of body direction. After interviewing a farmer, +a bar graph was used to simplify the representation. The +comparison with the normal state in the amount of movement +(shown on the right side) is calculated from the amount of +change in the positional coordinates of the cattle. Compared +with the normal state, the amount of movement of cattle +during this 30-minute can be quantified and interpreted as +the reasons for the prediction results of the movement-based +stream. Finally, a bird’s eye view of the cattle’s position is +displayed on the right side of the monitoring video, which +shows the trajectory of the cattle’s position as seen from +directly above the room. This allows us to understand the +general movement pattern without continuously watching the +video. +B. User Study +To evaluate the proposed framework, we experimentally +compared it with the end-to-end system were conducted. The +experiment took about 60-90 minutes in total and consisted +of instruction, practice, experiment part 1, part 2, and a post- +experiment survey (Fig. 3). First, the participants watched a + +保存箱No.4 +Characteristic�behaviors�related� +to�posture�and�rotation�were� +observed�before�calving. +Fig. 2. User interface of proposed system (System B). User interface presented to participants was in Japanese. +5-minute instructional video on the experiment. Afterwards, +we obtained written informed consent signed by the par- +ticipants before the experiments, and the participants were +granted a gift card of ¥ 5,000 JPY (roughly $50 USD). As +an exercise, the participants responded to two notifications +from the proposed system and the end-to-end system. In this +section, the participants were instructed to think aloud [26] +and practiced answering questions by actively commenting +on what they saw and thought during the experiment. The +experiment consisted of two parts with a break in between, +and the participants responded to 36 notifications in total. The +participants were divided into two experimental groups, and +each group was shown the user interface in a pseudo-random +order to account for the sequential effects of the interface +order. After the experiment, a qualitative post-experiment +survey was conducted using Google Forms. +In the experiment, the two user interfaces were presented +to the participants in a pseudo-random order on the basis +of the experimental group. The user interface of the end-to- +end system for comparison is shown in Fig. 4. Compared +with the user interface of the proposed system, the end- +to-end system presents only the posterior probability graph +of the prediction result. For the same 30-minute sequence, +the value of the posterior probability presented in the user +interface of the end-to-end system was the same value as +the average posterior probability of each stream in the user +interface of the proposed system. To avoid bias from the +system names, we designated the end-to-end system as system +A and the proposed system as system B when presenting the +user interfaces to the participants. +The unit of the notification was a 30-minute video sequence, +and a total of 18 sequences were prepared. A positive case was +defined as the time from three to zero hours before calving +and a negative case as the time from 24 to 27 hours before +calving. The prepared sequences consisted of six sequences +each of true positives and false positives, and three sequences +each of false negatives and true negatives. In other words, we +assume that the notification threshold is significantly lowered +and contains a certain number of false positives. +The participants are asked to answer the following two +questions (maximum of four) at the bottom of the interface: +Q1-1 Did you recognize calving signs in this 30-minutes +scene? (-3: Absolutely Not, ... 3: Absolutely Yes) +Q1-2 What are the reasons for your answer? (Verbal an- +swer) +The participants responded to the following prompts in Q1-2: +1) What did you see on the screen?, 2) What did you think?, +and 3) How did you reach your decision? The intention of +this format was to reveal as much of their thought process as +possible. After answering the above questions, if the answer to +Q1-1 was “neither” or “predictive” (0-3), the participant was +asked to answer the following questions as well: +Q2-1 What action would you take after seeing this user +interface? (1: Begin assisting immediately, 2: Start +making arrangements, 3: Do nothing) +Q2-2 What are the reasons for your answer? (Verbal an- +swer) +The candidate answers to Q2-1 are terms referring to common +decision-making behaviors for livestock farmers. The terms +are used in the contact-type sensor, Gyuonkei1, a calving +notification sensor widely used in Japan. Here, “Begin assist- +ing immediately” refers to the decision to immediately begin +assisting with calving, and “Start making arrangements” refers +to the preparations made about 24 hours before calving. These +questions are designed to encourage participants to use the +interface for their decision-making. Each participant is asked +1Gyuonkei, http://www.gyuonkei.jp + +2017/06/1406:54:18 +Probability of calvingsigns +Average +59.0 +Posture +86.5 +Rotation +55.1 +Movement +35.3 +0% +20% +40% +60% +80% +100% +Acharacteristic behavior related to +posture androtation was observed +before calving. +What posture does the cattle have? +Howmuchdoesthecattlerotate? +How far does the cattlemove? +Percentage of observed posture frequency +Amount of rotation +Amountof movement +0443052500 +Standing,w/o tail raised +Standing,w/ tail raised Lying +Amountofrotation +Can't tell +Depth of the room +30-minute value +Normal average value +30-minute value +Normal average value06/14 06:32 - 07:00: Calving sign was detected.5 +1.�Instruction +2.�Practice +3.Experiment� +(Part�1) +4.�Experiment +(Part�2) +5.�Post-experiment� +Survey +2 tasks +… +18�tasks +18�tasks +18 tasks +18�tasks +A +… +B +B +B +A +A +B +B +A +A +A +B +… +… +• Consent +• Watch� +instructional� +video�(5�min) +A +B +A +B +E2E�system +Proposed system +Group�Ⅰ +P2,�P3,�P5 +Group�Ⅱ +P1,�P4,�P6 +Participants +Break +Fig. 3. Schematic diagram of experimental procedure. Order of tasks differed for each experimental group. +to respond to the above questions for a total of 36 notifications +(2 UIs x 18 sequences). The questions in the post-experiment +survey are as follows: +Q’1 +Which AI system would you like to use in the future? +(1. System A, 2. System B) +Q’2 +Why did you choose that system? (Orally, if you +prefer.) +Q’3 +How useful was the information presented in System +A? (-2. Useless, ..., 2. Useful) +a +Graph of the probability of calving signs. +Q’4 +How useful was the information presented in System +B? (-2. Useless, ..., 2. Useful) +a +Graph of the probability of calving signs. +b +Graph of the posture frequency. +c +Graph of the amount of rotation. +d +Graph of the amount of movement. +e +Bird’s eye view (position of the cattle as +seen from directly above). +Q’5 +Which +system +provided +more +accurate +predic- +tions?(1. System A, 2. System B, 3. Neither) +Q’6 +What are the advantages of the system you chose in +Q’1 over the other? (Orally, if you prefer.) +Q’7 +What improvements (if any) would you make to +System B? (Orally, if you prefer.) +Six people who involved in livestock farming participated in +the user experiment. Age, sex, and years of experience in the +livestock industry of the participants are shown in Table I. The +participants belonged to two groups: farmers (P1-4) and people +from an agricultural college (P5-6). Here, P5 is a professor, +and P6 is a student of animal science. +The experiment was administered on a computer, and the +participants accessed the URL to the experiment page provided +in advance. We used a web conference application (Zoom) to +record the participants’ responses. To record the participant’s +speech during their think-aloud, we recorded online on Zoom +and locally as a backup. +C. Results and Findings +Participants judged whether a behavior was a calving sign +from the monitoring video, but the behavior of their responses +TABLE I +DETAILS OF EXPERIMENT PARTICIPANTS. FOUR ARE FARMERS (P1-P4), +AND OTHER TWO STUDIES ARE IN AGRICULTURAL COLLEGE (P5-P6). +ID +Group +Age +Sex +Years of experience +P1 +II +30’s +M +10-19 +P2 +I +50’s +M +20-29 +P3 +I +20’s +F +5-9 +P4 +II +40’s +M +20-29 +P5 +I +40’s +M +20-29 +P6 +II +10’s +F +0-1 +varied depending on the type of the user interface presented. +When the interface of the end-to-end system was presented, +most of the verbal comments were related to the video. +However, when the interface of the proposed system was +presented, some of the comments were related to the statistics. +One participant shared their judgment, referring to a graph +comparing the amount of movement with normal conditions. +P5 stated “I think the cattle is near to calving because the +data shows a lot of movement and an increase in rotation.”. +Given the trend of the responses, we believe that presenting the +internal state of the proposed system was effective for judging +whether a behavior was a calving sign. +The results of the responses to the post-experimental survey +are shown in Fig 5. Five out of six participants selected the +proposed system as the system they would like to use in the +future (Fig. 5, Q’1). They found that the internal state of +the system was helpful in identifying calving signs. P1 said +“It is easier to understand when the amount of movement, +which is usually judged by the senses, is visualized in a +graph.” P2 said “It’s helpful to see detailed information about +the cattle.” P3 said “The presented graphs were helpful in +determining whether a behavior was a calving sign.” , and +P5 said “Because the data is presented in detail, I think less +experienced farmers can make decisions with more certainty.” +In addition, some of the comments fit the hypothesis that +the user interface of the end-to-end system, which is a black- +box, is insufficient for decision-making. P1 stated, “The [end- +to-end system’s] predicted probability alone does not tell us + +6 +Fig. 4. User interface of end-to-end system (System A). User interface presented to participants was in Japanese. +what we should do. I don’t know what to do with it.”. In +contrast, P4, who was the only one to choose the end-to-end +system in this question, said, “I did not find the indicators +in the proposed system to be judgmental because there were +many situations where the information presented differed from +my perceptions.” P4 was a farmer who was asked to assist +in the interviews to collect domain knowledge about calving +signs, and he stated that the information presented was less +accurate than he expected. One shortcoming is that a user +evaluation should have been conducted in the second phase of +the framework design. Because we did not receive sufficient +feedback from experts in this phase, the extracted features +were not always sufficiently useful indicators. In addition, it +was suggested that actively presenting the internal state of the +system, including the errors, may cause noise in judgment and +cause participants to distrust the system. +Next, we discuss which information on the user interface +was useful (Fig. 5, Q’3, Q’4). We found that the trend in +the probability of calving signs varied between participants. +Only P4 responded that the probability of the end-to-end +system was more useful, while the other participants found +the proposed system to be more useful or about the same. +We now turn our attention to the results of the user interface +of each statistic on the posture, rotation, and movement of +the proposed system (Fig. 5, Q’4). Two participants from +the agricultural college (P5, P6) responded that the graph of +the posture frequency was useful, while the remaining four +farmers responded that it was generally not useful. This may +be because the farmers could view the raw data to obtain more +accurate information on posture frequency, so there was less +of a need to refer to the information presented by the system. +On the other hand, participants from an agricultural college +with an academic background found it useful to be able to +quantitatively visualize the posture frequency. +A higher percentage of participants found the graph of +the amount of movement to be more useful than the posture +frequency graph because the meta-information, which cannot +be captured from the raw data, is useful for judgment. In fact, +many of the participants said that the comparison with normal +conditions was helpful during their responses. P1 said “Before, +I judged the amount of rotation and movement intuitively, so +it’s easier to see when graphed.” and P6 said “The graph +[of the amount of movement] makes it easier to notice the +degree of change compared with the normal conditions.” The +rotation graph tended to be less useful than the movement +graph because the participants paid attention to the intensity +of the cattle’s movement as one of the criteria for judging +whether it was a calving sign, and the amount of rotation was +not directly related to their judgment. Thus, it is important to +present features in a way that users can easily understand, and +a more abstract expression such as “intensity of movement” +may be more suitable. +In their responses to the last question (Q’7), participants +pointed out issues related to the accuracy of the presented +information and that the user interface made the video smaller +when information is presented. For the former issue, P4 said +“I felt that the graph was not directly linked to the calving +signs.” and P6 said “I was worried when the AI identified an +indeterminable posture in many cases. I would prefer to judge +it myself in such situations.” For the latter issue, P5 stated, +“Readability. I don’t want the video to be too small.” +IV. DISCUSSION +Black-box AI systems are considered inadequate for sup- +porting human decision-making. One farmer pointed out that +the end-to-end system lacked instructions on what they should +do after seeing the value for the probability of calving signs. +Several farmers pointed out the advantages of our proposed +system, stating that the detailed information would help novice +farmers make correct decisions confidently and the reason +visualization was helpful. In particular, statistical information +including meta-information that cannot be obtained from the +raw data tended to be particularly useful. In contrast, we found +that the reliability of the system is adversely affected when +the presented information is incorrect. This has been reported +in several studies as algorithm aversion [27], [28], which is +a phenomenon in which users stop trusting algorithms after +seeing its mistakes. To present more information to users, the + +06/2403:51-04:18:Calvingsignwasdetected +2017/06/2403:56:55 +Probability of calving signs +100 +96.1% +80 +70 +30 +20 +10 +-00:287 +Q’1.�Which�AI�system�would�you�like�to�use�in�the�future? +Q’3.�How�useful�was�the�information� +presented�in�end-to-end�system?� +P6 +P3,�P4 +P1,�P2,�P5 +P5,�P6 +P1,�P3 +P4 +P2 +P4 +P3 +P2 +P1 +P5,�P6 +P4 +P1,�P3 +P2, P5,�P6 +P2 +P1,�P3,�P4,�P5,�P6 +P2 +P1 +P6 +P3,�P4 +P5 +Q’5.�Which�AI�system�provided�more�accurate�predictions? +Q’4.�How�useful�was�the�information� +presented�in�proposed�system?� +Fig. 5. Responses to select questions in post-experiment survey. +proposed framework needs to take into account the accuracy +of the information. In addition, we should have conducted a +user evaluation in the second phase of the framework design +because the extracted features may not have been sufficiently +useful indicators for users. In the participants’ feedback, they +noted that the user interface became more complicated as more +information was presented. When considering development for +actual use, it is necessary to consider elderly users as well +as smartphones which may make the screen more difficult to +view. A more sophisticated user interface design is needed, +such as the separation of the summary and detailed analysis +screens. We anticipate that our evaluation of an XAI frame- +work through user studies will contribute to the integration of +XAI in various industrial domains. +V. CONCLUSION +In this study, we proposed a framework for a video +surveillance system that incorporates experts’ decision-making +processes into the architecture such that the reasons for the +prediction results can be interpreted. We evaluated the calving +sign detection systems based on our proposed framework +through a user study with people involved in livestock farming +(N=6). The proposed framework was compared with an end- +to-end system, and five out of six participants selected the +proposed framework as the system they would like to use in the +future. In addition, the proposed framework is used to present +the internal state of the system, which can be used to help users +make decisions and identify system errors. However, we found +that presenting an inaccurate internal state of the system could +interfere with the user’s judgment and cause them to distrust +the system. In future work, we intend to study the accuracy +of the information presented to users. +ACKNOWLEDGMENT +We would like to thank the farmers who participated in the +experiment and Kagoshima Prefecture Agricultural College. +We also thank Kagoshima Brain Center for fruitful discus- +sions. +REFERENCES +[1] E. Beede, E. Baylor, F. Hersch, A. Iurchenko, L. Wilcox, P. Ruamvi- +boonsuk, and L. M. Vardoulakis, “A human-centered evaluation of a +deep learning system deployed in clinics for the detection of diabetic +retinopathy,” in Proceedings of the 2020 CHI Conference on Human +Factors in Computing Systems (CHI ’20), 2020, pp. 1—-12. +[2] A. Abdul, J. Vermeulen, D. Wang, B. Y. Lim, and M. Kankanhalli, +“Trends and trajectories for explainable, accountable and intelligible +systems: An hci research agenda,” in Proceedings of the 2018 CHI +Conference on Human Factors in Computing Systems (CHI’18), 2018, +pp. 1–18. +[3] F. Doshi-Velez and B. 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Peddinti, “Mean temporal distance: +Predicting ASR error from temporal properties of speech signal,” in +International Conference on Acoustics, Speech and Signal Processing +(ICASSP 2013), 2013, pp. 7423–7426. +[24] T. Ogawa, S. Mallidi, E. Dupoux, J. Cohen, N. Feldman, and H. Her- +mansky, “A new efficient measure for accuracy prediction and its appli- +cation to multistream-based unsupervised adaptation,” in International +Conference on Pattern Recognition (ICPR 2017), 2017, pp. 2222–2227. +[25] J. Redmon and A. Farhadi, “Yolov3: An incremental improvement,” +CoRR, vol. abs/1804.02767, 2018. [Online]. Available: http://arxiv.org/ +abs/1804.02767 +[26] M. Jaspers, T. Steen, C. Bos, and M. Geenen, “The think aloud method: +A guide to user interface design,” International journal of medical +informatics, vol. 73, pp. 781–95, 12 2004. +[27] B. Dietvorst, J. Simmons, and C. Massey, “Algorithm aversion: People +erroneously avoid algorithms after seeing them err,” Journal of experi- +mental psychology. General, vol. 144, 11 2014. +[28] M. Dzindolet, L. Pierce, H. Beck, and L. Dawe, “The perceived utility +of human and automated aids in a visual detection task,” Human factors, +vol. 44, pp. 79–94, 2 2002. + diff --git a/s9E2T4oBgHgl3EQffgcE/content/tmp_files/load_file.txt b/s9E2T4oBgHgl3EQffgcE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..639ff8642c0c6f08edd2a309292f2710c7a7f80a --- /dev/null +++ b/s9E2T4oBgHgl3EQffgcE/content/tmp_files/load_file.txt @@ -0,0 +1,512 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf,len=511 +page_content='1 Video Surveillance System Incorporating Expert Decision-making Process: A Case Study on Detecting Calving Signs in Cattle Ryosuke Hyodo∗, Susumu Saito∗†, Teppei Nakano∗†, Makoto Akabane∗, Ryoichi Kasuga‡, Tetsuji Ogawa∗ ∗ Waseda University, Tokyo, Japan † Intelligent Framework Lab, Tokyo, Japan ‡ Farmers Support, Kagoshima, Japan Abstract—Through a user study in the field of livestock farming, we verify the effectiveness of an XAI framework for video surveillance systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' The systems can be made inter- pretable by incorporating experts’ decision-making processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' AI systems are becoming increasingly common in real-world applications, especially in fields related to human decision- making, and its interpretability is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' However, there are still relatively few standard methods for assessing and addressing the interpretability of machine learning-based systems in real- world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' In this study, we examine the framework of a video surveillance AI system that presents the reasoning behind predictions by incorporating experts’ decision-making processes with rich domain knowledge of the notification target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' While general black-box AI systems can only present final probability values, the proposed framework can present information relevant to experts’ decisions, which is expected to be more helpful for their decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' In our case study, we designed a system for detecting signs of calving in cattle based on the proposed framework and evaluated the system through a user study (N=6) with people involved in livestock farming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' A comparison with the black-box AI system revealed that many participants referred to the presented reasons for the prediction results, and five out of six participants selected the proposed system as the system they would like to use in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' It became clear that we need to design a user interface that considers the reasons for the prediction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Index Terms-XAI, machine learning, user study, precision livestock farming I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' INTRODUCTION In recent years, artificial intelligence (AI) and machine learning (ML) have been used in areas related to human decision-making, including the medical field [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' When devel- oping AI systems, an end-to-end approach is generally con- sidered to be an indispensable technique due to its simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' However, it has been pointed out that this approach is black- box in its internal behavior and is not a sufficient system for supporting human decision-making [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Meanwhile, there has been growing interest in the interpretability of AI systems as explainable AI (XAI) [3], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Representative technologies of XAI include visualization techniques such as CAM [5] and Grad-CAM [6], and post-hoc explanations such as local interpretable model agnostic explanations (LIME) [7] and Shapley additive explanations (SHAP) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' However, many developments in AI and ML tend to suffer from a lack of usability and practical interpretability for real decision- makers [3], [4], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' In recent years, the human-computer interaction (HCI) community has recognized the importance of human-centered evaluation, which incorporates user evaluation into the inter- pretability of AI systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' There has been increasing research on the user evaluation of these explanatory techniques using experimental data sets [10], [11], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' However, when imple- menting AI systems for highly specialized tasks in industry, the aforementioned generic explanatory techniques and the find- ings on general user evaluations using experimental datasets do not lead to interpretability for experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Thus, system design and empirical experiments in highly specialized tasks are essential;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' however, there is still a lack of research on XAI frameworks for industrial use and their validation through user studies [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' This study examines the framework of a video surveillance system that can provide relevant explanations for expert judg- ments by incorporating their decision-making processes into the neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' The experts watch a video and determine whether there are any abnormalities after considering vari- ous attributes based on their domain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' We design a neural network to incorporate this process into anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Specifically, we extract different features related to an anomaly based on domain knowledge, and each stream with extracted features as input determine whether the features are anomalous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Here, explicitly extracting the features that experts use to make judgments enables them to be presented as interpretable information for the experts The above framework is applied to video-based detection for calving signs in cattle thorough a user study with livestock farmers (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Since calf deaths during calving can be very costly for farmers [14], accurately detecting the signs of calving is important for livestock management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Although contact-type sensors, which are attached directly to cattle, are widely used, a detection system using a camera, which is a non-contact-type sensor, is valuable in terms of man- agement efficiency and animal welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' First, to apply the above framework, we reviewed the literature on animal studies and conducted farmer interviews to collect domain knowledge about calving signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Using our findings, we designed a system that explicitly extracts statistical information on the posture, rotation, and movement of cattle relevant to calving and then identifies the signs from these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' We designed the user interface of the notification screen using the information provided by the feature extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' In the user interface, the fre- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='03926v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='HC] 10 Jan 2023 2 quency of posture, amount of rotation, amount of movement, and the pattern of movement (bird’s eye view) of cattle are presented to farmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Compared with the user interface of the end-to-end system, which only presents the probability values of calving signs, the presentation of the internal state of the system is expected to be more interpretable for farmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' The user interface of the end-to-end system and the proposed sys- tem were evaluated by people who were involved in livestock farming (N=6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' We anticipate that our findings will be useful in developing human-centered explainable AI-based systems that effectively incorporate experts’ knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' In Section II, we briefly explain the framework which enables us to under- stand the reasoning behind predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Section III describes the case study on a calving detection system in livestock farming based on our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' In Section IV, we discuss the findings from the case study and future work of the proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' We conclude in Section V with a brief summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' FRAMEWORK The proposed framework incorporates the users’ decision- making process into a prediction network and allows for interpretation on the basis of their domain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' A typical machine leaning-based system uses an end-to-end approach with experts’ annotations to model the notification target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' In contrast, the proposed framework uses a human- centric approach and models the notification target through user interviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' This section describes the following four phases of the design procedure for the proposed framework: Interview with an expert Designing streams to extract the information that charac- terizes what is being monitored Designing detectors by integrating information from mul- tiple streams Designing the notification interface In the first phase, we interview experts on the notification target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' In interviews, we ask them to verbalize their decision- making processes when detecting the target, and determine the features they focus on in daily operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Then we divide those features into more detailed features that can be judged even by non-experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' In the next phase, we extract more detailed features from the ones that experts focus on when they detect the notification target, and then we develop a stream network based on the extracted features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Dividing the features into more detailed features that can be judged even by non-experts, enables the addition of crowd-sourced annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' This is important in terms of making the surveillance system sustainable, as it allows us to increase the amount of training data without having to rely on experts in the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' In the third phase, we develop a network that integrates multiple stream networks on the basis of different attribute features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' The users are notified based on the posterior proba- bility of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Here, even if the detection of each stream network is not necessarily effective, if the predictions of each stream are complementary, the detection can be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Finally, we design the user interface that is presented to the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' In addition to the final posterior probability of the no- tification target, multiple features extracted from each stream network can be presented as the reasons for the prediction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' These features are designed with consideration for the responses in the user interviews, so they provide information necessary for users’ decision-making processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' CASE STUDY In this section, we verify the effectiveness of the proposed framework for detecting signs of calving in cattle through a user study with livestock farmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' First, we describe the system design based on our framework introduced in Section II, and then we describe the procedure for the user experiments and the results and findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' System Design In this section, we describe the design of our calving detec- tion system based on the proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' We describe the domain knowledge of calving signs and the network structure that explicitly extracts features relevant to calving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Finally, we explain the user interface that presents the farmers with the reasons for the prediction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' 1) Calving Signs Observable from Video: This section presents an overview of the domain knowledge about calving signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' We interviewed farmers and reviewed the animal sci- ence literature to identify cattle behaviors related to calving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Changes in postures and behaviors related to calving have been extensively investigated in animal science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' This part corresponds to the implementation of the first phase of the proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' The following are typical posture-based calving signs that can be observed from images: Switching between standing and lying postures: About two to six hours before calving, the number of posture changes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=', switching between standing and lying) become more frequent [15], [16], [17] and the time spent lying increases two hours before calving [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Tail raising: About four to six hours before calving, tail raising becomes more frequent [17] and the position of the tail before calving is elevated [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' The following action-based calving signs were observed in the video: Increase in the number of rotations and turns: Char- acteristic walking patterns (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=', rotations and turns) can be observed four hours before calving and become more frequent three hours before calving [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Increase in aimless walking time: The duration of walking on the calving day increases [16], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Aimless walking time apparently increases about 140 minutes before calving [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' 2) System Architecture: We designed a multi-stream net- work [21] that extracts statistical information on posture, rotation, and movement based on calving signs identified in prior studies and integrates the three streams depending on the situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' This part corresponds to the implementation 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' User study of livestock farmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' of the second and third phases of the proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Specifically, each stream identifies calving signs for each 30- minute input video based on features as follows: Posture-based feature: The appearance of a cattle stand- ing, lying, and raising its tail are captured for each video frame using ResNet-50 [22] and then accumulated into the relevant frequencies using temporal pooling tech- niques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Rotation-based feature: Information on body direction is extracted from each video frame using ResNet-50 and accumulated into a statistic on the cattle’s rotation by measuring the changes in the body direction using the M-measure [23], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Movement-based feature: The region of the cattle’s body is detected in each video frame using YOLOv3 [25] and differences in locations across frames are accumu- lated into a statistic on the cattle’s movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' The calving-relevant features are designed to be extracted from information that can be judged by non-experts, such as posture, neck and tail positions, and positional coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' This makes it possible to collect data using crowdsourcing, and re-training the feature extraction mechanism becomes easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' 3) User Interface of Notification Screen: In this section, we describe the design of the user interface of the proposed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' This part corresponds to the implementation of the fourth phase of the proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' In addition to the final posterior probability of calving signs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' the proposed frame- work provides the following information related to calving signs for each frame: Posterior probability of cattle’s posture,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' 1) standing cattle with tail raised,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' 2) standing cattle without tail raised,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' 3) lying cattle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' and 4) can’t tell Heatmaps of cattle’s body direction Position coordinates of cattle Displaying these data in an easy-to-understand representation will help farmers to estimate when cattle start calving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' 2 shows the user interface designed using the infor- mation described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' In addition to the monitoring video and the posterior probability values of the system in this scene, the interface also displays information on the frequency of posture, amount of rotation, amount of movement, and trajectory of the cattle as seen from directly above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' The upper right bar graph shows the posterior probability of calving signs which the system output using the 30-minute video frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Four graphs are presented, each with posterior probabilities and their mean values based on the statistics on posture, rotation, and movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Users can check which information the system considers to be a calving sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' The statistics on posture, rotation, and movement are presented at the bottom of the screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' The circle graph on the left shows the frequency of these 30-minute postures calculated from the posterior probability of the posture classification obtained by the feature extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' This graph can be interpreted as the reasons for the prediction results of the posture-based stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' The comparison with the normal state in the amount of rotation (shown in the center) is calculated from the estimated heatmaps of the cattle’s body direction obtained from the feature extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' It is possible to quantify how much the cattle turn during these 30 minutes compared with their default state, which can then be interpreted as reasons for the prediction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' In the prototype user interface, we visualized the time-series changes in the angle of body direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' After interviewing a farmer, a bar graph was used to simplify the representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' The comparison with the normal state in the amount of movement (shown on the right side) is calculated from the amount of change in the positional coordinates of the cattle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Compared with the normal state, the amount of movement of cattle during this 30-minute can be quantified and interpreted as the reasons for the prediction results of the movement-based stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Finally, a bird’s eye view of the cattle’s position is displayed on the right side of the monitoring video, which shows the trajectory of the cattle’s position as seen from directly above the room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' This allows us to understand the general movement pattern without continuously watching the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' User Study To evaluate the proposed framework, we experimentally compared it with the end-to-end system were conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' The experiment took about 60-90 minutes in total and consisted of instruction, practice, experiment part 1, part 2, and a post- experiment survey (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' First, the participants watched a 保存箱No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='4 Characteristic�behaviors�related� to�posture�and�rotation�were� observed�before�calving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' User interface of proposed system (System B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' User interface presented to participants was in Japanese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' 5-minute instructional video on the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Afterwards, we obtained written informed consent signed by the par- ticipants before the experiments, and the participants were granted a gift card of ¥ 5,000 JPY (roughly $50 USD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' As an exercise, the participants responded to two notifications from the proposed system and the end-to-end system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' In this section, the participants were instructed to think aloud [26] and practiced answering questions by actively commenting on what they saw and thought during the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' The experiment consisted of two parts with a break in between, and the participants responded to 36 notifications in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' The participants were divided into two experimental groups, and each group was shown the user interface in a pseudo-random order to account for the sequential effects of the interface order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' After the experiment, a qualitative post-experiment survey was conducted using Google Forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' In the experiment, the two user interfaces were presented to the participants in a pseudo-random order on the basis of the experimental group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' The user interface of the end-to- end system for comparison is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Compared with the user interface of the proposed system, the end- to-end system presents only the posterior probability graph of the prediction result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' For the same 30-minute sequence, the value of the posterior probability presented in the user interface of the end-to-end system was the same value as the average posterior probability of each stream in the user interface of the proposed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' To avoid bias from the system names, we designated the end-to-end system as system A and the proposed system as system B when presenting the user interfaces to the participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' The unit of the notification was a 30-minute video sequence, and a total of 18 sequences were prepared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' A positive case was defined as the time from three to zero hours before calving and a negative case as the time from 24 to 27 hours before calving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' The prepared sequences consisted of six sequences each of true positives and false positives, and three sequences each of false negatives and true negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' In other words, we assume that the notification threshold is significantly lowered and contains a certain number of false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' The participants are asked to answer the following two questions (maximum of four) at the bottom of the interface: Q1-1 Did you recognize calving signs in this 30-minutes scene?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' (-3: Absolutely Not, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' 3: Absolutely Yes) Q1-2 What are the reasons for your answer?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' (Verbal an- swer) The participants responded to the following prompts in Q1-2: 1) What did you see on the screen?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=', 2) What did you think?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=', and 3) How did you reach your decision?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' The intention of this format was to reveal as much of their thought process as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' After answering the above questions, if the answer to Q1-1 was “neither” or “predictive” (0-3), the participant was asked to answer the following questions as well: Q2-1 What action would you take after seeing this user interface?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' (1: Begin assisting immediately, 2: Start making arrangements, 3: Do nothing) Q2-2 What are the reasons for your answer?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' (Verbal an- swer) The candidate answers to Q2-1 are terms referring to common decision-making behaviors for livestock farmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' The terms are used in the contact-type sensor, Gyuonkei1, a calving notification sensor widely used in Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Here, “Begin assist- ing immediately” refers to the decision to immediately begin assisting with calving, and “Start making arrangements” refers to the preparations made about 24 hours before calving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' These questions are designed to encourage participants to use the interface for their decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Each participant is asked 1Gyuonkei, http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='gyuonkei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='jp 2017/06/1406:54:18 Probability of calvingsigns Average 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='0 Posture 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='5 Rotation 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='1 Movement 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='3 0% 20% 40% 60% 80% 100% Acharacteristic behavior related to posture androtation was observed before calving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' What posture does the cattle have?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Howmuchdoesthecattlerotate?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' How far does the cattlemove?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=" Percentage of observed posture frequency Amount of rotation Amountof movement 0443052500 Standing,w/o tail raised Standing,w/ tail raised Lying Amountofrotation Can't tell Depth of the room 30-minute value Normal average value 30-minute value Normal average value06/14 06:32 - 07:00: Calving sign was detected." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='�Instruction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='�Practice 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='Experiment� (Part�1) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='�Experiment (Part�2) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='�Post-experiment� Survey 2 tasks … 18�tasks 18�tasks 18 tasks 18�tasks A … B B B A A B B A A A B … … Consent Watch� instructional� video�(5�min) A B A B E2E�system Proposed system Group�Ⅰ P2,�P3,�P5 Group�Ⅱ P1,�P4,�P6 Participants Break Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Schematic diagram of experimental procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Order of tasks differed for each experimental group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' to respond to the above questions for a total of 36 notifications (2 UIs x 18 sequences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' The questions in the post-experiment survey are as follows: Q’1 Which AI system would you like to use in the future?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' System A, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' System B) Q’2 Why did you choose that system?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' (Orally, if you prefer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=') Q’3 How useful was the information presented in System A?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' (-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Useless, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=', 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Useful) a Graph of the probability of calving signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Q’4 How useful was the information presented in System B?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' (-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Useless, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=', 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Useful) a Graph of the probability of calving signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' b Graph of the posture frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' c Graph of the amount of rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' d Graph of the amount of movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' e Bird’s eye view (position of the cattle as seen from directly above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Q’5 Which system provided more accurate predic- tions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' System A, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' System B, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Neither) Q’6 What are the advantages of the system you chose in Q’1 over the other?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' (Orally, if you prefer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=') Q’7 What improvements (if any) would you make to System B?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' (Orally, if you prefer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=') Six people who involved in livestock farming participated in the user experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Age, sex, and years of experience in the livestock industry of the participants are shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' The participants belonged to two groups: farmers (P1-4) and people from an agricultural college (P5-6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Here, P5 is a professor, and P6 is a student of animal science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' The experiment was administered on a computer, and the participants accessed the URL to the experiment page provided in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' We used a web conference application (Zoom) to record the participants’ responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' To record the participant’s speech during their think-aloud, we recorded online on Zoom and locally as a backup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Results and Findings Participants judged whether a behavior was a calving sign from the monitoring video, but the behavior of their responses TABLE I DETAILS OF EXPERIMENT PARTICIPANTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' FOUR ARE FARMERS (P1-P4), AND OTHER TWO STUDIES ARE IN AGRICULTURAL COLLEGE (P5-P6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' ID Group Age Sex Years of experience P1 II 30’s M 10-19 P2 I 50’s M 20-29 P3 I 20’s F 5-9 P4 II 40’s M 20-29 P5 I 40’s M 20-29 P6 II 10’s F 0-1 varied depending on the type of the user interface presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' When the interface of the end-to-end system was presented, most of the verbal comments were related to the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' However, when the interface of the proposed system was presented, some of the comments were related to the statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' One participant shared their judgment, referring to a graph comparing the amount of movement with normal conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' P5 stated “I think the cattle is near to calving because the data shows a lot of movement and an increase in rotation.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Given the trend of the responses, we believe that presenting the internal state of the proposed system was effective for judging whether a behavior was a calving sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' The results of the responses to the post-experimental survey are shown in Fig 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Five out of six participants selected the proposed system as the system they would like to use in the future (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' 5, Q’1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' They found that the internal state of the system was helpful in identifying calving signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' P1 said “It is easier to understand when the amount of movement,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' which is usually judged by the senses,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' is visualized in a graph.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' P2 said “It’s helpful to see detailed information about the cattle.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' P3 said “The presented graphs were helpful in determining whether a behavior was a calving sign.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' and P5 said “Because the data is presented in detail,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' I think less experienced farmers can make decisions with more certainty.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' In addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' some of the comments fit the hypothesis that the user interface of the end-to-end system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' which is a black- box,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' is insufficient for decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' P1 stated, “The [end- to-end system’s] predicted probability alone does not tell us 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' User interface of end-to-end system (System A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' User interface presented to participants was in Japanese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' what we should do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' I don’t know what to do with it.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' In contrast, P4, who was the only one to choose the end-to-end system in this question, said, “I did not find the indicators in the proposed system to be judgmental because there were many situations where the information presented differed from my perceptions.” P4 was a farmer who was asked to assist in the interviews to collect domain knowledge about calving signs, and he stated that the information presented was less accurate than he expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' One shortcoming is that a user evaluation should have been conducted in the second phase of the framework design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Because we did not receive sufficient feedback from experts in this phase, the extracted features were not always sufficiently useful indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' In addition, it was suggested that actively presenting the internal state of the system, including the errors, may cause noise in judgment and cause participants to distrust the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Next, we discuss which information on the user interface was useful (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' 5, Q’3, Q’4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' We found that the trend in the probability of calving signs varied between participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Only P4 responded that the probability of the end-to-end system was more useful, while the other participants found the proposed system to be more useful or about the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' We now turn our attention to the results of the user interface of each statistic on the posture, rotation, and movement of the proposed system (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' 5, Q’4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Two participants from the agricultural college (P5, P6) responded that the graph of the posture frequency was useful, while the remaining four farmers responded that it was generally not useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' This may be because the farmers could view the raw data to obtain more accurate information on posture frequency, so there was less of a need to refer to the information presented by the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' On the other hand, participants from an agricultural college with an academic background found it useful to be able to quantitatively visualize the posture frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' A higher percentage of participants found the graph of the amount of movement to be more useful than the posture frequency graph because the meta-information, which cannot be captured from the raw data, is useful for judgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' In fact, many of the participants said that the comparison with normal conditions was helpful during their responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' P1 said “Before,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' I judged the amount of rotation and movement intuitively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' so it’s easier to see when graphed.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' and P6 said “The graph [of the amount of movement] makes it easier to notice the degree of change compared with the normal conditions.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' The rotation graph tended to be less useful than the movement graph because the participants paid attention to the intensity of the cattle’s movement as one of the criteria for judging whether it was a calving sign,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' and the amount of rotation was not directly related to their judgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Thus, it is important to present features in a way that users can easily understand, and a more abstract expression such as “intensity of movement” may be more suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' In their responses to the last question (Q’7), participants pointed out issues related to the accuracy of the presented information and that the user interface made the video smaller when information is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' For the former issue, P4 said “I felt that the graph was not directly linked to the calving signs.” and P6 said “I was worried when the AI identified an indeterminable posture in many cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' I would prefer to judge it myself in such situations.” For the latter issue, P5 stated, “Readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' I don’t want the video to be too small.” IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' DISCUSSION Black-box AI systems are considered inadequate for sup- porting human decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' One farmer pointed out that the end-to-end system lacked instructions on what they should do after seeing the value for the probability of calving signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Several farmers pointed out the advantages of our proposed system, stating that the detailed information would help novice farmers make correct decisions confidently and the reason visualization was helpful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' In particular, statistical information including meta-information that cannot be obtained from the raw data tended to be particularly useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' In contrast, we found that the reliability of the system is adversely affected when the presented information is incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' This has been reported in several studies as algorithm aversion [27], [28], which is a phenomenon in which users stop trusting algorithms after seeing its mistakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' To present more information to users, the 06/2403:51-04:18:Calvingsignwasdetected 2017/06/2403:56:55 Probability of calving signs 100 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='1% 80 70 30 20 10 00:287 Q’1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='�Which�AI�system�would�you�like�to�use�in�the�future?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Q’3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='�How�useful�was�the�information� presented�in�end-to-end�system?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='� P6 P3,�P4 P1,�P2,�P5 P5,�P6 P1,�P3 P4 P2 P4 P3 P2 P1 P5,�P6 P4 P1,�P3 P2, P5,�P6 P2 P1,�P3,�P4,�P5,�P6 P2 P1 P6 P3,�P4 P5 Q’5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='�Which�AI�system�provided�more�accurate�predictions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Q’4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='�How�useful�was�the�information� presented�in�proposed�system?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='� Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Responses to select questions in post-experiment survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' proposed framework needs to take into account the accuracy of the information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' In addition, we should have conducted a user evaluation in the second phase of the framework design because the extracted features may not have been sufficiently useful indicators for users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' In the participants’ feedback, they noted that the user interface became more complicated as more information was presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' When considering development for actual use, it is necessary to consider elderly users as well as smartphones which may make the screen more difficult to view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' A more sophisticated user interface design is needed, such as the separation of the summary and detailed analysis screens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' We anticipate that our evaluation of an XAI frame- work through user studies will contribute to the integration of XAI in various industrial domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' CONCLUSION In this study, we proposed a framework for a video surveillance system that incorporates experts’ decision-making processes into the architecture such that the reasons for the prediction results can be interpreted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' We evaluated the calving sign detection systems based on our proposed framework through a user study with people involved in livestock farming (N=6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' The proposed framework was compared with an end- to-end system, and five out of six participants selected the proposed framework as the system they would like to use in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' In addition, the proposed framework is used to present the internal state of the system, which can be used to help users make decisions and identify system errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' However, we found that presenting an inaccurate internal state of the system could interfere with the user’s judgment and cause them to distrust the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' In future work, we intend to study the accuracy of the information presented to users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' ACKNOWLEDGMENT We would like to thank the farmers who participated in the experiment and Kagoshima Prefecture Agricultural College.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' We also thank Kagoshima Brain Center for fruitful discus- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' REFERENCES [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Beede, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Baylor, F.' metadata={'source': 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Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=', 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' 4765–4774.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' [9] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Miller, “Explanation in artificial intelligence: Insights from the social sciences,” 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=" Useless SomewhatUseless Neither SomewhatUseful ■Useful Posterior Probability 0 2 4 9Useless Somewhat Useless Neither Somewhat Useful ■ Useful Posterior probability Percentage of posture frequency Amount of rotation Amount of movement Bird's-eye view 0 2 6E2E system 16." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='7% 5 Proposed system 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='3%E2E system 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='7% Neither 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='3% 2 3 Proposed system 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content='0%8 [10] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E2T4oBgHgl3EQffgcE/content/2301.03926v1.pdf'} +page_content=' Kaur, H.' 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0000000000000000000000000000000000000000..668446c19a1c1861fa5404271023cef66c4944d5 --- /dev/null +++ b/u9E2T4oBgHgl3EQfgQcG/content/tmp_files/2301.03934v1.pdf.txt @@ -0,0 +1,1876 @@ +Dissipation losses limiting first-order phase transition materials in +cryogenic caloric cooling: A case study on all-d-metal Ni(-Co)-Mn-Ti +Heusler alloys +Benedikt Beckmanna,∗, David Kochb, Lukas Pfeuffera, Tino Gottschallc, Andreas Taubela, +Esmaeil Adabifiroozjaeid, Olga N. Miroshkinae, Stefan Riegga, Timo Niehoffc,f, Nagaarjhuna +A. Kania,d, Markus E. Grunere, Leopoldo Molina-Lunad, Konstantin P. Skokova and +Oliver Gutfleischa +aFunctional Materials, Institute of Materials Science, Technical University of Darmstadt, Darmstadt 64287, Germany +bStructure Research, Institute of Materials Science, Technical University of Darmstadt, Darmstadt 64287, Germany +cDresden High Magnetic Field Laboratory (HLD-EMFL), Helmholtz-Zentrum Dresden-Rossendorf (HZDR), Dresden 01328, Germany +dAdvanced Electron Microscopy (AEM), Institute of Materials Science, Technical University of Darmstadt, Darmstadt 64287, Germany +eFaculty of Physics and Center for Nanointegration Duisburg-Essen (CENIDE), University of Duisburg-Essen, Duisburg 47057, Germany +fInstitute of Solid State and Materials Physics, Technische Universität Dresden, Dresden 01069, Germany +A R T I C L E I N F O +Keywords: +Heusler alloys +Magnetostructural transformation +Martensitic transformation +Solid-state caloric cooling +Hydrogen +A B S T R A C T +Ni-Mn-based Heusler alloys, in particular all-d-metal Ni(-Co)-Mn-Ti, are highly promising materials +for energy-efficient solid-state refrigeration as large multicaloric effects can be achieved across their +magnetostructural martensitic transformation. However, no comprehensive study on the crucially +important transition entropy change Δ푠푡 exists so far for Ni(-Co)-Mn-Ti. Here, we present a systematic +study analyzing the composition and temperature dependence of Δ푠푡. Our results reveal a substantial +structural entropy change contribution of approximately 65 J(kgK)-1, which is compensated at lower +temperatures by an increasingly negative entropy change associated with the magnetic subsystem. +This leads to compensation temperatures 푇푐표푚푝 of 75 K and 300 K in Ni35Co15Mn50-yTiy and +Ni33Co17Mn50-yTiy, respectively, below which the martensitic transformations are arrested. In addi- +tion, we simultaneously measured the responses of the magnetic, structural and electronic subsystems +to the temperature- and field-induced martensitic transformation near 푇푐표푚푝, showing an abnormal +increase of hysteresis and consequently dissipation energy at cryogenic temperatures. Simultaneous +measurements of magnetization and adiabatic temperature change Δ푇푎푑 in pulsed magnetic fields +reveal a change in sign of Δ푇푎푑 and a substantial positive and irreversible Δ푇푎푑 up to 15 K at 15 K +as a consequence of increased dissipation losses and decreased heat capacity. Most importantly, this +phenomenon is universal, it applies to any first-order material with non-negligible hysteresis and any +stimulus, effectively limiting the utilization of their caloric effects for gas liquefaction at cryogenic +temperatures. +1. Introduction +In an age when anthropogenic climate change and deple- +tion of natural energy resources are two of the most urgent +world-wide societal and existential challenges, the search for +energy-efficient and environmentally friendly technologies +is of outermost importance. This development is strongly +intertwined with globally rising population and standard of +living [1], leading to an ever growing demand for cool- +ing [2, 3], which is predicted to exceed the energy con- +sumption for heating in this century [4]. Currently, most +room-temperature cooling devices are based on the 120- +year old vapor-compression refrigeration cycle, which lacks +in thermodynamic efficiency and has a profound negative +environmental impact [5]. In addition, the compression- +based cooling process used for the liquefaction of hydrogen, +which is critical as an alternative energy carrier for the +transition towards renewable energies [6, 7], is considered +to be inefficient as well [8, 9]. Therefore, future cooling +∗Corresponding author +benedikt.beckmann@tu-darmstadt.de ( Beckmann) +ORCID(s): 0000-0002-2232-1804 ( Beckmann) +technologies for room as well as cryogenic temperature +applications need to provide energy-efficient and environ- +mentally friendly alternatives. +As such an alternative, solid-state caloric cooling is +based on the response of caloric materials to the application +of external stimuli, such as electric or magnetic fields [10], +hydrostatic pressure [11, 12] and uniaxial stress [13]. Mul- +ticaloric materials are susceptible to more than one stimulus +[14, 15], allowing the design of advanced cooling cycles, +e.g. by subsequently utilizing magnetic field and uniax- +ial stress [16]. Depending on the selected thermodynamic +boundary conditions, the stimulus leads to an isothermal +entropy change Δ푠푇 or adiabatic temperature change Δ푇푎푑 +of the material. Conventional caloric materials respond with +Δ푠푇 < 0 and Δ푇푎푑 > 0 to the application of a given external +stimulus, whereas inverse caloric materials show Δ푠푇 > 0 +and Δ푇푎푑 < 0 [14]. Of all ferroic cooling technologies, +magnetocalorics is considered to be the best studied [17, 18] +and shows an improved thermodynamic efficiency compared +to the widely used compression-based technology [19, 20]. +B. Beckmann, D. Koch, L. Pfeuffer, et al.: Preprint submitted to Elsevier +Page 1 of 14 + +aHowever, the search for promising magnetocaloric materials +is still ongoing. +Since the discovery of the giant magnetocaloric effect +in Gd5Si2Ge2 [21], various material systems showing en- +hanced magnetocaloric properties due to first-order phase +transitions have been discovered [22]. The most promi- +nent material systems are La(Fe,Si)13 [23, 24], Fe2P-type +compounds [25–27], Ni-Mn-based Heusler alloys [10, 28– +31] and in some extend Fe-Rh [32–34]. In Ni-Mn-based +Heusler alloys, giant caloric effects are coupled to first-order +magnetostructural martensitic transformations between low- +temperature martensite and high-temperature austenite. In +this work, multicaloric all-d-metal Ni(-Co)-Mn-Ti Heusler +alloys are investigated as these alloys have recently attracted +particular interest in the fields of magneto- [35–38], elasto- +[39–42], baro- [43, 44] and multicalorics [45] due to an im- +proved mechanical stability [39, 40], large volume changes +[46] as well as good tuneability of the martensitic transfor- +mation temperature 푇푡 and austenite Curie temperature 푇 퐴 +퐶 +[38, 46, 47]. +In general, knowledge about composition and tempera- +ture dependencies of the transition entropy change Δ푠푡 in +any given caloric material is crucial to develop samples +with tailored properties for solid-state cooling applications. +Thereby, independent on the selected external stimulus to +drive the phase transition, the transition entropy change is of +universal importance and is expressed as +Δ푠푡 = Δ푠푙푎푡 + Δ푠푚푎푔 + Δ푠푒푙 +(1) +with Δ푠푙푎푡, Δ푠푚푎푔 and Δ푠푒푙 being the entropy change con- +tributions associated with the structural, magnetic and elec- +tronic subsystem, respectively. Kihara et al. showed that the +entropy change of martensitic transformations in Heusler +alloys is dominated by the structural subsystem, whereas the +contribution of the electronic subsystem is negligibly small +[48], i.e. |Δ푠푙푎푡| > |Δ푠푚푎푔| >> |Δ푠푒푙|. In inverse magne- +tocaloric Heusler alloys, such as Ni(-Co)-Mn-Ti, the domi- +nating Δ푠푙푎푡 is positive and Δ푠푚푎푔 is increasingly negative +towards lower temperatures. The opposing positive lattice +and negative magnetic entropy change contributions give +rise to the "dilemma of inverse magnetocaloric materials" +[49] and are compensated at the compensation temperature +푇푐표푚푝, below which the martensitic transformation is ther- +mally arrested due to the absence of a driving force. In the +literature, this thermodynamic effect is often called kinetic +arrest and is described in multiple classical Ni(-Co)-Mn-X +Heusler alloys with X being a main group element, namely +Sn [50], Sb [51], Al [52] and In [49, 53–55]. However, the +arrest phenomenon is not only limited to Heusler alloys but +also appears in other inverse magnetocaloric materials, such +as Fe-Rh compounds [34]. +In this study, we disentangle the contributions to the tran- +sition entropy change of the magnetostructural martensitic +transformation in multicaloric all-d-metal Ni(-Co)-Mn-Ti +Heusler alloys, since there is so far no comprehensive study +on Δ푠푡 in this material system, even though the transition en- +tropy change is a fundamental material property governing +all caloric effects. On this basis, we analyze the responses +of the magnetic, structural and electronic subsystems to the +temperature- and field-induced martensitic transformation, +showing an abnormally increased magnetic hysteresis width +at temperatures near and below 푇푐표푚푝. Based on this, we +reveal the detrimental effect of non-negligible hysteresis and +associated dissipation losses of first-order phase transitions +on the adiabatic temperature change, representing a crucial +limitation for recently emerging research interest in caloric +cooling applications at cryogenic temperatures, such as hy- +drogen liquefaction [56–58]. +2. Experimental Details +2.1. Synthesis +Various Ni50-xCoxMn50-yTiy (x=0,15,17 & 6≤y≤17) +samples have been synthesized by arc melting high-purity +elements in protective Ar atmosphere. Due to evaporation +losses, 3% excess Mn was added. Each 10 g sample was +turned and remolten at least five times to ensure a homo- +geneous distribution of elements. Based on the optimized +heat treatment conditions found in reference [38], the ingots +have been annealed at 1323 K for 96 h in sealed quartz +ampules with Ar atmosphere. Subsequently, the samples +were quenched by breaking the quartz tube in water. +2.2. Microstructural and structural +characterization +The crystal structure and phase-purity have been charac- +terized by powder X-Ray diffraction (XRD) in transmission +geometry. Room-temperature XRD has been carried out +with a Stoe Stadi P diffractometer using Mo K훼1 radiation +and a position sensitive detector in a 2휃 range from 5° +to 50° with an effective stepsize of 0.01°. Temperature- +dependent XRD has been performed with a purpose-built +diffractometer with Mo K훼1 radiation and a Dectris Mythen +1K R silicon strip detector in a 2휃 range from 7° to 58° +with an effective step size of 0.009°. A detailed description +of the device can be found in reference [59]. In this setup, +powder samples have been mixed with NIST SRM 640d +silicon standard powder and have been glued on graphite +foil. The annealed arc molten bulk samples have been milled +to powder of particle size <80 µm. To ensure the release +of milling-induced stresses, the powder was recrystallized +by annealing. Structural analysis has been performed by +Rietveld-refinement using FullProf software package [60] +for austenite and unmodulated martensite and JANA2006 +[61] for modulated martensite. Images of the crystal struc- +tures have been created with VESTA [62]. +The microstructure and chemical composition have been +analyzed with backscatter electron (BSE) imaging and energy- +dispersive X-Ray spectroscopy (EDX) using a Tescan Vega3 +scanning electron microscope (SEM). A Zeiss Axio Im- +ager.D2M has been used to obtain optical microscopy micro- +graphs of austenite and martensite. The nanostructure of se- +lected samples has been characterized at room-temperature +based on high-resolution transmission electron microscopy +B. Beckmann, D. Koch, L. Pfeuffer, et al.: Preprint submitted to Elsevier +Page 2 of 14 + +(HRTEM) images and selected area electron diffraction +(SAED) patterns obtained with the 200 kV JEOL JEM +2100-F transmission electron microscope (TEM). The TEM +lamellas have been prepared by a two-step ion milling +process at 80 K using a Gatan 691 Precision Ion Polishing +System (PIPS) following the procedure described in refer- +ence [63]. +2.3. Magnetometry +Magnetic measurements have been performed with a +LakeShore vibrating sample magnetometer (VSM) and Quan- +tum Design physical property measurement system (PPMS- +14T). Isofield measurements have been performed with a +heating and cooling rate of 2 Kmin-1. Isothermal measure- +ments have been carried out with a magnetic field ramp +rate of 5 mTs-1. To erase the remnants of the magnetic +field-induced phase transformation, a discontinuous tem- +perature protocol was used after each isothermal magnetic +measurement. In such a protocol, the sample is heated above +and subsequently cooled below the transition temperature +in zero-field. Based on the field-dependent magnetic mea- +surements, the isothermal entropy change Δ푠푇 (푇 , 퐻) of the +field-induced martensitic transformation has been calculated +following the guidelines in references [64, 65] using the +Maxwell-relation +Δ푠푇 (푇 , 퐻) = 휇0 ∫ +퐻 +0 +(휕푀 +휕푇 +) +퐻 푑퐻. +(2) +Based on isofield magnetic measurements, the transi- +tion entropy change Δ푠푡 has been estimated with Clausius- +Clapeyron (CC) equation +Δ푀 +Δ푠푡 += +푑푇푡 +휇0푑퐻 +(3) +using the magnetization change Δ푀 across the phase trans- +formation in 1 T as well as the sensitivity of the phase trans- +formation towards the magnetic field stimulus 푑푇푡∕휇0푑퐻, +determined based on 푀(푇 ) measurements in 0.1, 1, and 2 T. +The transition entropy changes estimated with the Clausius- +Clapeyron equation are labeled as Δ푠푡,퐶퐶 in the following. +To monitor the responses of all three material subsys- +tems to the martensitic transformation close to and below +the compensation temperature, simultaneous measurements +of magnetization 푀, strain Δ퐿∕퐿0 and electrical resistivity +휌 have been carried out using a purpose-built insert for +the PPMS-14T VSM option. To minimze the influence of +texture and to increase statistical robustness, the strain has +been determined as the average strain detected by two strain +gauges glued to the sample surface parallel and perpendic- +ular to the magnetic field direction. The temperature- and +field-induced response of the strain gauges itself has been +corrected by simultaneously measuring two reference strain +gauges glued to the quartz sample holder. The electrical +resistivity has been determined with the two-contact method. +A detailed description of the device can be found in reference +[66]. +2.4. Calorimetry +Magnetic field-dependent heat capacity 푐푝(푇 , 퐻) has +been measured with the PPMS-14T in magnetic fields of 0, +2, 5, and 14 T in a temperature range from 2 K to 395 K. The +total entropy 푠(푇 , 퐻) of the material has been calculated as +푠(푇 , 퐻) = ∫ +푇 +2퐾 +(푐푝 +푇 +) +퐻 +푑푇 . +(4) +Magnetic field-dependent transition entropy changes calcu- +lated based on this measurement are labeled Δ푠퐻푖 +푡,푐푝 in the +following. +Differential scanning calorimetry (DSC) measurements +have been performed with a Netzsch DSC 404 F1 Pegasus +in a temperature range from 150 K to 700 K and a heating +and cooling rate of 5 Kmin-1. For this purpose, Al crucibles +have been used in the silver-furnace setup, equipped with +a liquid nitrogen cooling system. The transition entropy +change Δ푠푡 of the reverse martensitic transformation, i.e. the +martensite to austenite transformation, has been determined +by integrating the baseline-corrected mass-specific heat flow +̇푄 over the phase transformation region +Δ푠푡 = ∫ +퐴푓 +퐴푠 +( ̇푄 − ̇푄퐵푎푠푒푙푖푛푒 +) 푇 −1 (푑푇 +푑푡 +)−1 +푑푇 +(5) +with 퐴푠 and 퐴푓 being the start and finish temperatures of +the transformation. This procedure was verified for selected +samples by calculating the zero-field heat capacity based on +additional calibration measurements using a sapphire as heat +capacity standard. To distinguish both methods, Δ푠0푇 +푡, ̇푄 and +Δ푠0푇 +푡,푐푝 are used in the following. +2.5. Adiabatic temperature change measurements +Adiabatic temperature changes Δ푇푎푑 associated with the +magnetocaloric effect have been simultaneously measured +with the magnetization of the sample in a solenoid magnet +at the Dresden High Magnetic Field Laboratory (HLD) in +pulsed magnetic fields of 10, 14, 20, 30, and 50 T. The +magnetic field was determined by means of a calibrated +pick-up coil. The maximum field strength of each pulse +was always reached after 13 ms. The adiabatic temperature +change has been measured with a thin type T thermocouple +glued between two pieces of the sample with silver epoxy +[67]. The magnetization of the sample has been measured +simultaneously with a compensated split coil wound around +the sample and a non-magnetic counterpart with opposite +winding (see supplementary material S1). The voltage signal +was fine compensated numerically by a small correction +using the field-induced signal of the pick coil and then +integrated to dimensionless magnetization. +Adiabatic temperature changes associated with the elas- +tocaloric effect have been detected in an Instron 5967 30 kN +universal testing machine, equipped with a temperature +chamber. Strain and force have been monitored with a strain +gauge extensometer attached to the compression platens +close to the specimen and load cell, respectively. To ensure +B. Beckmann, D. Koch, L. Pfeuffer, et al.: Preprint submitted to Elsevier +Page 3 of 14 + +Figure 1: Room-temperature XRD patterns of annealed nominal Ni35Co15Mn50-yTiy (a) and Ni33Co17Mn50-yTiy (d) powders +showing the gradual transition from cubic B2 austenite to a mixture of incommensurately modulated monoclinic and non- +modulated tetragonal L10 to single phase L10 martensite crystal structures (see vertical dashed lines) with decreasing Ti +content. The intensity of each pattern has been normalized. The patterns are labeled according to the Ti content determined +by EDX. HRTEM images (b, e) and SAED patterns (c, f) show 5M modulated and non-modulated L10 martensite for +Ni33.9Co14.6Mn38.1Ti13.4 (b, c) and Ni31.6Co16.6Mn39.7Ti12.1 (e, f), respectively. The SAED patterns of the 5M modulated and +L10 non-modulated martensite are obtained along the [420] and [110] zone axis, respectively. +quasi-adiabatic conditions, strain rates of 3 × 10−2 s-1 have +been used [63]. The Δ푇푎푑 has been measured with a K-type +thermocouple attached to the surface of the sample. +2.6. Density functional theory (DFT) +To take a closer look at the effect of Co on the mag- +netic properties, DFT calculations have been performed +for the ferromagnetic cubic B2 austenite with lattice pa- +rameters taken from XRD experiments. Chemical disor- +der was modeled analytically within the coherent poten- +tial approximation (CPA) allowing to simulate disordered +structures in small unit cells. The total magnetization and +exchange constants were obtained with the help of Kor- +ringa–Kohn–Rostoker (KKR) approach as implemented in +the Munich SPR-KKR code [68, 69] in full-potential mode +together with scalar relativistic corrections. The exchange- +correlation functional was treated within the generalized +gradient approximation (GGA) following the Perdew, Burke, +and Ernzerhof (PBE) scheme [70]. The angular momentum +expansion was carried out up to 푙푚푎푥 = 3 (푓-states). We as- +sumed electronic self-consistency to be reached when the +error in the potential functions dropped below 10−6. Bril- +louin zone integration was carried out using the special point +method with a regular 푘-point grid of 1540 points, which +corresponds to a 39×39×39 mesh in the full Brillouin zone. +The Heisenberg model exchange parameters 퐽푖푗 between +pairs of atoms 푖 and 푗 of all different chemical types and +positions within a cluster radius of four lattice constants were +calculated following Liechtenstein’s approach [71]. +3. Results and Discussion +3.1. Structural analysis +Room-temperature XRD patterns of nominal and chemi- +cally homogeneous all-d-metal Ni35Co15Mn50-yTiy (9≤y≤17) +and Ni33Co17Mn50-yTiy (9≤y≤15) Heusler alloys are shown +in figure 1 (a) and (d), respectively. In good agreement +with [47], a gradual transition from cubic B2 austenite to +a mixture of incommensurately modulated monoclinic and +non-modulated tetragonal L10 to single phase L10 marten- +site is observable with decreasing Ti content in both series. +This indicates the presence and tuneability of martensitic +transformations by composition in both series. No additional +phases besides martensite and austenite can be detected in all +samples. The increasing volume fraction of L10 martensite +with decreasing Ti content is identical to classical Heusler +B. Beckmann, D. Koch, L. Pfeuffer, et al.: Preprint submitted to Elsevier +Page 4 of 14 + +(a) +(d) +Ni,,Co15Mns +Ni2,Co1-Mn50- Tiv +Ti +50-y +50-y +35 +15 +33 +5M (1210) +T = 296 K +T = 296 K +(200) +(211) +5M (1210) +Intensity [a.u.] +(211) +Intensity [a.u.] +Ti9.2 ++B2 +Ti10.3 +Ti11.0 +Ti9.2 +1 +Ti10.1 +Ti12.1 +1 +Ti13.4 +Ti11.0 +Ti14.2 +Ti12.1 +Ti15.0 +Ti13.1 +Ti16.1 +Ti14.1 +Ti15.2 +Ti16.6 +15 +25 +30 +35 +40 +45 +5 +10 +15 +20 +25 +30 +35 +5 +10 +20 +50 +40 +45 +50 +20[°] +20[°] +(f) +(b) +(c) +Ti13.4 +Ti12.1 +Ti13.4 +(e) +121 +002 +112 +112 +121 +too +004 +112 +121 +002 +112 +121 +20nm +20nm +10 1/nm +10 1/nmalloys, such as Ni-Mn-Ga [72], Ni-Mn-Sn [73, 74] and +Ni-Mn-In [75]. The B2 ordered austenite is determined by +the absence and presence of the (111) and (200) reflections, +respectively. The presence of B2 order is in good agreement +with recent results of neutron diffraction experiments [76] +and is seen as the origin of the dependence of the austenite +Curie temperature 푇 퐴 +퐶 on the Ti content in this material +system [38]. +Figure 1 (b, e) and (c, f) show HRTEM images and +corresponding SAED patterns of one martensitic sample +of each Co-series at room-temperature, respectively. The +Ni33.9Co14.6Mn38.1Ti13.4 sample (see figure 1 (b, c)) ex- +hibits 5M modulated martensite as indicated by the satellite +reflexes caused by the modulation. The TEM analysis of +Ni31.6Co16.6Mn39.7Ti12.1 (see figure 1 (e, f)) reveals non- +modulated L10 martensite. Therefore, both SAED patterns +are in good agreement with the powder XRD results. +3.2. Transition entropy change +To study the influence of compositional variations on +the magnetostructural martensitic transformations in all- +d-metal Ni(-Co)-Mn-Ti Heusler alloys in detail, we per- +formed isofield magnetization measurements in 1 T of nom- +inal Ni35Co15Mn50-yTiy (9≤y≤17) and Ni33Co17Mn50-yTiy +(9≤y≤15) samples, shown in figure 2 (a) and (d), respec- +tively. The martensitic transformations are clearly observ- +able due to the magnetization change accompanying the +transition between low-temperature weak-magnetic marten- +site and high-temperature para- or ferromagnetic austenite +and are in good agreement with room-temperature XRD re- +sults (see figure 1 (a) and (d)). Larger magnetization changes +can be achieved at lower temperatures within each Co-series +as the austenite saturation magnetization rises with decreas- +ing temperature and 푇 퐴 +퐶 − 푇푡 grows with increasing Ti con- +tent. Since DFT calculations show that the addition of Co en- +hances saturation magnetization and ferromagnetic coupling +(see supplementary material S2), larger Δ푀 are present in +Ni33Co17Mn50-yTiy compared to Ni35Co15Mn50-yTiy at any +given transition temperature. As it will be shown later, the +increased low-temperature magnetization of samples with a +low transition temperature is linked to the presence of resid- +ual austenite. The composition dependencies of the marten- +sitic transformation temperature 푇푡 and austenite Curie tem- +perature 푇 퐴 +퐶 can be well described with the average number +of valence electrons per atom 푒∕푎 at 푇 ≥ 푇푐표푚푝, as demon- +strated in reference [38]. However, the estimation of 푇푡 based +on the 푒∕푎 ratio breaks down at 푇 = 푇푐표푚푝 as no temperature- +induced martensitic transformations can be observed below +approximately 75 K and 300 K in samples with a Ti content +larger than 16.1 at.% and 12.1 at.% in the Ni35Co15Mn50-yTiy +and Ni33Co17Mn50-yTiy series, respectively. This is in good +agreement with [47] in which no transformation has been +observed for nominal Ni33Co17Mn35Ti15. +Zero-field DSC measurements have been performed to +analyze the temperature dependence of the transition en- +tropy change associated with the martensitic transforma- +tion in Ni(-Co)-Mn-Ti. The heat flow data is shown for +Ni35Co15Mn50-yTiy and Ni33Co17Mn50-yTiy in figure 2 (b) +and (e), respectively. The forward and reverse martensitic +transformation are clearly visible upon cooling and heating +as exo- and endothermic peaks, respectively. The transi- +tion entropy change is calculated based on the area below +the heating curves (see equation 5). Qualitatively, one can +clearly see an increase in Δ푠푡 with an increasing martensitic +transformation temperature and consequently reduction of Ti +content, Δ푀 and 푇 퐴 +퐶 − 푇푡 in both series. +The transition entropy change of the martensite to austen- +ite transformations are shown for Ni35Co15Mn50-yTiy and +Ni33Co17Mn50-yTiy in figure 2 (c) and (f), respectively. +In both series, one can observe a very good agreement +between the various methods used to determine Δ푠푡 as well +as literature values [35, 37, 39, 42, 44]. The competition +of positive lattice entropy change Δ푠푙푎푡 and towards lower +temperatures increasingly negative magnetic entropy change +Δ푠푚푎푔 leads to the overall decrease of Δ푠푡 = Δ푠푙푎푡 + Δ푠푚푎푔 +for 푇푡 ≤ 푇 퐴 +퐶 . This competition leads to the compensation of +both entropy change contributions at 푇푐표푚푝, which is reached +at around 75 K and 300 K for the Ni35Co15Mn50-yTiy +and Ni33Co17Mn50-yTiy series, respectively. These com- +pensation temperatures are in excellent agreement with the +isofield magnetization curves shown in figure 2 (a, d), as +no transformations can be observed below the respective +푇푐표푚푝. The substantial difference in 푇푐표푚푝 between both +series originates from the larger negative magnetic entropy +change contribution in Ni33Co17Mn50-yTiy at any given +temperature due to increased ferromagnetic coupling and +saturation magnetization in the austenite phase caused by +the increased Co content. +If 푇푡 ≥ 푇 퐴 +퐶 , Δ푠푚푎푔 approaches zero due to the absence of +ferromagnetic order in the austenite phase and the transition +entropy change is purely defined by the structural subsystem +contribution Δ푠푙푎푡 [49], showing a value of approximately +65 J(kgK)-1. Since Δ푠푡 is equivalent for both series at +푇푡 ≥ 푇 퐴 +퐶 , Δ푠푙푎푡 is independent on such minor adjustments in +chemical composition. In order to estimate Δ푠푙푎푡 for an ex- +tended composition range in Ni(-Co)-Mn-Ti Heusler alloys, +Ni47.9Mn34.2Ti17.9 has been analyzed (see supplementary +material S3), as the composition is substantially different to +the Ni35Co15Mn50-yTiy and Ni33Co17Mn50-yTiy series and +the absence of Co guarantees non-ferromagnetic order in +martensite and austenite state [47]. The detected transition +entropy change of 66.2 J(kgK)-1 indicates the independence +of Δ푠푙푎푡 on the Co content and represents thereby the upper +limit of Δ푠푡 in all-d-metal Ni(-Co)-Mn-Ti Heusler alloys. +This value is also in good agreement with 66.1 J(kgK)-1 +reported for nominal Ni50Mn31.75Ti18.25 in reference [40]. +Based on the apparent independence of Δ푠푙푎푡 on such large +differences in Co and Ti content, the transition entropy +change at temperatures above 푇 퐴 +퐶 can be assumed to be also +constant in the Ni35Co15Mn50-yTiy and Ni33Co17Mn50-yTiy +series. However, the exact behavior remains unclear as +samples with a lower Ti content than approximately 9 at.% +form two distinctively different Ti-rich and Ti-poor phases +with individual martensitic transformation temperatures +B. Beckmann, D. Koch, L. Pfeuffer, et al.: Preprint submitted to Elsevier +Page 5 of 14 + +Figure 2: Isofield magnetization curves in 1 T (a, d), zero-field heat flow curves (b, e) and transition entropy changes (c, f) of +the nominal Ni35Co15Mn50-yTiy (a, b, c) and Ni33Co17Mn50-yTiy (d, e, f) series. All samples are labeled with the composition +determined by EDX. (b, e) Shaded areas under the heat flow curves qualitatively illustrate the transition entropy change. (c, f) The +transition entropy change is determined by PPMS-14T 푐푝 measurements (blue), sapphire-calibrated DSC 푐푝 measurements (green), +DSC heat flow measurements (red) and the Clausius-Clapeyron equation (open circles). Literature values [35, 37, 39, 42, 44] +(crosses) are given for comparison. Dashed lines are drawn to guide the eye. The compensation temperatures 푇푐표푚푝 can be estimated +as 75 K and 300 K for the Ni35Co15Mn50-yTiy and Ni33Co17Mn50-yTiy series, respectively. The estimated temperature at which +the martensitic transformation temperature 푇푡 is equal to the austenite Curie temperature 푇 퐴 +퐶 is given as vertical solid line. Shaded +areas at low and high temperatures illustrate regions with arrested martensitic transformations and chemically inhomogeneous +dual phase microstructures, respectively. +B. Beckmann, D. Koch, L. Pfeuffer, et al.: Preprint submitted to Elsevier +Page 6 of 14 + +(a) 180 +(d) 180 +Ni35Co15Mn. +T +160 +160 +16.6 +μ.H= 1 T +μ.H= 1 T +Ni.1 +140 +140 +Mn. +Magnetization [Am’kg"] +.......... +120 +120 +Ni.2 +Mna6Ti, +Vi +Mn. +100 +100 +MnzTi +11 +Ni +80 +80 +Ni +.Mnao.Ti, +5Mn 41.8Ti10.1 +60 +60 +Mn +Ni +40 +40 +Ni +20 +20 +0 +100 +700 +800 +0 +0 +200 +300 +400 +500 +600 +100 +200 +300 +400 +500 +600 +700 +800 +Temperature [K] +Temperature [K] +(b) 500 +(e) 500 +Mr +Ti +400 +400 +μ.H=OT +μ.H= O T +300 +300 +Exo! +Exo +200 +200 +Heat flow [Wkg" +Heat flow [Wkg" +100 +100 +0 +0 +-100 +-100 +-200 +-200 +-300 +-300 +-400 +-400 +-500 +-500 +0 +100 +200 +300 +400 +500 +600 +700 +800 +0 +100 +200 +300 +400 +500 +600 +700 +800 +Temperature [K] +Temperature [K] +(c) +(f) +90 +90 +Mn +.△sH +80 +80 +Arrested martensitic transformations +O △st. cc +T >T.A +Entropy change [(kgK)1 +70 +70 +Entropy change [(kgk)" +T 5T +250 +0 +ntropy +5 +1000 +Heat capacity [ +14T + 200 +800 +110 +120 +130 +100 +140 +150 +Temperature [K] +600 +14 T +工 +400 +c +200 +0 +0 +50 +100 +150 +200 +250 +300 +350 +400 +Temperature [K]Figure 5: (a) Adiabatic temperature change Δ푇푎푑 in Ni33.7Co14.8Mn35.4Ti16.1 at 15, 90, and 170 K. The Δ푇푎푑 values are based +on direct measurements in pulsed magnetic fields (solid circles) and indirect determinations with 푠(푇 ) based on 푐푝 measurements +(Δ푇푠(푇 ), open circles). For clarity, Δ푇푎푑 values are only given for the maximum magnetic field strength of each pulse. Dashed lines +are drawn to guide the eye. The inset shows the magnetic field dependence of the simultaneously measured magnetization (top +panel) and adiabatic temperature change (bottom panel) in a pulsed magnetic field of 20 T at 15, 90, and 170 K. (b) Temperature +dependence of the adiabatic temperature change measured with pulsed magnetic fields (solid circles) and determined with 푠(푇 ) +based on 푐푝 measurements (Δ푇푠(푇 ), open circles) in a magnetic field change of 14 T. The estimated effect of dissipation losses +Δ푇푑푖푠푠 on the resulting adiabatic temperature change is given by Δ푇푎푑 = Δ푇푠(푇 ) + Δ푇푑푖푠푠. The upper and lower limit (red dashed +lines) of the estimation is calculated with 푞푑푖푠푠 and 푐푝 values at the start and finish conditions of the adiabatic field-induced +martensite to austenite transformation. Boiling temperatures of H2 and N2 are given by 푇 퐻2 +퐿→퐺 and 푇 푁2 +퐿→퐺, respectively (dotted +vertical lines). The martensite to austenite transformation temperature in 0 T is given by 푇 0푇 +푡 +(dashed vertical line). +effect Δ푇푠(푇 ), which can be estimated with the 푠(푇 ) diagram +based on 푐푝 measurements, and a large positive dissipation +loss Δ푇푑푖푠푠. +Figure 5 (b) demonstrates that the measured adiabatic +temperature change of Ni33.7Co14.8Mn35.4Ti16.1 can be very +well described by Δ푇푎푑 = Δ푇푠(푇 ) + Δ푇푑푖푠푠. At low temper- +atures, Δ푇푑푖푠푠 rises drastically, causing the estimation of +the temperature change solely based on indirect methods +using datasets obtained under isothermal or isofield condi- +tions to break down as Δ푇푑푖푠푠 dramatically influences Δ푇푎푑. +This is especially critical as it is common practice in low- +temperature magnetocalorics to estimate Δ푇푎푑 either based +on the 푠(푇 ) diagram [84] or on Δ푠푇 and 푐푝 [85, 86]. There- +fore, it is imperative to directly measure and consequently +verify Δ푇푎푑 for materials showing field-induced first-order +phase transitions with non-negligible hysteresis at such low +temperatures in the future. However, it should be noted that +the precise determination of Δ푇푑푖푠푠 and Δ푇푎푑 is challenging +at 푇 ≤ 35 K as only minor deviations in 푐푝 dramatically +influence Δ푇푑푖푠푠 (see figure 5 (b)). According to the formula +used to estimate Δ푇푑푖푠푠, the assumption of a constant 푐푝 +itself ultimately leads to a divergent behavior of the upper +limit of Δ푇푑푖푠푠 at 푇 → 0, since Δ푇푑푖푠푠 → ∞ if 푐푝 → 0. +Nevertheless, taking into account the conditions prior to as +well as subsequently to the adiabatic magnetic field-induced +martensite to austenite transformation, it is possible to give +a reasonable estimation of Δ푇푎푑 (see figure 5 (b)). +The origin of the positive Δ푇푎푑 is therefore not linked +to a change from inverse to conventional magnetocaloric +effect, which could be proposed based on a linear extrap- +olation of Δ푠푡(푇 ) to 푇 < 푇푐표푚푝 in figure 2 (c), resulting +in a reversed sign of Δ푠푡 and consequently Δ푇푎푑. This +scenario is unlikely as the adiabatic temperature change +upon field removal, i.e. the austenite to martensite trans- +formation, is positive and has not changed sign (see fig- +ure 5 (a), inset). Furthermore, the same argument should +hold true not only for the magneto- but also elastocaloric +effect and not only in the Ni35Co15Mn50-yTiy but also +in the Ni33Co17Mn50-yTiy series. However, the uniaxial +stress-induced austenite to martensite transformation in +Ni31.9Co16.6Mn37.4Ti14.1, which shows an arrested trans- +formation below 푇푐표푚푝 of 300 K (see figure 2 (d)), clearly +exhibits an as-expected conventional elastocaloric effect +under adiabatic conditions at 220.9 K (see supplementary +material S8). +The change of sign and irreversibility of Δ푇푎푑 at low +temperatures is not only interesting for the fundamental un- +derstanding of the intertwined intrinsic and extrinsic param- +eters determining the thermal response of caloric materials +to an applied stimulus, but also raise an important question +B. Beckmann, D. Koch, L. Pfeuffer, et al.: Preprint submitted to Elsevier +Page 10 of 14 + +(a) 20 +(b) 20 +Ni. +Co +Mn +Ni +o Pulsed field +Co +Ti +Mn. +33.7 +14.8 +35.4 +16.1 +33.7 +14.8 +35.4 +16.1 +18 +18 +O△T +s(T) +△μ.H = 14 T +16 +16 +14 +14 +Pulsed field + Upper limit +[a.u.] +170 K +12 +12 +IV +s(T) +M +15 K +Lower limit: +10 +△T +IV + +s(T) + diss +8 +25 +1 +△Y +6 +6 +TN2 +1 +→G +4 +5E +4 +1 +0 +1 +2 +170K +2 +TH, +5 +10 +15 +20 +0 +L→G +00000 +Magnetic field [T] +0 +0 +1 +-2 +-2 +00 +90 K +0 +-4 +-4 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +55 +0 20 40 60 80 100 120 140 160 180 +Magnetic field [T] +Temperature TKiwith respect to recently emerging research interest in the uti- +lization of magnetocaloric materials for gas liquefaction at +low temperatures [56, 58]. Currently, one of the most promi- +nent materials for such an application are RT2-based Laves +phases, with T and R being transition metal (T) and critical +rare-earth (R) elements, respectively. Intermetallics such as +RCo2 [84, 85] and RAl2 [87, 88] show highly tuneable phase +transitions down to temperatures of only 10 K. Equivalent to +room-temperature magnetocalorics, the utilization of giant +magnetocaloric effects coupled to first-order magnetic or +magnetostructural phase transitions would enhance perfor- +mance and thereby overall feasibility of the technology, +assuming the availability of sufficiently large magnetic field +changes required to overcome the non-negligible hysteresis. +However, the here reported and so far overlooked aspect +of hysteresis for low-temperature magnetocalorics, leading +to an irreversible and substantial absolute increase of Δ푇푎푑 +due to dissipation losses, significantly limits such an utiliza- +tion, as Δ푇푑푖푠푠 clearly dominates the thermal response at +푇 ≤ 50 K (see figure 5 (b)). In combination with similar +irreversible, positive and directly measured Δ푇푎푑 signals +in inverse magnetocaloric Fe-Rh [34] at elevated temper- +atures, inverse elastocaloric Co-Cr-Al-Si [89] and conven- +tional elastocaloric Ti-Ni-Cu-Al [90] as well as Ti-Ni-based, +Cu-based and Ni-Mn-based shape memory alloys [91], we +expect the effect to be neither material nor stimulus specific +but fundamentally linked to phase transitions with hys- +teresis, underlining the universal importance of mastering +hysteresis [92] in all caloric materials showing first-order +phase transitions, especially at such low temperatures. +4. Conclusion +We present a systematic study on the transition en- +tropy change Δ푠푡 in multicaloric all-d-metal Ni(-Co)-Mn-Ti +Heusler alloys. The obtained composition dependence of +Δ푠푡 is fundamental to the design of materials with tai- +lored phase transition properties for any given future caloric +cooling application. The transition entropy change associ- +ated with the structural subsystem Δ푠푙푎푡 of 65 J(kgK)-1 +has been isolated by tuning the Ni/Co and Mn/Ti ratio and +thereby modifying the martensitic transition temperature 푇푡 +and austenite Curie temperature 푇 퐴 +퐶 , so that 푇푡 ≥ 푇 퐴 +퐶 . Since +the here reported Δ푠푡 values are substantially larger com- +pared to classical Heusler alloys, all-d-metal Ni(-Co)-Mn-Ti +will perform superior in caloric cooling applications har- +nessing the full potential of the phase transition due to +enhanced saturated caloric effects. For 푇푡 ≤ 푇 퐴 +퐶 , we have +identified the competition of positive lattice and negative +magnetic entropy change contributions, leading to lower +Δ푠푡 with decreasing temperature. This intrinsic competi- +tion leads to compensation temperatures 푇푐표푚푝 of 75 K +and 300 K, below which the martensitic transformations +are arrested in Ni35Co15Mn50-yTiy and Ni33Co17Mn50-yTiy, +respectively. +Based on the composition dependence of Δ푠푡, we simul- +taneously measured the responses of the magnetic, structural +and electronic subsystems close to and below the compen- +sation temperature 푇푐표푚푝 in Ni33.7Co14.8Mn35.4Ti16.1 and +Ni31.6Co16.6Mn39.7Ti12.1. The temperature dependence of +all subsystems shows the stabilization by magnetic field +of the high-temperature ferromagnetic austenite, leading +to the gradual shift of 푇푡 below 푇푐표푚푝 until the marten- +sitic transformation becomes fully arrested in sufficiently +large fields. Isofield measurements show a substantially in- +creased hysteresis and consequently dissipation energy at +cryogenic temperatures, primarily due to the shift of the for- +ward martensitic transformation towards smaller magnetic +fields with decreasing temperature. +Simultaneous magnetization and adiabatic temperature +change measurements of Ni33.7Co14.8Mn35.4Ti16.1 in pulsed +magnetic fields reveal a change in sign of Δ푇푎푑 with sub- +stantial positive and irreversible values up to 15 K at 15 K +due to the combination of increased dissipation energy and +decreased heat capacity. Therefore, the here reported irre- +versibility of Δ푇푎푑 demonstrates a so far overlooked lim- +itation of utilizing phase transitions with non-negligible +hysteresis at cryogenic temperatures for magnetocaloric gas +liquefaction. Furthermore, as dissipation losses are funda- +mentally linked to first-order phase transitions with hys- +teresis, the observed effect is of universal importance for +all caloric cooling applications of first-order materials at +cryogenic temperatures. +Acknowledgements +We acknowledge financial support by the Deutsche +Forschungsgemeinschaft (DFG) within the CRC/TRR 270 +(Project-ID 405553726) and BEsT (Project-ID 456263705) +as well as by the European Research Council (ERC) under +the European Union’s Horizon 2020 research and innova- +tion programme (Grant No. 743116). We acknowledge the +support of the HLD at HZDR, member of the European +Magnetic Field Laboratory (EMFL). +References +[1] United Nations, Department of Economic and Social Affairs, Popula- +tion Division, World Population Prospects 2019: Highlights, United +Nations, 2019. +[2] D. Coulomb, J. L. Dupont, A. 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Pfeuffer, et al.: Preprint submitted to Elsevier +Page 14 of 14 + diff --git a/u9E2T4oBgHgl3EQfgQcG/content/tmp_files/load_file.txt b/u9E2T4oBgHgl3EQfgQcG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1ac22fc0a53f5d92055bd23596f05e6abf92dc27 --- /dev/null +++ b/u9E2T4oBgHgl3EQfgQcG/content/tmp_files/load_file.txt @@ -0,0 +1,1942 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf,len=1941 +page_content='Dissipation losses limiting first-order phase transition materials in cryogenic caloric cooling: A case study on all-d-metal Ni(-Co)-Mn-Ti Heusler alloys Benedikt Beckmanna,∗, David Kochb, Lukas Pfeuffera, Tino Gottschallc, Andreas Taubela, Esmaeil Adabifiroozjaeid, Olga N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Miroshkinae, Stefan Riegga, Timo Niehoffc,f, Nagaarjhuna A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Kania,d, Markus E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Grunere, Leopoldo Molina-Lunad, Konstantin P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Skokova and Oliver Gutfleischa aFunctional Materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Institute of Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Technical University of Darmstadt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Darmstadt 64287,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Germany bStructure Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Institute of Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Technical University of Darmstadt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Darmstadt 64287,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Germany cDresden High Magnetic Field Laboratory (HLD-EMFL),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Helmholtz-Zentrum Dresden-Rossendorf (HZDR),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Dresden 01328,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Germany dAdvanced Electron Microscopy (AEM),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Institute of Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Technical University of Darmstadt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Darmstadt 64287,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Germany eFaculty of Physics and Center for Nanointegration Duisburg-Essen (CENIDE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' University of Duisburg-Essen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Duisburg 47057,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Germany fInstitute of Solid State and Materials Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Technische Universität Dresden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Dresden 01069,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Germany A R T I C L E I N F O Keywords: Heusler alloys Magnetostructural transformation Martensitic transformation Solid-state caloric cooling Hydrogen A B S T R A C T Ni-Mn-based Heusler alloys,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' in particular all-d-metal Ni(-Co)-Mn-Ti,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' are highly promising materials for energy-efficient solid-state refrigeration as large multicaloric effects can be achieved across their magnetostructural martensitic transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' However, no comprehensive study on the crucially important transition entropy change Δ푠푡 exists so far for Ni(-Co)-Mn-Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Here, we present a systematic study analyzing the composition and temperature dependence of Δ푠푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Our results reveal a substantial structural entropy change contribution of approximately 65 J(kgK)-1, which is compensated at lower temperatures by an increasingly negative entropy change associated with the magnetic subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' This leads to compensation temperatures 푇푐표푚푝 of 75 K and 300 K in Ni35Co15Mn50-yTiy and Ni33Co17Mn50-yTiy, respectively, below which the martensitic transformations are arrested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' In addi- tion, we simultaneously measured the responses of the magnetic, structural and electronic subsystems to the temperature- and field-induced martensitic transformation near 푇푐표푚푝, showing an abnormal increase of hysteresis and consequently dissipation energy at cryogenic temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Simultaneous measurements of magnetization and adiabatic temperature change Δ푇푎푑 in pulsed magnetic fields reveal a change in sign of Δ푇푎푑 and a substantial positive and irreversible Δ푇푎푑 up to 15 K at 15 K as a consequence of increased dissipation losses and decreased heat capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Most importantly, this phenomenon is universal, it applies to any first-order material with non-negligible hysteresis and any stimulus, effectively limiting the utilization of their caloric effects for gas liquefaction at cryogenic temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Introduction In an age when anthropogenic climate change and deple- tion of natural energy resources are two of the most urgent world-wide societal and existential challenges, the search for energy-efficient and environmentally friendly technologies is of outermost importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' This development is strongly intertwined with globally rising population and standard of living [1], leading to an ever growing demand for cool- ing [2, 3], which is predicted to exceed the energy con- sumption for heating in this century [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Currently, most room-temperature cooling devices are based on the 120- year old vapor-compression refrigeration cycle, which lacks in thermodynamic efficiency and has a profound negative environmental impact [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' In addition, the compression- based cooling process used for the liquefaction of hydrogen, which is critical as an alternative energy carrier for the transition towards renewable energies [6, 7], is considered to be inefficient as well [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Therefore, future cooling ∗Corresponding author benedikt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='beckmann@tu-darmstadt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='de ( Beckmann) ORCID(s): 0000-0002-2232-1804 ( Beckmann) technologies for room as well as cryogenic temperature applications need to provide energy-efficient and environ- mentally friendly alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' As such an alternative, solid-state caloric cooling is based on the response of caloric materials to the application of external stimuli, such as electric or magnetic fields [10], hydrostatic pressure [11, 12] and uniaxial stress [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Mul- ticaloric materials are susceptible to more than one stimulus [14, 15], allowing the design of advanced cooling cycles, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' by subsequently utilizing magnetic field and uniax- ial stress [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Depending on the selected thermodynamic boundary conditions, the stimulus leads to an isothermal entropy change Δ푠푇 or adiabatic temperature change Δ푇푎푑 of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Conventional caloric materials respond with Δ푠푇 < 0 and Δ푇푎푑 > 0 to the application of a given external stimulus, whereas inverse caloric materials show Δ푠푇 > 0 and Δ푇푎푑 < 0 [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Of all ferroic cooling technologies, magnetocalorics is considered to be the best studied [17, 18] and shows an improved thermodynamic efficiency compared to the widely used compression-based technology [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Beckmann, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Koch, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Pfeuffer, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 1 of 14 aHowever, the search for promising magnetocaloric materials is still ongoing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Since the discovery of the giant magnetocaloric effect in Gd5Si2Ge2 [21], various material systems showing en- hanced magnetocaloric properties due to first-order phase transitions have been discovered [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The most promi- nent material systems are La(Fe,Si)13 [23, 24], Fe2P-type compounds [25–27], Ni-Mn-based Heusler alloys [10, 28– 31] and in some extend Fe-Rh [32–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' In Ni-Mn-based Heusler alloys, giant caloric effects are coupled to first-order magnetostructural martensitic transformations between low- temperature martensite and high-temperature austenite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' In this work, multicaloric all-d-metal Ni(-Co)-Mn-Ti Heusler alloys are investigated as these alloys have recently attracted particular interest in the fields of magneto- [35–38], elasto- [39–42], baro- [43, 44] and multicalorics [45] due to an im- proved mechanical stability [39, 40], large volume changes [46] as well as good tuneability of the martensitic transfor- mation temperature 푇푡 and austenite Curie temperature 푇 퐴 퐶 [38, 46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' In general, knowledge about composition and tempera- ture dependencies of the transition entropy change Δ푠푡 in any given caloric material is crucial to develop samples with tailored properties for solid-state cooling applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Thereby, independent on the selected external stimulus to drive the phase transition, the transition entropy change is of universal importance and is expressed as Δ푠푡 = Δ푠푙푎푡 + Δ푠푚푎푔 + Δ푠푒푙 (1) with Δ푠푙푎푡, Δ푠푚푎푔 and Δ푠푒푙 being the entropy change con- tributions associated with the structural, magnetic and elec- tronic subsystem, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Kihara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' showed that the entropy change of martensitic transformations in Heusler alloys is dominated by the structural subsystem, whereas the contribution of the electronic subsystem is negligibly small [48], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' |Δ푠푙푎푡| > |Δ푠푚푎푔| >> |Δ푠푒푙|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' In inverse magne- tocaloric Heusler alloys, such as Ni(-Co)-Mn-Ti, the domi- nating Δ푠푙푎푡 is positive and Δ푠푚푎푔 is increasingly negative towards lower temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The opposing positive lattice and negative magnetic entropy change contributions give rise to the "dilemma of inverse magnetocaloric materials" [49] and are compensated at the compensation temperature 푇푐표푚푝, below which the martensitic transformation is ther- mally arrested due to the absence of a driving force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' In the literature, this thermodynamic effect is often called kinetic arrest and is described in multiple classical Ni(-Co)-Mn-X Heusler alloys with X being a main group element, namely Sn [50], Sb [51], Al [52] and In [49, 53–55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' However, the arrest phenomenon is not only limited to Heusler alloys but also appears in other inverse magnetocaloric materials, such as Fe-Rh compounds [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' In this study, we disentangle the contributions to the tran- sition entropy change of the magnetostructural martensitic transformation in multicaloric all-d-metal Ni(-Co)-Mn-Ti Heusler alloys, since there is so far no comprehensive study on Δ푠푡 in this material system, even though the transition en- tropy change is a fundamental material property governing all caloric effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' On this basis, we analyze the responses of the magnetic, structural and electronic subsystems to the temperature- and field-induced martensitic transformation, showing an abnormally increased magnetic hysteresis width at temperatures near and below 푇푐표푚푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Based on this, we reveal the detrimental effect of non-negligible hysteresis and associated dissipation losses of first-order phase transitions on the adiabatic temperature change, representing a crucial limitation for recently emerging research interest in caloric cooling applications at cryogenic temperatures, such as hy- drogen liquefaction [56–58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Experimental Details 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Synthesis Various Ni50-xCoxMn50-yTiy (x=0,15,17 & 6≤y≤17) samples have been synthesized by arc melting high-purity elements in protective Ar atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Due to evaporation losses, 3% excess Mn was added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Each 10 g sample was turned and remolten at least five times to ensure a homo- geneous distribution of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Based on the optimized heat treatment conditions found in reference [38], the ingots have been annealed at 1323 K for 96 h in sealed quartz ampules with Ar atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Subsequently, the samples were quenched by breaking the quartz tube in water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Microstructural and structural characterization The crystal structure and phase-purity have been charac- terized by powder X-Ray diffraction (XRD) in transmission geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Room-temperature XRD has been carried out with a Stoe Stadi P diffractometer using Mo K훼1 radiation and a position sensitive detector in a 2휃 range from 5° to 50° with an effective stepsize of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='01°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Temperature- dependent XRD has been performed with a purpose-built diffractometer with Mo K훼1 radiation and a Dectris Mythen 1K R silicon strip detector in a 2휃 range from 7° to 58° with an effective step size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='009°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' A detailed description of the device can be found in reference [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' In this setup, powder samples have been mixed with NIST SRM 640d silicon standard powder and have been glued on graphite foil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The annealed arc molten bulk samples have been milled to powder of particle size <80 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' To ensure the release of milling-induced stresses, the powder was recrystallized by annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Structural analysis has been performed by Rietveld-refinement using FullProf software package [60] for austenite and unmodulated martensite and JANA2006 [61] for modulated martensite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Images of the crystal struc- tures have been created with VESTA [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The microstructure and chemical composition have been analyzed with backscatter electron (BSE) imaging and energy- dispersive X-Ray spectroscopy (EDX) using a Tescan Vega3 scanning electron microscope (SEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' A Zeiss Axio Im- ager.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='D2M has been used to obtain optical microscopy micro- graphs of austenite and martensite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The nanostructure of se- lected samples has been characterized at room-temperature based on high-resolution transmission electron microscopy B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Beckmann, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Koch, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Pfeuffer, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 2 of 14 (HRTEM) images and selected area electron diffraction (SAED) patterns obtained with the 200 kV JEOL JEM 2100-F transmission electron microscope (TEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The TEM lamellas have been prepared by a two-step ion milling process at 80 K using a Gatan 691 Precision Ion Polishing System (PIPS) following the procedure described in refer- ence [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Magnetometry Magnetic measurements have been performed with a LakeShore vibrating sample magnetometer (VSM) and Quan- tum Design physical property measurement system (PPMS- 14T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Isofield measurements have been performed with a heating and cooling rate of 2 Kmin-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Isothermal measure- ments have been carried out with a magnetic field ramp rate of 5 mTs-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' To erase the remnants of the magnetic field-induced phase transformation, a discontinuous tem- perature protocol was used after each isothermal magnetic measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' In such a protocol, the sample is heated above and subsequently cooled below the transition temperature in zero-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Based on the field-dependent magnetic mea- surements, the isothermal entropy change Δ푠푇 (푇 , 퐻) of the field-induced martensitic transformation has been calculated following the guidelines in references [64, 65] using the Maxwell-relation Δ푠푇 (푇 , 퐻) = 휇0 ∫ 퐻 0 (휕푀 휕푇 ) 퐻 푑퐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' (2) Based on isofield magnetic measurements, the transi- tion entropy change Δ푠푡 has been estimated with Clausius- Clapeyron (CC) equation Δ푀 Δ푠푡 = 푑푇푡 휇0푑퐻 (3) using the magnetization change Δ푀 across the phase trans- formation in 1 T as well as the sensitivity of the phase trans- formation towards the magnetic field stimulus 푑푇푡∕휇0푑퐻, determined based on 푀(푇 ) measurements in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='1, 1, and 2 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The transition entropy changes estimated with the Clausius- Clapeyron equation are labeled as Δ푠푡,퐶퐶 in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' To monitor the responses of all three material subsys- tems to the martensitic transformation close to and below the compensation temperature, simultaneous measurements of magnetization 푀, strain Δ퐿∕퐿0 and electrical resistivity 휌 have been carried out using a purpose-built insert for the PPMS-14T VSM option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' To minimze the influence of texture and to increase statistical robustness, the strain has been determined as the average strain detected by two strain gauges glued to the sample surface parallel and perpendic- ular to the magnetic field direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The temperature- and field-induced response of the strain gauges itself has been corrected by simultaneously measuring two reference strain gauges glued to the quartz sample holder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The electrical resistivity has been determined with the two-contact method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' A detailed description of the device can be found in reference [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Calorimetry Magnetic field-dependent heat capacity 푐푝(푇 , 퐻) has been measured with the PPMS-14T in magnetic fields of 0, 2, 5, and 14 T in a temperature range from 2 K to 395 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The total entropy 푠(푇 , 퐻) of the material has been calculated as 푠(푇 , 퐻) = ∫ 푇 2퐾 (푐푝 푇 ) 퐻 푑푇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' (4) Magnetic field-dependent transition entropy changes calcu- lated based on this measurement are labeled Δ푠퐻푖 푡,푐푝 in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Differential scanning calorimetry (DSC) measurements have been performed with a Netzsch DSC 404 F1 Pegasus in a temperature range from 150 K to 700 K and a heating and cooling rate of 5 Kmin-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' For this purpose, Al crucibles have been used in the silver-furnace setup, equipped with a liquid nitrogen cooling system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The transition entropy change Δ푠푡 of the reverse martensitic transformation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' the martensite to austenite transformation, has been determined by integrating the baseline-corrected mass-specific heat flow ̇푄 over the phase transformation region Δ푠푡 = ∫ 퐴푓 퐴푠 ( ̇푄 − ̇푄퐵푎푠푒푙푖푛푒 ) 푇 −1 (푑푇 푑푡 )−1 푑푇 (5) with 퐴푠 and 퐴푓 being the start and finish temperatures of the transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' This procedure was verified for selected samples by calculating the zero-field heat capacity based on additional calibration measurements using a sapphire as heat capacity standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' To distinguish both methods, Δ푠0푇 푡, ̇푄 and Δ푠0푇 푡,푐푝 are used in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Adiabatic temperature change measurements Adiabatic temperature changes Δ푇푎푑 associated with the magnetocaloric effect have been simultaneously measured with the magnetization of the sample in a solenoid magnet at the Dresden High Magnetic Field Laboratory (HLD) in pulsed magnetic fields of 10, 14, 20, 30, and 50 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The magnetic field was determined by means of a calibrated pick-up coil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The maximum field strength of each pulse was always reached after 13 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The adiabatic temperature change has been measured with a thin type T thermocouple glued between two pieces of the sample with silver epoxy [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The magnetization of the sample has been measured simultaneously with a compensated split coil wound around the sample and a non-magnetic counterpart with opposite winding (see supplementary material S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The voltage signal was fine compensated numerically by a small correction using the field-induced signal of the pick coil and then integrated to dimensionless magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Adiabatic temperature changes associated with the elas- tocaloric effect have been detected in an Instron 5967 30 kN universal testing machine, equipped with a temperature chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Strain and force have been monitored with a strain gauge extensometer attached to the compression platens close to the specimen and load cell, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' To ensure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Beckmann, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Koch, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Pfeuffer, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 3 of 14 Figure 1: Room-temperature XRD patterns of annealed nominal Ni35Co15Mn50-yTiy (a) and Ni33Co17Mn50-yTiy (d) powders showing the gradual transition from cubic B2 austenite to a mixture of incommensurately modulated monoclinic and non- modulated tetragonal L10 to single phase L10 martensite crystal structures (see vertical dashed lines) with decreasing Ti content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The intensity of each pattern has been normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The patterns are labeled according to the Ti content determined by EDX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' HRTEM images (b, e) and SAED patterns (c, f) show 5M modulated and non-modulated L10 martensite for Ni33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='9Co14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='6Mn38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='1Ti13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='4 (b, c) and Ni31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='6Co16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='6Mn39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='7Ti12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='1 (e, f), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The SAED patterns of the 5M modulated and L10 non-modulated martensite are obtained along the [420] and [110] zone axis, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' quasi-adiabatic conditions, strain rates of 3 × 10−2 s-1 have been used [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The Δ푇푎푑 has been measured with a K-type thermocouple attached to the surface of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Density functional theory (DFT) To take a closer look at the effect of Co on the mag- netic properties, DFT calculations have been performed for the ferromagnetic cubic B2 austenite with lattice pa- rameters taken from XRD experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Chemical disor- der was modeled analytically within the coherent poten- tial approximation (CPA) allowing to simulate disordered structures in small unit cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The total magnetization and exchange constants were obtained with the help of Kor- ringa–Kohn–Rostoker (KKR) approach as implemented in the Munich SPR-KKR code [68, 69] in full-potential mode together with scalar relativistic corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The exchange- correlation functional was treated within the generalized gradient approximation (GGA) following the Perdew, Burke, and Ernzerhof (PBE) scheme [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The angular momentum expansion was carried out up to 푙푚푎푥 = 3 (푓-states).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' We as- sumed electronic self-consistency to be reached when the error in the potential functions dropped below 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Bril- louin zone integration was carried out using the special point method with a regular 푘-point grid of 1540 points, which corresponds to a 39×39×39 mesh in the full Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The Heisenberg model exchange parameters 퐽푖푗 between pairs of atoms 푖 and 푗 of all different chemical types and positions within a cluster radius of four lattice constants were calculated following Liechtenstein’s approach [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Results and Discussion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Structural analysis Room-temperature XRD patterns of nominal and chemi- cally homogeneous all-d-metal Ni35Co15Mn50-yTiy (9≤y≤17) and Ni33Co17Mn50-yTiy (9≤y≤15) Heusler alloys are shown in figure 1 (a) and (d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' In good agreement with [47], a gradual transition from cubic B2 austenite to a mixture of incommensurately modulated monoclinic and non-modulated tetragonal L10 to single phase L10 marten- site is observable with decreasing Ti content in both series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' This indicates the presence and tuneability of martensitic transformations by composition in both series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' No additional phases besides martensite and austenite can be detected in all samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The increasing volume fraction of L10 martensite with decreasing Ti content is identical to classical Heusler B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Beckmann, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Koch, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Pfeuffer, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 4 of 14 (a) (d) Ni,,Co15Mns Ni2,Co1-Mn50- Tiv Ti 50-y 50-y 35 15 33 5M (1210) T = 296 K T = 296 K (200) (211) 5M (1210) Intensity [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='] (211) Intensity [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='] Ti9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='2 +B2 Ti10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='3 Ti11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='0 Ti9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='2 1 Ti10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='1 Ti12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='1 1 Ti13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='4 Ti11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='0 Ti14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='2 Ti12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='1 Ti15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='0 Ti13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='1 Ti16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='1 Ti14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='1 Ti15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='2 Ti16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='6 15 25 30 35 40 45 5 10 15 20 25 30 35 5 10 20 50 40 45 50 20[°] 20[°] (f) (b) (c) Ti13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='4 Ti12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='1 Ti13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='4 (e) 121 002 112 112 121 too 004 112 121 002 112 121 20nm 20nm 10 1/nm 10 1/nmalloys, such as Ni-Mn-Ga [72], Ni-Mn-Sn [73, 74] and Ni-Mn-In [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The B2 ordered austenite is determined by the absence and presence of the (111) and (200) reflections, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The presence of B2 order is in good agreement with recent results of neutron diffraction experiments [76] and is seen as the origin of the dependence of the austenite Curie temperature 푇 퐴 퐶 on the Ti content in this material system [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Figure 1 (b, e) and (c, f) show HRTEM images and corresponding SAED patterns of one martensitic sample of each Co-series at room-temperature, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The Ni33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='9Co14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='6Mn38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='1Ti13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='4 sample (see figure 1 (b, c)) ex- hibits 5M modulated martensite as indicated by the satellite reflexes caused by the modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The TEM analysis of Ni31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='6Co16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='6Mn39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='7Ti12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='1 (see figure 1 (e, f)) reveals non- modulated L10 martensite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Therefore, both SAED patterns are in good agreement with the powder XRD results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Transition entropy change To study the influence of compositional variations on the magnetostructural martensitic transformations in all- d-metal Ni(-Co)-Mn-Ti Heusler alloys in detail, we per- formed isofield magnetization measurements in 1 T of nom- inal Ni35Co15Mn50-yTiy (9≤y≤17) and Ni33Co17Mn50-yTiy (9≤y≤15) samples, shown in figure 2 (a) and (d), respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The martensitic transformations are clearly observ- able due to the magnetization change accompanying the transition between low-temperature weak-magnetic marten- site and high-temperature para- or ferromagnetic austenite and are in good agreement with room-temperature XRD re- sults (see figure 1 (a) and (d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Larger magnetization changes can be achieved at lower temperatures within each Co-series as the austenite saturation magnetization rises with decreas- ing temperature and 푇 퐴 퐶 − 푇푡 grows with increasing Ti con- tent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Since DFT calculations show that the addition of Co en- hances saturation magnetization and ferromagnetic coupling (see supplementary material S2), larger Δ푀 are present in Ni33Co17Mn50-yTiy compared to Ni35Co15Mn50-yTiy at any given transition temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' As it will be shown later, the increased low-temperature magnetization of samples with a low transition temperature is linked to the presence of resid- ual austenite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The composition dependencies of the marten- sitic transformation temperature 푇푡 and austenite Curie tem- perature 푇 퐴 퐶 can be well described with the average number of valence electrons per atom 푒∕푎 at 푇 ≥ 푇푐표푚푝, as demon- strated in reference [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' However, the estimation of 푇푡 based on the 푒∕푎 ratio breaks down at 푇 = 푇푐표푚푝 as no temperature- induced martensitic transformations can be observed below approximately 75 K and 300 K in samples with a Ti content larger than 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='1 at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='% and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='1 at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='% in the Ni35Co15Mn50-yTiy and Ni33Co17Mn50-yTiy series, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' This is in good agreement with [47] in which no transformation has been observed for nominal Ni33Co17Mn35Ti15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Zero-field DSC measurements have been performed to analyze the temperature dependence of the transition en- tropy change associated with the martensitic transforma- tion in Ni(-Co)-Mn-Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The heat flow data is shown for Ni35Co15Mn50-yTiy and Ni33Co17Mn50-yTiy in figure 2 (b) and (e), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The forward and reverse martensitic transformation are clearly visible upon cooling and heating as exo- and endothermic peaks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The transi- tion entropy change is calculated based on the area below the heating curves (see equation 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Qualitatively, one can clearly see an increase in Δ푠푡 with an increasing martensitic transformation temperature and consequently reduction of Ti content, Δ푀 and 푇 퐴 퐶 − 푇푡 in both series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The transition entropy change of the martensite to austen- ite transformations are shown for Ni35Co15Mn50-yTiy and Ni33Co17Mn50-yTiy in figure 2 (c) and (f), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' In both series, one can observe a very good agreement between the various methods used to determine Δ푠푡 as well as literature values [35, 37, 39, 42, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The competition of positive lattice entropy change Δ푠푙푎푡 and towards lower temperatures increasingly negative magnetic entropy change Δ푠푚푎푔 leads to the overall decrease of Δ푠푡 = Δ푠푙푎푡 + Δ푠푚푎푔 for 푇푡 ≤ 푇 퐴 퐶 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' This competition leads to the compensation of both entropy change contributions at 푇푐표푚푝, which is reached at around 75 K and 300 K for the Ni35Co15Mn50-yTiy and Ni33Co17Mn50-yTiy series, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' These com- pensation temperatures are in excellent agreement with the isofield magnetization curves shown in figure 2 (a, d), as no transformations can be observed below the respective 푇푐표푚푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The substantial difference in 푇푐표푚푝 between both series originates from the larger negative magnetic entropy change contribution in Ni33Co17Mn50-yTiy at any given temperature due to increased ferromagnetic coupling and saturation magnetization in the austenite phase caused by the increased Co content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' If 푇푡 ≥ 푇 퐴 퐶 , Δ푠푚푎푔 approaches zero due to the absence of ferromagnetic order in the austenite phase and the transition entropy change is purely defined by the structural subsystem contribution Δ푠푙푎푡 [49], showing a value of approximately 65 J(kgK)-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Since Δ푠푡 is equivalent for both series at 푇푡 ≥ 푇 퐴 퐶 , Δ푠푙푎푡 is independent on such minor adjustments in chemical composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' In order to estimate Δ푠푙푎푡 for an ex- tended composition range in Ni(-Co)-Mn-Ti Heusler alloys, Ni47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='9Mn34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='2Ti17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='9 has been analyzed (see supplementary material S3), as the composition is substantially different to the Ni35Co15Mn50-yTiy and Ni33Co17Mn50-yTiy series and the absence of Co guarantees non-ferromagnetic order in martensite and austenite state [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The detected transition entropy change of 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='2 J(kgK)-1 indicates the independence of Δ푠푙푎푡 on the Co content and represents thereby the upper limit of Δ푠푡 in all-d-metal Ni(-Co)-Mn-Ti Heusler alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' This value is also in good agreement with 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='1 J(kgK)-1 reported for nominal Ni50Mn31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='75Ti18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='25 in reference [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Based on the apparent independence of Δ푠푙푎푡 on such large differences in Co and Ti content, the transition entropy change at temperatures above 푇 퐴 퐶 can be assumed to be also constant in the Ni35Co15Mn50-yTiy and Ni33Co17Mn50-yTiy series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' However, the exact behavior remains unclear as samples with a lower Ti content than approximately 9 at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='% form two distinctively different Ti-rich and Ti-poor phases with individual martensitic transformation temperatures B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Beckmann, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Koch, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Pfeuffer, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 5 of 14 Figure 2: Isofield magnetization curves in 1 T (a, d), zero-field heat flow curves (b, e) and transition entropy changes (c, f) of the nominal Ni35Co15Mn50-yTiy (a, b, c) and Ni33Co17Mn50-yTiy (d, e, f) series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' All samples are labeled with the composition determined by EDX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' (b, e) Shaded areas under the heat flow curves qualitatively illustrate the transition entropy change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' (c, f) The transition entropy change is determined by PPMS-14T 푐푝 measurements (blue), sapphire-calibrated DSC 푐푝 measurements (green), DSC heat flow measurements (red) and the Clausius-Clapeyron equation (open circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Literature values [35, 37, 39, 42, 44] (crosses) are given for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Dashed lines are drawn to guide the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The compensation temperatures 푇푐표푚푝 can be estimated as 75 K and 300 K for the Ni35Co15Mn50-yTiy and Ni33Co17Mn50-yTiy series, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' The estimated temperature at which the martensitic transformation temperature 푇푡 is equal to the austenite Curie temperature 푇 퐴 퐶 is given as vertical solid line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Shaded areas at low and high temperatures illustrate regions with arrested martensitic transformations and chemically inhomogeneous dual phase microstructures, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Beckmann, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Koch, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Pfeuffer, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 6 of 14 (a) 180 (d) 180 Ni35Co15Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' T 160 160 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='6 μ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='H= 1 T μ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='H= 1 T Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='1 140 140 Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Magnetization [Am’kg"] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='. 120 120 Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='2 Mna6Ti, Vi Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' 100 100 MnzTi 11 Ni 80 80 Ni .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='Mnao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='Ti, 5Mn 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='8Ti10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='1 60 60 Mn Ni 40 40 Ni 20 20 0 100 700 800 0 0 200 300 400 500 600 100 200 300 400 500 600 700 800 Temperature [K] Temperature [K] (b) 500 (e) 500 Mr Ti 400 400 μ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='H=OT μ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='H= O T 300 300 Exo!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' Exo 200 200 Heat flow [Wkg" Heat flow [Wkg" 100 100 0 0 100 100 200 200 300 300 400 400 500 500 0 100 200 300 400 500 600 700 800 0 100 200 300 400 500 600 700 800 Temperature [K] Temperature [K] (c) (f) 90 90 Mn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='△sH 80 80 Arrested martensitic transformations O △st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content=' cc T >T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E2T4oBgHgl3EQfgQcG/content/2301.03934v1.pdf'} +page_content='A Entropy change [(kgK)1 70 70 Entropy change [(kgk)" T 0. +Another ingredient of the proof is a construction of a number field with very small units compared +to its discriminant. A construction of Shapira following Cassels is a series of number fields K whose +unit groups are generated by elements u ∈ O× +K such that log |u| = O(log disc(K)) (which is excep- +tionally small compared to the bound given by the class number formula: log |u| = O(disc(K)1/d−1), +provided that the unit lattice is sufficiently balanced). Applying Conjecture 1.6 to these number +fields, we get that that the different measures µI for I ∈ ClOK are equidistributed. Using this +equidistribution we can approximate a generic point x ∈ Xn of our desired ergodic measure µ, +namely there is I ∈ ClOK such that d(x, x′) = O(disc−⋆(K)) for some x′ ∈ supp µI. Hence for +r = θ(log(disc(K))) ball in A, a ∈ BA(r) we have that ax is close to ax′ ∈ supp(µI). On the other +hand, the orbit ax′ is not much larger. The periods are controlled by the size of the units, which +3 + +are again O(log(disc(K))). This would imply that we can approximate every ball in an A-orbit +with measures µI for I in the class group of the special number field K. +To overcome the need to use Conjecture 1.6, we prove a weaker form on a special collec- +tion of orders. +We take a measure µI as above and apply to it a p-Hecke operator Tp. +On +the one hand, Hecke operators are known to have well-behaved equidistribution properties (See +[Clozel et al.(2001)Clozel, Oh, and Ullmo]). We expect to have f(TpµI, mXn) = O(p−⋆). On the +other hand, the resulting measure TpµI is the average of measures on compact orbits, corresponding +to modules of a sub-order R = Z + pOK ⊂ OK of index [OK : R] = pd−1. +While this guarantees that the orbits would be equidistributed, to bound the sizes of the orbits +we tweak the construction of K. Not only OK should have units of logarithmic size, but also R. +This guarantees that TpµI is an average of measures of logarithmically small compact orbits, and +enables the proof to work without Conjecture 1.6. +Remark 1.7. Conjecture 1.6 seems very complicated. It was proven in case n = 2 in [Duke(1988)]. +A non-effective version of it is proven for n = 3 under some splitting restriction in [Einsiedler et al.(2011)Einsiedler, Lindenstrauss, Michel, and Venkatesh]. +The proof uses a deep and nontrivial number theoretic result, namely, the subconvexity estimate of +L functions of cubic fields. +The proof of Theorem 1.5 has similar steps. We start with a compact orbit Ax0, constructed +using a variation on the techniques used in [Shapira(2016)]. As in [Shapira(2016)], this compact +orbit will lie in the cusp for most of its lifetime. +However, the points of the orbits distributes +continuously in the cusp, in the sense that some are further in the cusp then others. Then we take +some Hecke operator T . It pulls points out of the cusp in fixed rate, that is, if λ1(x) was very +small then on average over x′ ∈ T (x0) we have λ1(x′) = Θ(min(pαλ1(x0), 1)). We choose T such +that c proportion of T (Ax) will leave the cusp while the remaining 1 − c proportion of T (Ax) will +stay in the cusp. We will use the Hecke equidistribution to show that the part that leaves the +cusp equidistributes. A possible problem is that T (Ax0) is composed of possibly many A-orbits. +To solve that we arrange the following additional property: a positive proportion of x0-s T -Hecke +neighbors will be on the same A orbit. This implies that T (Ax0) contains a long A-orbit, Ax1. +Now we use ergodicity to deduce that the distribution estimate of T (Ax0) holds for its long A-orbit +subset Ax1. This orbits will give us Theorem 1.5. We further remark that the method of finding a +Hecke neighbor whose diagonal orbit occupies positive proportion of the entire Hecke neighbors is +also used in another paper by Uri Shapira and Menny Aka, [Aka and Shapira(2018)]. +Remark 1.8 (Comparison to [Shapira and Zheng(2021)]). Plugging c = 1 to our construction +yields examples similar in nature to those of [Shapira and Zheng(2021)]. Both are obtained from +Hecke neighbor of a Minkovski embedding of a number field. However our analysis is simpler in +this case, as a consequence of restricting the number field we use. +4 + +1.4 +Further Research +The first natural improvement of Theorem 1.1 is the full approximation. +Open Question 1.9. Let µ ∈ M(Xn)A +c . +Can it be the limit measure of other measures in +M(Xn)A +c ? In other words, is µ ∈ M(Xn)A +c \ {µ}? +It can be seen that simply improving the bounds we give in this paper cannot give a positive +answer to this question, for the following reason. If n = 3, Ay is a compact orbit and Ax is an +orbit. We can consider the set B = {a ∈ A : dX3(ax, Ay) < δ}. The connected components of B +are roughly hexagons. It can be seen that two such hexagons H1, H2 cannot have R long parts of +the boundaries which are δR close to one another, for some δ > 0 and all R > 0 sufficiently large. +1.5 +Acknowledgments +The second author would like to express his deep gratitude to Uri Shapira for his support and +encouragement. Without them, this paper would not exist. We thank Uri Shapira for bringing +the main questions answered in this paper into our attention and for many intriguing discussions +with him which contributed a lot to this paper. Moreover, the first author thanks Andreas Wieser +for many fruitful discussions. The second author acknowledges the support of ISF grants num- +ber 871/17. This work has received funding from the European Research Council (ERC) under +the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. +754475). +2 +Notation and Preliminaries +Definition 2.1 (O-notations). For two real functions f, g on a set A we write f ≪ g if there exists +a constant C independent on the parameters of f and g such that |f| ≤ Cg on A. The notation +O(g) will refer to some implicit function f which satisfies f ≪ g. The notation Θ(g) will refer to +some implicit function f which satisfies g ≪ f ≪ g. Whenever r is a parameter going to 0 or ∞, +the notation or(g) will refer to some implicit function f which satisfies f ≪ g · h, for some implicit +function h → 0 as r goes to 0 or ∞ respectively. In this case, we sometimes write f = O(g). +Whenever r is a parameter going to either ∞ or 0 and f +Fix ∥·∥ to denote the supremum norm on Rn. Let Xn denote the space of unimodular lattices +in Rn and let d(·, ·) denote the metric on Xn coming from the operator norm on linear operators +on (Rn, ∥·∥). +Let Rn−1 +0 += {v ∈ Rn : � +i vi = 0} and we abuse notations and define exp = +exp ◦ diag : Rn−1 +0 +→ A the standard parametrization. We denote by mXn probability measure on +Xn = SLn(R)/ SLn(Z) coming from the Haar measure on SLn(R). We now present formally the +5 + +relation between compact A-orbits in Xn and subgroups of number fields. Denote by L, U ⊂ SLn(R) +the subgroups of lower and upper triangular matrices with diagonal 1. +Definition 2.2 (Space of Measures). Let M(Xn) denote the space of finite measures on Xn endowed +with the topology induced by µk → µ if for any f ∈ Cc(Xn) it holds that µk(f) → µ(f). We define +a metric on Xn which induces this topology by letting, for any µ1, µ2 ∈ M(Xn): +d(µ1, µ2) = sup +ǫ>0 +sup{ǫ| +� +fdµ1 − +� +fdµ2| : f : Xn → R : f is 1-Lipschitz and is supported on Kǫ} +(2.1) +where Kǫ = {x ∈ Xn : λ1(x) > ǫ}. +The following definition is relevant to §4. +Definition 2.3 (Special subgroups of SLn(R)). Let +w0 = +�n + 1 − 2i +2 +�n +i=1 +∈ Rn−1 +0 +, +at = exp(tw0) for all t ∈ R, +(2.2) +and note that +L = {g ∈ SLn(R) : atga−t +t→∞ +−−−→ I}, +U = {g ∈ SLn(R) : a−tgat +t→∞ +−−−→ I}, +that is, L, U are the contracting and expanding horospheres with respect to at. +Definition 2.4. For every degree n, totally real number field K denote by LatK the set of free +Z-modules of rank n in K, where we identify two lattices Λ1, Λ2 ⊂ K if Λ1 = kΛ2 for some +k ∈ K×. For every rank n, Z-module Λ ⊆ K consider the lattice xΛ := σ(Λ)/cov(σ(Λ)) ∈ Xn, +where σi : K ֒→ R; i = 1, . . . , n is some ordering of the natural embeddings of K and let σ = +(σ1, . . . , σn) : K → Rn denote their concatenation. Denote by OΛ = {k ∈ K : kΛ ⊆ Λ}. This is +a ring. Denote by O×,>0 +Λ += {u ∈ O× +Λ : σi(u) > 0 : i = 1, . . . , n}. For every U ⊆ O×,>0 +K +denote +AU = {diag(σ1(u), σ2(u), ..., σn(u)) : u ∈ U}. +Theorem 2.5. Fix d ≥ 0. For every totally real number field K, [Λ] ∈ LatK, and choice of the +ordering of the real embeddings of K, the orbits AxΛ is compact, independent of the representative +Λ ⊂ K of [Λ] ∈ LatK and this is a one to one parametrization of all compact A-orbits in Xn. +This theorem is equivalent to [McMullen(2005)], apart from the part of this parametrization +being one-to-one, whsich is Folklore and we do not use in this paper. +Definition 2.6. For every point x ∈ Xn such that the orbit Ax is compact, denote µAx the A- +invariant measure on Ax. For every [Λ] ∈ LatK denote µΛ = µAxΛ. +6 + +3 +Effective Approximations +In this section, we prove an effective approximation theorem of points in Xn by points of compact +orbits. Formally, the goal is to prove the following lemma. +Lemma 3.1. For every compact set K ⊂ Xn there exists a sequence tk +k→∞ +−−−−→ ∞ such that for +every y ∈ Xn there exists a sequence (yk)k ⊂ Xn such that d(yk, y) < exp +� +− +⌊n/2⌋ +4n(n2−1)tk +� +such that +yk is stabilized by a subgroup exp(Λk) ⊆ A where Λk ⊆ Rn−1 +0 +is generated by vectors (v(i) +k )n−1 +i=1 such +that +���� +� +v(i) +k +� +j − (1 − δijn)tk +���� = O(1). +Remark 3.2. Lemma 3.1 implies in particular that Axk is compact with total volume nn−3/2tn−1 +k ++ +O(tn−2 +k +) in the Rn−1 +0 +parametrization, and in addition that for every v ∈ Rn−1 +0 +and for y, yk as in +the lemma, we have: +d((exp v).yk, (exp v).y) ≤ O +� +exp +� +max +1≤i,j≤n |vi − vj| − +⌊n/2⌋ +4n(n2 − 1)tk +�� +. +(3.1) +Since the set B0 = {v ∈ Rn +0 : max1≤i,j≤n |vi − vj| ≤ 1} has volume vol(B0) = +1 +√n, we deduce that a +portion of +� +1 +n +⌊n/2⌋ +4n(n2−1) +�n−1 ++ O(1/tk) of the orbit Ayk approximates the the orbit Ay. +The proof of Lemma 3.1 will be composed of two components. The first is an approximation of +lattices via Hecke neighbors. We will define Hecke neighbors in Definition 3.4 and obtain a good +approximation by them in Lemma 3.6. The second component generates infinitely many points +in Xn with compact A-orbits, with a control on the geometry of their stabilizers in A and the +stabilizers of their sublattices of a given index. This gives us the arsenal of points on which we +apply Lemma 3.6 to deduce Lemma 3.1. +3.1 +Hecke Density +The source of our good approximation comes from the quantitative version of the equidistribu- +tion of Hecke neighbors. There are many references to the equidistribution of Hecke neighbors +such as +[Clozel et al.(2001)Clozel, Oh, and Ullmo, Theorem 1.1], +[Gan and Oh(2003), Theorem +3.7], [Eskin and Oh(2006), Theorem 1.2]. In this section we will cite the result of Laurent Clozel, +Hee Oh and Emmanuel Ullmo [Clozel et al.(2001)Clozel, Oh, and Ullmo] and deduce the approxi- +mation result we need. +Definition 3.3 (Function space). Let L2 +0(Xn) = {f ∈ L2(Xn, mxn) : +� +Xn fdmXn = 0} be the +Hilbert space of L2 function on Xn with 0 mean. +7 + +Definition 3.4 (Definition of p-Hecke neighbors and Hecke operator). For every sequence of inte- +gers k1 ≤ k2 ≤ ... ≤ kn consider: +a = +1 +p(k1+···+kn)/n diag(pk1, pk2, . . . , pkn) ∈ SLn(R). +For every x = g SLn(Z) ∈ Xn denote Ta(x) = g SLn(Z)a SLn(Z). +This set is is finite since +a SLn(Z)a−1 is commensurable to SLn(Z). The size #Ta(x) = #(SLn(Z)a SLn(Z)/ SLn(Z)) de- +pends only on k1, ..., kn and not x. Equivalently, +Ta(x) = +� +1 +n� +cov(x′) +x′ : x′ ⊆ x with x/x′ ∼= Z/pk1Z ⊕ · · · ⊕ Z/pknZ +� +. +For every function f ∈ L2(Xn) define the Hecke action on functions +T F +a (f)(x) = +1 +#Ta(x) +� +x′∈Ta(x) +f(x′) ∈ L2 +0(Xn). +For every point x ∈ Xn, define the Hecke action on measures +T M +a (x) := +1 +#Ta(x) +� +x′∈Ta(x) +δx′ +and for every measure µ on Xn define +T M +a (µ) := +� +Xn +T M +a (x)dµ(x). +The following theorem is a particular case of [Clozel et al.(2001)Clozel, Oh, and Ullmo, Theorem +1.1], specialized for SLn as in [Clozel et al.(2001)Clozel, Oh, and Ullmo, Example 5.1]. +Theorem 3.5. For every prime p and k1 ≤ k2 ≤ ... ≤ kn, a ∈ SLn(R) as in Definition 3.4, the +operator norm of T F +a +�� +L2 +0(Xn) is bounded by: +���T F +a +�� +L2 +0(Xn) +��� ≤ +� +i≤n/2 +1 +p(kn+1−i−ki)/2 +(kn+1−i − ki)(p − 1) + (p + 1) +p + 1 +≤ p− 1 +2 +�n/2 +i=1(kn+1−i−ki)·C(k1, ..., kn), +where C(k1, ..., kn) depends polynomially on k1, ..., kn. +Lemma 3.6. For every compact subset K ⊂ Xn there exists C = C(K) > 0 such that for any +x, y ∈ K and p, k1 ≤ k2 ≤ ... ≤ kn, a ∈ SLn(R) as in Definition 3.4, there exists a Hecke neighbor +8 + +x′ ∈ Ta(x) such that: +d (x′, y) ≤ C(K) +���T F +a +�� +L2 +0(Xn) +��� +1/(n2−1) +, +(3.2) +where α > 0 is some constant depending only on n. +Proof. Recall the right invariant Riemannian metric dSLn(R) on SLn(R), and its descend to Xn, the +metric dXn. Let r0 < min(inj(y), inj(y)), where inj(x) is the injectivity radius of x, that is, the +maximal radius r such the translation map g �→ gx is injective on BSLn(R)(I; r), itself being the +radius r ball around the identity in SLn(R). Let fx = χBXn(x;r0) be the indicator of a radius r0 +ball of x. Then +� +Xn fxdmXn = vol(BSLn(R)(I; r0)) = Θ(rn2−1 +0 +), and a similar equality holds for +fy = χBXn(y;r0). Denote by vr0 = vol(BSLn(R)(I; r0)). Then ˜fx = fx − vr0 ∈ L2 +0(xn) and ˜fy = +fy − vr0 ∈ L2 +0(xn) have norm ∥fx∥2 = ∥fy∥2 = vr0(1 − vr0)2 + (1 − vr0)v2 +r0 = (1 − vr0)vr0. Consider +� +˜fx, T F +a ( ˜fy) +� +. On the one hand, it is at most +���T F +a +�� +L2 +0(Xn) +��� ∥fx∥∥fy∥ = +���T F +a +�� +L2 +0(Xn) +��� (1 − vr0)vr0. +On the other hand, consider the set: +T −1 +a (BXn(y, r0)) = {x′ ∈ Xn : Ta(x′) ∩ BXn(y, r0) ̸= ∅}. +and note that T F +a ( ˜fy)|(T −1 +a +(BXn(y,r0)))c ≡ −vr0. Assume that T −1 +a (BXn(y, r0)) ∩ BXn(x, r0) = ∅, +and get that T F +a (fy)|BXn(x,r0) ≡ −vr0. It follows that: +� +˜fx, T F +a ( ˜fy) +� += +� +Xn +˜fxT F +a ( ˜fy)dmXn = +� +Xn +fxT F +a ( ˜fy)dmXn = +� +BXn(x,r0) +T F +a ( ˜fy)dmXn += vr0 · (−vr0) = −v2 +r0. +We deduce that +v2 +r0 ≤ +���T F +a +�� +L2 +0(Xn) +��� · ∥ ˜fx∥ · ∥ ˜fy∥ = +���T F +a +�� +L2 +0(Xn) +��� (1 − vr0)vr0, +and hence vr0 ≤ +���T F +a +�� +L2 +0(Xn) +���. Using this logic in reverse, we deduce that if vr0 > +���T F +a +�� +L2 +0(Xn) +��� +then T −1 +a (BXn(y, r0)) ∩ BXn(x, r0) ̸= ∅, that is, there exist x′ = g0x, y′ = g1y such that y′ ∈ Ta(x′) +and g0, g1 ∈ BSLn(R)(I, r0). +Thus g−1 +0 y′ = g−1 +0 g1y ∈ g−1 +0 Ta(x′) = Ta(x). +On the other hand, +g−1 +0 y′ = g−1 +0 g1y satisfies dXn(g−1 +0 g1y, y) ≤ 2r0. +Altogether, we have proved that if +���T F +a +�� +L2 +0(Xn) +��� < r0 ≤ min(inj(x), inj(y)) then there exists +x′′ = g−1 +0 g1y with x′′ ∈ Ta(x) and dXn(x′′, y) ≤ 2r0. Now, let K ⊂ Xn be a compact set. Denote +the minimum of the injectivity radius on K by rK > 0. If vrK > +���T F +a +�� +L2 +0(Xn) +���, then we can find +r0 = Θ( +���T F +a +�� +L2 +0(Xn) +��� +1/(n2−1) +) with r0 < rK and vr0 > +���T F +a +�� +L2 +0(Xn) +���, and the desired follows. If +vrK ≤ +���T F +a +�� +L2 +0(Xn) +���, then the desired follows for C(K) = +diam(K) +(vrK )1/(n2−1) . +■ +9 + +Remark 3.7 (Composition of Hecke operators). The composition of Hecke operators is a linear +combination of different Hecke operators. This follows from the double-coset description, as the +multiplication of two finite double-cosets is again a double-coset. We mention explicitly the case of +the operators a, a′ defined with (−k, 0, 0, . . ., 0), (−l, 0, 0, . . ., 0) respectively, where 0 ≤ k ≤ l. The +composition is +T F +a ◦ T F +a′ = T F +a′ ◦ T F +a = pn−1(p − 1) +pn − 1 +T F +ak+l + +l−1 +� +i=1 +pn−1 − 1 +pn − 1 +· p − 1 +pi +· T F +ak+l−i,i + pn−1 − 1 +pn − 1 +· +1 +pl−1 T F +ak,l +Where ak′,l′ corresponds to −k′ ≤ −l′ ≤ 0 ≤ 0 ≤ · · · ≤ 0. A similar equality hold for T M. +3.2 +Construction of special number field +Theorem 3.8. For every prime number p ≡ 1 mod 2n sufficiently big as a function of n, one can +find a totally real number field K with the following properties: +(a) The unit group O× +K contains units u1, ..., un with u1u2 · · · un = 1. +(b) One can order the real embeddings σ1, ..., σn : K → R such that σi(uj) > 0 for all 1 ≤ i, j ≤ n, +and +log σi(uj) = + + + +−2d(d − 1) log p + O(1), +if i = j, +2d log p + O(1), +if i ̸= j. +(3.3) +(c) The units ui lie in the ring Z + pOK. +Proof. Since p ≡ 1 mod 2n, the polynomial xn + 1 has a n different solutions mod p. By Hensel’s +Lemma (See [Conrad(2015)]), it has n solutions mod pn, call them a′ +0, ..., a′ +n−1 ∈ {1, ..., pn −1}. Let +ai = a′ +i + 2ipn for i = 0, . . . , n − 1, and note that pn + 1 ≤ ai+1 − ai ≤ 3pn for all 0 ≤ i ≤ d − 2. +consider now the polynomial R(x) = +(px−a1)(px−a2)···(px−an)−1 +pn +. +It has integer coefficients since +by assumption, � +i1<... 0. Consequently, due to the mean value +theorem, there is xi0 ∈ [ai0/p, x′ +i0] with R(xi0) = 0. By Eq. (3.4) we deduce that: +0 = R(xi0) = − 1 +pn + (−1)n−i0αi0pn2−2n+1(xi0 − ai0/p) + O(pn2−3n+2)(xi0 − ai0/p)2. +(3.5) +This implies: +1 +pn = (xi0 − ai0/p) +� +(−1)n−i0αi0pn2−2n+1 + O(pn2−3n+2)(xi0 − ai0/p) +� += (xi0 − ai0/p)(−1)n−i0αi0pn2−2n+1 � +1 + O +� +p−n2�� +, +and hence, +xi0 = ai0/p + +(−1)n−i0 +pn2−n+1αi0 ++ O +� +1 +p2n2−n+1 +� +. +Since |ai0/p − xi0| < 1 we deduce that all xi are different, and hence these are the distinct roots +of R. Since |ai − aj| ≥ pn for all i ̸= j it follows that |ai/p − xj| ≥ pn−1 − 1 > 1. It follows that +R = �d +i=1(x − xi) is irreducible, as if R0 = � +i∈I(x − xi) is an integer polynomial i0 /∈ I then in the +ring Z[x]/(R0) the element px − ai0 is a unit, but it’s norm in Z[x]/(R0) is � +i∈I(pxi − ai0), which +is greater the 1 in absolute value. +11 + +The number field K = Q[α] where R(α) = 0 has units u′ +i = pα − ai. Let ui = (u′ +i)2. The real +embeddings of K are σi : α �→ xi, sends has log σi(uj) = 2 log |pxi − aj|. Now Eq. (3.3) follows +from properties of xi. +■ +3.3 +Application of Hecke operator to compact orbits +In this section, we introduce the following lemma, which describes how to verify that a Hecke +operator splits a compact orbit into sufficiently many compact diagonal orbits. +Lemma 3.9. Let K be a totally real number field of degree n, let xOK ∈ Xn the point with compact +orbit corresponding to the Z-module OK and let p, k1 ≤ · · · ≤ kn, a ∈ SLn(R) as in Definition 3.4. +Let U ⊆ O×,>0 +K +be a subgroup and AU ⊂ A the corresponding diagonal subgroup as in Definition +2.4. If U ⊆ Z + pkOK and 0 ≤ k1 ≤ kn ≤ k then AU ⊆ stabA(x′) for every x′ ∈ Ta(xOK). +Proof. Let u ∈ U. Since u ∈ Z + pkOK, there is m ∈ Z such that u ≡ m mod pk. Thus for every +¯b ∈ OK/(pk) we have u¯b = m¯b, that is, the multiplication by u action on OK/(pk) is in fact a +multiplication by a scalar. Hence, the element au = diag(σ1(u), σ2(u), . . . , σk(u)) ∈ AU preserves +xOK, pkxOK and acts on the quotient by multiplication by the scalar m. This implies that au acts +on the set of mid-groups {Λ : pkxOK ⊆ Λ ⊆ xOK} trivially. Indeed, there is an isomorphism +{Λ : pkxOK ⊆ Λ ⊆ xOK} ∼= {¯Λ ⊆ xOK/pkxOK}, +and u acts on the right-hand side as a multiplication by a scalar, which preserves all groups. This +implies that au preserves all Hecke neighbors, by the second construction (See Definition 3.4). +■ +Remark 3.10. Lemma 3.9 remains true if we replace xOK by xΛ for some Λ ∈ LatK and U by a +subgroup of OΛ satisfying U ⊆ Z + pkOΛ. The proof is similar. +Corollary 3.11. There is a compact subset C ⊂ Xn such that for any prime p ≡ 1 mod 2n large +enough as a function of n, there exists xp ∈ C, and a lattice Λ ⊂ Rn−1 +0 +such that, +1. For every Hecke operator Ta with 0 ≤ k1 ≤ · · · ≤ kn ≤ 1 and for every element y ∈ Ta(x) we +have that exp(Λ) ⊆ A stabilize y. +2. The lattice Λ is generated by vectors (v(i))n−1 +i=1 satisfying +��� +� +v(i)� +j − (2n − 2n2δij) log p +��� = O(1). +Proof. Fix p ≡ 1 mod 2n sufficiently big for Theorem 3.8. +Let K be the number field con- +structed in Theorem 3.8, and u1, . . . , un the corresponding units. +Let (σi)n +i=1 be the real em- +beddings of K. Let Λ be the subgroup of Rn−1 +0 +generated by (log σi(uj))n +i=1 for j = 1, . . . , n − 1. +The bounds on uj in Theorem 3.8 imply Part 2. +Consider the point xOK ∈ Xn. +It follows +from [Tomanov and Weiss(2003), Proposition A.1] that there is a compact set C ⊆ Xn which in- +tersects every A-orbit. Hence there is xp = mxOK ∈ C for some m ∈ A. Since the SLn(R) action +12 + +on Xn commutes with Ta, to prove Part 1 it is sufficient to prove that exp(Λ) ⊆ A stabilize y for +every y ∈ Ta(xOK). This follows from Lemma 3.9 and Property 3 of the unit u1, ..., uk. +■ +3.4 +Proof of Lemma 3.1 +The points we construct are indexed by primes p ≡ 1 mod 2n. +Fix such a prime p. +Denote +t′ +p = 2n log p and let xp and Λ be as in Corollary 3.11 with t′ +p instead of tp. Note that Corollary +3.11 implies the exact bounds on a set of generators of Λ we need. +Let k1 = · · · = k⌊n/2⌋ = +0, k⌊n/2⌋+1 = ... = kn = 1, and a ∈ A as in Definition 3.4. We claim that for some yp ∈ Taxp +we have d(y, yp) ≪ exp(− +⌊n/2⌋ +4n(n2−1)t′ +p). Indeed, by Theorem 3.5 we have +���T F +a +�� +L2 +0(Xn) +��� ≤ p−⌊n/2⌋/2. +Thus by Lemma 3.6, there is yp ∈ Ta(xp) with +d(yp, y) = O(p +− +⌊n/2⌋ +2(n2−1) ) = O +� +exp +� +t′ +p · +⌊n/2⌋ +4n(n2 − 1) +�� +. +We now choose tp = t′ +p = O(1) to be smaller then tp to ensure +d(yp, y) < exp +� +tp · +⌊n/2⌋ +4n(n2 − 1) +� +. +This preserves the bounds on the generators of Λ. +Note that the implicit constant was provided by Theorem 3.5 and depends only on the compact +set containing y. +■ +4 +Approximation of Invariant Measures +In this section, we prove Theorem 1.1. We approximate measures by approximating their generic +points. When we want to approximate a finite collection of measures simultaneously, we need to +find regions of A-orbits that approximate all measures on subregions. The following lemma is our +tool for this purpose: +Lemma 4.1 (Diagonal closing lemma). For every compact set K ⊆ Xd there are a compact sets +KA ⊂ A, KL ⊂ L, KU ⊂ U, such that for every x0, x1 ∈ K there are kA ∈ KA, kL ∈ KL, kU ∈ KU +such that kUkAkLx0 = x1. Here A, L, U are defined as in Definition 2.3. +Proof. First, we will show that for every x0, x1 ∈ Xd we have x0 ∈ LAUx1. Since U is the expanding +horosphere of at, almost every point in Ux1 is generic with respect to the forward at action and the +Haar measure mXn. Choose a at-generic point point u0x1 ∈ Ux1. The LU-decomposition implies +that the product map U × A × L → SLn(R) is one to one. Since U × A × L and SLn(R) are n2 − 1 +dimensional, this implies that UAL ⊆ SLn(R) is open. Hence UALx0 is an open set in Xn, and +13 + +hence of positive measure. Hence: +1 +T +� T +0 +χatu0x1∈UALx0dt +T →∞ +−−−−→ mXn(UALx0) > 0 +(4.1) +and hence for some t > 0 we have atu0x1 ∈ UALx0. Since AU = UA we deduce that x0 ∈ LAUx1. +Let ∥ · ∥A : A → [0, ∞), ∥ · ∥L : L → [0, ∞), ∥ · ∥U : U → [0, ∞) be the distance from the identity. +Define by f : K2 → [0, ∞) the map attaching +f(x1, x2) = inf{max(∥a∥A, ∥l∥L, ∥u∥U) : a ∈ A, l ∈ L, u ∈ U : x0 = laux1}. +Since LAU is open and the multiplication map L × A × U → LU is a homeomorphism, if we have +laux0 = x1 for a ∈ A, l ∈ L, u ∈ U, then for every sufficiently close ˆx0 ∼ x0, ˆx1 ∼ x1 there are +ˆa ∈ a, ˆl ∈ L, ˆu ∈ U arbitrarily close to a, l, u such that ˆuˆlˆaˆx0 = ˆx1. This implies that f is upper +semicontinuous. In particular, for every (x0, x1) ∈ K2 we get that f is bounded in a neighborhood +of (x1, x2). Since K2 is compact this implies that f is bounded everywhere, as desired. +■ +Definition 4.2. For v, v′ ∈ Rn−1 +0 +, we say that v ≺ v′ if for all i = 1, . . . , n − 1 we have vi+1 − vi ≤ +v′ +i+1 − v′ +i. For every v0 ∈ Rn−1 +0 +with 0 ≺ v0, define a box Sv0 = {v ∈ Rn−1 +0 +: 0 ≺ v ≺ v0}. A boxed +map is a map f : Sv0 → Xd of the form f(v) = exp(v).x0 for some x0 ∈ Xn. For every compact +set K ⊂ Xn, a boxed map f0 : Sv0 → Xd is said to be K-bounded if f(0), f(v0) ∈ K. Recall that +w0 = ( d+1−2i +2 +)n +i=1. +In the following lemma, we fix K, K′ and let R → ∞. The constants in the O-notations depend +on K, K′ and hold as R → ∞. +Corollary 4.3 (Gluing boxes). Let K ⊂ Xn be compact and K ⊂ K′ ⊂ Xn a compact neighborhood. +In this corollary all O notations depend on K, K′. Let R > 0. Fix any two K-bounded boxed maps +f1 : Sv1 → Xn, f2 : Sv2 → Xn with v1, v2 ≻ Rw0. Then there exists a K′-bounded boxed map +f : Sv3 → Xn with +(1) v3 = v1 + v2 + OK,K′(1); +(2) There are two points w1, w2 ∈ Sv3 with ∥w1∥ = oR(R), ∥w2−v1∥ = oR(R) and 0 < ρ = oR(R) +we have that for all i = 1, 2 and v ∈ Svi−2ρw0, +dXn (fi(ρw0 + v), f(wi + v)) = oR(1). +See Figure 1 for a visualization of these conditions. +14 + +f1 +f2 +f +Sv1 + w1 +Sv2 + w2 +Sv3 +w1 + v1 +w1 +w2 + v2 +w2 +0 +v3 +v1 +v1 + v2 +Figure 1: Plot of the boxes in Corollary 4.3. One can see Sv3, and in it Sv1 + w1, Sv2 + w2, and in +them the regions approximated by f1, f2. +Proof. Let KA ⊂ A, KL ⊂ L, KU ⊂ U as in Lemma 4.1, constructed for K. Assume KA = exp(K0) +for K0 ⊆ {w ∈ Rn−1 +0 +: |wi − wi+1| < C0 for all i = 1, . . . , n − 1}. Note that +for all T > 0, v ≥ T w0, u ∈ KU we have d(exp(−v)u exp(v), I) = O(exp(−T )), +(4.2) +and similarly, +for all T > 0, v ≥ T w0, l ∈ KL we have d(exp(v)l exp(−v), I) = O(exp(−T )). +Apply Lemma 4.1 for f1(v1), f2(0), we get that there are v0 ∈ Rn−1 +0 +, kl ∈ KL, ku ∈ KU such that +|(v0)i − (v0)i+1| < C0 for all i = 1, . . . , n − 1, +and +kl exp(v0)kuf1(v1) = f2(v2). +Let ρ = +√ +R, v3 = v1 + v2 + v0, x0 = exp(−v1)ku exp(v1)f1(0), f : Sv3 → Xn defined by +f(v) = exp(v)x0 for all v ∈ Sv3, v1 = ρw0 and v2 = v1 + ρw0 + v0. By Eq. (4.2) we get that +d(exp(−v1)ku exp(v1), I) = O(exp(−R)) = oR(1). +hence we deduce that for R sufficiently big, +x0 ∈ K′. +Moreover, for all v ∈ Sv1−2ρw0 we have +f(v0 + v) = exp(v0 + v)x0 = exp(v0 + v − v1)ku exp(v1 − (v0 + v))f1(v0 + v). +15 + +Since v ≤ v1 − 2ρw0, v0 = ρw0 we get that v1 − (v0 + v) ≥ ρw0, and hence +d (exp(v0 + v − v1)ku exp(v1 − (v0 + v)), I) = oR(1), +which implies that d(f(v0 + v), f1(ρw0 + v)) = oR(1). +Note that +f(v3) = exp(v1+v2+v0)x0 = exp(v2) exp(v0)kuf1(v1) = exp(v2)k−1 +l +f2(0) = exp(v2)k−1 +l +exp(−v2)f2(v2). +From here the proof of the remaining conditions for f2 and f(v3) are symmetric to the approximation +of f1 and f1(0). +■ +Corollary 4.4. For any µ1, . . . , µk ergodic A-invariant measures on Xn and any R > 0 there exists +xR ∈ Xn such that for every i = 1, . . . , k: +lim +R→∞ +1 +vol(SRw0) (v �→ exp (v + iRw0) .xR)∗ mRn−1 +0 +|SRw0 = µi +(4.3) +In addition, the set {xR : R > 0} is pre-compact. +Proof. Let K be a compact set with µi(K) > 0 for each i, and for each i = 1, . . . , k choose yi ∈ K +to be a generic point for the A action on µi. By ergodicity of µi: +lim +R→∞ +1 +vol(SRw0) +� +SRw0 +χexp(v)yi∈KdmRn−1 +0 += µi(K) > 0. +Denote +ρi = sup{ρ > 0 : exp(v)yi /∈ K, ∀v ∈ Rw0 − Sρw0}. +Since +� +Rw0−Sρw0 χexp(v)yi∈KdmRn−1 +0 += 0, we get that ρi +R +R→∞ +−−−−→ 0. By the Definition of ρi, there is +vi ∈ Rw0 − Sρiw0 such that exp(vi)yi ∈ K. Since ∥vi − Ri∥ = O(ρi) = oR(R) we deduce that +1 +vol(Svi) (v �→ exp(v)yi)∗ mRn−1 +0 +|Svi +R→∞ +−−−−→ µi. +The corollary follows now from iteratively applying Corollary 4.3 to glue the boxed maps (v �→ +exp(v)yi)|Svi to one boxed map f, using an increasing set of compact neighborhoods K ⊂ K1 ⊂ +K2 ⊂ · · · ⊂ Kk. +■ +As a corollary of Lemma 3.1 and Corollary 4.4 we can prove Theorem 1.1: +Proof of Theorem 1.1. Let µ1, µ2, . . . , µk be ergodic A-invariant measures on Xn. For any R > 0, +let xR ∈ Xn be the point guaranteed by Corollary 4.4. Let K the compact closure of {xR : R > 0}. +16 + +Lemma 3.1 guarantees a sequence (tm)∞ +m=0 such that tm +m→∞ +−−−−→ ∞ such that for all R > 0, m ≥ 0 +there is yR,m ∈ Xn such that d(xR, yR,m) < e−αtm, where α = +⌊n/2⌋ +4n(n2−1) > 0, and yR,m is stabilized +by a subgroup exp(ΛR,m) ⊆ A where ΛR,m ⊆ Rn−1 +0 +is generated by vectors (v(i) +R,m)n−1 +i=1 such that +���� +� +v(i) +R,m +� +j − (1 − δij)tm +���� = O(1). Choose Rm = +αtm +k(n−1) − √tm so that all v ∈ SkRmw0 we have +d(exp(v)xRm, exp(v)yRm,m) = om(1). This implies that +lim +m→∞ +1 +vol(SRmw0) (v �→ exp (v + iRmw0) .yRm,m)∗ mRn−1 +0 +|SRmw0 = µi. +(4.4) +The choice of Rm guarantees that SRw0 injects into Rn−1 +0 +/ΛRm,m and hence Eq. (4.4) implies that +limm→∞ µAyRm,m contains µi as an ergodic component. +■ +5 +Entire Mass Approximations +First, we prove Corollary 1.4 of Theorem 1.1 and [Shapira(2016), Theorem 1.1]. +Proof of Corollary 1.4. Let µ1, . . . , µk ∈ M(Xn)A +e . By [Shapira(2016), Theorem 1.1] we can find +(νm)m ⊂ M(Xn)A +c such that νm → 0 weakly. By Remark 1.2 of Theorem 1.1, for any m there +exists µ(m) ∈ M(Xn)A +c such that νm, µ1, . . . , µk all appear in µ(m)’s ergodic decomposition with +coefficients bounded below by Θ(k2−n). Any weak limit µ(∞) of (µ(m))m will belong to M(Xn)A +c +and satisfy, since νm → 0, that at least Θ(k2−n) of the mass of µ(m) escapes, namely µ(∞)(Xn) ≤ +1 − Θ(k2−n). On the other hand, for i = 1, . . . , k, µi will still appear in the ergodic decomposition +of µ(∞) with coefficient bounded below by Θ(k2−n) as desired. +■ +The remainder of this section will be dedicated to the proof of Theorem 1.5. The proof of this +theorem will follow the same lines as the proof of Theorem 1.1: We will first construct a special +number field and prove some result on its units. +Then we will connect these results to Hecke +operators and use the properties of Hecke operators to complete the proof. +In contrast to the +method in the previous section, here we will fix p to be the minimal prime which is congruent to 1 +mod 2n. +5.1 +Distribution of Hecke neighbors of points close to the cusp +The following corollary analyzes the distribution properties of Hecke operators where we begin with +a lattice high in the cusp. For that, we will use a result that follows from Theorem 3.5 but appeared +first at [Clozel and Ullmo(2004), Th´eor`eme 1.2]. +17 + +Corollary 5.1. Fix x ∈ Xn, and a prime number p. For k1 ≤ k2 ≤ ... ≤ kn, a ∈ SLn(R) as in +Definition 3.4 we have: +T M +a (x) +kn−k1→∞ +−−−−−−−→ mxn. +(5.1) +Remark 5.2. The convergence is uniform on compact sets. This follows from the fact that Ta(gx) = +gTa(x) for all g ∈ SLn(R), and the result for the particular case x = Zn ∈ Xn. +Recall the definition of Minkowski’s successive minima λi(x) for x ∈ Xn, i = 1, . . . , n from +[Cassels(2012), Chapter VIII]. They satisfy that a closed set U ⊆ Xn is compact if and only if +infx∈U λ1(x) > 0 if and only if supx∈U λn(x) < ∞. +Theorem 5.3. Fix a prime number p. Denote by a = diag(p−(n−1)/n, p1/n, p1/n, . . . , p1/n). For +every point x ∈ Xn and x′ ∈ Tak(x) we have +pk/nλ1(x) ≥ λ1(x′) +(5.2) +In addition let (xi)∞ +i=1 be a sequence of lattices and let ki +i→∞ +−−−→ ∞ be an integer sequence. Assume +that λ1(xi) ≥ p−(1−ε) ki +n for some ε > 0. Then for every δ > 0 there exists a compact set C(δ) such +that +lim inf +i→∞ T M +aki(xi)(C(δ)) > 1 − δ. +(5.3) +Remark 5.4. For different Hecke operators there are other thresholds, stated in terms of different +Minkowski successive minima. +Remark 5.5. Although we state no-escape-of-mass in Eq. (5.3), one can prove an equidistribution +result: +T M +aki(xi) +i→∞ +−−−→ mXn. +This result could simplify the proof of TODO, but its proof is too complicated and can be avoided. +The proof uses the fact that if k = k′ + s then T M +ak is a large component of T M +ak′ ◦ T M +as, and a p-adic +interpretation of Hecke operators. +Proof of Theorem 5.3. The bound on λ1 of every Hecke neighbor follows from the definition of +Hecke operators. Since x ⊂ p−k/nx′ we get λ1(x′) ≤ pk/nλ1(x). +Let x ∈ Xn be a lattice with λ1(x) ≥ p−(1−ε) k +n and x′ be a random point in Tak(x). We show that +for every δ > 0 there is a compact set C ⊂ Xn depending only on ε, δ such that P(x′ ∈ C) ≥ 1 − δ +provided that k is sufficiently large. +Let Cr = {y ∈ Xn : λ1(y) ≥ r} for some 1 > r > 0 which will be chosen later. To bound +18 + +P(x′ /∈ Cr), we will bound: +E(#(x′ ∩ B(r) \ {0})) ≥ P(x′ /∈ Cr). +Here B(r) is the radius r ball in Rn. Since p−k(n−1)/nx′ ⊆ x, we have: +E(#(x′ ∩ B(r) \ {0})) ≤ +� +v∈B(r)∩p−k(n−1)/nx +P(v ∈ x′). +To analyze the probability P(p−k(n−1)/nv ∈ x′) for v ∈ x consider pk(n−1)/nx′ ⊂ x as a kernel of a +random onto homomorphism ϕ : x → (Z/pk)n. Let ϕ : x → (Z/pk)n be a random onto map and +ϕ′ : x → (Z/pk)n be a random map which is not necessarily onto. One can see that for every v ∈ x +Pϕ(v ∈ ker ϕ) < Pϕ′(v ∈ ker ϕ′), and that for every v ∈ x we have: +Pϕ′(v ∈ ker ϕ′) = + + + +p−n(k−l), +if v ∈ plx \ pl+1x, for 0 ≤ l ≤ k − 1, +1, +if v ∈ pkx, +In particular, +� +v∈B(r)∩p−k(n−1)/nx +P(v ∈ x′) = +� +v∈B(pk(n−1)/nr)∩x +Pϕ(v ∈ ker ϕ) ≤ +� +v∈B(pk(n−1)/nr)∩x +Pϕ′(v ∈ ker ϕ′) += # +� +B(p(n−1)k/nr) ∩ pkx \ {0} +� ++ +k−1 +� +l=0 +p−(n−1)(k−l)# +� +B(pk(n−1)/nr) ∩ +� +plx \ pl+1x +� +\ {0} +� +≤ +k +� +l=0 +p−(n−1)(k−l)# +� +B +� +p +k(n−1) +n +−lr +� +∩ x \ {0} +� +We now need to analyze B(R) ∩ x for all R. +The following claim is a different wording of +[L. Liao and Tamam(2019), Lemma 3.5]. +Claim 5.6. For every lattice x, R > 0 we have that if R > λ1(x) we have +# (B(R) ∩ x \ 0) ≍ +n +max +i=1 +Ri +λ1(x) · · · λi(x) +By Minkowski’s theorem ( [Cassels(2012), Chapter VIII, Theorem V]), λ1(x) · · · λn(x) ≍ cov(x). +19 + +Thus, +k +� +l=0 +p−(n−1)(k−l)# +� +B +� +p +k(n−1) +n +−lr +� +∩ x \ {0} +� +≪ +k +� +l=0 +max +� +p−(n−1)(k−l) p +k(n−1)2 +n +−l(n−1)rn−1 +λ1(x)n−1 +, p−(n−1)(k−l)pk(n−1)−nlrn +� +≤ +k +� +l=0 +max +� +p−ε(n−1) k +n rn−1, p−lrn� +≤ +� +(k + 1)p−ε(n−1) k +n + 1 +� +rn−1. +For every ε > 0 there exists k0(ε) such that for all k ≥ k0(ε) we have (k + 1)p−ε(n−1) k +n ≤ 1. +Altogether we get +P(x′ /∈ Cr) ≪ rn−1. +For every δ > 0 choose r(δ) > 0 sufficiently small so that we have P(x′ ∈ Cr(δ)) > 1 − δ for all +k ≥ k0(ε). +■ +5.2 +Construction of Number Field (reprise) +Recall that a prime p ⊂ OK is said to be lying over a prime p in Z if p ∩ Z = pZ. +Definition 5.7 (totally ramified number fields). A degree n number field K is said to be totally +ramified at an integer prime p if there is exactly one prime p of OK lying over (p) and (p) = pn +over K. This implies that OK/p ∼= Fp. Indeed, OK/(p) = OK/pOk ∼= (Z/p)n. In addition, pk are +all different for 0 ≤ k ≤ n and hence #Ok/p0 < #Ok/p1 < ... < #Ok/pn = pn. Since all these +sizes divide pn we get that #Ok/pk = pk for all k = 0, ..., n. +Lemma 5.8. For every degree n number field over K which is totally ramified over p > 2n we have +(OK/pkOK)×/(Z/pkZ)× ∼= (Z/pk)n−1. +First, consider the ring Rk = OK/pk. Its only prime ideal is Ik = pOK/pkOK, hence R× +k = Rk\Ik. +The Ideal Ik satisfies Ik +k = {0} and p ∈ In +k . The size of the ring #Rk = pk as [OK : p] = p implies +that [OK : pk] = [OK : p]k = pk. Similarly, #pk = pk−1, and hence: +#R× +k = (p − 1)pk−1. +(5.4) +Consider the kernel ker(R× +k+n → R× +k ) for k ≥ 1. Since the map is onto, the kernel is of size pn. +20 + +Claim 5.9. Every element a ∈ ker(R× +k+n → R× +k ) satisfies ap = 1 ∈ R× +k+n. +This implies that +ker(R× +k+n → R× +k ) is a group of exponent p, and hence ker(R× +k+n → R× +k ) ∼= (Z/p)n. +Proof. Note that ap = (1 + (a − 1))p = 1 + �p−1 +i=1 +�p +i +� +(a − 1)i + (a − 1)p and p(a − 1) ∈ In +n+kIk +n+k = +In+k +n+k = {0}. Since p| +�p +i +� +for all i ≤ 1 ≤ p−1, we get that +�p +i +� +(a−1)i = 0. In addition, (a−1)p ∈ Ikp +n+k, +and since n ≤ p − 1, we have that kp = k + k(p − 1) ≥ k + n, and hence Ikp +n+k = {0}. Thus +ap = 1 + 0 + 0 = 1. +■ +Claim 5.9 implies that +(Rkn+1)× has exponent (p − 1)pk +(5.5) +Indeed, (Rkn+1)× has a filtration +R× +kn+1 ⊃ K0 ⊃ K1 +(5.6) +⊃ · · · ⊃ Kk = {1}, +(5.7) +where Ki = ker(R× +kn+1 → R× +in+1). Consider the quotients of consecutive pairs. The first quotient +R× +kn+1/K0 ∼= F× +p has exponent p − 1. The other quotients Ki/Ki+1 ∼= ker +� +R× +(i+1)n+1 → R× +in+1 +� +have exponent p. This implies that R× +kn+1 has exponent (p − 1)pk. Similar computation as in the +proof of Claim 5.9 shows the following claim: +Claim 5.10. Every element a ∈ ker(R× +k+n → R× +k ) is the p-power of the element (a − 1)/p + 1 ∈ +ker(R× +k+n → R× +k−n). +This implies that +Kp +0 = K1. +(5.8) +Indeed, by Claim 5.10 for every three consequent groups Ki+2 ⊂ Ki+1 ⊂ Ki we have (Ki/Ki+2)p = +Ki+1/Ki. Inductively this implies that Kp +0 contains all filtration elements less than ki for i = +k, . . . , 1. +Claim 5.11. R× +nk+1 ∼= Z/(p − 1) ⊕ (Z/pk)n. +Proof. Recall that R× +nk+1 has exponent (p − 1)pk and size (p − 1)pnk. The p − 1 part is arranged +in a cyclic group, as the map (Rkn+1)× → (R1)× ∼= F× +p is onto. Hence (Rkn+1)× ∼= Z/(p − 1)Z ⊕ +�r +i=1 Z/pαi for 1 ≤ αi ≤ k. Note that this isomorphism is of one group with a multiplicative +operation, and another with an additive notation. In particular K0 ∼= +�r +i=1 Z/pαi. Since K1 = Kp +0 +we get that K0/K1 ∼= (Z/p)r. On the other hand, the size computation implies that #K0/K1 = pn, +and hence r = n. This implies that �r +i=1 αi ≤ nk. On the other hand, #R× +nk+1 = pnk(p − 1) and +hence αi = k, as desired. +■ +21 + +Proof of Lemma 5.8. We need to compute R× +nk/(Z/pkZ)×. +Note that pnk+1 ∩ Z = pk+1Z and +pnk ∩ Z = pkOK ∩ Z = pkZ. This induces a commutative diagram of rings: +Z/pk+1Z +� +� � +� +Z/pkZ +� � +� +Rnk+1 +� Rnk +Size computation implies that R× +nk/(Z/pkZ)× ∼= R× +nk+1/(Z/pk+1Z)×. +Explicit computation of +(Z/pk+1Z)× ∼= Z/(p − 1) ⊕ Z/pk and Claim 5.11 implies the desired. +■ +Corollary 5.12. Let K totally ramified at p, and u1, ..., un ∈ O× +K elements whose images generate +(OK/pOK)×/(Z/pZ)×. Then for all k, the images generate (OK/pkOK)×/(Z/pkZ)×. +Proof. From Lemma 5.8 It is sufficient to show that U ⊆ (Z/pkZ)n−1 generates it if and only if its +image generates (Z/pZ)n−1. Indeed, if the image generates (Z/pZ)n−1 then there is u1, . . . , un−1 +such that det((u1, ..., un−1)) ̸= 0 mod p. +But then det((u1, ..., un−1)) is invertible (mod pk−1) +and hence the matrix (u1, ..., un−1) is invertible by Cramer’s rule, and hence u1, ..., un−1 generates +(Z/pZ)n−1. +■ +Lemma 5.13. Fix a prime number p ≡ 1 mod 2n. +Let a1, ..., an ∈ Z be numbers such that +{ai mod p : i = 1, . . . , n} ⊂ Fp is the set of roots of the polynomial xn + 1, and a1a2 · · · an ̸≡ (−1)n +mod p2. Let P(z) = (z − a1)(z − a2) · · · (z − an)− 1. Then K = Q[α], where P(α) = 0, is a number +field totally ramified at p. In addition, the index [OK : Z[α]] is not divisible by p and the units +(α − ai)n +i=1 generate (OK/pOK)×/(Z/pZ)×. +Proof. Let N denote the norm N : K → Q. To show that K is totally ramified at p, consider the +ideal factorization of the ideal I = (α) ⊆ OK. Since N(α) = (−1)np(0) = a1a2 · · · an − (−1)n, we +get that p|N(α) but p2 does not, and hence there is exactly one prime p over p that divides I. +Since P(z) ≡ zn mod p we get that 0 = P(α) ≡ αn mod p we get that p|αn and hence all +primes over p divide I. Hence there is a unique prime over p in OK. +We next claim that gcd([OK : Z[α]], p) = 1. Assume to the contrary that p|[OK : Z[α]]. Then +there is b ∈ Z[α] such that b/p ∈ OK \ Z[α]. Since b/p ∈ OK we have N(b/p) ∈ Z and hence +vp(N(b)) ≥ n, where vp : Z → {0, 1, . . .} is p-valuation. On the other hand, write b = �n−1 +i=0 biαi, +and since b/p /∈ Z[α], consider the smallest 0 ≤ m < n with bm ̸≡ 0 mod p. b = pb′+αm(bm+αb′′), +where b′ = �m−1 +i=0 biαi and b′′ = �n−1 +i=m+1 αi−m−1bi. We may replace b with b−pb′ and assume b′ = 0. +Then N(b) = N(α)mN(bm + αb′′). Since bm ̸≡ 0 mod p and α ∈ p we deduce that bm + αb′′ /∈ p +and hence p ̸ |N(bm + αb′′). We conclude that vp(N(b)) = mvp(N(α)) = m < n, which contradicts +the assumption. Consequently, gcd([OK : Z[α]], p) = 1, and hence OK/pOK ∼= Z[α]/pZ[α]. +22 + +It is left to show that (α − ai)n +i=1 generates (Z[α]/pZ[α])×/(Z/pZ)×. Denote R = Z[α]/pZ[α], +and note that the image ¯α of α in R satisfies ¯αn = 0. Hence one can think on R as being the ring +Fp[¯α] where ¯αn = 0. Denote by R× +1 = 1 + ¯αR ⊆ R× a subgroup of the multiplicative group. Note +that R× +1 ∼= R×/F× +p , and hence it is sufficient to show that (1 − ¯α/ai)n +i=1 generates R× +1 . +We now use the logarithmic map log : R× +1 → (R)+ defined by log(1 − ¯αa) = �n−1 +i=1 +1 +i (¯αa)i. It +is a homomorphism by the standard properties of the logarithm. We get that log(R× +1 ) = ¯αR from +the fact log(1 − ¯αi) = 1 +i ¯αi + ¯αi+1(?) generates ¯αiR/¯αi+1R for all i. Since #R× +1 = #¯αR = pn−1 +we get that the logarithm defines an isomorphism log : R× +1 +∼ +−→ R. Thus it is sufficient to show +that (log(1 − ¯α/ai))n +i=1 generates ¯αR. This follows from the following computation. When we +write (log(1 − ¯α/ai))n−1 +i=1 in the basis of (¯αj)n−1 +j=1 we get the Vandermonde matrix associated to +a−1 +1 , ..., a−1 +n−1 up to multiplication by scalars for the rows and columns, and is invertible. +This +concludes the proof. +■ +5.3 +Application of Hecke operator to compact orbits (reprise) +In this section, we introduce the following lemma, which describes what happens to a compact orbit +after we apply to it a Hecke operator. +Lemma 5.14. Let K be a totally real number field of degree n, R ⊆ OK an order, xR ∈ Xn the +corresponding point with compact orbit. Let k1 = 0 ≤ k2 = · · · = kn−1 = kn = k, for some k ≥ 0, +p, a ∈ SLn(R) as in Definition 2.4. Let U ⊆ R×,>0 be a subgroup and AU ⊂ A the corresponding +diagonal subgroup. Assume that U projects onto (R/(pk))×/(Z/pk)×. Then #(R/(pk))×/(Z/pk)× +elements of Ta(xR) lie on the same A-orbit, and no other point in Ta(xR) lies on this orbit. +Remark 5.15. Lemma 5.14 holds also in the case k1 = k2 = · · · = kn−1 = 0 ≤ kn = k, by duality. +Lemma 5.14 holds also for other ideal points xI, whenever R = OK and I ≤ OK is an ideal. The +proof follows from identifying I/pkI ∼= OK/(pk). However, the lemma cannot be extended for every +Λ ∈ LatK and U ⊆ R×. We will not use these extra results. +Proof. We claim that Hecke neighbors are classified by a quotient: +Ta(xR) ∼= {b ∈ R/(pk) : pk−1b ̸= 0}/(Z/pkZ)×. +(5.9) +Indeed, the Hecke neighbors are classified by subgroups Λ ⊆ I : I/Λ ∼= (Z/pk)n−1. Such a subgroup +Λ must contain pkI and we must have Λ/pkI ∼= Z/pk, which implies that Λ/pkI is generated by a +single element b ∈ R/(pk). Since the generator is well defined up to multiplication by (Z/pkZ)×, +we get Eq. (5.9). Note that the AU action on Ta(xR) corresponds to the +U ↷ {b ∈ R/(pk) : pk−1b ̸= 0}/(Z/pkZ)×, +(5.10) +23 + +and hence it is sufficient to prove that Action (5.10) has a # +� +(R/(pk))×/(Z/pk)×� +big orbit. +Since the projection of U to (R/(pk))×/(Z/pk)× is onto, and Action (5.10) factors through +(R/(pk))×/(Z/pk)× +↷ +{b ∈ R/(pk) : pk−1b ̸= 0}/(Z/pkZ)×, +(5.11) +it is sufficient to show that Action (5.11) has a # +� +(R/(pk))×/(Z/pk)×� +big orbit, which is clear. +■ +Proof of Theorem 1.5. Let c ∈ (0, 1]. Let p be the smallest prime number with p ≡ 1 mod 2n. Note +that p > 2n. Denote the roots of the polynomial xn + 1 in Fp by ¯a1, ¯a2, . . . , ¯an. Let M > 2np, and +for each i = 0, . . . , n−1 let ai be the smallest integer after iM/n such that ai ≡ ¯ai mod p. We have +0 ≤ ai − iM/n < p. If a1a2 · · · an ≡ (−1)n mod p2 Replace an by an + p. Now a1a2 · · · an ̸≡ (−1)n +mod p2, ai+1−ai > M/2n and 0 < a1 < a2 · · · < an < M. Let P(z) = (z−a1)(z−a2) · · · (z−an)−1 +and K = Q[α] where P(α) = 0. By Lemma 5.13, the number field K is totally ramified at p and +(α − ai)n +i=1 generates (OK/pOK)×/(Z/pZ)×. Let k > 0 be a parameter. By Corollary 5.12, we +get that the images of (α − ai)n +i=1 generate (OK/pkOK)×/(Z/pkZ)× for all k ≥ 0. Since p does +not divide [OK : Z[α]] we deduce that (OK/pkOK)×/(Z/pkZ)× ∼= (Z[α]/pkZ[α])×/(Z/pkZ)× and +hence the images of (α − ai)n +i=1 generate (Z[α]/pkZ[α])×/(Z/pkZ)× as well. +Let xZ[α] as in Definition 2.4. Let a be as in Theorem 5.3. Then by Lemma 5.14, we get that +there is a subset T0(xZ[α]) ⊆ Tak(xZ[α]) of size #T0(xZ[α]) = #(OK/(pk))×/(Z/pk)× such that for +all y ∈ T0(xZ[α]) we have Ay ∩Tak(xZ[α]) = T0(xZ[α]) . Note that T0(xZ[α]) is stabA(xZ[α])-invariant. +For all x′ = a′xZ[α], a′ ∈ A define T0(x′) = a′T0(xZ[α]). Since T0(xZ[α]) is stabA(xZ[α])-invariant, +this implies that T0(x′) depends only on x′ and not on a′. Hence for every y ∈ T0 +� +xZ[α] +� +we have +that Ay = � +x′∈Ax T0(x′), and: +µAy = +� +AxZ[α] +1 +#T0(x′) +� +y′∈T0(x′) +δy′dµZ[α]. +(5.12) +By Lemma 5.8 we get that #T0(xZ[α]) = p(n−1)k, and hence #T0(x′) = p(n−1)k for all a′ ∈ Ax. We +now apply a result of Shapira. The theorem we state is an accumulation of results and computations +given in Shapira [Shapira(2016)]. +Definition 5.16. For every π ∈ Sn (the permutation group), denote by Fπ ⊆ Rn−1 +0 +the set of +vectors Fπ = {v ∈ Rn−1 +0 +: vπ(i) ≥ vπ(i+1) − 1 : i = 1, . . . , n}. Here we use cyclic index notations and +π(n + 1) = π(1). The set Fπ is compact. Note that Fπ is well defined for π ∈ Sn/Cn where Cn is +the cyclic group of rotations. +Theorem 5.17 (Shapira [Shapira(2016)]). Let η > 0 be a fixed constant, then for all M > M0(η) +the following holds. +Let K = Q(α) such that P(α) = 0 where P(z) = �n +i=1(z − ai) − 1 for +24 + +a1 < a2 < · · · < an with |ai| < M for all i = 1, . . . , n and ai + ηM < ai+1 for all i = 1, . . . , n − 1. +Consider the point xZ[α] ∈ Xd. It is stabilized by exp Λ where Λ is generated by v1, ..., vn−1, vn +satisfying v1 + ... + vn = 0 and +���� +� +v(i)� +j − (2 − 2nδij) log M +���� = O(1). +Moreover there is a finite collection of points P0 ⊆ Rn−1 +0 +/Λ and a map π : P0 → Sn/Cn such +vol +�� +p∈P0 p + (1 − oM(1)) log(M)Fπ(p) +� +vol(Rn−1 +0 +/Λ) += 1 − oM(1), +and for all p ∈ P0, v ∈ (1 − oM(1)) log(M)Fπ(p) one has +dXn(exp(p + v)xZ[α], exp(v)Zn) = oM(1). +Denote by min-co : Rn−1 +0 +→ (−∞, 0] the minimum of the coordinates function, and note that +minFπ(min-co) = − d−1 +2 . This function is important since +λ1(exp(v)Zn) = exp(min-co(v)) +for all +v ∈ Rn−1 +0 +. +(5.13) +Let mFπ be the uniform probability measure on Fπ for all π ∈ Sn. Note that (min-co)∗mFπ is +independent of π ∈ Sn. We use the following corollary of Shapira’s result, which follows Eq. (5.13): +Corollary 5.18. In the setting of Theorem 5.17, we have +d +�� +δ +1 +log M log λ1(x′)dµZ[α](x′), (min-co)∗mF1 +� += oM(1). +Since mF1 is absolutely continuous with respect to Lebesgue, there is η < 0 such that mF1(min-co−1[η, 0]) = +c. Choose k such that +k log p +n log M = η(1 + oM(1)), and fix ε > 0. +Distinguish two parts of AxZ[α]: +(AxZ[α])− = {x′ ∈ AxZ[α] : λ1(x′) ≤ exp((−1 − ε)η log M)} +(5.14) +(AxZ[α])+ = {x′ ∈ AxZ[α] : λ1(x′) ≥ exp((−1 + ε)η log M)}. +(5.15) +Then by Corollary 5.18 we get that µZ[α]((AxZ[α])−) = 1−c+O(ε)+oM(1) and µZ[α]((AxZ[α])+) = +c + O(ε) + oM(1). Consequently, we get that +µZ[α](AxZ[α] \ ((AxZ[α])+ ∪ (AxZ[α])−)) = O(ε) + oM(1). +(5.16) +25 + +Fix y ∈ T0(xZ[α]). We will now analyze the measure µAy via Eq. (5.12) and the parts (AxZ[α])±. +First, for all x′ ∈ (AxZ[α])−, Theorem 5.3 implies that pk/nλ1(x′) ≥ λ1(y′) for all y′ ∈ Tak(x′). +By Eq. (5.14) and the choice of k, we get that λ1(x′) ≤ exp((−1−ε)η log M) = p−k/n(1+oM (1))(1+ε). +Hence we get an upper bound of +λ1(y′) ≤ p−k/n(ε+oM(1)). +Hence for all M sufficiently large as a function of ε we have +� +(AxZ[α])+ +1 +#T0(x′) +� +y′∈T0(x′) +χy′∈KεdµZ[α] = +� +(AxZ[α])+ +1 +#Tak(x′) +� +y′∈Tak(x′) +χy′∈KεdµZ[α] = 0. (5.17) +Here Kε is defined as in Definition 2.2. Now, let us analyze (AxZ[α])+. Let x′ ∈ (AxZ[α])+. By Eq. +(5.15) and the definition of k we get that +λ1(x′) ≥ p−k/n(1+oM(1))(1−ε) +Write k = k′ + s where λ1(x′) ≥ p−k′/n(1+oM(1))(1−ε/3) and s ≥ εk/3. By Theorem 5.3, we get that +Tak′(x′) has no escape of mass as M → ∞. By Corollary 5.1, we get that T M +as(T M +ak′ (x′)) +M→∞ +−−−−→ mXn. +Denote +� +as◦ak′ = +� +(AxZ[α])+ T M +as(T M +ak′ (x′))dµZ[α](x′), +� +ak = +� +(AxZ[α])+ T M +ak(x′)dµZ[α](x′), +� +0 += +� +(AxZ[α])+ +1 +#T0(x′) +� +y′∈T0(x′) +δy′dµZ[α](x′). +This implies that d +�� +asak′, µZ[α]((AxZ[α])+) · mXn +� += oM(1), and hence +d +�� +asak′, cmXn +� += oM(1) + O(ε). +(5.18) +Since we are interested in +� +0, we will think of µAy as an ergodic component of T M +akµZ[α], itself an +ergodic component of T M +as(T M +ak′ (µZ[α])) +M→∞ +−−−−→ mXn, and use the ergodicity of mXn. Formally, we +will take M → ∞ and then ε → 0 and consider the limits of this construction. +Eq (5.18) translates +to +lim +ε→0 lim +M→∞ +� +asak′ = cmXn. +By Eq. (5.17), and the negligibility of the complement of (AxZ[α])+ ∪ (AxZ[α])+, Eq. (5.16) we +26 + +deduce +lim +ε→0 lim +M→∞ µAy = lim +ε→0 lim +M→∞ +� +0 +. +(5.19) +By Eq. (5.18), we get that there is no escape of mass in limε→0 limM→∞ +� +asak′ . Note that by +the size estimate on T0(x′) ⊆ Tak(x′) we get that +� +0 +≤ #Tak(xZ[α]) +#T0(xZ[α]) +� +ak = +pn − 1 +pn−1(p − 1) +� +ak . +By Remark 3.7 we get a similar bound (for completely different reasons): +� +ak ≤ pn−1(p − 1) +pn − 1 +� +as◦ak′ . +Altogether we deduce that +� +0 +≤ p2n−2(p − 1)2 +(pn − 1)2 +� +as◦ak′ . +(5.20) +Since limε→0 limM→∞ +� +as◦ak′ has no escape of mass, we deduce that there is no escape of mass +in limε→0 limM→∞ +� +0, the RHS of Eq. (5.19) as well. In addition, taking the limit of Eq. (5.20) +implies that every partial limit in the RHS of Eq. (5.19) is absolutely continuous with respect mXn. +The A-ergodicity of mXn, [Moore(1966)], implies that every partial limit of the RHS of Eq. (5.19) +is a multiple of haar measure. The no escape of mass in the RHS of Eq. (5.19) implies that the +value of Eq. (5.19) is cmXn, as desired. +■ +Remark 5.19. In the proof above we use the composition of the Hecke operators T M +s ◦T M +k′ . 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Duke Mathematical Journal, 119(2):367–392, 2003. +29 + +This figure "geogebra_thumbnail.png" is available in "png"� format from: +http://arxiv.org/ps/2301.00721v1 + diff --git a/udAyT4oBgHgl3EQf0fnz/content/tmp_files/load_file.txt b/udAyT4oBgHgl3EQf0fnz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b92664e5832bcf3d27400ad3dbce54476c9b3817 --- /dev/null +++ b/udAyT4oBgHgl3EQf0fnz/content/tmp_files/load_file.txt @@ -0,0 +1,1126 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf,len=1125 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='00721v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='DS] 2 Jan 2023 Tori Approximation of Families of Diagonally Invariant Measures Omri Nisan Solan Yuval Yifrach Abstract We approximate any portion of any orbit of the full diagonal group A in the space of unimodular lattices in Rn using a fixed proportion of a compact A-orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Using those approxi- mations for the appropriate sequence of orbits, we prove the existence of non-ergodic measures which are also weak limits of compactly supported A-invariant measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' In fact, given any countably many A-invariant ergodic measures, our methods show that there exists a sequence of compactly supported periodic A-invariant measures such that the ergodic decomposition of its weak limit has these measures as factors with positive weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Using the same methods, we prove that any compactly supported A-invariant and ergodic measure is the weak limit of the restriction of different compactly supported periodic measures to a fixed proportion of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' In addition, for any c ∈ (0, 1] we find a sequence of compactly supported periodic A-invariant measures that converge weakly to cmXn where mXn denotes the Haar measure on Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' In particular, we prove the existence of partial escape of mass for compact A-orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' These results give affirmative answers to questions posed by Shapira in [Shapira(2016)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Our proofs are based on a modification of Shapira’s proof in [Shapira(2016)] and on a generalization of a construction of Cassels, as well as on effective equidistribution estimates of Hecke neighbors by Clozel et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' and a number theoretic construction of a special number field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' 1 Introduction Let (Xn, dXn) denote the metric space of unimodular lattices in Rn and let A < SLn(R) denote the subgroup of diagonal matrices with positive diagonal inside the group of real matrices with determinant 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Among the ergodic A-invariant probability measures on Xn, those which are sup- ported on compact orbits stand out due to their connection to number theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Indeed, there is a surjective map between the set of full Z-modules inside totally real number fields and the set of compact A-orbits in Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' For any totally real degree n number field K, let σi : K ֒→ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' , n be some ordering of the natural embeddings of K and let σ = (σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' , σn) : K → Rn denote their concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' The compact orbit corresponding to a full module M ⊂ K is then the A-orbit of the 1 unimodular lattice cov(σ(M))−1/nσ(M) where cov(Λ) denotes the co-volume of a lattice Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' This orbit is indeed compact, and every compact A-orbit is given in this manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Denote: M(Xn) = {finite measures on Xn};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' M(Xn)A e = {ergodic A-invariant probability measures on Xn};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' M(Xn)A c = {ergodic A-invariant probability measures on Xn supported on compact orbits}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' We endow M(Xn) with the weak topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' In this paper, we analyze the weak closure M(Xn)A c inside M(Xn) and show that it contains several families of natural and important measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='1 Measures with Predetermined Ergodic Factors The first family of measures we consider contains non-ergodic measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Given any finitely many elements µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' , µk ∈ M(Xn)A e , we find a measure in µ ∈ M(Xn)A c such that the ergodic decom- position of µ has positive weight on each of the µi’s for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' In [Shapira(2016), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='4 Q1, Q2], Shapira asks whether non-ergodic measures and measures with a given periodic A-invariant ergodic factor can be obtained as weak limits of M(Xn)A c elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' The following Theorem shows in particular that this is indeed the case: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Let µ1, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' , µk ∈ M(Xn)A e be a finite sequence of A-invariant ergodic measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Then there exists µ ∈ M(Xn)A c such that the ergodic decomposition of µ has positive weights on the µi’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' In particular for any µ ∈ M(Xn)A c there exists ˜µ ∈ M(Xn)A c \\ {µ} such that the ergodic decomposition of ˜µ, contains µ with positive weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='2 (The weights in the ergodic decomposition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' For a single measure µ1, we get an explicit lower bound on the weight, namely µ ≥ � ⌊n/2⌋ 4n2(n2−1) �n−1 µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' More generally, for a finite sequence of measures µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' , µk we get µ ≥ Θ(k1−n)µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='3 (Countably many measures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Using the same technics one can prove the same re- sult for countably many measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' For possibly infinite sequence µ1, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' of measures and every sequence of positive constants c1 + c2 + · · · = 1 we can we can construct µ with µ ≥ Θ(cn−1 i )µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='2 Partial Escape of Mass and Entire Mass Approximations The next measures we will find in M(Xn)A c are measures with total mass strictly between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' In his paper [Shapira(2016), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='4, Q2], Shapira proves that the zero measure is contained in M(Xn)A c and poses the question of whether measures with total mass strictly between 0 and 1 also lie there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' In [David and Shapira(2018)] David and Shapira find a sequence of orbits with a lower bound on the escape of mass, but did not prove that the limit is not zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' We answer Shapira’s question in the affirmative in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' The first is the following corollary of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='1: 2 Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' For every sequence µ1, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' , µk, · · · ∈ M(Xn)A e there exists µ ∈ M(Xn)A c such that µ(Xn) ∈ (0, 1) and the ergodic decomposition of µ has positive weights on µi for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' The next family of measures we find inside the closure of the compactly supported periodic measures is the family {cmXn : c ∈ (0, 1]} where mXn is the Haar probability measure on Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' This family of measures is substantially different from the measure families we approximated so far since we can describe their entire ergodic decomposition (which is simply cmXn) rather than describe some of the factors and have no control over the remaining factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' In [Shapira and Zheng(2021)], they show that mXn ∈ M(Xn)A c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Here we prove a more general theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Let c ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Then cmXn ∈ M(Xn)A c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' In particular, the above theorem gives another answer to Shapira’s question regarding the partial escape of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='3 Method of Proof We will first show a sketch of the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='1, for the particular case k = 1 where we approximate a single measure µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' We will first show how to do that depending on a conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Then we will show how to overcome the need to use the conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Let K be a number field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' For every I ≤ OK, we get a measure µI on a compact orbit associated to I (See Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' The measure µI depends only on the class of I in the class group ClOK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' It is believed that µK := 1 #ClOK � I∈ClOK µI disc(K) −−−−−→ mXn, where mXn is the Haar measure on Xn, and disc(K) is the discriminant of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Moreover, we expect that d(µK, mXn) = O(disc(K)−⋆), for ⋆ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Another ingredient of the proof is a construction of a number field with very small units compared to its discriminant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' A construction of Shapira following Cassels is a series of number fields K whose unit groups are generated by elements u ∈ O× K such that log |u| = O(log disc(K)) (which is excep- tionally small compared to the bound given by the class number formula: log |u| = O(disc(K)1/d−1), provided that the unit lattice is sufficiently balanced).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Applying Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='6 to these number fields, we get that that the different measures µI for I ∈ ClOK are equidistributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Using this equidistribution we can approximate a generic point x ∈ Xn of our desired ergodic measure µ, namely there is I ∈ ClOK such that d(x, x′) = O(disc−⋆(K)) for some x′ ∈ supp µI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Hence for r = θ(log(disc(K))) ball in A, a ∈ BA(r) we have that ax is close to ax′ ∈ supp(µI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' On the other hand, the orbit ax′ is not much larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' The periods are controlled by the size of the units, which 3 are again O(log(disc(K))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' This would imply that we can approximate every ball in an A-orbit with measures µI for I in the class group of the special number field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' To overcome the need to use Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='6, we prove a weaker form on a special collec- tion of orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' We take a measure µI as above and apply to it a p-Hecke operator Tp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' On the one hand, Hecke operators are known to have well-behaved equidistribution properties (See [Clozel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' (2001)Clozel, Oh, and Ullmo]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' We expect to have f(TpµI, mXn) = O(p−⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' On the other hand, the resulting measure TpµI is the average of measures on compact orbits, corresponding to modules of a sub-order R = Z + pOK ⊂ OK of index [OK : R] = pd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' While this guarantees that the orbits would be equidistributed, to bound the sizes of the orbits we tweak the construction of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Not only OK should have units of logarithmic size, but also R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' This guarantees that TpµI is an average of measures of logarithmically small compact orbits, and enables the proof to work without Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='6 seems very complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' It was proven in case n = 2 in [Duke(1988)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' A non-effective version of it is proven for n = 3 under some splitting restriction in [Einsiedler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' (2011)Einsiedler, Lindenstrauss, Michel, and Venkatesh].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' The proof uses a deep and nontrivial number theoretic result, namely, the subconvexity estimate of L functions of cubic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='5 has similar steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' We start with a compact orbit Ax0, constructed using a variation on the techniques used in [Shapira(2016)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' As in [Shapira(2016)], this compact orbit will lie in the cusp for most of its lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' However, the points of the orbits distributes continuously in the cusp, in the sense that some are further in the cusp then others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Then we take some Hecke operator T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' It pulls points out of the cusp in fixed rate, that is, if λ1(x) was very small then on average over x′ ∈ T (x0) we have λ1(x′) = Θ(min(pαλ1(x0), 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' We choose T such that c proportion of T (Ax) will leave the cusp while the remaining 1 − c proportion of T (Ax) will stay in the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' We will use the Hecke equidistribution to show that the part that leaves the cusp equidistributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' A possible problem is that T (Ax0) is composed of possibly many A-orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' To solve that we arrange the following additional property: a positive proportion of x0-s T -Hecke neighbors will be on the same A orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' This implies that T (Ax0) contains a long A-orbit, Ax1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Now we use ergodicity to deduce that the distribution estimate of T (Ax0) holds for its long A-orbit subset Ax1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' This orbits will give us Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' We further remark that the method of finding a Hecke neighbor whose diagonal orbit occupies positive proportion of the entire Hecke neighbors is also used in another paper by Uri Shapira and Menny Aka, [Aka and Shapira(2018)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='8 (Comparison to [Shapira and Zheng(2021)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Plugging c = 1 to our construction yields examples similar in nature to those of [Shapira and Zheng(2021)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Both are obtained from Hecke neighbor of a Minkovski embedding of a number field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' However our analysis is simpler in this case, as a consequence of restricting the number field we use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='4 Further Research The first natural improvement of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='1 is the full approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Open Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Let µ ∈ M(Xn)A c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Can it be the limit measure of other measures in M(Xn)A c ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' In other words, is µ ∈ M(Xn)A c \\ {µ}?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' It can be seen that simply improving the bounds we give in this paper cannot give a positive answer to this question, for the following reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' If n = 3, Ay is a compact orbit and Ax is an orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' We can consider the set B = {a ∈ A : dX3(ax, Ay) < δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' The connected components of B are roughly hexagons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' It can be seen that two such hexagons H1, H2 cannot have R long parts of the boundaries which are δR close to one another, for some δ > 0 and all R > 0 sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='5 Acknowledgments The second author would like to express his deep gratitude to Uri Shapira for his support and encouragement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Without them, this paper would not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' We thank Uri Shapira for bringing the main questions answered in this paper into our attention and for many intriguing discussions with him which contributed a lot to this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Moreover, the first author thanks Andreas Wieser for many fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' The second author acknowledges the support of ISF grants num- ber 871/17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' 754475).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' 2 Notation and Preliminaries Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='1 (O-notations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' For two real functions f, g on a set A we write f ≪ g if there exists a constant C independent on the parameters of f and g such that |f| ≤ Cg on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' The notation O(g) will refer to some implicit function f which satisfies f ≪ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' The notation Θ(g) will refer to some implicit function f which satisfies g ≪ f ≪ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Whenever r is a parameter going to 0 or ∞, the notation or(g) will refer to some implicit function f which satisfies f ≪ g · h, for some implicit function h → 0 as r goes to 0 or ∞ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' In this case, we sometimes write f = O(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Whenever r is a parameter going to either ∞ or 0 and f Fix ∥·∥ to denote the supremum norm on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Let Xn denote the space of unimodular lattices in Rn and let d(·, ·) denote the metric on Xn coming from the operator norm on linear operators on (Rn, ∥·∥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Let Rn−1 0 = {v ∈ Rn : � i vi = 0} and we abuse notations and define exp = exp ◦ diag : Rn−1 0 → A the standard parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' We denote by mXn probability measure on Xn = SLn(R)/ SLn(Z) coming from the Haar measure on SLn(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' We now present formally the 5 relation between compact A-orbits in Xn and subgroups of number fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Denote by L, U ⊂ SLn(R) the subgroups of lower and upper triangular matrices with diagonal 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='2 (Space of Measures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Let M(Xn) denote the space of finite measures on Xn endowed with the topology induced by µk → µ if for any f ∈ Cc(Xn) it holds that µk(f) → µ(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' We define a metric on Xn which induces this topology by letting, for any µ1, µ2 ∈ M(Xn): d(µ1, µ2) = sup ǫ>0 sup{ǫ| � fdµ1 − � fdµ2| : f : Xn → R : f is 1-Lipschitz and is supported on Kǫ} (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='1) where Kǫ = {x ∈ Xn : λ1(x) > ǫ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' The following definition is relevant to §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='3 (Special subgroups of SLn(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Let w0 = �n + 1 − 2i 2 �n i=1 ∈ Rn−1 0 , at = exp(tw0) for all t ∈ R, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='2) and note that L = {g ∈ SLn(R) : atga−t t→∞ −−−→ I}, U = {g ∈ SLn(R) : a−tgat t→∞ −−−→ I}, that is, L, U are the contracting and expanding horospheres with respect to at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' For every degree n, totally real number field K denote by LatK the set of free Z-modules of rank n in K, where we identify two lattices Λ1, Λ2 ⊂ K if Λ1 = kΛ2 for some k ∈ K×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' For every rank n, Z-module Λ ⊆ K consider the lattice xΛ := σ(Λ)/cov(σ(Λ)) ∈ Xn, where σi : K ֒→ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' , n is some ordering of the natural embeddings of K and let σ = (σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' , σn) : K → Rn denote their concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Denote by OΛ = {k ∈ K : kΛ ⊆ Λ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' This is a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Denote by O×,>0 Λ = {u ∈ O× Λ : σi(u) > 0 : i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' For every U ⊆ O×,>0 K denote AU = {diag(σ1(u), σ2(u), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=', σn(u)) : u ∈ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Fix d ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' For every totally real number field K, [Λ] ∈ LatK, and choice of the ordering of the real embeddings of K, the orbits AxΛ is compact, independent of the representative Λ ⊂ K of [Λ] ∈ LatK and this is a one to one parametrization of all compact A-orbits in Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' This theorem is equivalent to [McMullen(2005)], apart from the part of this parametrization being one-to-one, whsich is Folklore and we do not use in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' For every point x ∈ Xn such that the orbit Ax is compact, denote µAx the A- invariant measure on Ax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' For every [Λ] ∈ LatK denote µΛ = µAxΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' 6 3 Effective Approximations In this section, we prove an effective approximation theorem of points in Xn by points of compact orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Formally, the goal is to prove the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' For every compact set K ⊂ Xn there exists a sequence tk k→∞ −−−−→ ∞ such that for every y ∈ Xn there exists a sequence (yk)k ⊂ Xn such that d(yk, y) < exp � − ⌊n/2⌋ 4n(n2−1)tk � such that yk is stabilized by a subgroup exp(Λk) ⊆ A where Λk ⊆ Rn−1 0 is generated by vectors (v(i) k )n−1 i=1 such that ���� � v(i) k � j − (1 − δijn)tk ���� = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='1 implies in particular that Axk is compact with total volume nn−3/2tn−1 k + O(tn−2 k ) in the Rn−1 0 parametrization, and in addition that for every v ∈ Rn−1 0 and for y, yk as in the lemma, we have: d((exp v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='yk, (exp v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='y) ≤ O � exp � max 1≤i,j≤n |vi − vj| − ⌊n/2⌋ 4n(n2 − 1)tk �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='1) Since the set B0 = {v ∈ Rn 0 : max1≤i,j≤n |vi − vj| ≤ 1} has volume vol(B0) = 1 √n, we deduce that a portion of � 1 n ⌊n/2⌋ 4n(n2−1) �n−1 + O(1/tk) of the orbit Ayk approximates the the orbit Ay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' The proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='1 will be composed of two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' The first is an approximation of lattices via Hecke neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' We will define Hecke neighbors in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='4 and obtain a good approximation by them in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' The second component generates infinitely many points in Xn with compact A-orbits, with a control on the geometry of their stabilizers in A and the stabilizers of their sublattices of a given index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' This gives us the arsenal of points on which we apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='6 to deduce Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='1 Hecke Density The source of our good approximation comes from the quantitative version of the equidistribu- tion of Hecke neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' There are many references to the equidistribution of Hecke neighbors such as [Clozel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' (2001)Clozel, Oh, and Ullmo, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='1], [Gan and Oh(2003), Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='7], [Eskin and Oh(2006), Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' In this section we will cite the result of Laurent Clozel, Hee Oh and Emmanuel Ullmo [Clozel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' (2001)Clozel, Oh, and Ullmo] and deduce the approxi- mation result we need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='3 (Function space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Let L2 0(Xn) = {f ∈ L2(Xn, mxn) : � Xn fdmXn = 0} be the Hilbert space of L2 function on Xn with 0 mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' 7 Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='4 (Definition of p-Hecke neighbors and Hecke operator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' For every sequence of inte- gers k1 ≤ k2 ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' ≤ kn consider: a = 1 p(k1+···+kn)/n diag(pk1, pk2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' , pkn) ∈ SLn(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' For every x = g SLn(Z) ∈ Xn denote Ta(x) = g SLn(Z)a SLn(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' This set is is finite since a SLn(Z)a−1 is commensurable to SLn(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' The size #Ta(x) = #(SLn(Z)a SLn(Z)/ SLn(Z)) de- pends only on k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=', kn and not x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Equivalently, Ta(x) = � 1 n� cov(x′) x′ : x′ ⊆ x with x/x′ ∼= Z/pk1Z ⊕ · · · ⊕ Z/pknZ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' For every function f ∈ L2(Xn) define the Hecke action on functions T F a (f)(x) = 1 #Ta(x) � x′∈Ta(x) f(x′) ∈ L2 0(Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' For every point x ∈ Xn, define the Hecke action on measures T M a (x) := 1 #Ta(x) � x′∈Ta(x) δx′ and for every measure µ on Xn define T M a (µ) := � Xn T M a (x)dµ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' The following theorem is a particular case of [Clozel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' (2001)Clozel, Oh, and Ullmo, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='1], specialized for SLn as in [Clozel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' (2001)Clozel, Oh, and Ullmo, Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' For every prime p and k1 ≤ k2 ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' ≤ kn, a ∈ SLn(R) as in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='4, the operator norm of T F a �� L2 0(Xn) is bounded by: ���T F a �� L2 0(Xn) ��� ≤ � i≤n/2 1 p(kn+1−i−ki)/2 (kn+1−i − ki)(p − 1) + (p + 1) p + 1 ≤ p− 1 2 �n/2 i=1(kn+1−i−ki)·C(k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=', kn), where C(k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=', kn) depends polynomially on k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=', kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' For every compact subset K ⊂ Xn there exists C = C(K) > 0 such that for any x, y ∈ K and p, k1 ≤ k2 ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' ≤ kn, a ∈ SLn(R) as in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='4, there exists a Hecke neighbor 8 x′ ∈ Ta(x) such that: d (x′, y) ≤ C(K) ���T F a �� L2 0(Xn) ��� 1/(n2−1) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='2) where α > 0 is some constant depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Recall the right invariant Riemannian metric dSLn(R) on SLn(R), and its descend to Xn, the metric dXn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Let r0 < min(inj(y), inj(y)), where inj(x) is the injectivity radius of x, that is, the maximal radius r such the translation map g �→ gx is injective on BSLn(R)(I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' r), itself being the radius r ball around the identity in SLn(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Let fx = χBXn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='r0) be the indicator of a radius r0 ball of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Then � Xn fxdmXn = vol(BSLn(R)(I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' r0)) = Θ(rn2−1 0 ), and a similar equality holds for fy = χBXn(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='r0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Denote by vr0 = vol(BSLn(R)(I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' r0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Then ˜fx = fx − vr0 ∈ L2 0(xn) and ˜fy = fy − vr0 ∈ L2 0(xn) have norm ∥fx∥2 = ∥fy∥2 = vr0(1 − vr0)2 + (1 − vr0)v2 r0 = (1 − vr0)vr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Consider � ˜fx, T F a ( ˜fy) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' On the one hand, it is at most ���T F a �� L2 0(Xn) ��� ∥fx∥∥fy∥ = ���T F a �� L2 0(Xn) ��� (1 − vr0)vr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' On the other hand, consider the set: T −1 a (BXn(y, r0)) = {x′ ∈ Xn : Ta(x′) ∩ BXn(y, r0) ̸= ∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' and note that T F a ( ˜fy)|(T −1 a (BXn(y,r0)))c ≡ −vr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Assume that T −1 a (BXn(y, r0)) ∩ BXn(x, r0) = ∅, and get that T F a (fy)|BXn(x,r0) ≡ −vr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' It follows that: � ˜fx, T F a ( ˜fy) � = � Xn ˜fxT F a ( ˜fy)dmXn = � Xn fxT F a ( ˜fy)dmXn = � BXn(x,r0) T F a ( ˜fy)dmXn = vr0 · (−vr0) = −v2 r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' We deduce that v2 r0 ≤ ���T F a �� L2 0(Xn) ��� · ∥ ˜fx∥ · ∥ ˜fy∥ = ���T F a �� L2 0(Xn) ��� (1 − vr0)vr0, and hence vr0 ≤ ���T F a �� L2 0(Xn) ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Using this logic in reverse, we deduce that if vr0 > ���T F a �� L2 0(Xn) ��� then T −1 a (BXn(y, r0)) ∩ BXn(x, r0) ̸= ∅, that is, there exist x′ = g0x, y′ = g1y such that y′ ∈ Ta(x′) and g0, g1 ∈ BSLn(R)(I, r0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Thus g−1 0 y′ = g−1 0 g1y ∈ g−1 0 Ta(x′) = Ta(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' On the other hand, g−1 0 y′ = g−1 0 g1y satisfies dXn(g−1 0 g1y, y) ≤ 2r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Altogether, we have proved that if ���T F a �� L2 0(Xn) ��� < r0 ≤ min(inj(x), inj(y)) then there exists x′′ = g−1 0 g1y with x′′ ∈ Ta(x) and dXn(x′′, y) ≤ 2r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Now, let K ⊂ Xn be a compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Denote the minimum of the injectivity radius on K by rK > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' If vrK > ���T F a �� L2 0(Xn) ���, then we can find r0 = Θ( ���T F a �� L2 0(Xn) ��� 1/(n2−1) ) with r0 < rK and vr0 > ���T F a �� L2 0(Xn) ���, and the desired follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' If vrK ≤ ���T F a �� L2 0(Xn) ���, then the desired follows for C(K) = diam(K) (vrK )1/(n2−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' ■ 9 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='7 (Composition of Hecke operators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' The composition of Hecke operators is a linear combination of different Hecke operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' This follows from the double-coset description, as the multiplication of two finite double-cosets is again a double-coset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' We mention explicitly the case of the operators a, a′ defined with (−k, 0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=', 0), (−l, 0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=', 0) respectively, where 0 ≤ k ≤ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' The composition is T F a ◦ T F a′ = T F a′ ◦ T F a = pn−1(p − 1) pn − 1 T F ak+l + l−1 � i=1 pn−1 − 1 pn − 1 p − 1 pi T F ak+l−i,i + pn−1 − 1 pn − 1 1 pl−1 T F ak,l Where ak′,l′ corresponds to −k′ ≤ −l′ ≤ 0 ≤ 0 ≤ · · · ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' A similar equality hold for T M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='2 Construction of special number field Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' For every prime number p ≡ 1 mod 2n sufficiently big as a function of n, one can find a totally real number field K with the following properties: (a) The unit group O× K contains units u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=', un with u1u2 · · · un = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' (b) One can order the real embeddings σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=', σn : K → R such that σi(uj) > 0 for all 1 ≤ i, j ≤ n, and log σi(uj) = \uf8f1 \uf8f2 \uf8f3 −2d(d − 1) log p + O(1), if i = j, 2d log p + O(1), if i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='3) (c) The units ui lie in the ring Z + pOK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Since p ≡ 1 mod 2n, the polynomial xn + 1 has a n different solutions mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' By Hensel’s Lemma (See [Conrad(2015)]), it has n solutions mod pn, call them a′ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=', a′ n−1 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=', pn −1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' Let ai = a′ i + 2ipn for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' , n − 1, and note that pn + 1 ≤ ai+1 − ai ≤ 3pn for all 0 ≤ i ≤ d − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' consider now the polynomial R(x) = (px−a1)(px−a2)···(px−an)−1 pn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' It has integer coefficients since by assumption, � i1<.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAyT4oBgHgl3EQf0fnz/content/2301.00721v1.pdf'} +page_content=' i) absorb the divergent and the power counting violating terms appearing at order +O(Qj). +The LO potential V0 is regulated (the details are given in Sec. II B and in Appendix A) using a cutoff Λ to make +the iterations of V0 finite. We regard the cutoff value Λ (the largest cutoff among all cutoffs used in the LO potential) +to be of the order of the hard scale Λ ∼ Λb. Higher order potentials can be considered either regulated or unregulated +depending on a particular scheme, which will be discussed in the subsequent sections. +Note that to make some intermediate expressions mathematically well defined, one might need to introduce addi- +tional cutoffs that drop out from the final results after performing certain subtractions. Such cutoffs can be chosen +to be much larger than Λ (or even infinity large). + +4 +To make the formulation of the theory in terms of non-local (on the Lagrangian level) regularized potential contri- +butions completely equivalent to the original formulation in terms of local interactions, the regulator corrections δΛV +have to be taken into account: +δΛV = +� +i +δΛV (i), +δΛV (i) := V (i) +Λ=∞ − V (i) +Λ , +(6) +where V (i) +Λ=∞ is the unregulated potential at the chiral order i. One possibility, often implicitly used in practical +calculations, is to expand δΛV in powers of 1/Λ and absorb the resulting terms by higher order contact interactions. +This is possible if the potential does not contain non-locally regularized long-range contributions. Another approach +suggested in Ref. [16] is to keep the terms with δΛV explicitly and consider those as perturbation. This allows us to +reduce the cutoff dependence and extend the range of possible values of Λ, especially to smaller ones. +B. +LO and NLO potentials and regulators +Our treatment of the LO and NLO potentials is identical to Ref. [16]. +Weinberg’s power counting in Eq. (2) implies that the leading-order O(Q0) potential V0(⃗p ′, ⃗p ) is represented by +the sum of the regulated static one-pion-exchange potential and the short-range part: +V0(⃗p ′, ⃗p ) = V (0) +1π,Λ(⃗p ′, ⃗p ) + V (0) +short,Λ(⃗p ′, ⃗p ), +(7) +where the short-range part V (0) +short,Λ may contain momentum-independent contact terms as well as the contact terms +quadratic in momentum. The latter are formally of order O(Q2), as follows from Eq. (2). Nevertheless, it is known +that in some channels, e.g., 1S0 and 3P0, their promotion to leading order can be motivated by phenomenological +arguments, see, e.g., Refs. [18–21]. +For the sake of generality, we allow for different forms of regulators: power-like local, power-like non-local, Gaussian +local and Gaussian non-local regulators as well as all possible combinations of those. In Ref. [16], we argued that for +a local part of the LO potential V0,local(⃗q ), the regulator (if it is also local) can be rather “mild”. If the regulated LO +potential behaves as +V0,local(⃗q ) ∼ +1 +|⃗q |2 , +for |⃗q| → ∞, +(8) +both LO and NLO amplitudes turn finite after renormalization even if the NLO potential is not regulated. The reason +for that is a milder ultraviolet behavior of local structures after performing subtractions. Such a mild regulator cannot +be chosen for the non-local parts of the LO potential. +Equation (8) implies that in the spin-triplet channels the one-pion-exchange potential can be regulated by a dipole +form factor, +Fq,1π,Λ,1 = Λ2 − M 2 +π +q2 + Λ2 , +(9) +whereas for the spin-singlet channels it can even be left unregulated. +Although in practical calculations one typically implements Gaussian or even sharper regulators to guarantee the +finiteness of all integrals, we consider separately the above mentioned situation with a local part of the LO potential +having the ultraviolet asymptotics as in Eq. (8) and say that such a potential has a “mild” regulator in contrast to +“standard” regulators, i.e. all other cases. This is done to keep the analysis general and to clarify the difference between +perturbative and non-perturbative regimes. Moreover, such an analysis is useful to understand the cutoff dependence +of the NN amplitude: the milder regulator can be chosen, the weaker cutoff dependence should be expected. +For completeness, we provide the explicit expressions for the LO potential and the corresponding regulators in +Appendix A. +The next-to-leading-order potential V2(⃗p ′, ⃗p ) contains the short-range part, the two-pion-exchange potential and +the regulator corrections to the leading-order potential: +V2(⃗p ′, ⃗p ) = V (2) +2π (⃗p ′, ⃗p) + V (2) +short(⃗p ′, ⃗p ) + δΛV (0)(⃗p ′, ⃗p ) . +(10) +In Ref. [16], we found that one does not need to regularize the NLO potential to perform the renormalization of the +NLO amplitude. Or, equivalently, one can introduce a cutoff ΛNLO ≫ Λ. On the other hand in practical calculations, + +5 +one can choose ΛNLO ∼ Λb if it improves efficiency of a computational scheme. Both approaches are formally equivalent +because the regulator corrections δΛV (2) appear at order O(Q4) in accordance with the dimensional power counting. +It turns out, that the situation is slightly different in the general non-perturbative case, where for the choice of the +“mild” LO regulator we need to keep ΛNLO finite. It can still be larger than Λ, but not arbitrarily large, see discussion +in Sec. V. +The explicit expressions for the NLO potential can be found in Appendix B. +C. +NN amplitudes and contour rotation +In the present study we work predominantly in the partial wave lsj basis, which makes the analysis of the non- +perturbative effects more efficient. In the lsj basis, the potential and the amplitude are nPW × nPW matrices, where +nPW = 1 (nPW = 2) for the uncoupled (coupled) partial waves. The series for the partial wave LO amplitude and for +the unrenormalized NLO amplitude are given by +T0 = +∞ +� +n=0 +T [n] +0 , +T [n] +0 += V0Kn = ¯KnV0, +(11) +T2 = +∞ +� +m,n=0 +T [m,n] +2 +, +T [m,n] +2 += ¯KmV2Kn, +(12) +where G is the free two-nucleon propagator and +K = GV0, +¯K = V0G. +(13) +In the non-perturbative case these equations generalize to +T0 = V0R = ¯RV0, +(14) +T2 = ¯RV2R, +(15) +where R ( ¯R) is the resolvent of the Lippmann-Schwinger equation (LSE) +R = +1 +1 − K , +¯R = +1 +1 − ¯K . +(16) +The renormalized expression for the NLO amplitude R(T2) is obtained by adding the relevant counter term, see +Sec. V for details: +R(T2) = ¯R +� +V2 + δV (2) +0 +� +R. +(17) +The explicit form of the LSE, T0 = V0 + V0GT0, reads +(T0)l′l (p′, p; pon) = +� +l′′ +� p′′2dp′′ +(2π)3 (V0)l′l′′ (p′, p′′)G(p′′; pon) (T0)l′′l (p′′, p; pon), +G(p′′; pon) = +mN +p2on − p′′2 + iϵ. +(18) +The indices l, l′, l′′ denote the orbital angular momentum of the NN system, pon is the on-shell c.m. nucleon momentum +and p (p′) are the initial (final) off-shell c.m. momenta. +It turns out useful to modify the integration path over the off-shell momentum p′′ and rotate the contour into the +complex plane [22–24]. The new integration contour C is defined by p′′ = |p′′|e−iαC. Our choice for the rotation angle +αC is determined by the location of singularities of the LO potential in the complex plane [16]: +αC = 1 +2 arctan +Mπ +(pon)max +, +(19) +where (pon)max is the maximal considered on-shell momentum. +The contour rotation enables us to perform direct estimations of the bounds on the partial wave amplitudes avoiding +principal value integrals. + +6 +D. +Bounds on the potentials and the NN propagator +By analogy with Ref. [16], we use certain upper bounds for the potentials and the NN propagator that are valid for +off-shell momenta lying on the complex contour C and for the allowed real on-shell momenta. These bounds allow us +to estimate the nucleon-nucleon LO and NLO amplitudes and to verify the corresponding power counting. +Following Ref. [16], in the bounds considered below, we introduce dimensionless constants named Mi: MV0, MG, +etc., which are supposed to be of order one. Analogous constants appear in our final estimates for the amplitudes. +Some of the inequalities should be modified compared to Ref. [16] to be better suited for the non-perturbative +analysis. In particular, for the LO potential V0(p′, p), we need bounds that are separable in momenta p and p′. +The inequalities listed below are meant to hold for all matrix elements of the partial wave potentials V0(p′, p) and +V2(p′, p) in l , l′ space. Their derivation can be found in Appendices C and D. +The LO partial-wave potential obeys the following bounds: +|V0(p′, p)| ≤ MV0V0,max g(p′)h(p), +|V0(p′, p)| ≤ MV0V0,max h(p′)g(p), +(20) +with +V0,max = +8π2 +mNΛV +, +(21) +where the exact form of the functions g and h (and the value of MV0) depends on the partial wave and on the form +of a regulator. For l = 0 (for the coupled partial waves, we mean by l the lowest possible orbital angular momentum), +g and h are given by +g(p) = λlog(p/Λ) , h(p) = 1 , +(22) +for the “mild” regulator, and by +g(p) = [λ(p/Λ)]2 , h(p) = [λ(p/Λ)]−1 , +(23) +for the “standard” regulators with the functions λ and λlog defined as +λ(ξ) = θ(1 − |ξ|) + θ(|ξ| − 1) 1 +|ξ|2 , +λlog(ξ) = θ(1 − |ξ|) + θ(|ξ| − 1)1 + ln |ξ| +|ξ|2 +. +(24) +For higher partial waves, l ≥ 1, we adopt the bounds +g(p) = λlog(p/Λ)/|p| , h(p) = |p|. +(25) +Notice that while in the latter case one could use a stronger bound and replace λlog with λ for the “standard” regulator, +this would not affect our conclusions. Therefore, we prefer to employ this unified bound. +For spin-singlet partial waves without a short-range LO contribution, one can improve the above bounds and replace +in Eq. (25) λlog(p/Λ) with λlog(p/Mπ). However, in all such channels the perturbative regime for the LO potential is +realized, which has already been analyzed in Ref. [16] and will not be considered here. +Note that for |p| ≤ Λ, and, in particular, for the on-shell momentum |p| = pon, we have in all cases g(p) = h(p) = 1. +It is convenient also to introduce the functions +v0(p′, p) = V0(p′, p) [MV0V0,max h(p′)g(p)]−1 , +¯v0(p′, p) = V0(p′, p) [MV0V0,max g(p′)h(p)]−1 , +(26) +for which the following bounds hold: +|v0(p′, p)| ≤ 1 , +|¯v0(p′, p)| ≤ 1 . +(27) +For the unregulated NLO potential, we adopt the bounds from Ref. [16]. In particular, for l = 0: +|V2(p′, p)| ≤ MV2,0 +� +|p|2 + |p′|2� ˜flog(p′, p), +(28) + +7 +with +˜flog(p′, p) = +8π2 +mNΛV Λ2 +b +flog(p′, p) , +flog(p′, p) = θ(|p| − Mπ) ln |p| +Mπ ++ θ(|p′| − Mπ) ln |p′| +Mπ ++ 1 , +(29) +where we have dropped the log Λ/Mπ term in the definition of flog, which is unnecessary and was introduced in +Ref. [16] for convenience. +Note that in Ref. [16], the NLO potential V2 was split into two parts +V2(p ′, p) = ˆV2(p ′, p) + ˜V2(p ′, p), +(30) +with +ˆV2(p ′, p) = V2(0, 0) , +˜V2(p ′, p) = V2(p ′, p) − V2(0, 0), +(31) +and the inequality in Eq. (28) is, strictly speaking, valid for ˜V2. However, in the present work, we use most of the +time the scheme with V2(0, 0) = 0. Therefore, in what follows, we will always assume that ˜V2 = V2 unless specified +otherwise. For alternative schemes, we also provide the bound for ˆV2: +��� ˆV2(p′, p) +��� ≤ ˆ +MV2,0 +8π2 +mNΛV +M 2 +π +Λ2 +b +. +(32) +For higher partial waves l > 0, it is sufficient to implement the p-wave bound: +|V2(p′, p)| ≤ MV2,1|p′||p| ˜flog(p′, p). +(33) +For the regularized NLO potential with the cutoff ΛNLO, the bounds in Eq. (28) are modified as follows (see +Sec. D 3 a): +|V2(p′, p)| ≤ MV2,0 +� +|p|2 + |p′|2� ˜flog(p′, p)λlog(p′/ΛNLO) , or +|V2(p′, p)| ≤ MV2,0 +� +|p|2 + |p′|2� ˜flog(p′, p)λlog(p/ΛNLO) . +(34) +For the two-nucleon propagator G(p; pon) = mN/(p2 +on − p2), we use the same bound as in Ref. [16]: +|G(p; pon)| ≤ MG +mN +|p2| , +(35) +with MG = 1/ sin(2αC). +III. +LEADING-ORDER LIPPMANN-SCHWINGER EQUATION +In this section we outline the Fredholm method for solving integral equations and derive the bounds on the resolvents +of the LSE and on the LO amplitude in the non-perturbative case. The resolvents R and ¯R of the partial-wave LSE, +see Eq. (16), can be represented by means of the Fredholm formula [25, 26] as: +R = (1 − K)−1 = 1 + Y +D, +¯R = (1 − ¯K)−1 = 1 + +¯Y +D, +(36) +where the Fredholm determinant D is a number and depends only on the on-shell momentum D = D(pon), whereas +the minor Y ( ¯Y ) is a matrix in the l, l′ space and an operator in the space of the off-shell momenta: Y = Yji(p′, p; pon). +The quantities Y , ¯Y and D can be expanded into convergent series in powers of the LO potential V0: +Y = +∞ +� +n=1 +Y [n], +¯Y = +∞ +� +n=1 +¯Y [n], +D = +∞ +� +n=0 +D[n]. +(37) + +8 +In what follows, we will consider the resolvent R and the minor Y . The results are trivially generalized for ¯R and +¯Y . +The terms D[n] and Y [n] can be written as [25, 26] +D[n](pon) = (−1)n +n! +� +i1,...,in +� +n +� +k=1 +p2 +kdpk +(2π)3 [detD,n(K)]i1,... in (p1, . . . , pn; pon), +(38) +and +Y [n+1] +i′i +(p′, p; pon) =(−1)n +n! +� +i1,...,in +� +n +� +k=1 +p2 +kdpk +(2π)3 [detY,n+1(K)]i,i1,... in,i′ (p, p1, . . . , pn, p′; pon), +(39) +where the matrix indices i, i1, ..in and i′ correspond to the orbital angular momentum l = j ± 1 for coupled partial +waves and l = j for uncoupled partial waves. In the above equations, the determinants for an operator X with matrix +elements X(p′, p; pon) (or X(p′, p) if it is independent of pon) are defined as: +[detD,n(X)]i1,...,in (p1, . . . , pn; pon) = +������ +Xi1,i1(p1, p1; pon) · · · Xin,i1(p1, pn; pon) +. . . +. . . . . . +Xi1,in(pn, p1; pon) · · · Xin,in(pn, pn; pon) +������ +, +(40) +and +[detY,n+1(X)]i,i1,...,in,i′ (p, p1, . . . , pn, p′; pon) = +������� +Xi′i(p′, p; pon) +Xi1i(p1, p; pon) +· · · Xini(pn, p; pon) +Xi′i1(p′, p1; pon) Xi1i1(p1, p1; pon) · · · Xini1(pn, p1; pon) +· · · +· · · +· · · · · · +Xi′in(p′, pn; pon) Xi1in(p1, pn; pon) · · · Xinin(pn, pn; pon) +������� +. +(41) +Rescaling V0 as in Eq. (26), we obtain: +Ki′i(p′, p; pon) = (v0)i′i(p′, p)MV0V0,max g(p′)h(p)G(p′; pon), +(42) +so that +D[n](pon) = (−1)n +n! +(MV0V0,max)n +� +i1,...,in +� � n +� +k=1 +p2 +kdpk +(2π)3 g(pk)h(pk)G(pk; pon) +� +[detD,n(v0)]i1,...,in (p1, . . . , pn), +(43) +and +Y [n+1] +i′i +(p′, p; pon) = (−1)n +n! +(MV0V0,max)n+1 g(p′)h(p)G(p′; pon) +× +� +i1,...,in +� � n +� +k=1 +p2 +kdpk +(2π)3 g(pk)h(pk)G(pk; pon) +� +[detY,n+1(v0)]i,i1,...,in,i′ (p, p1, . . . , pn, p′). +(44) +A. +Upper bounds for the Fredholm determinant +First, we analyze the series for the Fredholm determinant D. Since the matrix elements v0;ji(p′, p) are bounded by +(see Eq. (27)) +|v0;ji(p′, p)| ≤ 1 , +(45) +the Hadamard’s inequality for determinants gives [25, 26] +|detD,n(v0)| ≤ nn/2. +(46) +Therefore, using Stirling’s formula, we can estimate D[n] as follows: +���D[n]��� ≤ 1 +n!Σnnn/2 ≤ +1 +√ +2πn +� eΣ +√n +�n += +1 +√ +2πeΣ +� eΣ +√n +�n+1 += +1 +√ +2πeΣ exp +� +− (n + 1) ln +√n +eΣ +� +=: MD,n. +(47) + +9 +where Σ is defined as +MV0V0,maxnPW +���� +� +p2dp +(2π)3 g(p)h(p)G(p; pon) +���� ≤ MV0MG +ΛV +nPW +� d|p| +π g(p)h(p) =: Σ . +(48) +Since g(p) and h(p) depend only on the ratio p/Λ, we can write +Σ = MΣ +Λ +ΛV +, +(49) +where the numerical value of the constant MΣ depends on a particular form of g(p) and h(p). +If we assume Λ ∼ ΛV , then Σ ∼ 1 up to a numerical factor. The situation when Σ < 1 corresponds to a convergent +series for the LO amplitude in terms of V0. In contrast, for the non-perturbative regime that we consider, we have +Σ ≥ 1. +The maximal value of D[n] is achieved at some n = nDmax and can be estimated by differentiating Eq. (47) with +respect to n: +nDmax ≈ eΣ2 , +|D[n]| ≤ MD[n],max ≈ eeΣ2/2 +√ +2πeΣ , +(50) +which is formally a number of order one, but it grows very rapidly with Σ. +The whole series for D is also bounded by a constant of order one: +|D| ≤ MD, +(51) +which can be estimated by replacing the sum with an integral and using Laplace’s method: +MD = +∞ +� +n=0 +MD,n ≈ +� ∞ +0 +dtMD,t ≈ +√ +2π +� +−∂2 ln MD,t +∂t2 +�−1/2 +MD,t +��� +t=nDmax +≈ +√ +2eeΣ2/2 , +(52) +which agrees rather well with the series summed numerically (see Eq. (47)). For example, for Σ = 1, both results give +MD ≈ 5. +The bounds (47) and (52) are rather weak and very conservative. If Σ is not close to one, the numerical values +for MD become very large. However, in realistic calculations, we can see that D does actually not exceed the values +of order one. Clearly, one can always perform a numerical check in order to verify whether our approach to the +renormalizability of the NN amplitude based on the Fredholm method is reliable. Note also that for the 1S0 and +3S1 − 3D1 NN channels, one can expect Σ to be close to one (ignoring the fine-tuning between attractive and repulsive +forces) because the first (quasi) bound states in these channels are very shallow. This is roughly confirmed by an +analysis of the Weinberg eigenvalues in Ref. [9]. +There are particular cases when the estimate in Eq. (47) can be readily improved. For example, for purely local +LO potentials, the quantities D and Y correspond to the Jost function and the regular solution of the Schr¨odinger +equation in configuration space and the terms in their expansion, D[n] and Y [n], decrease as 1/n!. On the other hand, +if the LO consists of only a short-range separable potential (or is dominated by such a contribution), the series for +D and Y contain a finite number of terms. However, in our general discussion, we will simply assume that Eq. (51) +holds. +We will also need an estimate for the series remainder: +δnD = +∞ +� +k=n+1 +D[n]. +(53) +From Eq. (47), we can conclude that for sufficiently large n, +n > n0 ≡ ˜ +MδD, +(54) +the terms D[n] and, therefore, also the remainder δn∆ decrease faster than exponential +δnD ≤ e−MδD n, +(55) + +10 +with any MδD, which we will use in our further estimates. The value +˜ +MδD depends on MδD and on Σ. Based on +Eq. (47), we can conclude that the exponential decrease starts only for +˜ +MδD > (eΣ)2, +(56) +which, being formally a number of order one, becomes extremely large unless Σ ≈ 1. However, as follows from the +discussion above, in realistic calculations, such an exponentially suppressed regime can be reached much earlier. In +fact in the numerical calculation presented in Sec. VI, the relative error δnD/D becomes less than one percent in most +cases for n = 3 or 4. +B. +Bounds for the minor Y +By analogy with the Fredholm determinant D, we can perform the same analysis for the minor Y starting from the +definition in Eq. (44). Using again the Hadamard’s inequality, +|detY,n(v0)| ≤ nn/2, +(57) +we get the bound for Y [n]: +���Y [n] +ji (p′, p; pon) +��� ≤ MY,n |G(p′, pon)| 8π2MV0 +mNΛV +g(p′)h(p) +(58) +with +MY,n = +1 +(n − 1)!Σn−1nn/2 ≤ +e +√ +2π +� eΣ +√n +�n−1 +. +(59) +Further, taking into account the bound for the propagator in Eq. (35), we obtain +���Y [n] +ji (p′, p; pon) +��� ≤ MY,n +8π2MV0MG +ΛV |p′|2 +g(p′)h(p) =: 8π2MY +ΛV |p′|2 MY,n g(p′)h(p). +(60) +Analogously to Eq. (52), the whole series for Y can be estimated to be +|Yji(p′, p; pon)| ≤ 8π2MY +ΛV |p′|2 Ymax g(p′)h(p) =: 8π2MYmax +ΛV |p′|2 +g(p′)h(p), +(61) +where +Ymax = +∞ +� +k=0 +MY,n ≤ +√ +2e Σ eeΣ2/2. +(62) +The remainder δnYmax, defined as +δnYmax = +∞ +� +k=n+1 +MY,n, +(63) +can be bounded similarly to δnD by an exponent with an arbitrary base: +δnYmax ≤ e−MδY n, +for n > ˜ +MδY , +(64) +with some +˜ +MδY . As in the case of δnD, the estimated value of +˜ +MδY ∼ (eΣ)2 becomes very large for Σ significantly +larger than one. However, in the actual calculations, its numerical value is typically much more natural, see the +discussion in the previous subsection. The same comment applies also to the bound in Eq. (62) for Ymax. +The remainder δnY (p′, p; pon) follows from Eq. (64): +|δnYji(p′, p; pon)| = +����� +∞ +� +k=n +Y [n] +ji (p′, p) +����� ≤ 8π2MY +ΛV |p′|2 δnYmax g(p′)h(p) =: 8π2NδnY +ΛV |p′|2 g(p′)h(p) . +(65) +The bounds for ¯Y (p′, p; pon) are obtained from Eqs. (61) and (65) by interchanging p ↔ p′. + +11 +C. +Bounds for the LO amplitude +After these preparations, we are finally in the position to deduce the bounds for the on-shell LO amplitude, which +can be represented as +T0 = V0R = N0 +D , +N0 = V0D + V0Y. +(66) +First, consider the quantity N0 defined explicitly as follows: +(N0)ji(pon) = (V0)ji(pon, pon)D(pon) ++ +� +i′ +� p′2dp′ +(2π)3 (V0)ji′(pon, p′)Yi′i(p′, pon; pon). +(67) +Applying the bounds from Eqs. (20), (51) and (61) , we obtain +|(N0)ji(pon)| ≤ MV0V0,max +� +MD + nPWMYmax +ΛV +� d|p| +π g(p)h(p) +� +≤ MV0V0,max +� +MD + MYmaxΣ +MV0MG +� +=: MN0V0,max. +(68) +Now, we can analyze the bounds for the LO amplitude T0. Since T0 is the ratio of N0 and D, it is important how +the Fredholm determinant D is bounded from below. From the definition in Eq. (38), it follows that all terms D[n] +should be in general of order O(Q0). However, in a realistic situation, there might be certain cancellations among +terms in the series, and the actual numerical value of D(pon) might turn out to be very small. This can happen when +there is a shallow bound or quasibound state, which leads to an enhancement of the amplitude at threshold. Such a +situation only takes place in the 1S0 of NN scattering. Therefore, in our analysis for higher partial waves with l ≥ 1, +we regard the Fredholm determinant as being “natural”: +|D(pon)| ≥ MD,min, +(69) +where MD,min is a constant of order one. From Eqs. (68) and (69), we conclude that for l ≥ 1, the LO amplitude is +bounded by +|(T0)ji| ≤ MT0V0,max, +(70) +and satisfies the same power counting as V0, i.e. is of order O(Q0). +For the S-wave channels, we allow for the real part of D to be small, while still bounded from below at least at +threshold. Moreover, we assume that the imaginary part of D, which is proportional to pon, is not a subject to +additional cancellations. In particular, we exclude the situation when both N and D are equal to zero, i.e. the +presence of a Castillejo-Dalitz-Dyson (CDD) pole [27, 28]. We combine these conditions into the following constraint: +|D(pon)| ≥ MD,min +� +κ + pon +ΛV +� +, +(71) +where κ > 0 is not necessarily of order one, but can be numerically small. The factor 1/ΛV in front of pon follows +from the upper bound for the imaginary part of D. +The LO amplitude T0 is enhanced compared to V0, which can be written as +|(T0)ji| ≤ MT0κ−1V0,max, +(72) +or +|(T0)ji| ≤ MT0 +ΛV +pon +V0,max, +(73) +depending on the value of the on-shell momentum pon. The latter bound is in fact a unitary limit for the LO amplitude +up to a numerical factor of order one, which justifies the coefficient 1/ΛV in Eq. (71), see the definition of V0,max +in Eq. (D21). Equation (73) means that the LO amplitude becomes effectively of order O(Q−1) in agreement with +findings of Refs. [29, 30]. +To summarize, we have applied the Fredholm method to decompose the resolvent of the LS equation and derived +the bounds for the Fredholm determinant D, the minor Y and the on-shell LO amplitude. +The bounds involve +undetermined dimensionless constants of order one, which can be calculated for each particular situation. + +12 +IV. +NEXT-TO-LEADING ORDER AMPLITUDE IN THE NON-PERTURBATIVE CASE. P- AND +HIGHER PARTIAL WAVES +In this section we consider the on-shell (p = p′ = pon) NLO amplitude T2 for orbital angular momenta l ≥ 1 and +derive the corresponding bounds in the non-perturbative regime. We represent the amplitude T2 using the Fredholm +decomposition of the resolvent in Eq. (36) as follows: +T2 = ¯RV2R = V2 + T2,Y /D + T2, ¯Y /D + T2, ¯Y Y /D2 =: N2 +D2 , +(74) +with +T2,Y = V2Y , +T2, ¯Y = ¯Y V2 , +T2, ¯Y Y = ¯Y V2Y , +(75) +or more explicitly: +T2,Y (p′, p; pon) = +� p2 +1dp1 +(2π)3 V2(p′, p1)Y (p1, p; pon) , +T2, ¯Y (p′, p; pon) = +� p′2 +1 dp′ +1 +(2π)3 ¯Y (p′, p′ +1; pon)V2(p′ +1, p) , +T2, ¯Y Y (p′, p; pon) = +� p2 +1dp1 +(2π)3 +p′2 +1 dp′ +1 +(2π)3 ¯Y (p′, p′ +1; pon)V2(p′ +1, p1)Y (p1, p; pon). +(76) +First, consider T2,Y . The bounds for V2 and Y in Eqs. (33) and (61) give +|T2,Y (p′, p; pon)| ≤ +� |p1|2d|p1| +(2π)3 +|V2(p′, p1)||Y (p1, p; pon)| +≤ MV2,1nPW +8π2MYmax +ΛV +|p′|h(p) +� |p1|d|p1| +(2π)3 +˜flog(p′, p1)g(p1). +(77) +The functions g and h for P- and higher partial waves are given in Eq. (25), which results in the following inequality: +|T2,Y (p′, p; pon)| ≤ MV2,1nPW +8π2MYmax +ΛV +|p′||p| +� +d|p1| +(2π)3 ˜flog(p′, p1)λlog(p1/Λ) += MV2,1nPW +8π2MYmax +ΛV +8π2 +mNΛV Λ2 +b +|p′||p| +� +d|p1| +(2π)3 flog(p′, p1)λlog(p1/Λ) += MV2,1nPWMYmax +ΛV +8π2 +mNΛV Λ2 +b +|p′||p| +�� +1 + θ(|p′| − Mπ) ln |p′| +Mπ +� +Iλlog,1 + Iλlog,2 +� +, +(78) +where the typical integrals Iλlog,1 and Iλlog,2 are defined and estimated in Appendix F and we have used Eq. (29). +Using those estimates, we obtain: +|T2,Y (p′, p; pon)| ≤ M2,Y +8π2 +mNΛV Λ2 +b +|p′||p| Λ +ΛV +� +1 + θ(|p′| − Mπ) ln |p′| +Mπ ++ ln Λ +Mπ +� +, +(79) +which reduces to +|T2,Y (pon)| ≤ M2,Y ;on +8π2 +mNΛV Λ2 +b +Λ +ΛV +p2 +on ln Λ +Mπ +, +(80) +for the on-shell momenta p = p′ = pon. The bounds for T2, ¯Y are the same as for T2,Y . +Next, we analyze T2, ¯Y Y : +��T2, ¯Y Y (p′, p; pon) +�� ≤ +� |p1|2d|p1| +(2π)3 +||p′ +1|2d|p′ +1| +(2π)3 +| ¯Y (p′, p′ +1; pon)||V2(p′ +1, p1)||Y (p1, p; pon)| +≤ MV2,1n2 +PW +�8π2MYmax +ΛV +�2 +h(p′)h(p) +� |p1|d|p1| +(2π)3 +|p′ +1|d|p′ +1| +(2π)3 +˜flog(p′, p1)g(p′ +1)g(p1) . +(81) + +13 +The integrals over p1 and p′ +1 factorize, giving rise to the same set of integrals as in T2,Y . The analog of Eq. (80) for +T2, ¯Y Y in the on-shell kinematics is given by +��T2, ¯Y Y (pon) +�� ≤ M2, ¯Y Y ;on +8π2 +mNΛV Λ2 +b +Λ2 +Λ2 +V +p2 +on ln Λ +Mπ +. +(82) +Combining the bounds for V2, T2,Y , T2, ¯Y and T2, ¯Y Y and setting Λ ∼ ΛV , we obtain +|T2(pon)| ≤ ˜ +M2 +8π2 +mNΛV Λ2 +b +p2 +on ln Λ +Mπ +� +1 + D(pon)−1 + D(pon)−2� +. +(83) +Since we assume that for the P- and higher partial waves the Fredholm determinant is bounded from below by a +constant of order one, see Eq. (69), equation (83) takes the form +|T2(pon)| ≤ M2 +8π2 +mNΛV Λ2 +b +p2 +on ln Λ +Mπ +. +(84) +Thus, the NLO amplitude is of order O(Q2) up to a factor ln Λ/Mπ, which agrees with the dimensional power counting. +This result reproduces the one obtained in Ref. [16] for the case of a perturbative LO interaction. +A. +Promoting a contact term to leading order +In this subsection we consider separately the scenario with promoting leading P-wave contact terms to the LO +potential. As already discussed in Sec. II B, phenomenological arguments may require a promotion of contact interac- +tions quadratic in momenta to the LO potential, even though they are formally of order O(Q2). A typical example is +the 3P0 partial wave, where the promotion of the contact interaction to leading order is often considered as necessary. +Below, we discuss the subtlety related to the freedom of choosing the renormalization condition, i.e., deciding what +part of the considered contact interaction should be included into the LO potential and what part of it should be left +in the NLO potential. +The LO partial wave contact interaction in the P-wave channel i is given by +V (0) +short,Λ,i(p ′, p) = Ci VCi,Λ(p ′, p), +(85) +where VCi,Λ(p ′, p) is the partial wave projection of the regulated contact term (see Appendix A) relevant for the +considered channel. The corresponding NLO contact interaction has the same structure: +V (2) +short,Λ,i(p ′, p) = C2,i VCi,Λ(p ′, p). +(86) +In our estimates, we always assume that the LO low energy constants (LECs) are of natural size, +Ci = MCi +Λ2 +b +8π2 +mNΛV +, +(87) +see Appendix of Ref. [16] (the factor of 4π corresponds to the partial-wave basis), so that the contact interactions +quadratic in momenta are of order ∼ p2/Λ2 ∼ O(Q2) and are suppressed for small momenta. As a consequence, +the regulator corrections to the contact interactions quadratic in momenta are effects of order O(Q4) and can be +neglected in the present study. This is why we adopt the same regulator for V (0) +short,Λ,i and V (2) +short,Λ,i even though, in +principle, one could employ a larger cutoff for the NLO terms or even use the unregulated potential. Nevertheless, if +the contact interactions quadratic in momenta are promoted to leading order, their contribution in the iterations of +the LO potential at momenta p ∼ Λ is of the same order as those of the momentum-independent contact interactions +and of the one-pion-exchange potential as long as we treat Λ ∼ Λb. +The freedom to choose the renormalization scheme manifests itself schematically as follows: if we perform the +transformation +Ci → Ci + δCi, +C2,i → C2,i − δCi, +δCi ≪ Ci , +(88) +and expand the LO and NLO amplitudes in Eqs. (14) and (15) in δC, then the linear in δC terms cancel: +δT0 ≈ −δT2 ≈ δC ¯RVCi,ΛR, +(89) + +14 +where we have neglected higher order effects, such as the terms proportional simultaneously to δC and the NLO +potential. +As was shown in this section, there are no power counting breaking contributions in P-waves at NLO stemming +from the iterations of the LO potential. This means that C2,i is the renormalized quantity, where we assume that +the divergent contributions to the two-pion-exchange diagrams are subtracted within some scheme, e.g. as is done +for our choice of the non-polynomial two-pion-exchange contribution, see Eq. (B2). Then, one obvious choice for the +renormalization condition is +C2,i = 0. +(90) +However, at higher orders, power counting breaking terms will appear also in P-waves, and one will have to absorb +them by performing renormalization of the same contact interaction. Therefore, to be consistent with our subtraction +scheme for the S-waves, we impose the renormalization condition on Ci and C2,i by requiring that the NLO amplitude +in channels with l = 1 vanishes at threshold faster than p2 +on: +(T2)11(pon)/p2 +on +��� +pon=0 = 0. +(91) +Instead of the threshold point pon = 0, one can also take another renormalization point below or above threshold +within the applicability of our approach. +A potential problem related to the above renormalization condition was discussed in great detail in Ref. [31] when +studying schemes with large or infinite cutoffs. It arises near “exceptional” cutoff values for which the contribution +of the contact interaction to the NLO amplitude is unnaturally small: +� ¯RVCi,short,ΛR +� +(pon)/p2 +on +��� +pon=0 ≈ 0, +(92) +which, in turn, leads to an unnaturally large value of C2,i. In such a case, the power counting is violated unless the +zero of the function on the left-hand side of Eq. (92) is factorizable (i.e., it appears at all energies). The condition in +Eq. (92) can take place, e.g., in the spin-triplet channels with attractive one-pion-exchange potential such as 3P0 if +the adopted cutoff value is too large. Then one starts to feel the singular nature of the one-pion-exchange potential, +which is reflected in oscillations of the scattering wave function at short distances. Note that this effect does not +directly correspond to the appearance of spurious bound states, although the two issues are related to each other. +In Ref. [31], several particular cases were discussed when the condition in Eq. (92) can be avoided or the corre- +sponding zero is factorizable. However, we are interested in the general case, in which the practical solution of the +problem would be to explicitly verify that the LO potential is chosen in such a way that the condition in Eq. (92) is +not fulfilled. In such a case, the NLO amplitude will satisfy the expected power counting. In fact, for the regulators +mentioned in the discussion in Sec. VI and many other choices tested by us, if the cutoff value is of the order of the +hard scale, Eq. (92) is never fulfilled. A simple indication that the cutoff of the LO potential is not “exceptional” is +the naturalness of the renormalized NLO low energy constants. +To summarize, we have shown that the P-wave NLO amplitudes formally satisfy the dimensional power counting +in the non-perturbative regime. This holds also for the case when a contact interaction quadratic in momenta is +promoted to LO if one makes sure that a certain condition on the LO potential is satisfied. + +15 +V. +NON-PERTURBATIVE RENORMALIZATION OF THE AMPLITUDE AT NLO. S-WAVES. +In this section we consider the renormalization of the NLO amplitude in the non-perturbative regime for S-waves. +As in the perturbative case considered in Ref. [16], subtractions have to be made in order to absorb contributions that +violate power counting. We will start with generalizing the perturbative result of Ref. [16] and then analyze under +which conditions a particular power counting can be established. +A. +General formula +Analogously to the situation discussed in Sec. IV A, there is freedom to choose the momentum-independent part of +the NLO potential +ˆV2(p′, p) = V2(0, 0), +(93) +because it can be partly or completely absorbed by the LO potential. In the perturbative case, the NLO amplitude +corresponding to ˆV2 does not contain any power counting breaking contributions in contrast to the remaining part ˜T2 +that is generated by +˜V2(p′, p) = V2(p′, p) − ˆV2(p′, p). +(94) +In what follows, we will mostly consider the scheme with ˆV2(p′, p) = 0, which is well suited for compensating possible +threshold enhancement of the LO amplitude due to non-perturbative effects. +Alternative schemes will be briefly +discussed separately. Therefore, when using the results of Ref. [16], we will assume +˜V2 = V2, +˜T2 = T2. +(95) +First, we recall some notation from Ref. [16]. For an operator X = Xl′l(p′, p; pon), where l(l′) is the initial (final) +orbital angular momentum, we define the subtraction operation T: +T(X) = X00(0, 0, 0)Vct, +(96) +where the contact term is given by +Vct = |χ⟩⟨χ|, +⟨p, lsj|χ⟩ = δl,0. +(97) +(98) +We assume that the counter term is unregulated or regulated with some Λct ≫ Λ. Analogously, we introduce the +subtraction operation Tmi,ni for subdiagrams (mi, ni) of the diagram (m, n) corresponding to T [m,n] +2 +. We follow the +Bogoliubov-Parasiuk-Hepp-Zimmermann (BPHZ) subtraction scheme [32–34] and represent the renormalized ampli- +tude via the forest formula: +R(T [m,n] +2 +) = T [m,n] +2 ++ +� +Uk∈Fm,n +� +� +(mi,ni)∈Uk +−Tmi,ni +� +T [m,n] +2 +, +(99) +where Fm,n represents the set of all forests, i.e, the set of all possible distinct sequences of nested subdiagrams (mi, ni): +Uk = ((mk;1, nk;1), (mk;2, nk;2), . . . ) , +m ≥ mk;i+1 ≥ mk;i ≥ 0 , +n ≥ nk;i+1 ≥ nk;i ≥ 0 , +n + m > 0. +(100) +In Ref. [16], it was proved that each term in the expansion in V0 of the renormalized NLO amplitude satisfies the +dimensional power counting and is bounded by +���R(T [m,n] +2 +)(pon) +��� ≤ 8π2MT2 +mNΛV +Σm+n +2,0 +p2 +on +Λ2 +b +ln Λ +Mπ +, +(101) +where +Σ2,0 = 2Mmax +Λ +ΛV +(102) + +16 +is a quantity of order one (Σ2,0 ≥ 1 in the non-perturbative case). +To resum the series +R(T2)(pon) = +∞ +� +m,n=0 +R(T [m,n] +2 +)(pon), +(103) +we perform some rearrangement of Eq. (99), as explained below. +It is convenient to introduce the following notation: +| ¯ψ⟩ = ¯R|χ⟩, +⟨ψ| = ⟨χ|R, +ψl(p; pon) = ⟨ψ|p, lsj⟩ = ⟨p, lsj| ¯ψ⟩. +(104) +For on-shell momenta p = pon, the explicit form of ψl reads +ψl(pon) := ψl(pon; pon) = δl,0 + +� +p2dp +(2π)3 G(p; pon)(T0)0,l(p, pon; pon), +(105) +and it coincides with the scattering wave function at the origin (r = 0). +Now, consider the sum of all unrenormalized diagrams: +T2 = ¯RV2R, +(106) +and perform first all single overall subtractions: +δT (1),overall +2 += −T(T2) = −(T2)00(0, 0; 0)|χ⟩⟨χ|, +(107) +where the superscript (1) denotes the number of subtractions. +If we add all possible rescatterings with the LO potential, we will obtain all terms with single subtractions in +subdiagrams: +δT (1) +2 += ¯RδT (1),overall +2 +R = −(T2)00(0, 0; 0) ¯R|χ⟩⟨χ|R = −(T2)00(0, 0; 0)⟩| ¯ψ⟩⟨ψ| . +(108) +Analogously, the sum of all double nested subtractions (one of which is an overall subtraction) is given by +δT (2),overall +2 += −T +� +δT (1) +2 +− δT (1),overall +2 +� += (T2)00(0, 0; 0) +� +ψ0(0)2 − 1 +� +|χ⟩⟨χ| = − +� +ψ0(0)2 − 1 +� +δT (1),overall +2 +, +(109) +where the constant term δT (1) +2 +was already subtracted in the previous step and should be excluded. All terms with +double nested subtractions in subdiagrams are obtained by adding the rescattering contributions: +δT (2) +2 += ¯RδT (2),overall +2 +R = − +� +ψ0(0)2 − 1 +� +δT (1) +2 +. +(110) +Continuing with further multiple nested subtractions, we obtain recursion relations: +δT (n+1),overall +2 += −T +� +δT (n) +2 +− δT (n),overall +2 +� += − +� +ψ0(0)2 − 1 +� +δT (n),overall +2 +, +(111) +and +δT (n+1) +2 += − +� +ψ0(0)2 − 1 +� +δT (n) +2 +, +(112) +where the superscripts (n) and (n + 1) denote the number of nested subtractions. The terms T (n) +2 +can be summed up +to +δT2 = +∞ +� +n=1 +δT (n) +2 += δT (1) +2 +∞ +� +n=0 +� +1 − ψ0(0)2�n += δT (1) +2 +1 +ψ0(0)2 . +(113) + +17 +Finally, +R(T2) = T2 + δT2 = T2 − (T2)00(0, 0; 0) +ψ0(0)2 +| ¯ψ⟩⟨ψ| . +(114) +Taking the on-shell matrix elements of R(T2), we obtain: +R(T2)l′l(pon) = (T2)l′l(pon) + δCψl′(pon)ψl(pon), +(115) +with the counter term constant +δC = −(T2)00(0) +ψ0(0)2 +. +(116) +Equation (115) can also be obtained directly without referring to the perturbative result from the renormalization +condition: +R(T2)l′l(0) = 0. +(117) +Therefore, the perturbative and non-perturbative results match in the regime where both are applicable. +Similarly to the analysis of higher partial waves in Sec. IV, we use the Fredholm decomposition of the resolvent of +the LS equation and introduce the quantities N2 and ν, +(T2)l′l(pon) = (N2)l′l(pon) +D(pon)2 +, +ψl(pon) = νl(pon) +D(pon). +(118) +The counter term constant can be expressed as +δC = −(N2)00(0) +ν0(0)2 +. +(119) +Then, the renormalized amplitude R(T2) reads +R(T2)l′l(pon) = +1 +D(pon)2 +� +(N2)l′l(pon) + δC νl′(pon)νl(pon) +� += R(N2)l′l(pon) +D(pon)2 += R( ˜N2)l′l(pon) +D(pon)2 ν0(0)2 , +(120) +where, for convenience, the following quantities have been introduced: +R(N2)l′l(pon) = (N2)l′l(pon) + δC νl′(pon)νl(pon), +(121) +R( ˜N2)l′l(pon) = (N2)l′l(pon)ν0(0)2 − (N2)00(0)νl′(pon)νl(pon). +(122) +B. +Power counting with the naturalness condition for ν0(0) +In this subsection we analyze the expression for the renormalized NLO amplitude R(T2) in Eq. (120) and determine +what power counting it satisfies under which conditions. +Considering different constraints on various quantities +entering R(T2), we can understand to what extent the renormalizability of the amplitude depends on details of the +short-range dynamics. +We assume that the Fredholm determinant D(pon) satisfies the bound in Eq. (71), which includes also the case of +a shallow (quasi-) bound state. For the function D(pon)2, we can write +|D(pon)2| ≥ M2 +D,minκ2, +(123) +or, if κ is very small, +|D(pon)2| ≥ M2 +D,min +p2 +on +Λ2 +V +. +(124) + +18 +We will also need the upper bound for the quantity νl(pon), see Eq. (E21): +νl(pon) ≤ Mν. +(125) +First, we consider the “natural” case when the quantity ν0(0) is bounded not only from above as in Eq. (125), but +also from below by some constant of order one: +ν0(0) ≥ Mν,min, +(126) +which also implies the natural value of the counter term constant δC, see Eq. (119), similarly to the condition of the +absence of “exceptional” cutoffs in Sec. IV A. Then as follows from Eq. (120), to analyze the power counting that the +renormalized amplitude R(T2) satisfies, it is sufficient to find bounds for R( ˜N2). +As we show in Appendix E, the quantity R( ˜N2) can be expanded into a convergent series in terms of V0: +R( ˜N2)(pon) = +∞ +� +m,n=0 +� +R( ˜N2)(pon) +�[m,n] += +nmax +� +m,n=0 +� +R( ˜N2)(pon) +�[m,n] ++ δnmax +� +R( ˜N2)(pon) +� +=: S ˜ +N2,nmax(pon) + δnmax +� +R( ˜N2)(pon) +� +, +(127) +and the remainder δn +� +R( ˜N2)(pon) +� +decreases faster than exponential with any base Mδ ˜ +N2 starting with some n = +˜ +Mδ ˜ +N2 (see Eq. (E22)): +|δn[R( ˜N2)]| ≤ +8π2 +mNΛV +N ˜ +N2e−Mδ ˜ +N2n, +for n > ˜ +Mδ ˜ +N2. +(128) +The prefactor N ˜ +N2 is given by +N ˜ +N2 = Λ2 +Λ2 +b +ln Λ +Mπ +(129) +in the case of the “standard” regulators of the LO potential. For the “mild” regulator, it depends also on the regulator +of the NLO potential ΛNLO: +N ˜ +N2 = ΛΛNLO +Λ2 +b +ln ΛNLO +Λ +ln ΛNLO +Mπ +, +(130) +and, in contrast to the perturbative regime, the regulator ΛNLO cannot be set to infinity (in general) but can be chosen +ΛNLO ≫ Λ. Note that we do not consider the choice ΛNLO ∼ Λ for the “mild” LO regulator because in such a case, +we would simply reproduce the variant with the “standard” regulators. The appearance of ΛNLO in the expression +for N ˜ +N2 is an indication of a potentially stronger cutoff dependence of the NLO amplitude in the non-perturbative +regime. +The general conservative estimate for +˜ +Mδ ˜ +N2 yields +˜ +Mδ ˜ +N2 ≳ (eΣ)2, which is rather large. In realistic calculations, +it turns out to be much smaller, see the discussion in Sec. III A and the numerical results in Sec. VI. +On the other hand, expanding Eq. (120) in V0 gives +� +R( ˜N2)(pon) +�[m,n] += +m +� +m1=0 +m−m1 +� +m2=0 +n +� +n1=0 +n−n1 +� +n2=0 +D[m−m1−m2](pon)D[n−n1−n2](pon) +× ν0(0)[m2]ν0(0)[n2]R(T [m1,n1] +2 +)(pon). +(131) +Using the perturbative bounds on R(T [m,n] +2 +) in Eq. (101) and Eqs. (51), and (125), we obtain +� +R( ˜N2)(pon) +�[m,n] +≤ M2 +DM2 +ν +m +� +m1=0 +n +� +n1=0 +���R(T [m1,n1] +2 +)(pon) +��� +≤ 8π2MT2M2 +DM2 +ν +mNΛV +p2 +on +Λ2 +b +ln Λ +Mπ +m +� +m1=0 +n +� +n1=0 +Σm1+n1 +2,0 +. +(132) + +19 +Performing the summation up to n = nmax, we obtain +���S ˜ +N2,nmax(pon) +��� ≤ +8π2MT2M2 +D[n],max +mNΛV +p2 +on +Λ2 +b +ln Λ +Mπ +nmax +� +m,n=0 +m +� +m1=0 +n +� +n1=0 +Σm1+n1 +2,0 +≤ 8π2MN2;2 +mNΛV +p2 +on +Λ2 +b +ln Λ +Mπ +n4 +maxΣ2nmax +2,0 +=: 8π2MS +mNΛV +p2 +on +Λ2 +b +Φlog. +(133) +Given that the remainder δn[R( ˜N2)] can be made arbitrarily small by choosing a sufficiently large nmax, e.g. +|δn[R( ˜N2)]| ≤ +8π2 +mNΛV +M 2 +πκ2 +Λ2 +b +, +(134) +whereas the sum in Eq. (133) has the bound similar to the one for the perturbative amplitude up to numerical +constants of order one and possible factors logarithmic in Λ, Φlog, we can conclude that R( ˜N2) is bounded as: +���R( ˜N2)(pon) +��� ≤ 8π2M ˜ +N2 +mNΛV +�p2 +on +Λ2 +b +Φlog + M 2 +π +Λ2 +b +κ2 +� +. +(135) +Whether this picture is indeed realized for the realistic NN interaction, i.e., whether M ˜ +N2 is really (and not only +formally) of the order of one, is straightforward to verify by explicit numerical checks of the series for R( ˜N2) as we +do partly in Sec. VI. +For completeness, we show below that Eq. (135) holds formally in the chiral limit, i.e. for the expansion parameter +Q ≪ 1. What we have to prove is that there exists such a value of nmax that the remainder δnmax[R( ˜N2)] satisfies +Eq. (134), and, at the same time, the prefactor +χ = n4 +maxΣ2nmax +2,0 +(136) +in Eq. (133) does not contain inverse powers of Q and, therefore, does not destroy the power counting. +The choice +nmax ≥ max(k0, ¯k0), +(137) +with +k0 = ˜ +Mδ ˜ +N2, +¯k0 = − +1 +Mδ ˜ +N2 +ln M 2 +πκ2 +Λ2 +bN ˜ +N2 +, +(138) +guarantees that Eq. (134) holds, as follows from Eq. (128). Note that the inequality ¯k0 > k0 holds only for extremely +small Q = Mπ/Λb. However, in the actual calculations, this can happen also for physical values of Q. +The factor χ is then given by +χ = ˜ +M4 +δ ˜ +N2Σ +2 ˜ +Mδ ˜ +N2 +2,0 +(139) +if nmax = k0, and by +χ = +1 +M4 +δ ˜ +N2 +� +ln M 2 +πκ2 +Λ2 +bN ˜ +N2 +�4 � +M 2 +πκ2 +Λ2 +bN ˜ +N2 +�−2 +ln Σ2,0 +Mδ ˜ +N2 +, +(140) +if nmax = ¯k0. In the latter case, if Mδ ˜ +N2 is chosen to be Mδ ˜ +N2 ≫ ln Σ2,0, the factor +� +M 2 +πκ2 +Λ2 +bN ˜ +N2 +�−2 +ln Σ2,0 +Mδ ˜ +N2 can be neglected. +Thus, we conclude that Eq. (135) holds with +Φlog = +� +� +� +Mlog ln +Λ +Mπ , +nmax = k0 +Mlog ln +Λ +Mπ +� +ln M 2 +πκ2 +Λ2 +bN ˜ +N2 +�4 +, +nmax = ¯k0 +(141) + +20 +Now we come back to the expression for the renormalized NLO amplitude in Eq. (120). For small on-shell momenta +pon, i.e., when +���S ˜ +N2,nmax(pon) +��� ≤ |δn[R( ˜N2)]|, +(142) +Eqs. (123), (126) and (134) give: +|R(T2)l′l(pon)| ≤ 8π2MT2,low +mNΛV +M 2 +π +Λ2 +b +, +(143) +which means that in this energy region, R(T2) is of order O(Q2). +As the on-shell momentum increases, i.e., +���S ˜ +N2,nmax(pon) +��� ≥ |δn[R( ˜N2)]|, +(144) +we should use Eq. (133) instead of Eq. (134) to obtain +|R(T2)l′l(pon)| ≤ 8π2MT2,high +mNΛV +p2 +on +Λ2 +b +Φlog +κ2 , +(145) +which is enhanced compared to O(Q2) by a factor 1/κ2. In the worst case of the unitary limit, we obtain from +Eq. (124): +|R(T2)l′l(pon)| ≤ 8π2MT2,high +mNΛV +Λ2 +V +Λ2 +b +Φlog, +(146) +which corresponds effectively to R(T2) ∼ O(Q0). This is still one order higher than the LO amplitude O(Q−1), see +Eq. (73), but the convergence rate is rather low in this case. A natural way to reduce the effect of the numerical +enhancement of the LO amplitude and to improve convergence is to promote some part of the NLO potential to +leading order, which will make the numerical constant MT2,high smaller. The simplest recipe would be to promote +the contact interactions quadratic in momentum. As already mentioned, this approach is suggested, e.g., for the 1S0 +partial wave. We will discuss this possibility in Sec. V D. +C. +Local LO potential in a spin-singlet channel and analogous cases +Above, we considered the general case of the LO potential under an additional assumption on its short-range part +formulated in Eq. (126) in terms of the naturalness of ν0(0). It is instructive to consider one particular case, when +the LO potential in a spin-singlet channel is fully local. Then, this condition is satisfied automatically. Moreover, for +a local LO potential, the following identity holds: +ν0(pon) ≡ 1, +(147) +which follows from the fact that the scattering wave function at the origin ψpon coincides with the inverse of the Jost +function f(pon) and the inverse of the Fredholm determinant [25]: +ψ(pon) = f(pon)−1 = D(pon)−1, +(148) +and the definition (118). Therefore, we have (see the definitions in Eqs. (122) and (121)) +R( ˜N2)(pon) = R(N2)(pon) = ∆N2(pon) = N2(pon) − N2(0). +(149) +The whole discussion in the previous subsection applies for the case of a local LO potential, except the absence +of the additional condition (126). In the general case, when the constraint in Eq. (126) is not satisfied, we still can +have a situation similar to the local single-channel potential if we assume that the series for R(N2) (not for R( ˜N2)) +converges and the bound analogous to Eq. (135) holds: +|R(N2)(pon)| ≤ 8π2M ˜ +N2 +mNΛV +�p2 +on +Λ2 +b +Φlog + M 2 +π +Λ2 +b +κ2 +� +. +(150) +This is possible if the smallness of ν0(0) in the denominator of R(N2) is compensated by a corresponding small factor +in the numerator, see Eq. (121). Whether this indeed takes place can be verified numerically in any particular case. +From Eq. (150), we can deduce the same bounds for the renormalized NLO amplitude as in Eqs. (143), (145) and (146). +We made this comment to emphasize that the naturalness constraint on ν0(0) is not necessary to guarantee renor- +malizability of the NLO amplitude, but is the most simple one from the practical point of view. + +21 +D. +Promoting a momentum dependent contact term to leading order +In this subsection we analyze the situation when it is necessary to promote the momentum dependent S-wave contact +term to leading order. For definiteness, we consider the 1S0 partial wave, where such a promotion has been shown +to significantly improve the convergence of the chiral EFT expansion, see Refs. [18, 35]. Since this is a spin-singlet +channel, we omit the l, l′ indices in this subsection. We also omit all channel indices. +The whole analysis in the preceding subsections remains valid in this case, except that similarly to the promotion +of the subleading term in the P-waves considered in Sec. IV A, there is freedom choosing what part of such a contact +term should be included in LO potential V0 and what part remains in the NLO potential V2. +We rewrite Eq. (121) by explicitly separating the part with the contact term quadratic in momenta: +R(N2)(pon) = N2(pon) + δCν(pon)2 +=: ∆N2(pon) + δCν(pon)2 + C2NC2(pon), +(151) +with +NC2(pon) = [ ¯RVCR](pon)D(pon)2. +(152) +The potential VC is the contact interaction quadratic in momenta that projects onto the 1S0 partial wave. This +potential can remain regulated because the regulator corrections to it are of higher order. +Following our subtraction scheme at pon = 0, we introduce two renormalization conditions to fix δC and C2: +R(N2)(0) = 0, +d2R(N2)(pon) +dp2on +���� +pon=0 += 0. +(153) +Note that N2 is an analytic function of p2 +on at pon = 0. +Of course, Eq. (153) can be also formulated in terms of the amplitudes: +R(T2)(0) = 0, +d2R(T2)(pon) +dp2on +���� +pon=0 += 0. +(154) +Analogously to the situation in P-waves, the above renormalization conditions can lead to a problem for “exceptional” +cutoffs when Eqs. (153) become inconsistent, which happens not only when ν(0) = 0 but also when the following +equation is satisfied [31]: +�d2NC2(pon) +dp2on +− 2NC2(pon)ν(pon)d2ν(pon) +dp2on +����� +pon=0 += 0. +(155) +As in the case of the P-waves, an indirect indication that the cutoff is not close to an “exceptional” value is the +naturalness of the NLO LECs. In our numerical calculation in Sec. VI, we found no “exceptional” cutoffs for the +cutoff values of the order or below the hard scale. +E. +Other subtraction schemes +In all analyses of the non-perturbative regime, we have always adopted the prescription to perform subtractions +at threshold, see Eq. (117). In this subsection we briefly discuss other possibilities. Choosing different subtraction +points, e.g., the deuteron pole position for the 3S1 − 3D1 channel, is equivalent to setting, in contrast to Eq. (93), +ˆV2 ̸= 0: +ˆV2(p′, p) = ˆκ2 +8π2 +mNΛV +M 2 +π +Λ2 +b +, +(156) +where ˆκ is a constant of order one, see Eq. (D30). Since this potential is just an S-wave contact term, the corresponding +NLO amplitude is given by +( ˆT2)l′l(pon) = ˆκ2 +8π2 +mNΛV +M 2 +π +Λ2 +b +ψl′(pon)ψl(pon) = ˆκ2 +8π2 +mNΛV +M 2 +π +Λ2 +b +νl′(pon)νl(pon) +D(pon)2 +. +(157) + +22 +From Eqs. (123) and (125), we obtain the following bound: +|( ˆT2)l′l(pon)| ≤ +8π2 +mNΛV +M2 +ν +M2 +D,min +M 2 +π +Λ2 +b +ˆκ2 +κ2 . +(158) +For the perturbative case considered in Ref. [16] and for the case without an enhancement of the LO amplitude, the +amplitude ˆT2 satisfies the dimensional power counting: T2 ∼ O(Q2). However, in the situation when the LO amplitude +is enhanced, the additional factor ˆκ2/κ in Eq. (158) relative to Eq. (72) spoils convergence even at threshold. We will +have the worst situation in the unitary limit with κ ≪ 1. +Thus, we conclude that for a reasonable convergence in the case of an enhanced LO amplitude, one should choose +a subtraction scheme not much different from ours, i.e., such that ˆκ/κ ∼ 1. +To summarize, we have shown that renormalization of the NLO amplitude for the S-waves can be done explicitly +also in the non-perturbative regime by analyzing the Fredholm decomposition of the amplitudes. In contrast to the +perturbative case discussed in Ref. [16], additional constraints on the LO potential have to be fulfilled to ensure +renormalizability and convergence of the chiral expansion. Then, the power counting works also in the situation when +the LO amplitude is enhanced at threshold, although to make the scheme more efficient, it might be necessary to +promote certain contributions to leading order. +VI. +NUMERICAL RESULTS +In this section we illustrate our theoretical findings by explicit numerical calculation of the NLO NN amplitude +in the three channels where the LO interaction should be treated non-perturbatively: 3P0, 3S1 − 3D1 and 1S0. The +results for other channels were presented in Ref. [16]. +We adopt the same values for the numerical constants as in Ref. [16]: the pion decay constant Fπ = 92.1 MeV, +the isospin average nucleon and pion masses mN = 938.9 MeV, Mπ = 138.04 MeV and the effective nucleon axial +coupling constant gA = 1.29. The calculations have been performed using Mathematica [36]. +For the regularization of the LO and NLO potentials, we adopt the scheme similar to the one used in realistic +calculation in Ref. [9] at fifth order in the chiral expansion, which allows us to have a direct interpretation of the +numerical values of the cutoffs. In particular, we use the local Gaussian regulator for the one-pion-exchange potential +and the non-local Gaussian regulator for all contact interactions with the same cutoff Λ, see Appendix A. For the +sake of simplicity, we also employ the local Gaussian regulator in the form of the overall factor FΛNLO,exp(q) for the +two-pion-exchange potential. As in Ref. [16], the cutoff value ΛNLO is set to the hard scale ΛNLO = 600 MeV. This +choice for the chiral expansion breakdown scale is consistent with the recent studies in the few-nucleon sector [37–40]. +The momentum-independent contact interactions at NLO are included without a regulator in accordance with our +power counting. The contact interactions quadratic in momenta are regulated with the same cutoff ΛNLO at LO +and at NLO in contrast to our choice in Ref. [16], where, for simplicity, we left the corresponding NLO contact +terms unregulated. Both options are legitimate since the regulator corrections to the contact interactions quadratic +in momenta is an effect of a higher order, O(Q4) . By the same reason, the regulator corrections to the LO contact +interactions quadratic in momenta are not taken into account. +The cutoff values for the one-pion-exchange potential and for the momentum-independent LO contact interactions +are varied in the regions below and above Λ = 450 MeV, which was found to be the optimal cutoff value in Ref. [9]. +The lower region corresponds to extremely soft cutoffs, where explicit regulator corrections to the LO potential are +likely to be important. The upper region contains relatively hard (of the order of the hard scale) cutoffs as well as +cutoffs above Λb, for which we expect slower convergence in terms of the Fredholm expansion and, therefore, potential +problems with interpretation within our renormalization scheme. +The free parameters are determined by a fit to the empirical phase shifts from the Nijmegen partial wave analysis [41] +up to Elab = 150 MeV. The phase shifts and the mixing parameters are calculated through the following unitarization +procedure. First, the non-unitary NLO T-matrix is transformed to the S-matrix via +Sl′l(pon) = 1 − imNpon +8π2 +Tl′l(pon). +(159) +The diagonal phase shifts in the Stapp parametrization of the S-matrix [42] are determined as (modulo π) +δll = 1 +2 arg(Sll), +(160) +whereas the mixing parameter ϵl+1 is obtained from the off-diagonal element of the S-matrix: +Sl+2,l = i sin(2ϵl+1) exp(iδl + iδl+2). +(161) + +23 +The dependence of the results on a particular unitarization scheme is a higher-order effect, provided the chiral +expansion for the amplitude is convergent. +The numerical analysis we perform does not aim at achieving a perfect description of the data as we work only +at next-to-leading order in the chiral expansion. Rather, we are interested in the convergence and renormalization +issues. In particular, we make sure that for the cutoff values we employ, no spurious bound states appear and no +“exceptional” cutoffs discussed in Secs. IV and V lie within this range. The latter fact manifests itself in the natural +values of the fitted next-to-leading-order LECs. The natural values of the NLO LECs are also an indication of the +“naturalness” of the quantity ν0(0), which is the simplest condition for the renormalizability of the S-wave NLO +amplitudes, see Sec. V B. The natural size is roughly given by +8π2 +mNΛb +, +(162) +for the LECs accompanying momentum-independent contact terms and by +8π2 +mNΛ3 +b +, +(163) +for the LECs of contact terms quadratic in momenta. Obviously, naturalness is not a mathematically strict criterion. +However a sign of potential problems would be a rapid growth with cutoff of one or several LECs. +Understanding the power counting for the renormalized amplitudes in terms of the convergence of the Fredholm +expansion is demonstrated by looking at the convergence of the Fredholm determinant expanded in terms of the +LO potential. Convergence of other elements of the Fredholm formulas for the LO and the NLO amplitudes can +be analyzed in a similar manner. Their convergence rates are typically comparable with the one for the Fredholm +determinant. An absolute value of Fredholm determinant much larger than 1 is also a problem for our interpretation +of the power counting, especially for the channels with the enhanced LO amplitude. In such a case, the numerators +in the Fredholm formulas N0, N2 will also be very large, contradicting the power counting that we suggest. On the +contrary, we expect the absolute value of the Fredholm determinant for those channels to be smaller than 1. +A. +3P0 channel +We begin our discussion with the 3P0 partial wave and first follow the dimensional power counting. That means +that at leading order, we include only the one-pion-exchange potential and no further terms are promoted. At next- +to-leading order, there is one free parameter C2,3P0 that determines the strength of the NLO contact interaction. The +results for the LO and NLO calculations are presented in Fig. 1. In contrast to other plots in this section, we restrict +ourselves to the values of the cutoff Λ ≤ 600 MeV because for larger cutoffs, the calculated phase shifts deviate too +strongly from the data points. +For soft cutoffs values below Λ = 450 MeV, the convergence of the chiral EFT expansion and the description of +the data are reasonable. Moreover for such cutoffs, the LO amplitude can be regarded as perturbative, in the sense +that the series in V0 converges very rapidly, and already a single iteration of the LO potential provides an accuracy +of one percent. Therefore, the analysis of Ref. [16] can be applied. One can also see that the band for next-to-leading +order corresponding to the variation of the cutoff gets considerably narrower if the regulator correction to the one- +pion-exchange potential is taken into account explicitly. Further discussion of the fully perturbative approach in the +3P0 channel can be found in Refs. [44, 45]. +As one increases the cutoff value, the convergence of expansion of the amplitude in powers of V0 becomes much +slower. This is not problem for our formalism as we formulated the power counting in the non-perturbative case in +Sec. IV. However, as one can see in Fig. 1, the disagreement with the data gets more severe and the convergence of +the chiral EFT expansion deteriorates. In fact, such a strong deviation of the LO phase shifts from the data leads +to a strong violation of unitarity. Another indication of the inefficiency of the resulting EFT expansion is a rather +small value of the Fredholm determinant. At threshold, it equals D ∼ 0.4 for Λ = 600 MeV compared to D ∼ 1 for +Λ = 300 − 450 MeV. +Large contributions from higher orders makes it more efficient to promote the NLO contact interaction to leading +order, see also Refs. [21] and [20]. In fact, the case of very soft cutoffs considered above, which shows a reasonable +convergence of the chiral expansion, can also be viewed as a modification of the short-range part of the LO potential +analogous to promotion of a contact interaction. Note that our motivation for promoting the NLO contact term is +not the requirement of the existence of an infinite cutoff limit as advocated, e.g., in Ref. [20], but rather a large +strength of the LO one-pion-exchange potential in this channel. Specifically, we demand that the difference between + +24 +0 +50 +100 +150 +200 +250 +Elab [MeV] +0 +10 +20 +30 +Phase Shift [deg] +without δΛV (0) +0 +50 +100 +150 +200 +250 +Elab [MeV] +with δΛV (0) +0 +50 +100 +150 +200 +250 +Elab [MeV] +without δΛV (0) +0 +50 +100 +150 +200 +250 +Elab [MeV] +with δΛV (0) +FIG. 1. The results of the leading-order (blue dashed lines) and next-to-leading-order (red solid lines) calculations for the +3P0 partial wave without promoting the contact interaction. The bands indicate the variation of the one-pion-exchange cutoff +within the range Λ1π ∈ (300, 450) MeV for two left plots and within the range Λ1π ∈ (450, 600) MeV for two right plots. The +second and fourth plots correspond to the NLO potential with the regulator correction δΛV (0), while the results in the first +and third plots are obtained without this term. The empirical phase shifts shown by black solid dots are from Ref. [41]. The +plots were created using Matplotlib [43]. +0 +50 +100 +150 +200 +250 +Elab [MeV] +−10 +0 +10 +20 +Phase Shift [deg] +without δΛV (0) +0 +50 +100 +150 +200 +250 +Elab [MeV] +with δΛV (0) +0 +50 +100 +150 +200 +250 +Elab [MeV] +without δΛV (0) +0 +50 +100 +150 +200 +250 +Elab [MeV] +with δΛV (0) +FIG. 2. The results of the leading-order (blue dashed lines) and next-to-leading-order (red solid lines) calculations for the 3P0 +partial wave with the contact term promoted to leading order. The bands indicate the variation of the one-pion-exchange cutoff +within the range Λ1π ∈ (300, 450) MeV for two left plots and within the range Λ1π ∈ (450, 800) MeV for two right plots. The +second and fourth plots correspond to the NLO potential with the regulator correction δΛV (0), while the results in the first +and third plots are obtained without this term. The empirical phase shifts shown by black solid dots are from Ref. [41]. +the LO results and empirical values of the phase shifts can be corrected by a perturbative inclusion of higher-order +interactions. +In the scheme with a contact term at LO, there is also one free parameter to be determined from the fit, namely +C3P0, whereas the NLO constant C2,3P0 is fixed by the renormalization condition in Eq. (91). The corresponding +results are shown in Fig. 2. As one can see, the convergence pattern when going from LO to NLO becomes much +better. Taking into account the regulator correction to the one-pion-exchange potential δΛV (0) explicitly leads to +narrower cutoff-variation bands at NLO, especially for soft cutoffs. +The expansion of the Fredholm determinant in powers of V0 converges rather rapidly for the cutoffs Λ ≤ 600 MeV: +at order (V0)3, a one-percent accuracy is achieved. For Λ ∼ 800 MeV, the same accuracy requires expansion up to +order (V4)4. The absolute value of the Fredholm determinant varies within the range 0.7 − 2.3 increasing for higher +values of the cutoff. The numerical values of the constant C2,3P0 in the units of Eq. (163) is reasonably natural for +the choice of the hard scale Λb = 600 MeV at least for lower Λ values. Specifically, C2,3P0 ∼ 2 for Λ ∼ 450 MeV but +increases to C2,3P0 ∼ 30 for Λ ∼ 800 MeV. +Combining the above results, we conclude that for the cutoffs below or of the order of the hard scale, the renor- +malization of the NLO amplitude can be understood within the approach developed in this paper. For higher values +of the cutoff, the renormalizability of the theory becomes questionable. + +25 +Elab [MeV] +0 +50 +100 +150 +3S1 phase shift [deg] +without δΛV (0) +Elab [MeV] +with δΛV (0) +Elab [MeV] +without δΛV (0) +Elab [MeV] +with δΛV (0) +Elab [MeV] +−20 +−10 +0 +3D1 phase shift [deg] +Elab [MeV] +Elab [MeV] +Elab [MeV] +0 +50 +100 +150 +200 +Elab [MeV] +−10 +−5 +0 +5 +Mixing parameter ϵ1 [deg] +0 +50 +100 +150 +200 +Elab [MeV] +0 +50 +100 +150 +200 +Elab [MeV] +0 +50 +100 +150 +200 +Elab [MeV] +FIG. 3. The results of the leading-order (blue dashed lines) and next-to-leading-order (red solid lines) calculations for the +3S1 − 3D1 channels with the contact term promoted to leading order. The bands indicate the variation of the cutoff of the LO +potential within the range Λ ∈ (300, 450) MeV for two left columns and within the range Λ ∈ (450, 800) MeV for two right +columns. The second and fourth columns correspond to the NLO potential with the regulator correction δΛV (0), while the +results in the first and third columns are obtained without this term. The data are as in Figs. 1 and 2. +B. +3S1 − 3D1 channel +Next, we consider the system of the coupled 3S1−3D1 partial waves. The LO potential is obviously non-perturbative +due to the presence of the shallow deuteron bound state. The enhancement of the LO amplitude at threshold is not as +strong as, e.g., in the 1S0 channel. Therefore, we assume that within the renormalization scheme specified in Eq. (117), +the dimensional power counting should work. That means that the LO potential contains only the one-pion-exchange +and the momentum-independent contact term contributions. +There are three parameters to be determined from the fit: the LO constant C3S1, the NLO constant at the diagonal +contact term quadratic in momenta, C2,3S1,p2, and the NLO constant accompanying the off-diagonal contact term +C2,ϵ1. The NLO momentum-independent contact term with the constant C2,3S1 is fixed from the renormalization +condition in Eq. (117). The above mentioned three parameters are determined by fitting the phase shifts in the +diagonal 3S1 channel and the mixing parameter ϵ1, i.e., the channels with contact terms in the potential. The 3D1 +phase shift comes out as a prediction. +The results of the fit for various cutoffs are shown in Fig. 3. In general, we observe a reasonable convergence of the +chiral expansion except for the ϵ1 channel where the LO as well as the full contributions are rather small. +As expected for soft cutoffs Λ ≤ 450 MeV, taking into account the explicit regulator corrections δΛV (0) for the one- + +26 +0 +50 +100 +150 +200 +250 +Elab [MeV] +0 +20 +40 +60 +Phase Shift [deg] +without δΛV (0) +0 +50 +100 +150 +200 +250 +Elab [MeV] +with δΛV (0) +0 +50 +100 +150 +200 +250 +Elab [MeV] +without δΛV (0) +0 +50 +100 +150 +200 +250 +Elab [MeV] +with δΛV (0) +FIG. 4. The results of the leading-order (blue dashed lines) and next-to-leading-order (red solid lines) calculations for the +1S0 partial wave without promoting the contact interaction quadratic in momentum. The bands indicate the variation of the +cutoff of the LO potential within the range Λ ∈ (300, 450) MeV for two left plots and within the range Λ ∈ (450, 800) MeV for +two right plots. The second and fourth plots correspond to the NLO potential with the regulator correction δΛV (0), while the +results in the first and third plots are obtained without this term. The data are as in Figs. 1 and 2. +pion-exchange potential and the leading contact term significantly reduces the cutoff dependence at next-to-leading +order. +Given the relatively large number of free parameters and possible fine-tuning, it is necessary to explicitly verify the +renormalizability criteria specified above. +First, we check the naturalness of the NLO LECs in the units specified in Eqs. (162) and (163). The absolute +values of the constants C2,3S1,p2 and C2,ϵ1 do not exceed 12 for all considered values of the cutoffs. The maximal +absolute value of the constant C3S1,p2 is about 6 for Λ ≤ 600 MeV, but it starts rising very fast and reaches the value +of C3S1,p2 ∼ 20 for Λ = 800 MeV (and continues rising rapidly). +The Fredholm determinant converges with a one-percent accuracy at orders (V0)3 − (V0)5 for Λ ≤ 600 MeV and +at order (V0)6 for Λ = 800 MeV. The absolute value of the Fredholm determinant at threshold (at Elab = 250 MeV) +varies in the range 0.6 − 0.8 (1.8 − 3.6) for Λ ≤ 600 MeV and is as large as 1.6 (7.5 ) for Λ = 800 MeV. +Summarizing the above observations, our numerical results confirm the renormalizability of the NLO amplitude in +the 3S1 − 3D1 channels for the cutoffs below or of the order of the hard scale. For higher values of the cutoffs, the +renormalizability in the sense discussed in the present paper is not guaranteed. +C. +1S0 channel +Finally, we discuss the 1S0 partial wave. The enhancement of the LO amplitude due to the extremely shallow +quasibound state is very strong. Nevertheless, we start with trying to adopt the dimensional power counting and +do not promote any additional contact interaction to leading order. +Therefore, the LO potential consists of the +one-pion-exchange contribution and the leading contact term. Two parameters are determined from the fit: the LO +constant C1S0 and the NLO constant C2,1S0,p2 corresponding to the contact term quadratic in momenta. The NLO +constant C2,1S0 is fixed from the renormalization condition in Eq. (117). The results are shown in Fig. 4. As in +the case of the 3P0 partial wave, the convergence of the chiral expansion is acceptable only for small values of the +cutoff Λ ≤ 450 MeV. For larger values of the cutoffs, the LO contribution is too large compared to the data, which +leads to a strong violation of unitarity. The regulator corrections to the one-pion-exchange potential and the leading +contact term practically do not affect the size of the bands corresponding to the variations of the cutoff, which is also +a sign of a slow convergence. As the cutoff increases, the Fredholm determinant at threshold changes from 0.7 to +0.3. Therefore, the slow convergence of the chiral expansion for the NLO amplitude is expected from our analysis in +Sec. V. Nevertheless, the series for the Fredholm determinant converges rapidly: the one-percent accuracy is obtained +at order (V0)3. The naturalness of the NLO LECs in the units of Eqs. (162) and (163) is also reasonably fulfilled: the +absolute value of the constant C2,1S0 does not exceed 2, and the absolute value of the constant C2,1S0,p2 is below 25. +A large deviation of the LO results from the data is a motivation for promoting the subleading contact interaction +to leading order (as the simplest solution), see Refs. [18, 19]. As we argued in the discussion of the 3P0 partial wave, +adopting soft values of the cutoff Λ ≤ 450 MeV in the scheme with one contact term at leading order is a sizable +modification of the short-range part of the LO potential and is, to some extent, equivalent to the promotion of an +additional contact term. +Now, we consider the scheme with the contact interaction quadratic in momenta being promoted to the LO potential. + +27 +0 +50 +100 +150 +200 +250 +Elab [MeV] +0 +20 +40 +60 +Phase Shift [deg] +without δΛV (0) +0 +50 +100 +150 +200 +250 +Elab [MeV] +with δΛV (0) +0 +50 +100 +150 +200 +250 +Elab [MeV] +without δΛV (0) +0 +50 +100 +150 +200 +250 +Elab [MeV] +with δΛV (0) +FIG. 5. The results of the leading-order (blue dashed lines) and next-to-leading-order (red solid lines) calculations for the 1S0 +partial wave with the contact interaction quadratic in momentum promoted to leading order. The bands indicate the variation +of the cutoff of the LO potential within the range Λ ∈ (300, 450) MeV for two left plots and within the range Λ ∈ (450, 800) MeV +for two right plots. The second and fourth plots correspond to the NLO potential with the regulator correction δΛV (0), while +the results in the first and third plots are obtained without this term. The data are as in Figs. 1 and 2. +There are still two parameters to fit: C1S0 and C1S0,p2. +The constants C2,1S0 and C2,1S0,p2 are fixed from the +renormalization conditions in Eq. (154). The results for the scheme with two contact terms in the LO potential are +presented in Fig. 5. For higher Λ values, the convergence pattern for the EFT expansion in this scheme is significantly +better than in the scheme without promotion of the momentum-dependent contact term. The cutoff dependence is +weak for the cutoff values Λ ≥ 450 MeV. For soft cutoffs, it may seem that explicit regulator corrections makes the +cutoff dependence stronger. However, this is probably accidental because, as one can see, the cutoff dependence for +the case without regulator corrections is nonlinear and varies nontrivially with momentum. This is caused by various +cancellations due to the fine-tuning of two contact terms. +The absolute value of the Fredholm determinant at threshold is D ∼ 0.1 for all considered cutoffs, which is in +agreement with our expectations for the strongly enhanced LO amplitude. The expansion of the Fredholm determinant +in powers of the LO potential approaches an accuracy of one percent at order (V0)3 for the cutoffs below or equal to +the hard scale and at order (V0)4 for Λ = 800 MeV. +For all analyzed cutoffs, the naturalness constraint for the NLO constants is reasonably well satisfied without an +obvious tendency to its violation, which can be explained by a regular behaviour of the spin-singlet one-pion-exchange +potential at short distances. +To summarize, the numerical calculations for the channels 3P0, 3S1 − 3D1 and 1S0 are in agreement with our +theoretical analysis of the renormalization of the NLO amplitude with a finite cutoff. We observed a reasonable +convergence of the chiral EFT expansion. However, for the 3P0 and 1S0 partial waves a more efficient scheme within +the considered EFT formulation is obtained when the subleading contact interactions are promoted to leading order, +as has already been discussed in the literature. The naturalness constraints on the NLO LECs and on the value of +the Fredholm determinants are fulfilled for the cutoff values below or of the order of the hard scale. The convergence +rate of the Fredholm determinants in powers of the LO potential also appears to be sufficiently rapid for such values +of the cutoffs. This allows us to interpret the renormalizability of the NLO amplitude within the method developed +in the current paper. When the cutoff approaches the value Λ ∼ 800 MeV or higher, the renormalizability constraints +are not clearly fulfilled anymore, even though the convergence of the amplitude might still be reasonable. +Thus, we conclude that the preferable choice of the cutoff values is roughly Λ ≲ 600 MeV. For very soft cutoffs +Λ = 300 − 450 MeV, the regulator corrections to the LO potential should be explicitly taken into account to remove +the regulator artifacts. + +28 +VII. +SUMMARY +We have extended our previous study in Ref. [16] and analyzed the renormalization of the nucleon-nucleon amplitude +at NLO in chiral EFT in the case when the LO interaction is non-perturbative. Our scheme is based on the formulation +of chiral EFT with a finite cutoff derived from the effective Lagrangian. +In the previous paper, the power counting for the renormalized NLO amplitude was justified for the case when +the series for the iterations of the LO potential are (rapidly enough) convergent, i.e., for the perturbative case. The +corresponding subtractions in the form of the LO S-wave contact terms that absorb the power counting breaking +contributions were identified. Starting from P-waves, the NLO amplitudes were found not to require any subtractions +in agreement with dimensional power counting. +The method of analysis of the power counting in the non-perturbative regime relies on the Fredholm formula +for the solution of the integral equations, which represents the numerators and denominators of the amplitudes as +individually convergent series in powers of the LO potential. To implement the Fredholm decomposition, we first had +to derive stronger bounds on the LO potential compared to the ones used in the perturbative case. In contrast to the +perturbative regime, it turned out that the minimal “mild” regulator can, in general, not be employed if the NLO +potential remains unregulated. This implies a potentially stronger cutoff dependence in the non-perturbative case. +The results for the P- and higher partial waves in the NN system reproduce to a large extend our previous findings. +The dimensional power counting for the LO and NLO amplitudes is formally satisfied without subtractions unless +there is an enhancement of the LO amplitude due to the presence of a shallow (quasi-)bound state, which is not +the case for the physical channels. +Nevertheless, in some cases, the promotion of NLO contact terms to leading +order can be motivated by phenomenological arguments as, e.g., in the 3P0 channel. In the latter situation, however, +one has to choose the LO potential in such a way as to avoid the appearance of “exceptional” cutoffs, for which +the renormalization breaks down. The simplest way to verify that the adopted value of the cutoff is not close to +“exceptional” is to make sure that the NLO LECs are of a natural size. +For the S-waves, we have shown that the series for the subtractions at next-to-leading order, obtained in the +perturbative case, can be resummed in a closed form. Such a resummation is equivalent to the condition for the +renormalized NLO amplitude to vanish at threshold. Using the Fredholm formula allowed us to analyze also the case +when the LO amplitude is enhanced at threshold compared to the dimensional power counting estimate. This happens +in the 3S1 − 3D1 and 1S0 channels where shallow bound and quasibound states are present. The dimensional power +counting for the NLO amplitudes is still valid in those cases if certain additional constraints on the LO potential are +fulfilled. Again, these constraints eventually reveal themselves in the naturalness of the NLO LECs. However, the +convergence of the chiral expansion in the channels with enhanced LO amplitude may become significantly slower, +especially in the 1S0 channel, where the enhancement is most pronounced. To improve the convergence, one can, +analogously to the 3P0 partial wave, promote a subleading contact term to the LO potential with the same warning +regarding “exceptional” cutoffs. +Finally, we have illustrated our theoretical findings by numerical calculations of the NN phase shifts at next-to- +leading order by fitting the unknown free parameters to the empirical data. We considered three channels with non- +perturbative dynamics, namely 3P0, 3S1−3D1 and 1S0, and varied the LO cutoff in the range of Λ = 300−800 MeV. We +observed reasonable convergence of the chiral expansion, especially when the subleading contact terms are promoted +in the 3P0 and 1S0 channels. +As criteria for the interpretation of the renormalizability of the NLO amplitude in terms of the Fredholm expansion, +we used the naturalness of the NLO LECs and of the Fredholm determinant as well as the convergence rate of the +expansion of the latter in powers of the LO potential. It turns out that all these constraints are fulfilled as long +as the cutoff values are chosen below or of the order of the hard scale Λb ∼ 600 MeV. For particularly soft cutoffs +Λ = 300 − 450 MeV, taking into account explicit regulator corrections to the LO potential compensates for the +regulator artifacts and reduces the cutoff dependence. +When the cutoff increases beyond the hard scale, the renormalizability constraints start being violated. Therefore, +we conclude that the cutoff values Λ ≲ Λb are preferable from the point of view of the renormalization of EFT. +Further development of our approach goes in the direction of extending it beyond next-to-leading order in the +chiral expansion. It is also important to generalize the scheme to few-nucleon systems and the processes involving +electro-weak interactions. +ACKNOWLEDGMENTS +We would like to thank Jambul Gegelia for helpful discussions and for useful comments on the manuscript. This work +was supported by DFG (Grant No. 426661267), by BMBF (contract No. 05P21PCFP1), by ERC AdG NuclearTheory + +29 +(grant No. 885150) and by the EU Horizon 2020 research and innovation programme (STRONG-2020, grant agreement +No. 824093). + +30 +Appendix A: Leading-order potential +The short-range part of the leading-order potential in its general form can be chosen to include the momentum- +independent contact interactions and contact terms quadratic in momenta (altogether 9 terms), multiplied by the +power-like non-local form factor of an appropriate power n: +V (0) +short,Λ(⃗p ′, ⃗p ) = +� +i +Ci VCi FΛi,ni(p ′, p) , +(A1) +where VCi is any basis for the contact terms, e.g. the partial wave basis, and the regulators are given by +FΛ,n(p ′, p) = FΛ,n(p ′)FΛ,n(p) , +FΛ,n(p) = [FΛ(p)]n , +FΛ(p) = +Λ2 +p2 + Λ2 . +(A2) +One can also introduce a regulator of a Gaussian form by replacing FΛ,n(p) with +FΛ,exp(p) = exp (−p2/Λ2) . +(A3) +Alternatively, one could introduce local short-range interactions (for the terms that depend only on ⃗q, except for +the spin-orbit term) using the appropriate basis [46] and the local regulator +Fq,Λ,n(⃗p ′, ⃗p ) = [FΛ(q)]n = +� +Λ2 +q2 + Λ2 +�n +, +(A4) +or with the regulator in the Gaussian form FΛ,exp(q). +The long-range part of the LO potential is represent by the one-pion-exchange contribution, which is split into the +triplet, singlet, and contact parts +V (0) +1π = − +� gA +2Fπ +�2 +τ1 · τ2 +⃗σ1 · ⃗q ⃗σ2 · ⃗q +q2 + M 2π +=: V (0) +1π,t + V (0) +1π,s + V (0) +1π,ct , +(A5) +with +V (0) +1π,s = +� gA +2Fπ +�2 +τ1 · τ2 +(⃗σ1 · ⃗σ2 − 1) +4 +M 2 +π +q2 + M 2π +, +V (0) +1π,ct = − +� gA +2Fπ +�2 +τ1 · τ2 +(⃗σ1 · ⃗σ2 − 1) +4 +. +(A6) +All three parts, if necessary, are regularized individually. The contact part V1π,ct can be absorbed by the leading-order +1S0 contact term and thus needs not be considered separately. The triplet and singlet potentials can be regularized +by means of the non-local form factor (see Eq. (A2)): +V (0) +1π,Λ(⃗p ′, ⃗p ) = V (0) +1π,s(⃗p ′, ⃗p )FΛs,ns(p ′, p) + V (0) +1π,t(⃗p ′, ⃗p )FΛt,nt(p ′, p) , +(A7) +or by means of the local regulator: +V (0) +1π,Λ(⃗p ′, ⃗p ) = V (0) +1π,s(⃗p ′, ⃗p )Fq,1π,Λs(⃗p ′, ⃗p ) + V (0) +1π,t(⃗p ′, ⃗p )Fq,1π,Λt(⃗p ′, ⃗p ) , +(A8) +with +Fq,1π,Λs(⃗p ′, ⃗p ) = +�Λ2 +s − M 2 +π +q2 + Λ2s +�ns +, +Fq,1π,Λt(⃗p ′, ⃗p ) = +�Λ2 +t − M 2 +π +q2 + Λ2 +t +�nt +. +(A9) +Note that in Ref. [16], a more general form of the local regulator was considered. +The spin-singlet part of the one-pion-exchange potential can, in principle, be left unregulated. This is, however, +only relevant for the spin-singlet channels without short-range interactions. All such channels can be regarded as +having perturbative LO potential and were already analyzed in Ref. [16]. For the spin-singlet channel considered in +this work, 1S0, the effects of a regulator will be driven by the contact interaction in any case. +To regularize the spin-triplet part of the one-pion-exchange potential in the LO Lippmann-Schwinger equation, it +is sufficient to introduce a dipole (nt = 1) regulator, which we refer to as the “mild” regulator. All other options, i.e., +nt ≥ 2 are referred to as the “standard” regulators. +One can also adopt the local Gaussian regulator for the one-pion-exchange potential: +Fq,1π,exp,Λ(⃗p ′, ⃗p ) = exp +� +−(q2 + M 2 +π)/Λ2� +. +(A10) + +31 +Appendix B: Next-to-leading-order potential +The short-range part of the next-to-leading-order potential is given by the sum of contact terms analogous to +Eq. (A1): +V (2) +short(⃗p ′, ⃗p ) = +� +i +C2,i VCi . +(B1) +The non-polynomial part of the two-pion-exchange potential is given by (it is equivalent to the one provided in +Ref. [47] up to polynomial terms) +V (2) +2π (⃗p ′, ⃗p ) = − τ1 · τ2 +384π2F 4π +˜L(q) +� +4M 2 +π(5g4 +A − 4g2 +A − 1) + q2(23g4 +A − 10g2 +A − 1) + 48g4 +AM 4 +π +4M 2π + q2 +� ++ τ1 · τ2 +8π2F 4π +g4 +AM 2 +πq2 +4M 2π + q2 − +3g4 +A +64π2F 4π +˜L(q) +� +⃗σ1 · ⃗q⃗σ2 · ⃗q − q2⃗σ1 · ⃗σ2 +� +, +(B2) +where +˜L(q) := L(q) − L(0) = L(q) − 1 , +L(q) = 1 +q +� +4M 2π + q2 log +� +4M 2π + q2 + q +2Mπ +. +(B3) +The regulator of the NLO potential, not shown explicitly in the above expressions, can be a combination of any +local or non-local forms. For the two-pion-exchange potential, one can also employ a spectral function regularization +by introducing a finite upper limit in the dispersion representation of ˜L(q): +˜L(q) = q2 +� Λρ +2Mπ +dµ +µ2 +� +µ2 − 4M 2π +q2 + µ2 +. +(B4) +Appendix C: Bounds on the plane-wave potential +1. +Bounds on the substructures +Below, we list the inequalities for the building blocks of the LO and NLO potentials obtained in Ref. [16]. +The components of the initial and final nucleon c.m. momenta p and p′ are defined as +⃗p = p +� +� +0 +0 +1 +� +� , ⃗p ′ = p′ +� +� +sin θ cos φ +sin θ sin φ +cos θ +� +� , +(C1) +where p is either p = pon or lies on the complex contour p ∈ C: p = |p| exp(−iαC), and p′ is either p′ = pon or +p′ = |p′| exp(−iαC). +For the function +fµ(p′, p, x) = +1 +q2 + µ2 = +1 +p′2 + p2 − 2pp′x + µ2 , +(C2) +with µ ≥ Mπ, the following bounds hold +|fµ(p′, p, x)| ≤ +Mf +|p|2 + |p′|2 − 2|p||p′|x + µ2 , +(C3) +|qiqjfµ(p′, p, x)| ≤ Mf, +(C4) +���(⃗k × ⃗q)i fµ(p′, p, x) +��� ≤ Mf +� +1 − x2�−1/2 , +i, j = 1, 2, 3 . +(C5) +The subtraction remainders defined as +∆(n) +p f(p′, p) = f(p′, p) − +n +� +i=0 +∂if(p′, p) +i!(∂p)i +���� +p=0 +pi, +∆(n) +p′ f(p′, p) = f(p′, p) − +n +� +i=0 +∂if(p′, p) +i!(∂p′)i +���� +p′=0 +(p′)i, +(C6) + +32 +satisfy the following inequalities: +���∆(n) +p fµ(p′, p) +��� ≤ Mf,n +���� +p +p′ +���� +n+1 +|fµ(p′, p)| , +if |p′| > |p|, +���∆(n) +p′ fµ(p′, p) +��� ≤ Mf,n +���� +p′ +p +���� +n+1 +|fµ(p′, p)| +if |p| > |p′|. +(C7) +For a more general structure +Ψk,m,{µi}(p′, p, x) = Qk(p′, p, x)FΛ,m(p′, p)f{µi}(p′, p , x), +(C8) +where the form factor FΛ,m is defined in Eq. (A2), f{µi} is a product of several fµ +f{µi}(p′, p, x) = +� +i=1,r +fµi(p′, p, x), +(C9) +and Qk is a homogeneous polynomial of degree k, one can deduce the bounds for derivatives: +�����pn ∂nΨk,m,{µi}(p′, p, x) +∂pn +���� +p=0 +����� ≤ Mk,n +∂Ψ +���p′kFΛ,m− n+1 +2 (p′)f{µi}(p′, 0, x) +��� +���� +p +p′ +���� +n +, +�����(p′)n ∂nΨk,m,{µi}(p′, p, x) +∂(p′)n +���� +p′=0 +����� ≤ Mk,n +∂Ψ +���pkFΛ,m− n+1 +2 (p)f{µi}(0, p, x) +��� +���� +p′ +p +���� +n +, +n ≥ 0 , +(C10) +and for the subtraction remainders: +���∆(n) +p Ψk,m,{µi}(p′, p, x) +��� ≤ Mk,n +Ψ +���� +p +p′ +���� +n+1 +× +� +|Ψk,m,{µi}(p′, p, x)| + +���p′kFΛ,m− n+1 +2 (p′)f{µi}(p′, 0, x) +��� +� +, +if |p′| > |p|, +���∆(n) +p′ Ψk,m,{µi}(p′, p, x) +��� ≤ Mk,n +Ψ +���� +p′ +p +���� +n+1 +× +� +|Ψk,m,{µi}(p′, p, x)| + +���pkFΛ,m− n+1 +2 (p)f{µi}(0, p, x) +��� +� +, +if |p| > |p′|. +(C11) +2. +Bounds on the plane-wave leading-order potential +In this section we provide bounds for the leading-order potential. We will need slightly stronger bounds than those +obtained in Ref. [16]. In particular, we will need bounds that factorize in initial and finale momenta in the partial +wave basis. In order to obtain them, we will partly keep the angular dependence in binding functions. +The derivation is only slightly different from that of Ref. [16], which we demonstrate for the case of the spin-triplet +one-pion-exchange potential. +The locally regularized one-pion exchange potential in the spin-triplet channels can be bounded using equations of +Appendix. C 1 by the following inequality: +���V (0) +1π,t(⃗p ′, ⃗p ) +��� ≤ +������ +g2 +A +4F 2π +� +i,j +Mt,ij +qiqj +q2 + M 2π +� +Λ2 +t − M 2 +π +q2 + Λ2 +t,1 +�nt������ +≤ 2πMt +mNΛV +FΛ,nt(|p′|, |p|, x) , +(C12) +where we have introduced the form factors +FΛ,n(|p′|, |p|, x) = (FΛ(|p′|, |p|, x))n , +FΛ(|p′|, |p|, x) = +Λ2 +|p|2 + |p′|2 − 2|p||p′|x + Λ2 . +(C13) + +33 +In Eq. (C12), we replaced Λt with the largest cutoff Λ among all regulators in the LO potential, which is possible due +to the inequality: +FΛ1(|p′|, |p|, x) < FΛ2(|p′|, |p|, x) +for Λ1 < Λ2 . +(C14) +If the triplet one-pion exchange potential is regularized by the non-local form factor, we obtain +���V (0) +1π,t(⃗p ′, ⃗p ) +��� ≤ +������ +g2 +A +4F 2π +� +i,j +Mt,ij +qiqj +q2 + M 2π +� +Λ2 +p′2 + Λ2 +Λ2 +p2 + Λ2 +�nt +������ +≤ 2πMt +mNΛV +FΛ,nt(|p ′|)FΛ,nt(|p|). +(C15) +Analogously, we obtain bounds for other LO contributions as in Ref. [16] retaining the angular dependence of local +form factors and the powers of the form factors. +Finally, the full leading-order potential satisfies +|V0(⃗p ′, ⃗p )| ≤ MV0 +4π V0,max(p′, p, x), +|V0(⃗p ′, ⃗p )| ≤ MV0 +4π V0,max(p, p′, x) , +(C16) +where we have introduced +V0,max(p′, p, x) = +8π2 +mNΛV +� +FΛ,n(|p′|, |p|, x) + FΛ,n(|p′|) +� +, +(C17) +with n being the smallest power among all regulators in the LO potential. The cases of the “mild” and the “standard” +regulators correspond to n = 1 and n ≥ 2, respectively. The difference of Eq. (C17) from an analogous bound in +Ref. [16] is that the powers of both local and non-local form factors are retained and the x-dependence of the local +form factor is kept. The bounds for the Gaussian regulators can be reduced to the ones for the power-like regulators +as was shown in Ref. [16]. +For the spin-singlet channels without a short-range interaction, the bounds in Eq. (C17) can be improved by +replacing Λ with Mπ. However as mentioned above, those channels were already covered in our previous study. +The remainders ∆(n) +p V0(⃗p ′, ⃗p ) for |p′| > |p| can be estimated using Eq. (C11): +���∆(n) +p V0(⃗p ′, ⃗p ) +��� ≤ M∆V0,n +4π +���� +p +p′ +���� +n+1 +V0,max(p′, p, x) +if |p′| > |p| , +���∆(n) +p′ V0(⃗p ′, ⃗p ) +��� ≤ M∆V0,n +4π +���� +p′ +p +���� +n+1 +V0,max(p, p′, x) +if |p| > |p′| . +(C18) +From Eq. (C10) one obtains the estimates for the derivatives of the leading-order potential: +�����pm ∂mV0(⃗p ′, ⃗p ) +(∂p)m +���� +p=0 +����� ≤ 2πM∂V0,n +mNΛV +F˜Λ,n(|p′|) +���� +p +p′ +���� +m +≤ M∂V0,n +4π +���� +p +p′ +���� +m +V0,max(p′, p, x) , +(C19) +�����p′m ∂mV0(⃗p ′, ⃗p ) +(∂p′)m +���� +p′=0 +����� ≤ 2πM∂V0,n +mNΛV +F˜Λ,n(|p|) +���� +p′ +p +���� +m +≤ M∂V0,n +4π +���� +p′ +p +���� +m +V0,max(p, p′, x) , +(C20) +including the case m = 0, where we have used that the local form factor satisfies +FΛ(p′, 0, x) = FΛ(p′). +(C21) +Applying Eq. (C19) (Eq. (C20)) to the definition of ∆(n) +p V0(⃗p ′, ⃗p ) (∆(n) +p′ V0(⃗p ′, ⃗p )) in Eq. (C6) for |p| > |p′| (|p′| > +|p|), and combining it with Eq. (C18), we obtain the following bounds for the remainders: +���∆(n) +p V0(⃗p ′, ⃗p ) +��� ≤ M∆V0,n +4π +���� +p +p′ +���� +n+1 +V0,max(p′, p, x) , +���∆(n) +p′ V0(⃗p ′, ⃗p ) +��� ≤ M∆V0,n +4π +���� +p′ +p +���� +n+1 +V0,max(p, p′, x) , +(C22) + +34 +which are valid for all considered p and p′. All above general formulas do not include the case when the LO potential +contains a locally regulated spin-orbit short-range interaction such as +V (0) +C5 (⃗p ′, ⃗p ) = C5 +i +2(⃗σ1 + ⃗σ2) · (⃗k × ⃗q ) +� +Λ2 +5 +q2 + Λ2 +5 +�n5 +, +(C23) +with n5 > 1 (or with the Gaussian form factor). Following the arguments provided in Ref. [16], one can formulate the +same bounds as in Eqs. (C16) and (C22) for the quantity ˜V (0) +C5 defined as +V (0) +C5 (⃗p ′, ⃗p ) = ˜V (0) +C5 (⃗p ′, ⃗p ) i +2(⃗σ1 + ⃗σ2) · ⃗nφ/ sin θ, +⃗nφ = (− sin φ, cos φ, 0), +(C24) +which makes it possible, after the partial-wave projection, to treat this interaction on the same footing as all other +LO terms. +3. +Bounds on the plane-wave next-to-leading-order potential +For the NLO potential, we use the bounds obtained in Ref. [16]. The NLO potential is split into two parts: +V2(⃗p ′, ⃗p ) = ˆV2(⃗p ′, ⃗p ) + ˜V2(⃗p ′, ⃗p ) , +(C25) +with +ˆV2(⃗p ′, ⃗p ) = V2(0, 0) , +˜V2(⃗p ′, ⃗p ) = V2(⃗p ′, ⃗p ) − V2(0, 0) , +(C26) +which are bound as +��� ˆV2(⃗p ′, ⃗p ) +��� ≤ ˆ +MV2 +2π +mNΛV +M 2 +π +Λ2 +b +, +(C27) +and +��� ˜V2(⃗p ′, ⃗p ) +��� ≤ 2πMV2 +mNΛV +|p|2 + |p′|2 +Λ2 +b +flog(p′, p) = MV2 +4π +� +|p|2 + |p′|2� ˜flog(p′, p) , +(C28) +with +˜flog(p′, p) = +8π2 +mNΛV Λ2 +b +flog(p′, p) , +flog(p′, p) = θ(|p| − Mπ) ln |p| +Mπ ++ θ(|p′| − Mπ) ln |p′| +Mπ ++ ln +˜Λ +Mπ ++ 1 . +(C29) +In the function flog(p′, p), the term ln +˜Λ +Mπ was introduced for convenience so we can omit it (or set ˜Λ = Mπ). +We also allow for a regulator (local or non-local) for the NLO potential. We can introduce it simply as a factor, so +that the bounds in Eq. (C28) are modified as +��� ˜V2(⃗p ′, ⃗p ) +��� ≤ MV2 +4π +� +|p|2 + |p′|2� ˜flog(p′, p) +� +FΛNLO(|p′|, |p|, x) + FΛNLO(|p′|) +� +, +(C30) +where we combined local and non-local regulators into one factor. The first power (n = 1) of the form factors is +sufficient for our estimates. The cases of higher powers (or Gaussian cutoffs) are included automatically, because +FΛNLO,n(|p|) ≤ FΛNLO(|p|) , +FΛNLO,n(|p′|, |p|, x) ≤ FΛNLO(|p′|, |p|, x) , +(C31) +for n > 1. If different values of the cutoff are used for different NLO contributions, ΛNLO can be chosen to be the +largest value. + +35 +Appendix D: Bounds on the partial-wave potential +Below, we repeat the arguments of Ref. [16] for deriving the bounds on the partial-wave potential, but take into +account an angular dependence of the binding functions. +The partial-wave potential is obtained from the plain-wave potential via +V sj +l′,l(p′, p) = +� +λ1,λ2,λ′ +1,λ′ +2 +� +dΩ ⟨jl′s|λ′ +1λ′ +2⟩⟨λ′ +1λ′ +2|V (⃗p ′, ⃗p )|λ1λ2⟩⟨λ1λ2|jls⟩dj +λ1−λ2,λ′ +1−λ′ +2(θ) , +⟨λ1λ2|jls⟩ = +� 2l + 1 +2j + 1 +� 1 +2 +C(l , s , j; 0 , λ1 − λ2)C (1/2 , 1/2 , s; λ1, −λ2) , +(D1) +where λi, λ′ +i are the helicities of the corresponding nucleons. +Due to unitarity of the transformation, the following constraints hold: +|⟨λ1λ2|jls⟩| ≤ 1, +|⟨1/2 , sz|λ⟩| ≤ 1, +|dj +λ,λ′(θ)| ≤ 1 . +(D2) +Therefore, if the plain-wave potential is bounded by some angle-dependent function φ(p′, p, x): +|V (⃗p ′, ⃗p )| ≤ Mkφ(p′, p, x) , +(D3) +then, for the partial-wave potential, we obtain: +|V sj +l′,l(p′, p)| ≤ 2π ˜ +Mk +� 1 +−1 +dx φ(p′, p, x) . +(D4) +For the special case of the locally regulated spin-orbit contact interaction, a bound of the same type can be obtained +if one replaces |V (⃗p ′, ⃗p )| by | ˜V (⃗p ′, ⃗p )| = |V (⃗p ′, ⃗p )| +√ +1 − x2, see Appendix. C 2 and Ref. [16]. +1. +Bounds on the form factor Fµ,n(q) integrated over x +In this subsection we derive the bounds on the local form factors +Fµ(q) = +µ2 +q2 + µ2 , +Fµ,2(q) = Fµ(q)2, +(D5) +integrated over the angle variable x, which are relevant when considering bounds for the partial-wave potentials. The +form factors Fµ,n(q) with n > 2 satisfy (at least) the same bounds as Fµ,2(q), which is sufficient for our estimates. +The same is true for the form factors of the Gaussian form, which was analyzed in detail in Ref. [16]. +From Eq. (C3), it follows +��q2 + µ2�� ≥ M−1 +f +� +|p|2 + |p′|2 − 2|p||p′|x + µ2� += M−1 +f +� +(|p′|x − |p|)2 + |p′|2(1 − x2) + µ2� +≥ M−1 +f +� +|p′|2(1 − x)/2 + µ2� +. +(D6) +For |p′| ≥ µ, we obtain +���� +� 1 +−1 +Fµ(q)dx +���� = +���� +� 1 +−1 +µ2dx +q2 + µ2 +���� ≤ 2Mfµ2 +� 1 +−1 +dx +|p′|2(1 − x) + 2µ2 = 2Mfµ2 +|p′|2 +ln +� +1 + |p′|2/µ2� +≤ 2Mfµ2 +|p′|2 +ln 2|p′|2 +µ2 +< 2Mfµ2 +|p′|2 +� +1 + ln |p′|2 +µ2 +� +, +(D7) +and +���� +� 1 +−1 +Fµ,2(q)dx +���� = +����� +� 1 +−1 +µ4dx +(q2 + µ2)2 +����� ≤ 4Mfµ4 +� 1 +−1 +dx +[|p′|2(1 − x) + 2µ2]2 = 2Mfµ2 +|p′|2 + µ2 < 2Mfµ2 +|p′|2 +, +(D8) + +36 +whereas for |p′| < µ, we can simply use +���� +� 1 +−1 +dxFµ,n(q) +���� ≤ +� 1 +−1 +dx(Mf)n = 2(Mf)n , +n = 1, 2 . +(D9) +Combining Eq. (D9) with Eq. (D7) or Eq. (D8) and introducing the functions +λ(ξ) = θ(1 − |ξ|) + θ(|ξ| − 1) 1 +|ξ|2 , +λlog(ξ) = θ(1 − |ξ|) + θ(|ξ| − 1)1 + ln |ξ| +|ξ|2 +, +(D10) +we arrive at the following bounds (obviously symmetric under the interchange p ↔ p′): +���� +� 1 +−1 +Fµ(q)dx +���� ≤ MF,1λlog(p′/µ) , and the same for p ↔ p′ , +(D11) +and +���� +� 1 +−1 +Fµ,2(q)dx +���� ≤ MF,2λ(p′/µ) , and the same for p ↔ p′ . +(D12) +For the function Fµ,2(q), we can also obtain another bound: +���� +� 1 +−1 +Fµ,2(q)dx +���� ≤ MF,2λ(p′/µ)2/λ(p/µ) , and the same for p ↔ p′ . +(D13) +To prove Eq. (D13), we consider three cases. +1. |p′| ≤ µ. In this case, λ(p′/µ) = 1. Since λ(p/µ) ≤ 1, Eq. (D13) follows from Eq. (D12). +2. |p| ≥ |p′| > µ. In this case, λ(p′/µ) ≥ λ(p/µ) and Eq. (D12) yields Eq. (D13). +3. |p| < |p′| and |p′| > µ. Consider the definition of the subtraction remainder ∆(1) +p +in Eq. (C7): +Fµ,2(q) = Fµ,2(p′) + p∂Fµ,2(q) +∂p +���� +p=0 ++ ∆(1) +p Fµ,2(q). +(D14) +Now, we estimate the three terms in the last equation individually. +���� +� 1 +−1 +Fµ,2(p′)dx +���� ≤ +� 1 +−1 +|Fµ,2(p′)| dx ≤ 2µ4 +|p′|4 = 2λ(p′/µ)2 ≤ 2λ(p′/µ)2/λ(p) . +(D15) +From the fact that ∂Fµ,2(q) +∂p +���� +p=0 +∝ x, it follows +� 1 +−1 +p∂Fµ,2(q) +∂p +���� +p=0 +dx = 0. +(D16) +The bound from Eq. (C7) gives +���� +� 1 +−1 +∆(1) +p Fµ,2(q) +���� dx ≤ Mf,1 +|p|2 +|p′|2 +� 1 +−1 +|Fµ,2(q)| dx , +(D17) +which (see Eq. (D8)) leads to +���� +� 1 +−1 +∆(1) +p Fµ,2(q) +���� dx ≤ 2Mf,1 +|p|2µ2 +|p′|4 += 2Mf,1 +|p|2 +µ2 λ(p′/µ)2 ≤ 2Mf,1λ(p′/µ)2/λ(p/µ) . +(D18) +Finally, +���� +� 1 +−1 +Fµ,2(q)dx +���� ≤ +���� +� 1 +−1 +Fµ,2(p′)dx +���� + +���� +� 1 +−1 +∆(1) +p Fµ,2(q) +���� dx ≤ 2(Mf,1 + 1)λ(p′/µ)2/λ(p/µ) . +(D19) +Combining all three cases, we obtain Eq. (D13). + +37 +2. +Bounds on the partial-wave leading-order potential +We represent the bounds for the partial-wave LO potential in the separable form: +|V0(p′, p)| ≤ MV0V0,max g(p′)h(p) , +|V0(p′, p)| ≤ MV0V0,max h(p′)g(p) , +(D20) +with +V0,max = +8π2 +mNΛV +, +(D21) +where the exact form of functions g and h (and the value of MV0) depends on the partial wave and on the form of a +regulator. +Introducing the functions +v0(p′, p) = V0(p′, p) [MV0V0,max h(p′)g(p)]−1 , +¯v0(p′, p) = V0(p′, p) [MV0V0,max g(p′)h(p)]−1 , +(D22) +we obtain the bounds +|v0(p′, p)| ≤ 1 , +|¯v0(p′, p)| ≤ 1 . +(D23) +The above inequalities are meant to hold for all matrix elements of V0(p′, p) in the l , l′ space. +a. +S-wave +Using the bounds for the plane-wave leading-order potential in Eq. (C16) and performing the partial-wave projection +according to Eqs. (D4), (D11), (D13), we obtain for l = 0 (for the coupled partial waves, we mean by l the lowest +orbital angular momentum): +g(p) = λlog(p/Λ), +h(p) = 1, +(D24) +for the “mild” regulator, and +g(p) = [λ(p/Λ)]2 , +h(p) = [λ(p/Λ)]−1 , +(D25) +for the “standard” regulators. +Note that for |p| ≤ Λ, in particular, for the on-shell momentum |p| = pon, we have g(p) = h(p) = 1. +b. +Higher partial waves +For l > 0, we can use the fact that for m < l, +∂mV0(p ′, p) +(∂p)m +���� +p=0 += ∂mV0(p ′, p) +(∂p′)m +���� +p′=0 += 0, +(D26) +and thus +∆(m) +p +V0(p ′, p) = ∆(m) +p′ V0(p ′, p) = V0(p ′, p). +(D27) +For the case of the “mild” regulator utilizing Eq. (C22) and performing the partial-wave projection according to +Eqs. (D4) and (D11), we derive +g(p) = λlog(p/Λ)/|p| +˜l, +h(p) = |p| +˜l, +(D28) +with ˜l ≤ l. Since +λ(p/Λ) ≤ λlog(p/Λ), +(D29) +the same bounds can be used for the “standard” regulators, see Eq. (D12). +For the purposes of the present paper, it is sufficient to choose ˜l = 1. + +38 +3. +Bounds on the partial-wave next-to-leading-order potential +a. +S-wave +For l = 0, the bounds on the NLO partial-wave potential are the same as in Ref. [16]: +��� ˆV2(p′, p) +��� ≤ ˆ +MV2,0 +8π2 +mNΛV +M 2 +π +Λ2 +b +, +(D30) +and +��� ˜V2(p′, p) +��� ≤ MV2,0 +� +|p|2 + |p′|2� ˜flog(p′, p), +(D31) +when one employs the “standard” regulators for the LO potentials. +In the case of the “mild” regulator of the LO potential, we use the partial-wave projected regularized expression, +applying Eq. (D11) to Eq. (C30): +��� ˜V2(p′, p) +��� ≤ MV2,0 +� +|p|2 + |p′|2� ˜flog(p′, p)λlog(p′/ΛNLO), or +��� ˜V2(p′, p) +��� ≤ MV2,0 +� +|p|2 + |p′|2� ˜flog(p′, p)λlog(p/ΛNLO). +(D32) +b. +Higher partial waves +For l ≥ 1, we simply adopt the bounds from Ref. [16] +��� ˜V2(p′, p) +��� ≤ MV2,˜l +���� +p +p′ +���� +˜l +|p′|2 ˜flog(p′, p), +(D33) +��� ˜V2(p′, p) +��� ≤ MV2,˜l +���� +p′ +p +���� +˜l +|p|2 ˜flog(p′, p), +(D34) +where 0 ≤ ˜l ≤ l. +For ˜l = 1, both above equations coincide: +��� ˜V2(p′, p) +��� ≤ MV2,1|p′||p| ˜flog(p′, p). +(D35) +For the purposes of the present paper, it is sufficient to take the choice ˜l = 1. +Appendix E: Bounds on various parts of the S-wave NLO amplitude +In this appendix we provide bounds for various parts of the unrenormalized and renormalized S-wave NLO amplitude +and their series remainders. +The unrenormalized NLO amplitude is decomposed by factoring out the Fredholm +determinant as in Eq. (74): +T2(p′, p; pon) = N2(p′, p; pon)/D(pon)2, +N2 = V2D2 + T2,Y D + T2, ¯Y D + T2, ¯Y Y , +(E1) +with +T2,Y (p′, p; pon) = +� p2 +1dp1 +(2π)3 V2(p′, p1)Y (p1, p; pon) , +T2, ¯Y (p′, p; pon) = +� p′2 +1 dp′ +1 +(2π)3 ¯Y (p′, p′ +1; pon)V2(p′ +1, p) , +T2, ¯Y Y (p′, p; pon) = +� p2 +1dp1 +(2π)3 +p′2 +1 dp′ +1 +(2π)3 ¯Y (p′, p′ +1; pon)V2(p′ +1, p1)Y (p1, p; pon). +(E2) +Below, we derive the bounds for the quantities T2,Y , T2, ¯Y and T2, ¯Y Y for the cases of the “standard” and the “mild” +regulators of the LO potential. + +39 +1. +“Standard” regulator +For the “standard” regulators of the LO potential, in particular, for the local regulators of the spin-triplet part of +the one-pion-exchange potential of power n ≥ 2, the binding functions g and h have the form (see Eq. (D25)) +g(p1) = λ(p1/Λ)2 , +h(p) = 1 , if p < Λ . +(E3) +From the bounds on V2 (Eq. (D31)) and V0 (Eq. (D20)), we obtain +|T2,Y (p′, p; pon)| ≤ MV2,0nPW +8π2MYmax +ΛV +� (|p1|2 + |p′|2)d|p1| +(2π)3 +˜flog(p′, p1)λ(p1/Λ)2 += MV2,0nPW +8π2MYmax +mNΛ2 +V Λ2 +b +� d|p1| +π +(|p1|2 + |p′|2)flog(p′, p1)λ(p1/Λ)2 += MV2,0nPW +8π2MYmax +mNΛ2 +V Λ2 +b +� � +|p′|2Iλ,1a + Iλ,1b +� � +1 + θ(|p′| − Mπ) ln |p′| +Mπ +� ++ |p′|2Iλ,2a + Iλ,2b +� +, +(E4) +where the typical integrals Ii are defined and estimated in Appendix F. Setting all external momenta on shell, +p = p′ = pon, and using pon ≪ Λ, gives +|T2,Y (pon)| ≤ 8π2MT2,Y MYmax +mNΛ2 +V Λ2 +b +Λ3 ln Λ +Mπ +, +(E5) +or, assuming Λ ∼ ΛV , +|T2,Y (pon)| ≤ 8π2 ˜ +MT2,Y MYmax +mNΛV +Λ2 +Λ2 +b +ln Λ +Mπ +. +(E6) +Symmetrically, the same bound holds for T2, ¯Y (p′, p; pon). +Next, we consider the contribution T2, ¯Y Y : +��T2, ¯Y Y (p′, p; pon) +�� ≤ MV2,0 +�8π2nPWMYmax +ΛV +�2 +× +� +d|p1| +(2π)3 +d|p′ +1| +(2π)3 (|p1|2 + |p′2 +1 |) ˜flog(p′ +1, p1)λ(p1/Λ)2λ(p′ +1/Λ)2 += MV2,0n2 +PW +8π2M2 +Ymax +mNΛ3 +V Λ2 +b +� d|p1|d|p′ +1| +π2 +(|p1|2 + |p′ +1|2)flog(p′, p1)λ(p1/Λ)2λ(p′ +1/Λ)2 += MV2,0n2 +PW +8π2M2 +Ymax +mNΛ3 +V Λ2 +b +2 (Iλ,1aIλ,1b + Iλ,2aIλ,1b + Iλ,2bIλ,1a) . +(E7) +Setting all external momenta on shell, p = p′ = pon, and using pon ≪ Λ, we obtain +��T2, ¯Y Y (pon) +�� ≤ 8π2MT2, ¯Y Y M2 +Ymax +mNΛ3 +V Λ2 +b +Λ4 ln Λ +Mπ +, +(E8) +or, assuming Λ ∼ ΛV : +��T2, ¯Y Y (pon) +�� ≤ 8π2 ˜ +MT2, ¯Y Y M2 +Ymax +mNΛV +Λ2 +Λ2 +b +ln Λ +Mπ +. +(E9) +2. +“Mild” regulator +For the “mild” regulator of the LO potential, including the case when the spin-triplet one-pion-exchange contribution +is regularized by the local dipole regulator, the binding functions g and h have the form (see Eq. (D24)) +g(p1) = λlog(p1/Λ) , +h(p) = 1 , if p < Λ . +(E10) + +40 +By analogy with Eq. (E4) from the bounds on the regularized V2 (Eq. (D32)) and V0 (Eq. (D20)), we obtain +|T2,Y (p′, p; pon)| ≤ MV2,0nPW +8π2MYmax +mNΛ2 +V Λ2 +b +� d|p1| +π +(|p1|2 + |p′|2) +× flog(p′, p1)λlog(p1/Λ)λlog(p1/ΛNLO) += MV2,0nPW +8π2MYmax +mNΛ2 +V Λ2 +b +� � +|p′|2Iλlog,1a + Iλlog,1b +� � +1 + θ(|p′| − Mπ) ln |p′| +Mπ +� ++ |p′|2Iλlog,2a + Iλlog,2b +� +, +(E11) +where the typical integrals Ii are defined and estimated in Appendix F. Setting all external momenta on shell, +p = p′ = pon, and using pon ≪ Λ ≪ ΛNLO, yields +|T2,Y (pon)| ≤ 8π2MT2,Y MYmax +mNΛ2 +V Λ2 +b +Λ2ΛNLO ln ΛNLO +Λ +ln ΛNLO +Mπ +, +(E12) +or, assuming Λ ∼ ΛV : +|T2,Y (pon)| ≤ 8π2 ˜ +MT2,Y MYmax +mNΛV +ΛΛNLO +Λ2 +b +ln ΛNLO +Λ +ln ΛNLO +Mπ +. +(E13) +Symmetrically, the same bound holds for T2, ¯Y (p′, p; pon). +Analogously to Eq. (E7), the following bound holds for T2, ¯Y Y : +��T2, ¯Y Y (p′, p; pon) +�� ≤ MV2,0n2 +PW +8π2M2 +Ymax +mNΛ3 +V Λ2 +b +× +� d|p1|d|p′ +1| +π2 +flog(p′ +1, p1)λlog(p1/Λ)λlog(p′ +1/Λ) +× +� +|p1|2λlog(p1/ΛNLO) + |p′ +1|2λlog(p′ +1/ΛNLO) +� += MV2,0n2 +PW +8π2M2 +Ymax +mNΛ3 +V Λ2 +b +× 2 +� +Iλlog,1Iλlog,1b + Iλlog,2Iλlog,1b + Iλlog,2bIλlog,1 +� +. +(E14) +Setting all external momenta on shell, p = p′ = pon, and using pon ≪ Λ ≪ ΛNLO, we obtain +��T2, ¯Y Y (pon) +�� ≤ 8π2MT2, ¯Y Y M2 +Ymax +mNΛ3 +V Λ2 +b +Λ3ΛNLO ln ΛNLO +Λ +ln ΛNLO +Mπ +, +(E15) +or, assuming Λ ∼ ΛV : +��T2, ¯Y Y (pon) +�� ≤ 8π2 ˜ +MT2, ¯Y Y M2 +Ymax +mNΛV +ΛΛNLO +Λ2 +b +ln ΛNLO +Λ +ln ΛNLO +Mπ +. +(E16) +3. +Bounds on the function ν(pon) +In this subsection we provide bounds on the function νl(pon), defined in Eq. (118). We introduce another function +νY,l as follows: +νl(pon) = D(pon) [δl,0 + νY,l(pon)] , +(E17) +which equals (see Eq. (105)) +νY,l(pon) = +� p2 +1dp1 +(2π)3 Y0,l(p1, pon; pon). +(E18) + +41 +Using Eq. (61), we derive the following bound for the function nY,l in the case of the “standard” regulator of the LO +potential (see Appendix D 2 a): +|νY,l(pon)| ≤ MYmax +ΛV +� d|p1| +π +g(p1/Λ)h(pon) += MYmax +ΛV +� d|p1| +π +λ(p1/Λ)2 = MYmax +ΛV +Iλ,1a = MYmaxMλ +Λ +ΛV +, +(E19) +where we have utilized the bounds for typical integrals provided in Appendix F. +Assuming Λ ∼ ΛV yields +|νY,l(pon)| ≤ MYmax ˜ +Mλ. +(E20) +For the “mild” regulator of the LO potential, one should replace λ(p1/Λ)2 with λlog(p1/Λ) and Iλ,1a with Iλlog,1 in +Eq. (E19). Since our bounds for Iλlog,1 and Iλ,1a are the same, see Eqs. (F2) and (F4), equation (E20) holds also for +the “mild” regulator. +Since the Fredholm determinant D is bounded by a constant of order one (Eq. (51)), the same is true for the +function νl(pon): +νl(pon) ≤ Mν, +(E21) +as follows from Eqs. (E20) and (E17). +4. +Series remainders +From the bounds on the matrix elements of the operator Y ( ¯Y ) and its series remainders (Eqs. (61) and (65)) as well +as the bounds on the Fredholm determinant D and its series remainders (Eqs. (51) and (53)), it is straightforward to +deduce also the bounds for the series remainders of the quantities T2,Y , T2, ¯Y , T2, ¯Y Y and νY by just replacing MYmax +with NδnY = MY δnYmax and M2 +Ymax with 2MYmaxNδnY + N 2 +δnY . Being proportional to δnYmax or (δnYmax)2, T2,Y , +T2, ¯Y , T2, ¯Y Y and νY decrease faster than exponential with any base, see Eq. (64). The series remainder of the Fredholm +determinant possesses the same property, see Eq. (55). Therefore, from Eq. (E1) we conclude that N2 also decreases +faster than exponential as well as the renormalized quantity R( ˜N2) (Eq. (122)), because those are polynomials in +T2,Y , T2, ¯Y , T2, ¯Y Y , νY and D. +To be specific, the following bound holds: +|δn[R( ˜N2)]| = +��� +∞ +� +k1,k2=0 +R( ˜N2)[k1,k2] − +n +� +k1,k2=0 +R( ˜N2)[k1,k2]��� +≤ +8π2 +mNΛV +N ˜ +N2e−Mδ ˜ +N2n, +for n > ˜ +Mδ ˜ +N2, +(E22) +where +˜ +Mδ ˜ +N2 is of order +˜ +Mδ ˜ +N2 ≳ (eΣ)2 in the general case but is typically much smaller in realistic calculations. The +prefactors N ˜ +N2 follow from Eqs. (E6), (E9), (E13), (E16), (E21) and (51): +N ˜ +N2 = Λ2 +Λ2 +b +ln Λ +Mπ +(E23) +in the case of the “standard” regulators of the LO potential and +N ˜ +N2 = ΛΛNLO +Λ2 +b +ln ΛNLO +Λ +ln ΛNLO +Mπ +(E24) +in the case of the “mild” regulator. +Appendix F: Bounds on typical integrals +In this appendix we provide the bounds for typical integrals that appear in the course of evaluation of various +amplitudes. + +42 +The integrals +Iλlog,1 = +� d|p| +π λlog(p/Λ), +Iλlog,1a = +� d|p| +π λlog(p/ΛNLO)λlog(p/Λ), +Iλlog,2 = +� d|p| +π λlog(p/Λ)θ(|p| − Mπ) ln |p| +Mπ +, +Iλlog,2a = +� d|p| +π λlog(p/ΛNLO)λlog(p/Λ)θ(|p| − Mπ) ln |p| +Mπ +, +(F1) +with functions λ and λlog defined in Eq. (D10) can be bounded as follows: +Iλlog,1 = Λ +� dξ +π λlog(ξ) =: MλΛ, +Iλlog,1a < Iλlog,1 = MλΛ, +Iλlog,2 = 1 +π +� +2 + Λ + 2Λ ln Λ +Mπ +� +≤ Mλ,2Λ ln Λ +Mπ +, +Iλlog,2a < Iλlog,2 ≤ Mλ,2Λ ln Λ +Mπ +. +(F2) +Analogously, for the integrals +Iλ,1 = +� d|p| +π λ(p/Λ), +Iλ,1a = +� d|p| +π λ(p/Λ)2, +Iλ,1b = +� |p|2d|p| +π +λ(p/Λ)2, +Iλ,2 = +� d|p| +π λ(p/Λ)θ(|p| − Mπ) ln |p| +Mπ +, +Iλ,2a = +� d|p| +π λ(p/Λ)2θ(|p| − Mπ) ln |p| +Mπ +Iλ,2b = +� |p|2d|p| +π +λ(p/Λ)2, θ(|p| − Mπ) ln |p| +Mπ +, +(F3) +we obtain the following bounds: +Iλ,1 = Λ +� dξ +π λ(ξ) < Λ +� dξ +π λlog(ξ) = MλΛ, +Iλ,1a < Iλ,1 ≤ MλΛ, +Iλ,1b = Λ3 +� ξ2dξ +π +λ(ξ)2 < Λ3 +� dξ +π λ(ξ) < Λ3 +� dξ +π λlog(ξ) = MλΛ3, +Iλ,2 < Iλlog,2 ≤ Mλ,2Λ ln Λ +Mπ +, +Iλ,2a < Iλ,2 ≤ Mλ,2Λ ln Λ +Mπ +, +Iλ,2b < Λ2Iλ,2 ≤ Mλ,2Λ3 ln Λ +Mπ +. +(F4) +Next, we estimate the integral +Iλlog,1b = +� |p|2d|p| +π +λlog(p/ΛNLO)λlog(p/Λ). +(F5) + +43 +Direct estimation under the assumption ΛNLO ≫ Λ gives +Iλlog,1b = 2 +π Λ2ΛNLO ln ΛNLO +Λ ++ O(ΛNLO) ≤ Mλ,1aΛ2ΛNLO ln ΛNLO +Λ +. +(F6) +Finally, we derive a bound for the integral +Iλlog,2b = +� |p|2d|p| +π +λlog(p/ΛNLO)λlog(p/Λ)θ(|p| − Mπ) ln |p| +Mπ +. +(F7) +Direct calculation yields +Iλlog,2b = 1 +π Λ2ΛNLO ln ΛNLO +Λ +ln ΛNLO +Mπ ++ O(ΛNLO ln ΛNLO/Mπ) +≤ Mλ,1aΛ2ΛNLO ln ΛNLO +Λ +ln ΛNLO +Mπ +. +(F8) +[1] S. 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Epelbaum, W. Gl¨ockle, and U. -G. Meißner, Nucl.Phys. A747, 362 (2005), nucl-th/0405048. + diff --git a/vNFPT4oBgHgl3EQfODR9/content/tmp_files/load_file.txt b/vNFPT4oBgHgl3EQfODR9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9b57a495fabb3aaa3f5f96e0be8e1a56b8ebaa25 --- /dev/null +++ b/vNFPT4oBgHgl3EQfODR9/content/tmp_files/load_file.txt @@ -0,0 +1,1642 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf,len=1641 +page_content='Renormalization of nuclear chiral effective field theory with non-perturbative leading order interactions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Gasparyan1, ∗ and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Epelbaum1, † 1Ruhr-Universit¨at Bochum, Fakult¨at f¨ur Physik und Astronomie, Institut f¨ur Theoretische Physik II, D-44780 Bochum, Germany We extend the renormalizability study of the formulation of chiral effective field theory with a finite cutoff, applied to nucleon-nucleon scattering, by taking into account non-perturbative effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' We consider the nucleon-nucleon interaction up to next-to-leading order in the chiral expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The leading-order interaction is treated non-perturbatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In contrast to the previously considered case when the leading-order interaction was assumed to be perturbative, new features related to the renormalization of the effective field theory are revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In particular, more severe constraints on the leading-order potential are formulated, which can enforce the renormalizability and the correct power counting for the next-to-leading order amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' To illustrate our theoretical findings, several partial waves in the nucleon-nucleon scattering, 3P0, 3S1 − 3D1 and 1S0 are analyzed numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The cutoff dependence and the convergence of the chiral expansion for those channels are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' ∗ Email: ashot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='gasparyan@rub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='de † Email: evgeny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='epelbaum@rub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='de arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='13032v1 [nucl-th] 30 Jan 2023 2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' INTRODUCTION Over the last decades, the effective field theory (EFT) approach has become a standard tool in studies of the nucleon-nucleon (NN), few-nucleon and many-nucleon systems due to the possibility to perform systematically im- provable calculations in accordance with the chiral power counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The chiral power counting implies an expansion of observables in terms of the ratio of the soft and the hard scales Q = q/Λb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The soft scale is given by the pion mass Mπ and the external particle 3-momenta |⃗p |, whereas the hard scale Λb is the breakdown scale of the EFT expansion of the order of the ρ-meson mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Starting with the seminal work by Weinberg [1, 2], a lot of progress has been achieved in this field, see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [3–8] for reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In realistic calculations, one has to deal with regularization of an infinite number of divergent Feynman diagrams originating from the field theoretic treatment of non-perturbative amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' One of the most practical approaches is related to introducing a finite (of the order of the hard scale Λb) cutoff Λ in momentum space (or a corresponding short distance cutoff in coordinate space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The success of such a scheme is reflected in very accurate calculations at high orders in the chiral expansion, see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [9–11] for recent applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' A justification of such an approach from the fundamental point of view is complicated by the issue of renormal- ization and power counting violation due to the appearance of positive powers of the cutoff in the amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Such contributions are generated by loop momenta of the order of the cutoff Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' There exists a qualitative understanding in the literature [12–15] that such positive powers of Λ in the leading-order (LO) amplitude get compensated by the negative powers of the scale ΛV stemming from the LO potential, which is also regarded to be of the order of the hard scale Λb: ΛV ∼ Λ ∼ Λb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Further, one believes that at higher chiral orders, the power counting breaking terms can be absorbed by a renormalization (shift) of lower order contact interactions [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' However, until recently, a rigorous treatment of these problems and a systematic analysis of conditions under which the renormalization program can be carried out has been missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Such a rigorous treatment is extremely important within the EFT approach, where systematic power counting is utilized to estimate theoretical uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' We addressed this issues in our study in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In particular, we considered the LO potential consisting of the long-range one-pion-exchange term and a set of contact interactions that are momentum-independent or quadratic in momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The LO potential was regularized by various types of the form factors in momentum space, including local and non-local regulators both power-like and Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' This covers most of the schemes considered in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16], it was assumed that the iterations of the leading-order potential V0 can be treated perturbatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' More precisely, the series of the LO and the next-to-leading-order (NLO) amplitude in powers of V0 were assumed to be convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' However, the convergence rate of the expansion in V0 might still be slower than the convergence rate of the chiral EFT expansion, which makes it necessary to sum up all (or many) iterations of V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' On the other hand, the NLO potential needs not be iterated in the NLO amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In the physical case of the NN scattering, such a perturbative regime is realized in most of the partial waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The prominent exceptions are the 1S0, 3S1 − 3D1 and 3P0 channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Under the above rather general assumptions, we proved the following statements: The LO amplitude satisfies the dimensional power counting at each order in V0 and is of chiral order O(Q0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' If necessary, contact interactions quadratic in momenta can be promoted to leading order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The NLO amplitude in P- and higher waves satisfies the dimensional power counting at each order in V0 and is of chiral order O(Q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The unrenormalized NLO amplitudes in the S-waves (including the 3S1 − 3D1 channel) violate the power counting and are of order O(Q0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' To absorb the power-counting breaking terms, we employed the Bogoliubov- Parasiuk-Hepp-Zimmermann (BPHZ) renormalization procedure and performed the overall subtractions in the diagrams as well as subtractions in all nested subdiagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' As a result, the renormalized NLO amplitude was shown to satisfy the dimensional power counting and being of chiral order O(Q2) up to corrections logarithmic in the cutoff at each order in V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In the current work, we extend our analysis to the non-perturbative case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' to the situation when the series in the LO potential V0 do not converge for the LO and/or NLO amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' This will allow us to consider the above mentioned non-perturbative channels in NN scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Our analysis is based on the application of the Fredholm method of solving integral equations, which enables us to match the perturbative and non-perturbative regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Our paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' II, we briefly describe our formalism based on the effective Lagrangian, the corresponding effective potential and the way the amplitude is constructed in the non-perturbative case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' III, we explain the application of the Fredholm method for the LO Lippmann-Schwinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' IV, we demonstrate the renormalization of the nucleon-nucleon interaction in P-waves and higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The renormalization 3 in the S-waves is addressed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Numerical results that illustrate our formal considerations are presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The paper ends with a summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Bounds on the effective potential and various integrals are collected in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' FORMALISM A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Effective Lagrangian and potential In this section we briefly describe the formalism of chiral EFT used in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Some details are omitted and can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The starting point is the effective chiral Lagrangian represented as a series of all possible terms consistent with the symmetries of the underlying theory [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The expansion of the Lagrangian is performed in terms of the quark masses and field derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The effective Lagrangian contains purely pionic terms, single nucleon terms, two-nucleon interactions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' : Leff = L(2) π + L(4) π + L(1) πN + L(2) πN + L(0) NN + L(2) NN + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' , (1) where the superscripts denote chiral orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The chiral expansion of the NN amplitude in terms of the small parameter Q is performed according to the Weinberg power counting [2] (with possible modifications based on phenomenological arguments, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' promotion of certain higher order contributions to lower orders).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The power of Q for a potential (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' two-nucleon-irreducible) contribution is determined by a sum over all vertices i in the diagram: D = 2L + � i � di + ni 2 − 2 � , (2) where L is the number of loops, ni is the number of nucleon lines at vertex i and di is the number of derivatives and pion-mass insertions at vertex i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The chiral order of a 2N-reducible diagram is equal to the sum of the orders of its irreducible components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Since the LO contributions appear at order O(Q0), the corresponding potential terms have to be iterated an infinite number of times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' To implement this procedure on a formal level and to regularize multiple-loop integrals, it is convenient to reformulate the effective Lagrangian of two-nucleon interactions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (1) in terms of the non-local regularized potential contributions of the form (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16] for details) LV (x) = − � d⃗y d⃗y ′ 1 2N † j1(x0, ⃗x − ⃗y ′/2)N † j2(x0, ⃗x + ⃗y ′/2)V (⃗y ′, ⃗y)j1,j2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='i1,i2Ni2(x0, ⃗x + ⃗y/2)Ni1(x0, ⃗x − ⃗y/2) , (3) where i1, i2, j1, j2 are the combined spin and isospin indices of the corresponding nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' This formulation is customary for the few-body and nuclear physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The full potential is organized as a series according to the chiral expansion: V = V (0) + V (2) + V (3) + V (4) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (4) Bare potentials V (i) are split into the renormalized parts Vi and the counter terms δVi: V (i) = Vi + δVi , δVi = δV (2) i + δV (3) i + δV (4) i + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (5) The counter terms δV (j) i (j > i) absorb the divergent and the power counting violating terms appearing at order O(Qj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The LO potential V0 is regulated (the details are given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' II B and in Appendix A) using a cutoff Λ to make the iterations of V0 finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' We regard the cutoff value Λ (the largest cutoff among all cutoffs used in the LO potential) to be of the order of the hard scale Λ ∼ Λb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Higher order potentials can be considered either regulated or unregulated depending on a particular scheme, which will be discussed in the subsequent sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Note that to make some intermediate expressions mathematically well defined, one might need to introduce addi- tional cutoffs that drop out from the final results after performing certain subtractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Such cutoffs can be chosen to be much larger than Λ (or even infinity large).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 4 To make the formulation of the theory in terms of non-local (on the Lagrangian level) regularized potential contri- butions completely equivalent to the original formulation in terms of local interactions, the regulator corrections δΛV have to be taken into account: δΛV = � i δΛV (i), δΛV (i) := V (i) Λ=∞ − V (i) Λ , (6) where V (i) Λ=∞ is the unregulated potential at the chiral order i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' One possibility, often implicitly used in practical calculations, is to expand δΛV in powers of 1/Λ and absorb the resulting terms by higher order contact interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' This is possible if the potential does not contain non-locally regularized long-range contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Another approach suggested in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16] is to keep the terms with δΛV explicitly and consider those as perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' This allows us to reduce the cutoff dependence and extend the range of possible values of Λ, especially to smaller ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' LO and NLO potentials and regulators Our treatment of the LO and NLO potentials is identical to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Weinberg’s power counting in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (2) implies that the leading-order O(Q0) potential V0(⃗p ′, ⃗p ) is represented by the sum of the regulated static one-pion-exchange potential and the short-range part: V0(⃗p ′, ⃗p ) = V (0) 1π,Λ(⃗p ′, ⃗p ) + V (0) short,Λ(⃗p ′, ⃗p ), (7) where the short-range part V (0) short,Λ may contain momentum-independent contact terms as well as the contact terms quadratic in momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The latter are formally of order O(Q2), as follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Nevertheless, it is known that in some channels, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=', 1S0 and 3P0, their promotion to leading order can be motivated by phenomenological arguments, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [18–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For the sake of generality, we allow for different forms of regulators: power-like local, power-like non-local, Gaussian local and Gaussian non-local regulators as well as all possible combinations of those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16], we argued that for a local part of the LO potential V0,local(⃗q ), the regulator (if it is also local) can be rather “mild”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' If the regulated LO potential behaves as V0,local(⃗q ) ∼ 1 |⃗q |2 , for |⃗q| → ∞, (8) both LO and NLO amplitudes turn finite after renormalization even if the NLO potential is not regulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The reason for that is a milder ultraviolet behavior of local structures after performing subtractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Such a mild regulator cannot be chosen for the non-local parts of the LO potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Equation (8) implies that in the spin-triplet channels the one-pion-exchange potential can be regulated by a dipole form factor, Fq,1π,Λ,1 = Λ2 − M 2 π q2 + Λ2 , (9) whereas for the spin-singlet channels it can even be left unregulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Although in practical calculations one typically implements Gaussian or even sharper regulators to guarantee the finiteness of all integrals, we consider separately the above mentioned situation with a local part of the LO potential having the ultraviolet asymptotics as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (8) and say that such a potential has a “mild” regulator in contrast to “standard” regulators, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' all other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' This is done to keep the analysis general and to clarify the difference between perturbative and non-perturbative regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Moreover, such an analysis is useful to understand the cutoff dependence of the NN amplitude: the milder regulator can be chosen, the weaker cutoff dependence should be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For completeness, we provide the explicit expressions for the LO potential and the corresponding regulators in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The next-to-leading-order potential V2(⃗p ′, ⃗p ) contains the short-range part, the two-pion-exchange potential and the regulator corrections to the leading-order potential: V2(⃗p ′, ⃗p ) = V (2) 2π (⃗p ′, ⃗p) + V (2) short(⃗p ′, ⃗p ) + δΛV (0)(⃗p ′, ⃗p ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (10) In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16], we found that one does not need to regularize the NLO potential to perform the renormalization of the NLO amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Or, equivalently, one can introduce a cutoff ΛNLO ≫ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' On the other hand in practical calculations, 5 one can choose ΛNLO ∼ Λb if it improves efficiency of a computational scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Both approaches are formally equivalent because the regulator corrections δΛV (2) appear at order O(Q4) in accordance with the dimensional power counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' It turns out, that the situation is slightly different in the general non-perturbative case, where for the choice of the “mild” LO regulator we need to keep ΛNLO finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' It can still be larger than Λ, but not arbitrarily large, see discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The explicit expressions for the NLO potential can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' NN amplitudes and contour rotation In the present study we work predominantly in the partial wave lsj basis, which makes the analysis of the non- perturbative effects more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In the lsj basis, the potential and the amplitude are nPW × nPW matrices, where nPW = 1 (nPW = 2) for the uncoupled (coupled) partial waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The series for the partial wave LO amplitude and for the unrenormalized NLO amplitude are given by T0 = ∞ � n=0 T [n] 0 , T [n] 0 = V0Kn = ¯KnV0, (11) T2 = ∞ � m,n=0 T [m,n] 2 , T [m,n] 2 = ¯KmV2Kn, (12) where G is the free two-nucleon propagator and K = GV0, ¯K = V0G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (13) In the non-perturbative case these equations generalize to T0 = V0R = ¯RV0, (14) T2 = ¯RV2R, (15) where R ( ¯R) is the resolvent of the Lippmann-Schwinger equation (LSE) R = 1 1 − K , ¯R = 1 1 − ¯K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (16) The renormalized expression for the NLO amplitude R(T2) is obtained by adding the relevant counter term, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' V for details: R(T2) = ¯R � V2 + δV (2) 0 � R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (17) The explicit form of the LSE, T0 = V0 + V0GT0, reads (T0)l′l (p′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) = � l′′ � p′′2dp′′ (2π)3 (V0)l′l′′ (p′, p′′)G(p′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) (T0)l′′l (p′′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon), G(p′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) = mN p2on − p′′2 + iϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (18) The indices l, l′, l′′ denote the orbital angular momentum of the NN system, pon is the on-shell c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' nucleon momentum and p (p′) are the initial (final) off-shell c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' It turns out useful to modify the integration path over the off-shell momentum p′′ and rotate the contour into the complex plane [22–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The new integration contour C is defined by p′′ = |p′′|e−iαC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Our choice for the rotation angle αC is determined by the location of singularities of the LO potential in the complex plane [16]: αC = 1 2 arctan Mπ (pon)max , (19) where (pon)max is the maximal considered on-shell momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The contour rotation enables us to perform direct estimations of the bounds on the partial wave amplitudes avoiding principal value integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 6 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Bounds on the potentials and the NN propagator By analogy with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16], we use certain upper bounds for the potentials and the NN propagator that are valid for off-shell momenta lying on the complex contour C and for the allowed real on-shell momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' These bounds allow us to estimate the nucleon-nucleon LO and NLO amplitudes and to verify the corresponding power counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16], in the bounds considered below, we introduce dimensionless constants named Mi: MV0, MG, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=', which are supposed to be of order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Analogous constants appear in our final estimates for the amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Some of the inequalities should be modified compared to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16] to be better suited for the non-perturbative analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In particular, for the LO potential V0(p′, p), we need bounds that are separable in momenta p and p′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The inequalities listed below are meant to hold for all matrix elements of the partial wave potentials V0(p′, p) and V2(p′, p) in l , l′ space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Their derivation can be found in Appendices C and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The LO partial-wave potential obeys the following bounds: |V0(p′, p)| ≤ MV0V0,max g(p′)h(p), |V0(p′, p)| ≤ MV0V0,max h(p′)g(p), (20) with V0,max = 8π2 mNΛV , (21) where the exact form of the functions g and h (and the value of MV0) depends on the partial wave and on the form of a regulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For l = 0 (for the coupled partial waves, we mean by l the lowest possible orbital angular momentum), g and h are given by g(p) = λlog(p/Λ) , h(p) = 1 , (22) for the “mild” regulator, and by g(p) = [λ(p/Λ)]2 , h(p) = [λ(p/Λ)]−1 , (23) for the “standard” regulators with the functions λ and λlog defined as λ(ξ) = θ(1 − |ξ|) + θ(|ξ| − 1) 1 |ξ|2 , λlog(ξ) = θ(1 − |ξ|) + θ(|ξ| − 1)1 + ln |ξ| |ξ|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (24) For higher partial waves, l ≥ 1, we adopt the bounds g(p) = λlog(p/Λ)/|p| , h(p) = |p|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (25) Notice that while in the latter case one could use a stronger bound and replace λlog with λ for the “standard” regulator, this would not affect our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Therefore, we prefer to employ this unified bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For spin-singlet partial waves without a short-range LO contribution, one can improve the above bounds and replace in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (25) λlog(p/Λ) with λlog(p/Mπ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' However, in all such channels the perturbative regime for the LO potential is realized, which has already been analyzed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16] and will not be considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Note that for |p| ≤ Λ, and, in particular, for the on-shell momentum |p| = pon, we have in all cases g(p) = h(p) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' It is convenient also to introduce the functions v0(p′, p) = V0(p′, p) [MV0V0,max h(p′)g(p)]−1 , ¯v0(p′, p) = V0(p′, p) [MV0V0,max g(p′)h(p)]−1 , (26) for which the following bounds hold: |v0(p′, p)| ≤ 1 , |¯v0(p′, p)| ≤ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (27) For the unregulated NLO potential, we adopt the bounds from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In particular, for l = 0: |V2(p′, p)| ≤ MV2,0 � |p|2 + |p′|2� ˜flog(p′, p), (28) 7 with ˜flog(p′, p) = 8π2 mNΛV Λ2 b flog(p′, p) , flog(p′, p) = θ(|p| − Mπ) ln |p| Mπ + θ(|p′| − Mπ) ln |p′| Mπ + 1 , (29) where we have dropped the log Λ/Mπ term in the definition of flog, which is unnecessary and was introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16] for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Note that in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16], the NLO potential V2 was split into two parts V2(p ′, p) = ˆV2(p ′, p) + ˜V2(p ′, p), (30) with ˆV2(p ′, p) = V2(0, 0) , ˜V2(p ′, p) = V2(p ′, p) − V2(0, 0), (31) and the inequality in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (28) is, strictly speaking, valid for ˜V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' However, in the present work, we use most of the time the scheme with V2(0, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Therefore, in what follows, we will always assume that ˜V2 = V2 unless specified otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For alternative schemes, we also provide the bound for ˆV2: ��� ˆV2(p′, p) ��� ≤ ˆ MV2,0 8π2 mNΛV M 2 π Λ2 b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (32) For higher partial waves l > 0, it is sufficient to implement the p-wave bound: |V2(p′, p)| ≤ MV2,1|p′||p| ˜flog(p′, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (33) For the regularized NLO potential with the cutoff ΛNLO, the bounds in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (28) are modified as follows (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' D 3 a): |V2(p′, p)| ≤ MV2,0 � |p|2 + |p′|2� ˜flog(p′, p)λlog(p′/ΛNLO) , or |V2(p′, p)| ≤ MV2,0 � |p|2 + |p′|2� ˜flog(p′, p)λlog(p/ΛNLO) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (34) For the two-nucleon propagator G(p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) = mN/(p2 on − p2), we use the same bound as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16]: |G(p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon)| ≤ MG mN |p2| , (35) with MG = 1/ sin(2αC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' LEADING-ORDER LIPPMANN-SCHWINGER EQUATION In this section we outline the Fredholm method for solving integral equations and derive the bounds on the resolvents of the LSE and on the LO amplitude in the non-perturbative case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The resolvents R and ¯R of the partial-wave LSE, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (16), can be represented by means of the Fredholm formula [25, 26] as: R = (1 − K)−1 = 1 + Y D, ¯R = (1 − ¯K)−1 = 1 + ¯Y D, (36) where the Fredholm determinant D is a number and depends only on the on-shell momentum D = D(pon), whereas the minor Y ( ¯Y ) is a matrix in the l, l′ space and an operator in the space of the off-shell momenta: Y = Yji(p′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The quantities Y , ¯Y and D can be expanded into convergent series in powers of the LO potential V0: Y = ∞ � n=1 Y [n], ¯Y = ∞ � n=1 ¯Y [n], D = ∞ � n=0 D[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (37) 8 In what follows, we will consider the resolvent R and the minor Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The results are trivially generalized for ¯R and ¯Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The terms D[n] and Y [n] can be written as [25, 26] D[n](pon) = (−1)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' � i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=',in � n � k=1 p2 kdpk (2π)3 [detD,n(K)]i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' in (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' , pn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon), (38) and Y [n+1] i′i (p′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) =(−1)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' � i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=',in � n � k=1 p2 kdpk (2π)3 [detY,n+1(K)]i,i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' in,i′ (p, p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' , pn, p′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon), (39) where the matrix indices i, i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='.in and i′ correspond to the orbital angular momentum l = j ± 1 for coupled partial waves and l = j for uncoupled partial waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In the above equations, the determinants for an operator X with matrix elements X(p′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) (or X(p′, p) if it is independent of pon) are defined as: [detD,n(X)]i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=',in (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' , pn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) = ������ Xi1,i1(p1, p1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) · · · Xin,i1(p1, pn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Xi1,in(pn, p1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) · · · Xin,in(pn, pn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) ������ , (40) and [detY,n+1(X)]i,i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=',in,i′ (p, p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' , pn, p′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) = ������� Xi′i(p′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) Xi1i(p1, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) · · Xini(pn, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) Xi′i1(p′, p1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) Xi1i1(p1, p1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) · · · Xini1(pn, p1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) · · · · · · · · · Xi′in(p′, pn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) Xi1in(p1, pn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) · · · Xinin(pn, pn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) ������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (41) Rescaling V0 as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (26), we obtain: Ki′i(p′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) = (v0)i′i(p′, p)MV0V0,max g(p′)h(p)G(p′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon), (42) so that D[n](pon) = (−1)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (MV0V0,max)n � i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=',in � � n � k=1 p2 kdpk (2π)3 g(pk)h(pk)G(pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) � [detD,n(v0)]i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=',in (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' , pn), (43) and Y [n+1] i′i (p′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) = (−1)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (MV0V0,max)n+1 g(p′)h(p)G(p′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) × � i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=',in � � n � k=1 p2 kdpk (2π)3 g(pk)h(pk)G(pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) � [detY,n+1(v0)]i,i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=',in,i′ (p, p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' , pn, p′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (44) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Upper bounds for the Fredholm determinant First, we analyze the series for the Fredholm determinant D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Since the matrix elements v0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='ji(p′, p) are bounded by (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (27)) |v0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='ji(p′, p)| ≤ 1 , (45) the Hadamard’s inequality for determinants gives [25, 26] |detD,n(v0)| ≤ nn/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (46) Therefore, using Stirling’s formula, we can estimate D[n] as follows: ���D[n]��� ≤ 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='Σnnn/2 ≤ 1 √ 2πn � eΣ √n �n = 1 √ 2πeΣ � eΣ √n �n+1 = 1 √ 2πeΣ exp � − (n + 1) ln √n eΣ � =: MD,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (47) 9 where Σ is defined as MV0V0,maxnPW ���� � p2dp (2π)3 g(p)h(p)G(p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) ���� ≤ MV0MG ΛV nPW � d|p| π g(p)h(p) =: Σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (48) Since g(p) and h(p) depend only on the ratio p/Λ, we can write Σ = MΣ Λ ΛV , (49) where the numerical value of the constant MΣ depends on a particular form of g(p) and h(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' If we assume Λ ∼ ΛV , then Σ ∼ 1 up to a numerical factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The situation when Σ < 1 corresponds to a convergent series for the LO amplitude in terms of V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In contrast, for the non-perturbative regime that we consider, we have Σ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The maximal value of D[n] is achieved at some n = nDmax and can be estimated by differentiating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (47) with respect to n: nDmax ≈ eΣ2 , |D[n]| ≤ MD[n],max ≈ eeΣ2/2 √ 2πeΣ , (50) which is formally a number of order one, but it grows very rapidly with Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The whole series for D is also bounded by a constant of order one: |D| ≤ MD, (51) which can be estimated by replacing the sum with an integral and using Laplace’s method: MD = ∞ � n=0 MD,n ≈ � ∞ 0 dtMD,t ≈ √ 2π � −∂2 ln MD,t ∂t2 �−1/2 MD,t ��� t=nDmax ≈ √ 2eeΣ2/2 , (52) which agrees rather well with the series summed numerically (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (47)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For example, for Σ = 1, both results give MD ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The bounds (47) and (52) are rather weak and very conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' If Σ is not close to one, the numerical values for MD become very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' However, in realistic calculations, we can see that D does actually not exceed the values of order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Clearly, one can always perform a numerical check in order to verify whether our approach to the renormalizability of the NN amplitude based on the Fredholm method is reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Note also that for the 1S0 and 3S1 − 3D1 NN channels, one can expect Σ to be close to one (ignoring the fine-tuning between attractive and repulsive forces) because the first (quasi) bound states in these channels are very shallow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' This is roughly confirmed by an analysis of the Weinberg eigenvalues in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' There are particular cases when the estimate in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (47) can be readily improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For example, for purely local LO potentials, the quantities D and Y correspond to the Jost function and the regular solution of the Schr¨odinger equation in configuration space and the terms in their expansion, D[n] and Y [n], decrease as 1/n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='. On the other hand, if the LO consists of only a short-range separable potential (or is dominated by such a contribution), the series for D and Y contain a finite number of terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' However, in our general discussion, we will simply assume that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (51) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' We will also need an estimate for the series remainder: δnD = ∞ � k=n+1 D[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (53) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (47), we can conclude that for sufficiently large n, n > n0 ≡ ˜ MδD, (54) the terms D[n] and, therefore, also the remainder δn∆ decrease faster than exponential δnD ≤ e−MδD n, (55) 10 with any MδD, which we will use in our further estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The value ˜ MδD depends on MδD and on Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (47), we can conclude that the exponential decrease starts only for ˜ MδD > (eΣ)2, (56) which, being formally a number of order one, becomes extremely large unless Σ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' However, as follows from the discussion above, in realistic calculations, such an exponentially suppressed regime can be reached much earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In fact in the numerical calculation presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' VI, the relative error δnD/D becomes less than one percent in most cases for n = 3 or 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Bounds for the minor Y By analogy with the Fredholm determinant D, we can perform the same analysis for the minor Y starting from the definition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Using again the Hadamard’s inequality, |detY,n(v0)| ≤ nn/2, (57) we get the bound for Y [n]: ���Y [n] ji (p′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) ��� ≤ MY,n |G(p′, pon)| 8π2MV0 mNΛV g(p′)h(p) (58) with MY,n = 1 (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='Σn−1nn/2 ≤ e √ 2π � eΣ √n �n−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (59) Further, taking into account the bound for the propagator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (35), we obtain ���Y [n] ji (p′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) ��� ≤ MY,n 8π2MV0MG ΛV |p′|2 g(p′)h(p) =: 8π2MY ΛV |p′|2 MY,n g(p′)h(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (60) Analogously to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (52), the whole series for Y can be estimated to be |Yji(p′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon)| ≤ 8π2MY ΛV |p′|2 Ymax g(p′)h(p) =: 8π2MYmax ΛV |p′|2 g(p′)h(p), (61) where Ymax = ∞ � k=0 MY,n ≤ √ 2e Σ eeΣ2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (62) The remainder δnYmax, defined as δnYmax = ∞ � k=n+1 MY,n, (63) can be bounded similarly to δnD by an exponent with an arbitrary base: δnYmax ≤ e−MδY n, for n > ˜ MδY , (64) with some ˜ MδY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' As in the case of δnD, the estimated value of ˜ MδY ∼ (eΣ)2 becomes very large for Σ significantly larger than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' However, in the actual calculations, its numerical value is typically much more natural, see the discussion in the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The same comment applies also to the bound in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (62) for Ymax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The remainder δnY (p′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (64): |δnYji(p′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon)| = ����� ∞ � k=n Y [n] ji (p′, p) ����� ≤ 8π2MY ΛV |p′|2 δnYmax g(p′)h(p) =: 8π2NδnY ΛV |p′|2 g(p′)h(p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (65) The bounds for ¯Y (p′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) are obtained from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (61) and (65) by interchanging p ↔ p′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 11 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Bounds for the LO amplitude After these preparations, we are finally in the position to deduce the bounds for the on-shell LO amplitude, which can be represented as T0 = V0R = N0 D , N0 = V0D + V0Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (66) First, consider the quantity N0 defined explicitly as follows: (N0)ji(pon) = (V0)ji(pon, pon)D(pon) + � i′ � p′2dp′ (2π)3 (V0)ji′(pon, p′)Yi′i(p′, pon;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (67) Applying the bounds from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (20), (51) and (61) , we obtain |(N0)ji(pon)| ≤ MV0V0,max � MD + nPWMYmax ΛV � d|p| π g(p)h(p) � ≤ MV0V0,max � MD + MYmaxΣ MV0MG � =: MN0V0,max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (68) Now, we can analyze the bounds for the LO amplitude T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Since T0 is the ratio of N0 and D, it is important how the Fredholm determinant D is bounded from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' From the definition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (38), it follows that all terms D[n] should be in general of order O(Q0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' However, in a realistic situation, there might be certain cancellations among terms in the series, and the actual numerical value of D(pon) might turn out to be very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' This can happen when there is a shallow bound or quasibound state, which leads to an enhancement of the amplitude at threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Such a situation only takes place in the 1S0 of NN scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Therefore, in our analysis for higher partial waves with l ≥ 1, we regard the Fredholm determinant as being “natural”: |D(pon)| ≥ MD,min, (69) where MD,min is a constant of order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (68) and (69), we conclude that for l ≥ 1, the LO amplitude is bounded by |(T0)ji| ≤ MT0V0,max, (70) and satisfies the same power counting as V0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' is of order O(Q0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For the S-wave channels, we allow for the real part of D to be small, while still bounded from below at least at threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Moreover, we assume that the imaginary part of D, which is proportional to pon, is not a subject to additional cancellations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In particular, we exclude the situation when both N and D are equal to zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' the presence of a Castillejo-Dalitz-Dyson (CDD) pole [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' We combine these conditions into the following constraint: |D(pon)| ≥ MD,min � κ + pon ΛV � , (71) where κ > 0 is not necessarily of order one, but can be numerically small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The factor 1/ΛV in front of pon follows from the upper bound for the imaginary part of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The LO amplitude T0 is enhanced compared to V0, which can be written as |(T0)ji| ≤ MT0κ−1V0,max, (72) or |(T0)ji| ≤ MT0 ΛV pon V0,max, (73) depending on the value of the on-shell momentum pon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The latter bound is in fact a unitary limit for the LO amplitude up to a numerical factor of order one, which justifies the coefficient 1/ΛV in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (71), see the definition of V0,max in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Equation (73) means that the LO amplitude becomes effectively of order O(Q−1) in agreement with findings of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' To summarize, we have applied the Fredholm method to decompose the resolvent of the LS equation and derived the bounds for the Fredholm determinant D, the minor Y and the on-shell LO amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The bounds involve undetermined dimensionless constants of order one, which can be calculated for each particular situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 12 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' NEXT-TO-LEADING ORDER AMPLITUDE IN THE NON-PERTURBATIVE CASE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' P- AND HIGHER PARTIAL WAVES In this section we consider the on-shell (p = p′ = pon) NLO amplitude T2 for orbital angular momenta l ≥ 1 and derive the corresponding bounds in the non-perturbative regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' We represent the amplitude T2 using the Fredholm decomposition of the resolvent in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (36) as follows: T2 = ¯RV2R = V2 + T2,Y /D + T2, ¯Y /D + T2, ¯Y Y /D2 =: N2 D2 , (74) with T2,Y = V2Y , T2, ¯Y = ¯Y V2 , T2, ¯Y Y = ¯Y V2Y , (75) or more explicitly: T2,Y (p′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) = � p2 1dp1 (2π)3 V2(p′, p1)Y (p1, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) , T2, ¯Y (p′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) = � p′2 1 dp′ 1 (2π)3 ¯Y (p′, p′ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon)V2(p′ 1, p) , T2, ¯Y Y (p′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) = � p2 1dp1 (2π)3 p′2 1 dp′ 1 (2π)3 ¯Y (p′, p′ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon)V2(p′ 1, p1)Y (p1, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (76) First, consider T2,Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The bounds for V2 and Y in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (33) and (61) give |T2,Y (p′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon)| ≤ � |p1|2d|p1| (2π)3 |V2(p′, p1)||Y (p1, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon)| ≤ MV2,1nPW 8π2MYmax ΛV |p′|h(p) � |p1|d|p1| (2π)3 ˜flog(p′, p1)g(p1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (77) The functions g and h for P- and higher partial waves are given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (25), which results in the following inequality: |T2,Y (p′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon)| ≤ MV2,1nPW 8π2MYmax ΛV |p′||p| � d|p1| (2π)3 ˜flog(p′, p1)λlog(p1/Λ) = MV2,1nPW 8π2MYmax ΛV 8π2 mNΛV Λ2 b |p′||p| � d|p1| (2π)3 flog(p′, p1)λlog(p1/Λ) = MV2,1nPWMYmax ΛV 8π2 mNΛV Λ2 b |p′||p| �� 1 + θ(|p′| − Mπ) ln |p′| Mπ � Iλlog,1 + Iλlog,2 � , (78) where the typical integrals Iλlog,1 and Iλlog,2 are defined and estimated in Appendix F and we have used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Using those estimates, we obtain: |T2,Y (p′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon)| ≤ M2,Y 8π2 mNΛV Λ2 b |p′||p| Λ ΛV � 1 + θ(|p′| − Mπ) ln |p′| Mπ + ln Λ Mπ � , (79) which reduces to |T2,Y (pon)| ≤ M2,Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='on 8π2 mNΛV Λ2 b Λ ΛV p2 on ln Λ Mπ , (80) for the on-shell momenta p = p′ = pon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The bounds for T2, ¯Y are the same as for T2,Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Next, we analyze T2, ¯Y Y : ��T2, ¯Y Y (p′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) �� ≤ � |p1|2d|p1| (2π)3 ||p′ 1|2d|p′ 1| (2π)3 | ¯Y (p′, p′ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon)||V2(p′ 1, p1)||Y (p1, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon)| ≤ MV2,1n2 PW �8π2MYmax ΛV �2 h(p′)h(p) � |p1|d|p1| (2π)3 |p′ 1|d|p′ 1| (2π)3 ˜flog(p′, p1)g(p′ 1)g(p1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (81) 13 The integrals over p1 and p′ 1 factorize, giving rise to the same set of integrals as in T2,Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The analog of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (80) for T2, ¯Y Y in the on-shell kinematics is given by ��T2, ¯Y Y (pon) �� ≤ M2, ¯Y Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='on 8π2 mNΛV Λ2 b Λ2 Λ2 V p2 on ln Λ Mπ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (82) Combining the bounds for V2, T2,Y , T2, ¯Y and T2, ¯Y Y and setting Λ ∼ ΛV , we obtain |T2(pon)| ≤ ˜ M2 8π2 mNΛV Λ2 b p2 on ln Λ Mπ � 1 + D(pon)−1 + D(pon)−2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (83) Since we assume that for the P- and higher partial waves the Fredholm determinant is bounded from below by a constant of order one, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (69), equation (83) takes the form |T2(pon)| ≤ M2 8π2 mNΛV Λ2 b p2 on ln Λ Mπ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (84) Thus, the NLO amplitude is of order O(Q2) up to a factor ln Λ/Mπ, which agrees with the dimensional power counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' This result reproduces the one obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16] for the case of a perturbative LO interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Promoting a contact term to leading order In this subsection we consider separately the scenario with promoting leading P-wave contact terms to the LO potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' As already discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' II B, phenomenological arguments may require a promotion of contact interac- tions quadratic in momenta to the LO potential, even though they are formally of order O(Q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' A typical example is the 3P0 partial wave, where the promotion of the contact interaction to leading order is often considered as necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Below, we discuss the subtlety related to the freedom of choosing the renormalization condition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=', deciding what part of the considered contact interaction should be included into the LO potential and what part of it should be left in the NLO potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The LO partial wave contact interaction in the P-wave channel i is given by V (0) short,Λ,i(p ′, p) = Ci VCi,Λ(p ′, p), (85) where VCi,Λ(p ′, p) is the partial wave projection of the regulated contact term (see Appendix A) relevant for the considered channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The corresponding NLO contact interaction has the same structure: V (2) short,Λ,i(p ′, p) = C2,i VCi,Λ(p ′, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (86) In our estimates, we always assume that the LO low energy constants (LECs) are of natural size, Ci = MCi Λ2 b 8π2 mNΛV , (87) see Appendix of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16] (the factor of 4π corresponds to the partial-wave basis), so that the contact interactions quadratic in momenta are of order ∼ p2/Λ2 ∼ O(Q2) and are suppressed for small momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' As a consequence, the regulator corrections to the contact interactions quadratic in momenta are effects of order O(Q4) and can be neglected in the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' This is why we adopt the same regulator for V (0) short,Λ,i and V (2) short,Λ,i even though, in principle, one could employ a larger cutoff for the NLO terms or even use the unregulated potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Nevertheless, if the contact interactions quadratic in momenta are promoted to leading order, their contribution in the iterations of the LO potential at momenta p ∼ Λ is of the same order as those of the momentum-independent contact interactions and of the one-pion-exchange potential as long as we treat Λ ∼ Λb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The freedom to choose the renormalization scheme manifests itself schematically as follows: if we perform the transformation Ci → Ci + δCi, C2,i → C2,i − δCi, δCi ≪ Ci , (88) and expand the LO and NLO amplitudes in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (14) and (15) in δC, then the linear in δC terms cancel: δT0 ≈ −δT2 ≈ δC ¯RVCi,ΛR, (89) 14 where we have neglected higher order effects, such as the terms proportional simultaneously to δC and the NLO potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' As was shown in this section, there are no power counting breaking contributions in P-waves at NLO stemming from the iterations of the LO potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' This means that C2,i is the renormalized quantity, where we assume that the divergent contributions to the two-pion-exchange diagrams are subtracted within some scheme, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' as is done for our choice of the non-polynomial two-pion-exchange contribution, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (B2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Then, one obvious choice for the renormalization condition is C2,i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (90) However, at higher orders, power counting breaking terms will appear also in P-waves, and one will have to absorb them by performing renormalization of the same contact interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Therefore, to be consistent with our subtraction scheme for the S-waves, we impose the renormalization condition on Ci and C2,i by requiring that the NLO amplitude in channels with l = 1 vanishes at threshold faster than p2 on: (T2)11(pon)/p2 on ��� pon=0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (91) Instead of the threshold point pon = 0, one can also take another renormalization point below or above threshold within the applicability of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' A potential problem related to the above renormalization condition was discussed in great detail in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [31] when studying schemes with large or infinite cutoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' It arises near “exceptional” cutoff values for which the contribution of the contact interaction to the NLO amplitude is unnaturally small: � ¯RVCi,short,ΛR � (pon)/p2 on ��� pon=0 ≈ 0, (92) which, in turn, leads to an unnaturally large value of C2,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In such a case, the power counting is violated unless the zero of the function on the left-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (92) is factorizable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=', it appears at all energies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (92) can take place, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=', in the spin-triplet channels with attractive one-pion-exchange potential such as 3P0 if the adopted cutoff value is too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Then one starts to feel the singular nature of the one-pion-exchange potential, which is reflected in oscillations of the scattering wave function at short distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Note that this effect does not directly correspond to the appearance of spurious bound states, although the two issues are related to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [31], several particular cases were discussed when the condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (92) can be avoided or the corre- sponding zero is factorizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' However, we are interested in the general case, in which the practical solution of the problem would be to explicitly verify that the LO potential is chosen in such a way that the condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (92) is not fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In such a case, the NLO amplitude will satisfy the expected power counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In fact, for the regulators mentioned in the discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' VI and many other choices tested by us, if the cutoff value is of the order of the hard scale, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (92) is never fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' A simple indication that the cutoff of the LO potential is not “exceptional” is the naturalness of the renormalized NLO low energy constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' To summarize, we have shown that the P-wave NLO amplitudes formally satisfy the dimensional power counting in the non-perturbative regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' This holds also for the case when a contact interaction quadratic in momenta is promoted to LO if one makes sure that a certain condition on the LO potential is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 15 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' NON-PERTURBATIVE RENORMALIZATION OF THE AMPLITUDE AT NLO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' S-WAVES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In this section we consider the renormalization of the NLO amplitude in the non-perturbative regime for S-waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' As in the perturbative case considered in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16], subtractions have to be made in order to absorb contributions that violate power counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' We will start with generalizing the perturbative result of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16] and then analyze under which conditions a particular power counting can be established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' General formula Analogously to the situation discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' IV A, there is freedom to choose the momentum-independent part of the NLO potential ˆV2(p′, p) = V2(0, 0), (93) because it can be partly or completely absorbed by the LO potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In the perturbative case, the NLO amplitude corresponding to ˆV2 does not contain any power counting breaking contributions in contrast to the remaining part ˜T2 that is generated by ˜V2(p′, p) = V2(p′, p) − ˆV2(p′, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (94) In what follows, we will mostly consider the scheme with ˆV2(p′, p) = 0, which is well suited for compensating possible threshold enhancement of the LO amplitude due to non-perturbative effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Alternative schemes will be briefly discussed separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Therefore, when using the results of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16], we will assume ˜V2 = V2, ˜T2 = T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (95) First, we recall some notation from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For an operator X = Xl′l(p′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon), where l(l′) is the initial (final) orbital angular momentum, we define the subtraction operation T: T(X) = X00(0, 0, 0)Vct, (96) where the contact term is given by Vct = |χ⟩⟨χ|, ⟨p, lsj|χ⟩ = δl,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (97) (98) We assume that the counter term is unregulated or regulated with some Λct ≫ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Analogously, we introduce the subtraction operation Tmi,ni for subdiagrams (mi, ni) of the diagram (m, n) corresponding to T [m,n] 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' We follow the Bogoliubov-Parasiuk-Hepp-Zimmermann (BPHZ) subtraction scheme [32–34] and represent the renormalized ampli- tude via the forest formula: R(T [m,n] 2 ) = T [m,n] 2 + � Uk∈Fm,n � � (mi,ni)∈Uk −Tmi,ni � T [m,n] 2 , (99) where Fm,n represents the set of all forests, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='e, the set of all possible distinct sequences of nested subdiagrams (mi, ni): Uk = ((mk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='1, nk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='1), (mk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='2, nk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' ) , m ≥ mk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='i+1 ≥ mk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='i ≥ 0 , n ≥ nk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='i+1 ≥ nk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='i ≥ 0 , n + m > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (100) In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16], it was proved that each term in the expansion in V0 of the renormalized NLO amplitude satisfies the dimensional power counting and is bounded by ���R(T [m,n] 2 )(pon) ��� ≤ 8π2MT2 mNΛV Σm+n 2,0 p2 on Λ2 b ln Λ Mπ , (101) where Σ2,0 = 2Mmax Λ ΛV (102) 16 is a quantity of order one (Σ2,0 ≥ 1 in the non-perturbative case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' To resum the series R(T2)(pon) = ∞ � m,n=0 R(T [m,n] 2 )(pon), (103) we perform some rearrangement of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (99), as explained below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' It is convenient to introduce the following notation: | ¯ψ⟩ = ¯R|χ⟩, ⟨ψ| = ⟨χ|R, ψl(p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) = ⟨ψ|p, lsj⟩ = ⟨p, lsj| ¯ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (104) For on-shell momenta p = pon, the explicit form of ψl reads ψl(pon) := ψl(pon;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) = δl,0 + � p2dp (2π)3 G(p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon)(T0)0,l(p, pon;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon), (105) and it coincides with the scattering wave function at the origin (r = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Now, consider the sum of all unrenormalized diagrams: T2 = ¯RV2R, (106) and perform first all single overall subtractions: δT (1),overall 2 = −T(T2) = −(T2)00(0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 0)|χ⟩⟨χ|, (107) where the superscript (1) denotes the number of subtractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' If we add all possible rescatterings with the LO potential, we will obtain all terms with single subtractions in subdiagrams: δT (1) 2 = ¯RδT (1),overall 2 R = −(T2)00(0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 0) ¯R|χ⟩⟨χ|R = −(T2)00(0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 0)⟩| ¯ψ⟩⟨ψ| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (108) Analogously, the sum of all double nested subtractions (one of which is an overall subtraction) is given by δT (2),overall 2 = −T � δT (1) 2 − δT (1),overall 2 � = (T2)00(0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 0) � ψ0(0)2 − 1 � |χ⟩⟨χ| = − � ψ0(0)2 − 1 � δT (1),overall 2 , (109) where the constant term δT (1) 2 was already subtracted in the previous step and should be excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' All terms with double nested subtractions in subdiagrams are obtained by adding the rescattering contributions: δT (2) 2 = ¯RδT (2),overall 2 R = − � ψ0(0)2 − 1 � δT (1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (110) Continuing with further multiple nested subtractions, we obtain recursion relations: δT (n+1),overall 2 = −T � δT (n) 2 − δT (n),overall 2 � = − � ψ0(0)2 − 1 � δT (n),overall 2 , (111) and δT (n+1) 2 = − � ψ0(0)2 − 1 � δT (n) 2 , (112) where the superscripts (n) and (n + 1) denote the number of nested subtractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The terms T (n) 2 can be summed up to δT2 = ∞ � n=1 δT (n) 2 = δT (1) 2 ∞ � n=0 � 1 − ψ0(0)2�n = δT (1) 2 1 ψ0(0)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (113) 17 Finally, R(T2) = T2 + δT2 = T2 − (T2)00(0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 0) ψ0(0)2 | ¯ψ⟩⟨ψ| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (114) Taking the on-shell matrix elements of R(T2), we obtain: R(T2)l′l(pon) = (T2)l′l(pon) + δCψl′(pon)ψl(pon), (115) with the counter term constant δC = −(T2)00(0) ψ0(0)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (116) Equation (115) can also be obtained directly without referring to the perturbative result from the renormalization condition: R(T2)l′l(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (117) Therefore, the perturbative and non-perturbative results match in the regime where both are applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Similarly to the analysis of higher partial waves in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' IV, we use the Fredholm decomposition of the resolvent of the LS equation and introduce the quantities N2 and ν, (T2)l′l(pon) = (N2)l′l(pon) D(pon)2 , ψl(pon) = νl(pon) D(pon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (118) The counter term constant can be expressed as δC = −(N2)00(0) ν0(0)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (119) Then, the renormalized amplitude R(T2) reads R(T2)l′l(pon) = 1 D(pon)2 � (N2)l′l(pon) + δC νl′(pon)νl(pon) � = R(N2)l′l(pon) D(pon)2 = R( ˜N2)l′l(pon) D(pon)2 ν0(0)2 , (120) where, for convenience, the following quantities have been introduced: R(N2)l′l(pon) = (N2)l′l(pon) + δC νl′(pon)νl(pon), (121) R( ˜N2)l′l(pon) = (N2)l′l(pon)ν0(0)2 − (N2)00(0)νl′(pon)νl(pon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (122) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Power counting with the naturalness condition for ν0(0) In this subsection we analyze the expression for the renormalized NLO amplitude R(T2) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (120) and determine what power counting it satisfies under which conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Considering different constraints on various quantities entering R(T2), we can understand to what extent the renormalizability of the amplitude depends on details of the short-range dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' We assume that the Fredholm determinant D(pon) satisfies the bound in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (71), which includes also the case of a shallow (quasi-) bound state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For the function D(pon)2, we can write |D(pon)2| ≥ M2 D,minκ2, (123) or, if κ is very small, |D(pon)2| ≥ M2 D,min p2 on Λ2 V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (124) 18 We will also need the upper bound for the quantity νl(pon), see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (E21): νl(pon) ≤ Mν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (125) First, we consider the “natural” case when the quantity ν0(0) is bounded not only from above as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (125), but also from below by some constant of order one: ν0(0) ≥ Mν,min, (126) which also implies the natural value of the counter term constant δC, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (119), similarly to the condition of the absence of “exceptional” cutoffs in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' IV A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Then as follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (120), to analyze the power counting that the renormalized amplitude R(T2) satisfies, it is sufficient to find bounds for R( ˜N2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' As we show in Appendix E, the quantity R( ˜N2) can be expanded into a convergent series in terms of V0: R( ˜N2)(pon) = ∞ � m,n=0 � R( ˜N2)(pon) �[m,n] = nmax � m,n=0 � R( ˜N2)(pon) �[m,n] + δnmax � R( ˜N2)(pon) � =: S ˜ N2,nmax(pon) + δnmax � R( ˜N2)(pon) � , (127) and the remainder δn � R( ˜N2)(pon) � decreases faster than exponential with any base Mδ ˜ N2 starting with some n = ˜ Mδ ˜ N2 (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (E22)): |δn[R( ˜N2)]| ≤ 8π2 mNΛV N ˜ N2e−Mδ ˜ N2n, for n > ˜ Mδ ˜ N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (128) The prefactor N ˜ N2 is given by N ˜ N2 = Λ2 Λ2 b ln Λ Mπ (129) in the case of the “standard” regulators of the LO potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For the “mild” regulator, it depends also on the regulator of the NLO potential ΛNLO: N ˜ N2 = ΛΛNLO Λ2 b ln ΛNLO Λ ln ΛNLO Mπ , (130) and, in contrast to the perturbative regime, the regulator ΛNLO cannot be set to infinity (in general) but can be chosen ΛNLO ≫ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Note that we do not consider the choice ΛNLO ∼ Λ for the “mild” LO regulator because in such a case, we would simply reproduce the variant with the “standard” regulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The appearance of ΛNLO in the expression for N ˜ N2 is an indication of a potentially stronger cutoff dependence of the NLO amplitude in the non-perturbative regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The general conservative estimate for ˜ Mδ ˜ N2 yields ˜ Mδ ˜ N2 ≳ (eΣ)2, which is rather large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In realistic calculations, it turns out to be much smaller, see the discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' III A and the numerical results in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' On the other hand, expanding Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (120) in V0 gives � R( ˜N2)(pon) �[m,n] = m � m1=0 m−m1 � m2=0 n � n1=0 n−n1 � n2=0 D[m−m1−m2](pon)D[n−n1−n2](pon) × ν0(0)[m2]ν0(0)[n2]R(T [m1,n1] 2 )(pon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (131) Using the perturbative bounds on R(T [m,n] 2 ) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (101) and Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (51), and (125), we obtain � R( ˜N2)(pon) �[m,n] ≤ M2 DM2 ν m � m1=0 n � n1=0 ���R(T [m1,n1] 2 )(pon) ��� ≤ 8π2MT2M2 DM2 ν mNΛV p2 on Λ2 b ln Λ Mπ m � m1=0 n � n1=0 Σm1+n1 2,0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (132) 19 Performing the summation up to n = nmax, we obtain ���S ˜ N2,nmax(pon) ��� ≤ 8π2MT2M2 D[n],max mNΛV p2 on Λ2 b ln Λ Mπ nmax � m,n=0 m � m1=0 n � n1=0 Σm1+n1 2,0 ≤ 8π2MN2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='2 mNΛV p2 on Λ2 b ln Λ Mπ n4 maxΣ2nmax 2,0 =: 8π2MS mNΛV p2 on Λ2 b Φlog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (133) Given that the remainder δn[R( ˜N2)] can be made arbitrarily small by choosing a sufficiently large nmax, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' |δn[R( ˜N2)]| ≤ 8π2 mNΛV M 2 πκ2 Λ2 b , (134) whereas the sum in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (133) has the bound similar to the one for the perturbative amplitude up to numerical constants of order one and possible factors logarithmic in Λ, Φlog, we can conclude that R( ˜N2) is bounded as: ���R( ˜N2)(pon) ��� ≤ 8π2M ˜ N2 mNΛV �p2 on Λ2 b Φlog + M 2 π Λ2 b κ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (135) Whether this picture is indeed realized for the realistic NN interaction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=', whether M ˜ N2 is really (and not only formally) of the order of one, is straightforward to verify by explicit numerical checks of the series for R( ˜N2) as we do partly in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For completeness, we show below that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (135) holds formally in the chiral limit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' for the expansion parameter Q ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' What we have to prove is that there exists such a value of nmax that the remainder δnmax[R( ˜N2)] satisfies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (134), and, at the same time, the prefactor χ = n4 maxΣ2nmax 2,0 (136) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (133) does not contain inverse powers of Q and, therefore, does not destroy the power counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The choice nmax ≥ max(k0, ¯k0), (137) with k0 = ˜ Mδ ˜ N2, ¯k0 = − 1 Mδ ˜ N2 ln M 2 πκ2 Λ2 bN ˜ N2 , (138) guarantees that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (134) holds, as follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (128).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Note that the inequality ¯k0 > k0 holds only for extremely small Q = Mπ/Λb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' However, in the actual calculations, this can happen also for physical values of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The factor χ is then given by χ = ˜ M4 δ ˜ N2Σ 2 ˜ Mδ ˜ N2 2,0 (139) if nmax = k0, and by χ = 1 M4 δ ˜ N2 � ln M 2 πκ2 Λ2 bN ˜ N2 �4 � M 2 πκ2 Λ2 bN ˜ N2 �−2 ln Σ2,0 Mδ ˜ N2 , (140) if nmax = ¯k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In the latter case, if Mδ ˜ N2 is chosen to be Mδ ˜ N2 ≫ ln Σ2,0, the factor � M 2 πκ2 Λ2 bN ˜ N2 �−2 ln Σ2,0 Mδ ˜ N2 can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Thus, we conclude that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (135) holds with Φlog = � � � Mlog ln Λ Mπ , nmax = k0 Mlog ln Λ Mπ � ln M 2 πκ2 Λ2 bN ˜ N2 �4 , nmax = ¯k0 (141) 20 Now we come back to the expression for the renormalized NLO amplitude in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (120).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For small on-shell momenta pon, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=', when ���S ˜ N2,nmax(pon) ��� ≤ |δn[R( ˜N2)]|, (142) Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (123), (126) and (134) give: |R(T2)l′l(pon)| ≤ 8π2MT2,low mNΛV M 2 π Λ2 b , (143) which means that in this energy region, R(T2) is of order O(Q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' As the on-shell momentum increases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=', ���S ˜ N2,nmax(pon) ��� ≥ |δn[R( ˜N2)]|, (144) we should use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (133) instead of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (134) to obtain |R(T2)l′l(pon)| ≤ 8π2MT2,high mNΛV p2 on Λ2 b Φlog κ2 , (145) which is enhanced compared to O(Q2) by a factor 1/κ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In the worst case of the unitary limit, we obtain from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (124): |R(T2)l′l(pon)| ≤ 8π2MT2,high mNΛV Λ2 V Λ2 b Φlog, (146) which corresponds effectively to R(T2) ∼ O(Q0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' This is still one order higher than the LO amplitude O(Q−1), see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (73), but the convergence rate is rather low in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' A natural way to reduce the effect of the numerical enhancement of the LO amplitude and to improve convergence is to promote some part of the NLO potential to leading order, which will make the numerical constant MT2,high smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The simplest recipe would be to promote the contact interactions quadratic in momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' As already mentioned, this approach is suggested, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=', for the 1S0 partial wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' We will discuss this possibility in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' V D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Local LO potential in a spin-singlet channel and analogous cases Above, we considered the general case of the LO potential under an additional assumption on its short-range part formulated in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (126) in terms of the naturalness of ν0(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' It is instructive to consider one particular case, when the LO potential in a spin-singlet channel is fully local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Then, this condition is satisfied automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Moreover, for a local LO potential, the following identity holds: ν0(pon) ≡ 1, (147) which follows from the fact that the scattering wave function at the origin ψpon coincides with the inverse of the Jost function f(pon) and the inverse of the Fredholm determinant [25]: ψ(pon) = f(pon)−1 = D(pon)−1, (148) and the definition (118).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Therefore, we have (see the definitions in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (122) and (121)) R( ˜N2)(pon) = R(N2)(pon) = ∆N2(pon) = N2(pon) − N2(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (149) The whole discussion in the previous subsection applies for the case of a local LO potential, except the absence of the additional condition (126).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In the general case, when the constraint in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (126) is not satisfied, we still can have a situation similar to the local single-channel potential if we assume that the series for R(N2) (not for R( ˜N2)) converges and the bound analogous to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (135) holds: |R(N2)(pon)| ≤ 8π2M ˜ N2 mNΛV �p2 on Λ2 b Φlog + M 2 π Λ2 b κ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (150) This is possible if the smallness of ν0(0) in the denominator of R(N2) is compensated by a corresponding small factor in the numerator, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (121).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Whether this indeed takes place can be verified numerically in any particular case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (150), we can deduce the same bounds for the renormalized NLO amplitude as in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (143), (145) and (146).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' We made this comment to emphasize that the naturalness constraint on ν0(0) is not necessary to guarantee renor- malizability of the NLO amplitude, but is the most simple one from the practical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 21 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Promoting a momentum dependent contact term to leading order In this subsection we analyze the situation when it is necessary to promote the momentum dependent S-wave contact term to leading order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For definiteness, we consider the 1S0 partial wave, where such a promotion has been shown to significantly improve the convergence of the chiral EFT expansion, see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [18, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Since this is a spin-singlet channel, we omit the l, l′ indices in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' We also omit all channel indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The whole analysis in the preceding subsections remains valid in this case, except that similarly to the promotion of the subleading term in the P-waves considered in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' IV A, there is freedom choosing what part of such a contact term should be included in LO potential V0 and what part remains in the NLO potential V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' We rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (121) by explicitly separating the part with the contact term quadratic in momenta: R(N2)(pon) = N2(pon) + δCν(pon)2 =: ∆N2(pon) + δCν(pon)2 + C2NC2(pon), (151) with NC2(pon) = [ ¯RVCR](pon)D(pon)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (152) The potential VC is the contact interaction quadratic in momenta that projects onto the 1S0 partial wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' This potential can remain regulated because the regulator corrections to it are of higher order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Following our subtraction scheme at pon = 0, we introduce two renormalization conditions to fix δC and C2: R(N2)(0) = 0, d2R(N2)(pon) dp2on ���� pon=0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (153) Note that N2 is an analytic function of p2 on at pon = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Of course, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (153) can be also formulated in terms of the amplitudes: R(T2)(0) = 0, d2R(T2)(pon) dp2on ���� pon=0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (154) Analogously to the situation in P-waves, the above renormalization conditions can lead to a problem for “exceptional” cutoffs when Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (153) become inconsistent, which happens not only when ν(0) = 0 but also when the following equation is satisfied [31]: �d2NC2(pon) dp2on − 2NC2(pon)ν(pon)d2ν(pon) dp2on ����� pon=0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (155) As in the case of the P-waves, an indirect indication that the cutoff is not close to an “exceptional” value is the naturalness of the NLO LECs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In our numerical calculation in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' VI, we found no “exceptional” cutoffs for the cutoff values of the order or below the hard scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Other subtraction schemes In all analyses of the non-perturbative regime, we have always adopted the prescription to perform subtractions at threshold, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (117).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In this subsection we briefly discuss other possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Choosing different subtraction points, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=', the deuteron pole position for the 3S1 − 3D1 channel, is equivalent to setting, in contrast to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (93), ˆV2 ̸= 0: ˆV2(p′, p) = ˆκ2 8π2 mNΛV M 2 π Λ2 b , (156) where ˆκ is a constant of order one, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Since this potential is just an S-wave contact term, the corresponding NLO amplitude is given by ( ˆT2)l′l(pon) = ˆκ2 8π2 mNΛV M 2 π Λ2 b ψl′(pon)ψl(pon) = ˆκ2 8π2 mNΛV M 2 π Λ2 b νl′(pon)νl(pon) D(pon)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (157) 22 From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (123) and (125), we obtain the following bound: |( ˆT2)l′l(pon)| ≤ 8π2 mNΛV M2 ν M2 D,min M 2 π Λ2 b ˆκ2 κ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (158) For the perturbative case considered in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16] and for the case without an enhancement of the LO amplitude, the amplitude ˆT2 satisfies the dimensional power counting: T2 ∼ O(Q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' However, in the situation when the LO amplitude is enhanced, the additional factor ˆκ2/κ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (158) relative to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (72) spoils convergence even at threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' We will have the worst situation in the unitary limit with κ ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Thus, we conclude that for a reasonable convergence in the case of an enhanced LO amplitude, one should choose a subtraction scheme not much different from ours, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=', such that ˆκ/κ ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' To summarize, we have shown that renormalization of the NLO amplitude for the S-waves can be done explicitly also in the non-perturbative regime by analyzing the Fredholm decomposition of the amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In contrast to the perturbative case discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16], additional constraints on the LO potential have to be fulfilled to ensure renormalizability and convergence of the chiral expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Then, the power counting works also in the situation when the LO amplitude is enhanced at threshold, although to make the scheme more efficient, it might be necessary to promote certain contributions to leading order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' NUMERICAL RESULTS In this section we illustrate our theoretical findings by explicit numerical calculation of the NLO NN amplitude in the three channels where the LO interaction should be treated non-perturbatively: 3P0, 3S1 − 3D1 and 1S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The results for other channels were presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' We adopt the same values for the numerical constants as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16]: the pion decay constant Fπ = 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='1 MeV, the isospin average nucleon and pion masses mN = 938.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='9 MeV, Mπ = 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='04 MeV and the effective nucleon axial coupling constant gA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The calculations have been performed using Mathematica [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For the regularization of the LO and NLO potentials, we adopt the scheme similar to the one used in realistic calculation in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [9] at fifth order in the chiral expansion, which allows us to have a direct interpretation of the numerical values of the cutoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In particular, we use the local Gaussian regulator for the one-pion-exchange potential and the non-local Gaussian regulator for all contact interactions with the same cutoff Λ, see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For the sake of simplicity, we also employ the local Gaussian regulator in the form of the overall factor FΛNLO,exp(q) for the two-pion-exchange potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' As in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16], the cutoff value ΛNLO is set to the hard scale ΛNLO = 600 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' This choice for the chiral expansion breakdown scale is consistent with the recent studies in the few-nucleon sector [37–40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The momentum-independent contact interactions at NLO are included without a regulator in accordance with our power counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The contact interactions quadratic in momenta are regulated with the same cutoff ΛNLO at LO and at NLO in contrast to our choice in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16], where, for simplicity, we left the corresponding NLO contact terms unregulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Both options are legitimate since the regulator corrections to the contact interactions quadratic in momenta is an effect of a higher order, O(Q4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' By the same reason, the regulator corrections to the LO contact interactions quadratic in momenta are not taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The cutoff values for the one-pion-exchange potential and for the momentum-independent LO contact interactions are varied in the regions below and above Λ = 450 MeV, which was found to be the optimal cutoff value in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The lower region corresponds to extremely soft cutoffs, where explicit regulator corrections to the LO potential are likely to be important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The upper region contains relatively hard (of the order of the hard scale) cutoffs as well as cutoffs above Λb, for which we expect slower convergence in terms of the Fredholm expansion and, therefore, potential problems with interpretation within our renormalization scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The free parameters are determined by a fit to the empirical phase shifts from the Nijmegen partial wave analysis [41] up to Elab = 150 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The phase shifts and the mixing parameters are calculated through the following unitarization procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' First, the non-unitary NLO T-matrix is transformed to the S-matrix via Sl′l(pon) = 1 − imNpon 8π2 Tl′l(pon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (159) The diagonal phase shifts in the Stapp parametrization of the S-matrix [42] are determined as (modulo π) δll = 1 2 arg(Sll), (160) whereas the mixing parameter ϵl+1 is obtained from the off-diagonal element of the S-matrix: Sl+2,l = i sin(2ϵl+1) exp(iδl + iδl+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (161) 23 The dependence of the results on a particular unitarization scheme is a higher-order effect, provided the chiral expansion for the amplitude is convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The numerical analysis we perform does not aim at achieving a perfect description of the data as we work only at next-to-leading order in the chiral expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Rather, we are interested in the convergence and renormalization issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In particular, we make sure that for the cutoff values we employ, no spurious bound states appear and no “exceptional” cutoffs discussed in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' IV and V lie within this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The latter fact manifests itself in the natural values of the fitted next-to-leading-order LECs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The natural values of the NLO LECs are also an indication of the “naturalness” of the quantity ν0(0), which is the simplest condition for the renormalizability of the S-wave NLO amplitudes, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' V B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The natural size is roughly given by 8π2 mNΛb , (162) for the LECs accompanying momentum-independent contact terms and by 8π2 mNΛ3 b , (163) for the LECs of contact terms quadratic in momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Obviously, naturalness is not a mathematically strict criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' However a sign of potential problems would be a rapid growth with cutoff of one or several LECs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Understanding the power counting for the renormalized amplitudes in terms of the convergence of the Fredholm expansion is demonstrated by looking at the convergence of the Fredholm determinant expanded in terms of the LO potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Convergence of other elements of the Fredholm formulas for the LO and the NLO amplitudes can be analyzed in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Their convergence rates are typically comparable with the one for the Fredholm determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' An absolute value of Fredholm determinant much larger than 1 is also a problem for our interpretation of the power counting, especially for the channels with the enhanced LO amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In such a case, the numerators in the Fredholm formulas N0, N2 will also be very large, contradicting the power counting that we suggest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' On the contrary, we expect the absolute value of the Fredholm determinant for those channels to be smaller than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 3P0 channel We begin our discussion with the 3P0 partial wave and first follow the dimensional power counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' That means that at leading order, we include only the one-pion-exchange potential and no further terms are promoted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' At next- to-leading order, there is one free parameter C2,3P0 that determines the strength of the NLO contact interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The results for the LO and NLO calculations are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In contrast to other plots in this section, we restrict ourselves to the values of the cutoff Λ ≤ 600 MeV because for larger cutoffs, the calculated phase shifts deviate too strongly from the data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For soft cutoffs values below Λ = 450 MeV, the convergence of the chiral EFT expansion and the description of the data are reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Moreover for such cutoffs, the LO amplitude can be regarded as perturbative, in the sense that the series in V0 converges very rapidly, and already a single iteration of the LO potential provides an accuracy of one percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Therefore, the analysis of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16] can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' One can also see that the band for next-to-leading order corresponding to the variation of the cutoff gets considerably narrower if the regulator correction to the one- pion-exchange potential is taken into account explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Further discussion of the fully perturbative approach in the 3P0 channel can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' As one increases the cutoff value, the convergence of expansion of the amplitude in powers of V0 becomes much slower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' This is not problem for our formalism as we formulated the power counting in the non-perturbative case in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' However, as one can see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 1, the disagreement with the data gets more severe and the convergence of the chiral EFT expansion deteriorates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In fact, such a strong deviation of the LO phase shifts from the data leads to a strong violation of unitarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Another indication of the inefficiency of the resulting EFT expansion is a rather small value of the Fredholm determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' At threshold, it equals D ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='4 for Λ = 600 MeV compared to D ∼ 1 for Λ = 300 − 450 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Large contributions from higher orders makes it more efficient to promote the NLO contact interaction to leading order, see also Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [21] and [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In fact, the case of very soft cutoffs considered above, which shows a reasonable convergence of the chiral expansion, can also be viewed as a modification of the short-range part of the LO potential analogous to promotion of a contact interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Note that our motivation for promoting the NLO contact term is not the requirement of the existence of an infinite cutoff limit as advocated, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=', in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [20], but rather a large strength of the LO one-pion-exchange potential in this channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Specifically, we demand that the difference between 24 0 50 100 150 200 250 Elab [MeV] 0 10 20 30 Phase Shift [deg] without δΛV (0) 0 50 100 150 200 250 Elab [MeV] with δΛV (0) 0 50 100 150 200 250 Elab [MeV] without δΛV (0) 0 50 100 150 200 250 Elab [MeV] with δΛV (0) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The results of the leading-order (blue dashed lines) and next-to-leading-order (red solid lines) calculations for the 3P0 partial wave without promoting the contact interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The bands indicate the variation of the one-pion-exchange cutoff within the range Λ1π ∈ (300, 450) MeV for two left plots and within the range Λ1π ∈ (450, 600) MeV for two right plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The second and fourth plots correspond to the NLO potential with the regulator correction δΛV (0), while the results in the first and third plots are obtained without this term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The empirical phase shifts shown by black solid dots are from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The plots were created using Matplotlib [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 0 50 100 150 200 250 Elab [MeV] −10 0 10 20 Phase Shift [deg] without δΛV (0) 0 50 100 150 200 250 Elab [MeV] with δΛV (0) 0 50 100 150 200 250 Elab [MeV] without δΛV (0) 0 50 100 150 200 250 Elab [MeV] with δΛV (0) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The results of the leading-order (blue dashed lines) and next-to-leading-order (red solid lines) calculations for the 3P0 partial wave with the contact term promoted to leading order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The bands indicate the variation of the one-pion-exchange cutoff within the range Λ1π ∈ (300, 450) MeV for two left plots and within the range Λ1π ∈ (450, 800) MeV for two right plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The second and fourth plots correspond to the NLO potential with the regulator correction δΛV (0), while the results in the first and third plots are obtained without this term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The empirical phase shifts shown by black solid dots are from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' the LO results and empirical values of the phase shifts can be corrected by a perturbative inclusion of higher-order interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In the scheme with a contact term at LO, there is also one free parameter to be determined from the fit, namely C3P0, whereas the NLO constant C2,3P0 is fixed by the renormalization condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (91).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The corresponding results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' As one can see, the convergence pattern when going from LO to NLO becomes much better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Taking into account the regulator correction to the one-pion-exchange potential δΛV (0) explicitly leads to narrower cutoff-variation bands at NLO, especially for soft cutoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The expansion of the Fredholm determinant in powers of V0 converges rather rapidly for the cutoffs Λ ≤ 600 MeV: at order (V0)3, a one-percent accuracy is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For Λ ∼ 800 MeV, the same accuracy requires expansion up to order (V4)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The absolute value of the Fredholm determinant varies within the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='7 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='3 increasing for higher values of the cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The numerical values of the constant C2,3P0 in the units of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (163) is reasonably natural for the choice of the hard scale Λb = 600 MeV at least for lower Λ values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Specifically, C2,3P0 ∼ 2 for Λ ∼ 450 MeV but increases to C2,3P0 ∼ 30 for Λ ∼ 800 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Combining the above results, we conclude that for the cutoffs below or of the order of the hard scale, the renor- malization of the NLO amplitude can be understood within the approach developed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For higher values of the cutoff, the renormalizability of the theory becomes questionable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 25 Elab [MeV] 0 50 100 150 3S1 phase shift [deg] without δΛV (0) Elab [MeV] with δΛV (0) Elab [MeV] without δΛV (0) Elab [MeV] with δΛV (0) Elab [MeV] −20 −10 0 3D1 phase shift [deg] Elab [MeV] Elab [MeV] Elab [MeV] 0 50 100 150 200 Elab [MeV] −10 −5 0 5 Mixing parameter ϵ1 [deg] 0 50 100 150 200 Elab [MeV] 0 50 100 150 200 Elab [MeV] 0 50 100 150 200 Elab [MeV] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The results of the leading-order (blue dashed lines) and next-to-leading-order (red solid lines) calculations for the 3S1 − 3D1 channels with the contact term promoted to leading order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The bands indicate the variation of the cutoff of the LO potential within the range Λ ∈ (300, 450) MeV for two left columns and within the range Λ ∈ (450, 800) MeV for two right columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The second and fourth columns correspond to the NLO potential with the regulator correction δΛV (0), while the results in the first and third columns are obtained without this term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The data are as in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 3S1 − 3D1 channel Next, we consider the system of the coupled 3S1−3D1 partial waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The LO potential is obviously non-perturbative due to the presence of the shallow deuteron bound state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The enhancement of the LO amplitude at threshold is not as strong as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=', in the 1S0 channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Therefore, we assume that within the renormalization scheme specified in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (117), the dimensional power counting should work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' That means that the LO potential contains only the one-pion-exchange and the momentum-independent contact term contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' There are three parameters to be determined from the fit: the LO constant C3S1, the NLO constant at the diagonal contact term quadratic in momenta, C2,3S1,p2, and the NLO constant accompanying the off-diagonal contact term C2,ϵ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The NLO momentum-independent contact term with the constant C2,3S1 is fixed from the renormalization condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (117).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The above mentioned three parameters are determined by fitting the phase shifts in the diagonal 3S1 channel and the mixing parameter ϵ1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=', the channels with contact terms in the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The 3D1 phase shift comes out as a prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The results of the fit for various cutoffs are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In general, we observe a reasonable convergence of the chiral expansion except for the ϵ1 channel where the LO as well as the full contributions are rather small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' As expected for soft cutoffs Λ ≤ 450 MeV, taking into account the explicit regulator corrections δΛV (0) for the one- 26 0 50 100 150 200 250 Elab [MeV] 0 20 40 60 Phase Shift [deg] without δΛV (0) 0 50 100 150 200 250 Elab [MeV] with δΛV (0) 0 50 100 150 200 250 Elab [MeV] without δΛV (0) 0 50 100 150 200 250 Elab [MeV] with δΛV (0) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The results of the leading-order (blue dashed lines) and next-to-leading-order (red solid lines) calculations for the 1S0 partial wave without promoting the contact interaction quadratic in momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The bands indicate the variation of the cutoff of the LO potential within the range Λ ∈ (300, 450) MeV for two left plots and within the range Λ ∈ (450, 800) MeV for two right plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The second and fourth plots correspond to the NLO potential with the regulator correction δΛV (0), while the results in the first and third plots are obtained without this term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The data are as in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pion-exchange potential and the leading contact term significantly reduces the cutoff dependence at next-to-leading order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Given the relatively large number of free parameters and possible fine-tuning, it is necessary to explicitly verify the renormalizability criteria specified above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' First, we check the naturalness of the NLO LECs in the units specified in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (162) and (163).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The absolute values of the constants C2,3S1,p2 and C2,ϵ1 do not exceed 12 for all considered values of the cutoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The maximal absolute value of the constant C3S1,p2 is about 6 for Λ ≤ 600 MeV, but it starts rising very fast and reaches the value of C3S1,p2 ∼ 20 for Λ = 800 MeV (and continues rising rapidly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The Fredholm determinant converges with a one-percent accuracy at orders (V0)3 − (V0)5 for Λ ≤ 600 MeV and at order (V0)6 for Λ = 800 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The absolute value of the Fredholm determinant at threshold (at Elab = 250 MeV) varies in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='6 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='8 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='8 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='6) for Λ ≤ 600 MeV and is as large as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='6 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='5 ) for Λ = 800 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Summarizing the above observations, our numerical results confirm the renormalizability of the NLO amplitude in the 3S1 − 3D1 channels for the cutoffs below or of the order of the hard scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For higher values of the cutoffs, the renormalizability in the sense discussed in the present paper is not guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 1S0 channel Finally, we discuss the 1S0 partial wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The enhancement of the LO amplitude due to the extremely shallow quasibound state is very strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Nevertheless, we start with trying to adopt the dimensional power counting and do not promote any additional contact interaction to leading order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Therefore, the LO potential consists of the one-pion-exchange contribution and the leading contact term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Two parameters are determined from the fit: the LO constant C1S0 and the NLO constant C2,1S0,p2 corresponding to the contact term quadratic in momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The NLO constant C2,1S0 is fixed from the renormalization condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (117).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' As in the case of the 3P0 partial wave, the convergence of the chiral expansion is acceptable only for small values of the cutoff Λ ≤ 450 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For larger values of the cutoffs, the LO contribution is too large compared to the data, which leads to a strong violation of unitarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The regulator corrections to the one-pion-exchange potential and the leading contact term practically do not affect the size of the bands corresponding to the variations of the cutoff, which is also a sign of a slow convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' As the cutoff increases, the Fredholm determinant at threshold changes from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='7 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Therefore, the slow convergence of the chiral expansion for the NLO amplitude is expected from our analysis in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Nevertheless, the series for the Fredholm determinant converges rapidly: the one-percent accuracy is obtained at order (V0)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The naturalness of the NLO LECs in the units of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (162) and (163) is also reasonably fulfilled: the absolute value of the constant C2,1S0 does not exceed 2, and the absolute value of the constant C2,1S0,p2 is below 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' A large deviation of the LO results from the data is a motivation for promoting the subleading contact interaction to leading order (as the simplest solution), see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' As we argued in the discussion of the 3P0 partial wave, adopting soft values of the cutoff Λ ≤ 450 MeV in the scheme with one contact term at leading order is a sizable modification of the short-range part of the LO potential and is, to some extent, equivalent to the promotion of an additional contact term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Now, we consider the scheme with the contact interaction quadratic in momenta being promoted to the LO potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 27 0 50 100 150 200 250 Elab [MeV] 0 20 40 60 Phase Shift [deg] without δΛV (0) 0 50 100 150 200 250 Elab [MeV] with δΛV (0) 0 50 100 150 200 250 Elab [MeV] without δΛV (0) 0 50 100 150 200 250 Elab [MeV] with δΛV (0) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The results of the leading-order (blue dashed lines) and next-to-leading-order (red solid lines) calculations for the 1S0 partial wave with the contact interaction quadratic in momentum promoted to leading order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The bands indicate the variation of the cutoff of the LO potential within the range Λ ∈ (300, 450) MeV for two left plots and within the range Λ ∈ (450, 800) MeV for two right plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The second and fourth plots correspond to the NLO potential with the regulator correction δΛV (0), while the results in the first and third plots are obtained without this term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The data are as in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' There are still two parameters to fit: C1S0 and C1S0,p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The constants C2,1S0 and C2,1S0,p2 are fixed from the renormalization conditions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (154).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The results for the scheme with two contact terms in the LO potential are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For higher Λ values, the convergence pattern for the EFT expansion in this scheme is significantly better than in the scheme without promotion of the momentum-dependent contact term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The cutoff dependence is weak for the cutoff values Λ ≥ 450 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For soft cutoffs, it may seem that explicit regulator corrections makes the cutoff dependence stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' However, this is probably accidental because, as one can see, the cutoff dependence for the case without regulator corrections is nonlinear and varies nontrivially with momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' This is caused by various cancellations due to the fine-tuning of two contact terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The absolute value of the Fredholm determinant at threshold is D ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='1 for all considered cutoffs, which is in agreement with our expectations for the strongly enhanced LO amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The expansion of the Fredholm determinant in powers of the LO potential approaches an accuracy of one percent at order (V0)3 for the cutoffs below or equal to the hard scale and at order (V0)4 for Λ = 800 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For all analyzed cutoffs, the naturalness constraint for the NLO constants is reasonably well satisfied without an obvious tendency to its violation, which can be explained by a regular behaviour of the spin-singlet one-pion-exchange potential at short distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' To summarize, the numerical calculations for the channels 3P0, 3S1 − 3D1 and 1S0 are in agreement with our theoretical analysis of the renormalization of the NLO amplitude with a finite cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' We observed a reasonable convergence of the chiral EFT expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' However, for the 3P0 and 1S0 partial waves a more efficient scheme within the considered EFT formulation is obtained when the subleading contact interactions are promoted to leading order, as has already been discussed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The naturalness constraints on the NLO LECs and on the value of the Fredholm determinants are fulfilled for the cutoff values below or of the order of the hard scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The convergence rate of the Fredholm determinants in powers of the LO potential also appears to be sufficiently rapid for such values of the cutoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' This allows us to interpret the renormalizability of the NLO amplitude within the method developed in the current paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' When the cutoff approaches the value Λ ∼ 800 MeV or higher, the renormalizability constraints are not clearly fulfilled anymore, even though the convergence of the amplitude might still be reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Thus, we conclude that the preferable choice of the cutoff values is roughly Λ ≲ 600 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For very soft cutoffs Λ = 300 − 450 MeV, the regulator corrections to the LO potential should be explicitly taken into account to remove the regulator artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 28 VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' SUMMARY We have extended our previous study in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16] and analyzed the renormalization of the nucleon-nucleon amplitude at NLO in chiral EFT in the case when the LO interaction is non-perturbative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Our scheme is based on the formulation of chiral EFT with a finite cutoff derived from the effective Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In the previous paper, the power counting for the renormalized NLO amplitude was justified for the case when the series for the iterations of the LO potential are (rapidly enough) convergent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=', for the perturbative case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The corresponding subtractions in the form of the LO S-wave contact terms that absorb the power counting breaking contributions were identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Starting from P-waves, the NLO amplitudes were found not to require any subtractions in agreement with dimensional power counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The method of analysis of the power counting in the non-perturbative regime relies on the Fredholm formula for the solution of the integral equations, which represents the numerators and denominators of the amplitudes as individually convergent series in powers of the LO potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' To implement the Fredholm decomposition, we first had to derive stronger bounds on the LO potential compared to the ones used in the perturbative case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In contrast to the perturbative regime, it turned out that the minimal “mild” regulator can, in general, not be employed if the NLO potential remains unregulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' This implies a potentially stronger cutoff dependence in the non-perturbative case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The results for the P- and higher partial waves in the NN system reproduce to a large extend our previous findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The dimensional power counting for the LO and NLO amplitudes is formally satisfied without subtractions unless there is an enhancement of the LO amplitude due to the presence of a shallow (quasi-)bound state, which is not the case for the physical channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Nevertheless, in some cases, the promotion of NLO contact terms to leading order can be motivated by phenomenological arguments as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=', in the 3P0 channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In the latter situation, however, one has to choose the LO potential in such a way as to avoid the appearance of “exceptional” cutoffs, for which the renormalization breaks down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The simplest way to verify that the adopted value of the cutoff is not close to “exceptional” is to make sure that the NLO LECs are of a natural size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For the S-waves, we have shown that the series for the subtractions at next-to-leading order, obtained in the perturbative case, can be resummed in a closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Such a resummation is equivalent to the condition for the renormalized NLO amplitude to vanish at threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Using the Fredholm formula allowed us to analyze also the case when the LO amplitude is enhanced at threshold compared to the dimensional power counting estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' This happens in the 3S1 − 3D1 and 1S0 channels where shallow bound and quasibound states are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The dimensional power counting for the NLO amplitudes is still valid in those cases if certain additional constraints on the LO potential are fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Again, these constraints eventually reveal themselves in the naturalness of the NLO LECs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' However, the convergence of the chiral expansion in the channels with enhanced LO amplitude may become significantly slower, especially in the 1S0 channel, where the enhancement is most pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' To improve the convergence, one can, analogously to the 3P0 partial wave, promote a subleading contact term to the LO potential with the same warning regarding “exceptional” cutoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Finally, we have illustrated our theoretical findings by numerical calculations of the NN phase shifts at next-to- leading order by fitting the unknown free parameters to the empirical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' We considered three channels with non- perturbative dynamics, namely 3P0, 3S1−3D1 and 1S0, and varied the LO cutoff in the range of Λ = 300−800 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' We observed reasonable convergence of the chiral expansion, especially when the subleading contact terms are promoted in the 3P0 and 1S0 channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' As criteria for the interpretation of the renormalizability of the NLO amplitude in terms of the Fredholm expansion, we used the naturalness of the NLO LECs and of the Fredholm determinant as well as the convergence rate of the expansion of the latter in powers of the LO potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' It turns out that all these constraints are fulfilled as long as the cutoff values are chosen below or of the order of the hard scale Λb ∼ 600 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For particularly soft cutoffs Λ = 300 − 450 MeV, taking into account explicit regulator corrections to the LO potential compensates for the regulator artifacts and reduces the cutoff dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' When the cutoff increases beyond the hard scale, the renormalizability constraints start being violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Therefore, we conclude that the cutoff values Λ ≲ Λb are preferable from the point of view of the renormalization of EFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Further development of our approach goes in the direction of extending it beyond next-to-leading order in the chiral expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' It is also important to generalize the scheme to few-nucleon systems and the processes involving electro-weak interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' ACKNOWLEDGMENTS We would like to thank Jambul Gegelia for helpful discussions and for useful comments on the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' This work was supported by DFG (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 426661267), by BMBF (contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 05P21PCFP1), by ERC AdG NuclearTheory 29 (grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 885150) and by the EU Horizon 2020 research and innovation programme (STRONG-2020, grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 824093).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 30 Appendix A: Leading-order potential The short-range part of the leading-order potential in its general form can be chosen to include the momentum- independent contact interactions and contact terms quadratic in momenta (altogether 9 terms), multiplied by the power-like non-local form factor of an appropriate power n: V (0) short,Λ(⃗p ′, ⃗p ) = � i Ci VCi FΛi,ni(p ′, p) , (A1) where VCi is any basis for the contact terms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' the partial wave basis, and the regulators are given by FΛ,n(p ′, p) = FΛ,n(p ′)FΛ,n(p) , FΛ,n(p) = [FΛ(p)]n , FΛ(p) = Λ2 p2 + Λ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (A2) One can also introduce a regulator of a Gaussian form by replacing FΛ,n(p) with FΛ,exp(p) = exp (−p2/Λ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (A3) Alternatively, one could introduce local short-range interactions (for the terms that depend only on ⃗q, except for the spin-orbit term) using the appropriate basis [46] and the local regulator Fq,Λ,n(⃗p ′, ⃗p ) = [FΛ(q)]n = � Λ2 q2 + Λ2 �n , (A4) or with the regulator in the Gaussian form FΛ,exp(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The long-range part of the LO potential is represent by the one-pion-exchange contribution, which is split into the triplet, singlet, and contact parts V (0) 1π = − � gA 2Fπ �2 τ1 · τ2 ⃗σ1 · ⃗q ⃗σ2 · ⃗q q2 + M 2π =: V (0) 1π,t + V (0) 1π,s + V (0) 1π,ct , (A5) with V (0) 1π,s = � gA 2Fπ �2 τ1 · τ2 (⃗σ1 · ⃗σ2 − 1) 4 M 2 π q2 + M 2π , V (0) 1π,ct = − � gA 2Fπ �2 τ1 · τ2 (⃗σ1 · ⃗σ2 − 1) 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (A6) All three parts, if necessary, are regularized individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The contact part V1π,ct can be absorbed by the leading-order 1S0 contact term and thus needs not be considered separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The triplet and singlet potentials can be regularized by means of the non-local form factor (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (A2)): V (0) 1π,Λ(⃗p ′, ⃗p ) = V (0) 1π,s(⃗p ′, ⃗p )FΛs,ns(p ′, p) + V (0) 1π,t(⃗p ′, ⃗p )FΛt,nt(p ′, p) , (A7) or by means of the local regulator: V (0) 1π,Λ(⃗p ′, ⃗p ) = V (0) 1π,s(⃗p ′, ⃗p )Fq,1π,Λs(⃗p ′, ⃗p ) + V (0) 1π,t(⃗p ′, ⃗p )Fq,1π,Λt(⃗p ′, ⃗p ) , (A8) with Fq,1π,Λs(⃗p ′, ⃗p ) = �Λ2 s − M 2 π q2 + Λ2s �ns , Fq,1π,Λt(⃗p ′, ⃗p ) = �Λ2 t − M 2 π q2 + Λ2 t �nt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (A9) Note that in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16], a more general form of the local regulator was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The spin-singlet part of the one-pion-exchange potential can, in principle, be left unregulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' This is, however, only relevant for the spin-singlet channels without short-range interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' All such channels can be regarded as having perturbative LO potential and were already analyzed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For the spin-singlet channel considered in this work, 1S0, the effects of a regulator will be driven by the contact interaction in any case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' To regularize the spin-triplet part of the one-pion-exchange potential in the LO Lippmann-Schwinger equation, it is sufficient to introduce a dipole (nt = 1) regulator, which we refer to as the “mild” regulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' All other options, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=', nt ≥ 2 are referred to as the “standard” regulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' One can also adopt the local Gaussian regulator for the one-pion-exchange potential: Fq,1π,exp,Λ(⃗p ′, ⃗p ) = exp � −(q2 + M 2 π)/Λ2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (A10) 31 Appendix B: Next-to-leading-order potential The short-range part of the next-to-leading-order potential is given by the sum of contact terms analogous to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (A1): V (2) short(⃗p ′, ⃗p ) = � i C2,i VCi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (B1) The non-polynomial part of the two-pion-exchange potential is given by (it is equivalent to the one provided in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [47] up to polynomial terms) V (2) 2π (⃗p ′, ⃗p ) = − τ1 · τ2 384π2F 4π ˜L(q) � 4M 2 π(5g4 A − 4g2 A − 1) + q2(23g4 A − 10g2 A − 1) + 48g4 AM 4 π 4M 2π + q2 � + τ1 · τ2 8π2F 4π g4 AM 2 πq2 4M 2π + q2 − 3g4 A 64π2F 4π ˜L(q) � ⃗σ1 · ⃗q⃗σ2 · ⃗q − q2⃗σ1 · ⃗σ2 � , (B2) where ˜L(q) := L(q) − L(0) = L(q) − 1 , L(q) = 1 q � 4M 2π + q2 log � 4M 2π + q2 + q 2Mπ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (B3) The regulator of the NLO potential, not shown explicitly in the above expressions, can be a combination of any local or non-local forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For the two-pion-exchange potential, one can also employ a spectral function regularization by introducing a finite upper limit in the dispersion representation of ˜L(q): ˜L(q) = q2 � Λρ 2Mπ dµ µ2 � µ2 − 4M 2π q2 + µ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (B4) Appendix C: Bounds on the plane-wave potential 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Bounds on the substructures Below, we list the inequalities for the building blocks of the LO and NLO potentials obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The components of the initial and final nucleon c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' momenta p and p′ are defined as ⃗p = p � � 0 0 1 � � , ⃗p ′ = p′ � � sin θ cos φ sin θ sin φ cos θ � � , (C1) where p is either p = pon or lies on the complex contour p ∈ C: p = |p| exp(−iαC), and p′ is either p′ = pon or p′ = |p′| exp(−iαC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For the function fµ(p′, p, x) = 1 q2 + µ2 = 1 p′2 + p2 − 2pp′x + µ2 , (C2) with µ ≥ Mπ, the following bounds hold |fµ(p′, p, x)| ≤ Mf |p|2 + |p′|2 − 2|p||p′|x + µ2 , (C3) |qiqjfµ(p′, p, x)| ≤ Mf, (C4) ���(⃗k × ⃗q)i fµ(p′, p, x) ��� ≤ Mf � 1 − x2�−1/2 , i, j = 1, 2, 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C5) The subtraction remainders defined as ∆(n) p f(p′, p) = f(p′, p) − n � i=0 ∂if(p′, p) i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (∂p)i ���� p=0 pi, ∆(n) p′ f(p′, p) = f(p′, p) − n � i=0 ∂if(p′, p) i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (∂p′)i ���� p′=0 (p′)i, (C6) 32 satisfy the following inequalities: ���∆(n) p fµ(p′, p) ��� ≤ Mf,n ���� p p′ ���� n+1 |fµ(p′, p)| , if |p′| > |p|, ���∆(n) p′ fµ(p′, p) ��� ≤ Mf,n ���� p′ p ���� n+1 |fµ(p′, p)| if |p| > |p′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C7) For a more general structure Ψk,m,{µi}(p′, p, x) = Qk(p′, p, x)FΛ,m(p′, p)f{µi}(p′, p , x), (C8) where the form factor FΛ,m is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (A2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' f{µi} is a product of several fµ f{µi}(p′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' x) = � i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='r fµi(p′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C9) and Qk is a homogeneous polynomial of degree k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' one can deduce the bounds for derivatives: �����pn ∂nΨk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='{µi}(p′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' x) ∂pn ���� p=0 ����� ≤ Mk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='n ∂Ψ ���p′kFΛ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='m− n+1 2 (p′)f{µi}(p′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' x) ��� ���� p p′ ���� n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' �����(p′)n ∂nΨk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='{µi}(p′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' x) ∂(p′)n ���� p′=0 ����� ≤ Mk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='n ∂Ψ ���pkFΛ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='m− n+1 2 (p)f{µi}(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' x) ��� ���� p′ p ���� n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' n ≥ 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C10) and for the subtraction remainders: ���∆(n) p Ψk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='{µi}(p′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' x) ��� ≤ Mk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='n Ψ ���� p p′ ���� n+1 × � |Ψk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='{µi}(p′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' x)| + ���p′kFΛ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='m− n+1 2 (p′)f{µi}(p′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' x) ��� � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' if |p′| > |p|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' ���∆(n) p′ Ψk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='{µi}(p′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' x) ��� ≤ Mk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='n Ψ ���� p′ p ���� n+1 × � |Ψk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='{µi}(p′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' x)| + ���pkFΛ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='m− n+1 2 (p)f{µi}(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' x) ��� � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' if |p| > |p′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C11) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Bounds on the plane-wave leading-order potential In this section we provide bounds for the leading-order potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' We will need slightly stronger bounds than those obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In particular, we will need bounds that factorize in initial and finale momenta in the partial wave basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In order to obtain them, we will partly keep the angular dependence in binding functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The derivation is only slightly different from that of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16], which we demonstrate for the case of the spin-triplet one-pion-exchange potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The locally regularized one-pion exchange potential in the spin-triplet channels can be bounded using equations of Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' C 1 by the following inequality: ���V (0) 1π,t(⃗p ′, ⃗p ) ��� ≤ ������ g2 A 4F 2π � i,j Mt,ij qiqj q2 + M 2π � Λ2 t − M 2 π q2 + Λ2 t,1 �nt������ ≤ 2πMt mNΛV FΛ,nt(|p′|, |p|, x) , (C12) where we have introduced the form factors FΛ,n(|p′|, |p|, x) = (FΛ(|p′|, |p|, x))n , FΛ(|p′|, |p|, x) = Λ2 |p|2 + |p′|2 − 2|p||p′|x + Λ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C13) 33 In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C12), we replaced Λt with the largest cutoff Λ among all regulators in the LO potential, which is possible due to the inequality: FΛ1(|p′|, |p|, x) < FΛ2(|p′|, |p|, x) for Λ1 < Λ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C14) If the triplet one-pion exchange potential is regularized by the non-local form factor, we obtain ���V (0) 1π,t(⃗p ′, ⃗p ) ��� ≤ ������ g2 A 4F 2π � i,j Mt,ij qiqj q2 + M 2π � Λ2 p′2 + Λ2 Λ2 p2 + Λ2 �nt ������ ≤ 2πMt mNΛV FΛ,nt(|p ′|)FΛ,nt(|p|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C15) Analogously, we obtain bounds for other LO contributions as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16] retaining the angular dependence of local form factors and the powers of the form factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Finally, the full leading-order potential satisfies |V0(⃗p ′, ⃗p )| ≤ MV0 4π V0,max(p′, p, x), |V0(⃗p ′, ⃗p )| ≤ MV0 4π V0,max(p, p′, x) , (C16) where we have introduced V0,max(p′, p, x) = 8π2 mNΛV � FΛ,n(|p′|, |p|, x) + FΛ,n(|p′|) � , (C17) with n being the smallest power among all regulators in the LO potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The cases of the “mild” and the “standard” regulators correspond to n = 1 and n ≥ 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The difference of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C17) from an analogous bound in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16] is that the powers of both local and non-local form factors are retained and the x-dependence of the local form factor is kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The bounds for the Gaussian regulators can be reduced to the ones for the power-like regulators as was shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For the spin-singlet channels without a short-range interaction, the bounds in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C17) can be improved by replacing Λ with Mπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' However as mentioned above, those channels were already covered in our previous study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The remainders ∆(n) p V0(⃗p ′, ⃗p ) for |p′| > |p| can be estimated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C11): ���∆(n) p V0(⃗p ′, ⃗p ) ��� ≤ M∆V0,n 4π ���� p p′ ���� n+1 V0,max(p′, p, x) if |p′| > |p| , ���∆(n) p′ V0(⃗p ′, ⃗p ) ��� ≤ M∆V0,n 4π ���� p′ p ���� n+1 V0,max(p, p′, x) if |p| > |p′| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C18) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C10) one obtains the estimates for the derivatives of the leading-order potential: �����pm ∂mV0(⃗p ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' ⃗p ) (∂p)m ���� p=0 ����� ≤ 2πM∂V0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='n mNΛV F˜Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='n(|p′|) ���� p p′ ���� m ≤ M∂V0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='n 4π ���� p p′ ���� m V0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='max(p′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' x) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C19) �����p′m ∂mV0(⃗p ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' ⃗p ) (∂p′)m ���� p′=0 ����� ≤ 2πM∂V0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='n mNΛV F˜Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='n(|p|) ���� p′ p ���� m ≤ M∂V0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='n 4π ���� p′ p ���� m V0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='max(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' p′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' x) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C20) including the case m = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' where we have used that the local form factor satisfies FΛ(p′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' x) = FΛ(p′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C21) Applying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C19) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C20)) to the definition of ∆(n) p V0(⃗p ′, ⃗p ) (∆(n) p′ V0(⃗p ′, ⃗p )) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C6) for |p| > |p′| (|p′| > |p|), and combining it with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C18), we obtain the following bounds for the remainders: ���∆(n) p V0(⃗p ′, ⃗p ) ��� ≤ M∆V0,n 4π ���� p p′ ���� n+1 V0,max(p′, p, x) , ���∆(n) p′ V0(⃗p ′, ⃗p ) ��� ≤ M∆V0,n 4π ���� p′ p ���� n+1 V0,max(p, p′, x) , (C22) 34 which are valid for all considered p and p′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' All above general formulas do not include the case when the LO potential contains a locally regulated spin-orbit short-range interaction such as V (0) C5 (⃗p ′, ⃗p ) = C5 i 2(⃗σ1 + ⃗σ2) · (⃗k × ⃗q ) � Λ2 5 q2 + Λ2 5 �n5 , (C23) with n5 > 1 (or with the Gaussian form factor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Following the arguments provided in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16], one can formulate the same bounds as in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C16) and (C22) for the quantity ˜V (0) C5 defined as V (0) C5 (⃗p ′, ⃗p ) = ˜V (0) C5 (⃗p ′, ⃗p ) i 2(⃗σ1 + ⃗σ2) · ⃗nφ/ sin θ, ⃗nφ = (− sin φ, cos φ, 0), (C24) which makes it possible, after the partial-wave projection, to treat this interaction on the same footing as all other LO terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Bounds on the plane-wave next-to-leading-order potential For the NLO potential, we use the bounds obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The NLO potential is split into two parts: V2(⃗p ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' ⃗p ) = ˆV2(⃗p ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' ⃗p ) + ˜V2(⃗p ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' ⃗p ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C25) with ˆV2(⃗p ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' ⃗p ) = V2(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 0) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' ˜V2(⃗p ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' ⃗p ) = V2(⃗p ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' ⃗p ) − V2(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 0) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C26) which are bound as ��� ˆV2(⃗p ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' ⃗p ) ��� ≤ ˆ MV2 2π mNΛV M 2 π Λ2 b ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C27) and ��� ˜V2(⃗p ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' ⃗p ) ��� ≤ 2πMV2 mNΛV |p|2 + |p′|2 Λ2 b flog(p′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' p) = MV2 4π � |p|2 + |p′|2� ˜flog(p′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' p) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C28) with ˜flog(p′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' p) = 8π2 mNΛV Λ2 b flog(p′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' p) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' flog(p′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' p) = θ(|p| − Mπ) ln |p| Mπ + θ(|p′| − Mπ) ln |p′| Mπ + ln ˜Λ Mπ + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C29) In the function flog(p′, p), the term ln ˜Λ Mπ was introduced for convenience so we can omit it (or set ˜Λ = Mπ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' We also allow for a regulator (local or non-local) for the NLO potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' We can introduce it simply as a factor, so that the bounds in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C28) are modified as ��� ˜V2(⃗p ′, ⃗p ) ��� ≤ MV2 4π � |p|2 + |p′|2� ˜flog(p′, p) � FΛNLO(|p′|, |p|, x) + FΛNLO(|p′|) � , (C30) where we combined local and non-local regulators into one factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The first power (n = 1) of the form factors is sufficient for our estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The cases of higher powers (or Gaussian cutoffs) are included automatically, because FΛNLO,n(|p|) ≤ FΛNLO(|p|) , FΛNLO,n(|p′|, |p|, x) ≤ FΛNLO(|p′|, |p|, x) , (C31) for n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' If different values of the cutoff are used for different NLO contributions, ΛNLO can be chosen to be the largest value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 35 Appendix D: Bounds on the partial-wave potential Below, we repeat the arguments of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16] for deriving the bounds on the partial-wave potential, but take into account an angular dependence of the binding functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The partial-wave potential is obtained from the plain-wave potential via V sj l′,l(p′, p) = � λ1,λ2,λ′ 1,λ′ 2 � dΩ ⟨jl′s|λ′ 1λ′ 2⟩⟨λ′ 1λ′ 2|V (⃗p ′, ⃗p )|λ1λ2⟩⟨λ1λ2|jls⟩dj λ1−λ2,λ′ 1−λ′ 2(θ) , ⟨λ1λ2|jls⟩ = � 2l + 1 2j + 1 � 1 2 C(l , s , j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 0 , λ1 − λ2)C (1/2 , 1/2 , s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' λ1, −λ2) , (D1) where λi, λ′ i are the helicities of the corresponding nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Due to unitarity of the transformation, the following constraints hold: |⟨λ1λ2|jls⟩| ≤ 1, |⟨1/2 , sz|λ⟩| ≤ 1, |dj λ,λ′(θ)| ≤ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D2) Therefore, if the plain-wave potential is bounded by some angle-dependent function φ(p′, p, x): |V (⃗p ′, ⃗p )| ≤ Mkφ(p′, p, x) , (D3) then, for the partial-wave potential, we obtain: |V sj l′,l(p′, p)| ≤ 2π ˜ Mk � 1 −1 dx φ(p′, p, x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D4) For the special case of the locally regulated spin-orbit contact interaction, a bound of the same type can be obtained if one replaces |V (⃗p ′, ⃗p )| by | ˜V (⃗p ′, ⃗p )| = |V (⃗p ′, ⃗p )| √ 1 − x2, see Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' C 2 and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Bounds on the form factor Fµ,n(q) integrated over x In this subsection we derive the bounds on the local form factors Fµ(q) = µ2 q2 + µ2 , Fµ,2(q) = Fµ(q)2, (D5) integrated over the angle variable x, which are relevant when considering bounds for the partial-wave potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The form factors Fµ,n(q) with n > 2 satisfy (at least) the same bounds as Fµ,2(q), which is sufficient for our estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The same is true for the form factors of the Gaussian form, which was analyzed in detail in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C3), it follows ��q2 + µ2�� ≥ M−1 f � |p|2 + |p′|2 − 2|p||p′|x + µ2� = M−1 f � (|p′|x − |p|)2 + |p′|2(1 − x2) + µ2� ≥ M−1 f � |p′|2(1 − x)/2 + µ2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D6) For |p′| ≥ µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' we obtain ���� � 1 −1 Fµ(q)dx ���� = ���� � 1 −1 µ2dx q2 + µ2 ���� ≤ 2Mfµ2 � 1 −1 dx |p′|2(1 − x) + 2µ2 = 2Mfµ2 |p′|2 ln � 1 + |p′|2/µ2� ≤ 2Mfµ2 |p′|2 ln 2|p′|2 µ2 < 2Mfµ2 |p′|2 � 1 + ln |p′|2 µ2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D7) and ���� � 1 −1 Fµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='2(q)dx ���� = ����� � 1 −1 µ4dx (q2 + µ2)2 ����� ≤ 4Mfµ4 � 1 −1 dx [|p′|2(1 − x) + 2µ2]2 = 2Mfµ2 |p′|2 + µ2 < 2Mfµ2 |p′|2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D8) 36 whereas for |p′| < µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' we can simply use ���� � 1 −1 dxFµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='n(q) ���� ≤ � 1 −1 dx(Mf)n = 2(Mf)n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' n = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D9) Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D9) with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D7) or Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D8) and introducing the functions λ(ξ) = θ(1 − |ξ|) + θ(|ξ| − 1) 1 |ξ|2 , λlog(ξ) = θ(1 − |ξ|) + θ(|ξ| − 1)1 + ln |ξ| |ξ|2 , (D10) we arrive at the following bounds (obviously symmetric under the interchange p ↔ p′): ���� � 1 −1 Fµ(q)dx ���� ≤ MF,1λlog(p′/µ) , and the same for p ↔ p′ , (D11) and ���� � 1 −1 Fµ,2(q)dx ���� ≤ MF,2λ(p′/µ) , and the same for p ↔ p′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D12) For the function Fµ,2(q), we can also obtain another bound: ���� � 1 −1 Fµ,2(q)dx ���� ≤ MF,2λ(p′/µ)2/λ(p/µ) , and the same for p ↔ p′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D13) To prove Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D13), we consider three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' |p′| ≤ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In this case, λ(p′/µ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Since λ(p/µ) ≤ 1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D13) follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' |p| ≥ |p′| > µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In this case, λ(p′/µ) ≥ λ(p/µ) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D12) yields Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' |p| < |p′| and |p′| > µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Consider the definition of the subtraction remainder ∆(1) p in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C7): Fµ,2(q) = Fµ,2(p′) + p∂Fµ,2(q) ∂p ���� p=0 + ∆(1) p Fµ,2(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D14) Now, we estimate the three terms in the last equation individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' ���� � 1 −1 Fµ,2(p′)dx ���� ≤ � 1 −1 |Fµ,2(p′)| dx ≤ 2µ4 |p′|4 = 2λ(p′/µ)2 ≤ 2λ(p′/µ)2/λ(p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D15) From the fact that ∂Fµ,2(q) ∂p ���� p=0 ∝ x, it follows � 1 −1 p∂Fµ,2(q) ∂p ���� p=0 dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D16) The bound from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C7) gives ���� � 1 −1 ∆(1) p Fµ,2(q) ���� dx ≤ Mf,1 |p|2 |p′|2 � 1 −1 |Fµ,2(q)| dx , (D17) which (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D8)) leads to ���� � 1 −1 ∆(1) p Fµ,2(q) ���� dx ≤ 2Mf,1 |p|2µ2 |p′|4 = 2Mf,1 |p|2 µ2 λ(p′/µ)2 ≤ 2Mf,1λ(p′/µ)2/λ(p/µ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D18) Finally, ���� � 1 −1 Fµ,2(q)dx ���� ≤ ���� � 1 −1 Fµ,2(p′)dx ���� + ���� � 1 −1 ∆(1) p Fµ,2(q) ���� dx ≤ 2(Mf,1 + 1)λ(p′/µ)2/λ(p/µ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D19) Combining all three cases, we obtain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 37 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Bounds on the partial-wave leading-order potential We represent the bounds for the partial-wave LO potential in the separable form: |V0(p′, p)| ≤ MV0V0,max g(p′)h(p) , |V0(p′, p)| ≤ MV0V0,max h(p′)g(p) , (D20) with V0,max = 8π2 mNΛV , (D21) where the exact form of functions g and h (and the value of MV0) depends on the partial wave and on the form of a regulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Introducing the functions v0(p′, p) = V0(p′, p) [MV0V0,max h(p′)g(p)]−1 , ¯v0(p′, p) = V0(p′, p) [MV0V0,max g(p′)h(p)]−1 , (D22) we obtain the bounds |v0(p′, p)| ≤ 1 , |¯v0(p′, p)| ≤ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D23) The above inequalities are meant to hold for all matrix elements of V0(p′, p) in the l , l′ space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' S-wave Using the bounds for the plane-wave leading-order potential in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C16) and performing the partial-wave projection according to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D4), (D11), (D13), we obtain for l = 0 (for the coupled partial waves, we mean by l the lowest orbital angular momentum): g(p) = λlog(p/Λ), h(p) = 1, (D24) for the “mild” regulator, and g(p) = [λ(p/Λ)]2 , h(p) = [λ(p/Λ)]−1 , (D25) for the “standard” regulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Note that for |p| ≤ Λ, in particular, for the on-shell momentum |p| = pon, we have g(p) = h(p) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Higher partial waves For l > 0, we can use the fact that for m < l, ∂mV0(p ′, p) (∂p)m ���� p=0 = ∂mV0(p ′, p) (∂p′)m ���� p′=0 = 0, (D26) and thus ∆(m) p V0(p ′, p) = ∆(m) p′ V0(p ′, p) = V0(p ′, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D27) For the case of the “mild” regulator utilizing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C22) and performing the partial-wave projection according to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D4) and (D11), we derive g(p) = λlog(p/Λ)/|p| ˜l, h(p) = |p| ˜l, (D28) with ˜l ≤ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Since λ(p/Λ) ≤ λlog(p/Λ), (D29) the same bounds can be used for the “standard” regulators, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For the purposes of the present paper, it is sufficient to choose ˜l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 38 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Bounds on the partial-wave next-to-leading-order potential a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' S-wave For l = 0, the bounds on the NLO partial-wave potential are the same as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16]: ��� ˆV2(p′, p) ��� ≤ ˆ MV2,0 8π2 mNΛV M 2 π Λ2 b , (D30) and ��� ˜V2(p′, p) ��� ≤ MV2,0 � |p|2 + |p′|2� ˜flog(p′, p), (D31) when one employs the “standard” regulators for the LO potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' In the case of the “mild” regulator of the LO potential, we use the partial-wave projected regularized expression, applying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D11) to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (C30): ��� ˜V2(p′, p) ��� ≤ MV2,0 � |p|2 + |p′|2� ˜flog(p′, p)λlog(p′/ΛNLO), or ��� ˜V2(p′, p) ��� ≤ MV2,0 � |p|2 + |p′|2� ˜flog(p′, p)λlog(p/ΛNLO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D32) b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Higher partial waves For l ≥ 1, we simply adopt the bounds from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [16] ��� ˜V2(p′, p) ��� ≤ MV2,˜l ���� p p′ ���� ˜l |p′|2 ˜flog(p′, p), (D33) ��� ˜V2(p′, p) ��� ≤ MV2,˜l ���� p′ p ���� ˜l |p|2 ˜flog(p′, p), (D34) where 0 ≤ ˜l ≤ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' For ˜l = 1, both above equations coincide: ��� ˜V2(p′, p) ��� ≤ MV2,1|p′||p| ˜flog(p′, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D35) For the purposes of the present paper, it is sufficient to take the choice ˜l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Appendix E: Bounds on various parts of the S-wave NLO amplitude In this appendix we provide bounds for various parts of the unrenormalized and renormalized S-wave NLO amplitude and their series remainders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The unrenormalized NLO amplitude is decomposed by factoring out the Fredholm determinant as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (74): T2(p′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) = N2(p′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon)/D(pon)2, N2 = V2D2 + T2,Y D + T2, ¯Y D + T2, ¯Y Y , (E1) with T2,Y (p′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) = � p2 1dp1 (2π)3 V2(p′, p1)Y (p1, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) , T2, ¯Y (p′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) = � p′2 1 dp′ 1 (2π)3 ¯Y (p′, p′ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon)V2(p′ 1, p) , T2, ¯Y Y (p′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) = � p2 1dp1 (2π)3 p′2 1 dp′ 1 (2π)3 ¯Y (p′, p′ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon)V2(p′ 1, p1)Y (p1, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (E2) Below, we derive the bounds for the quantities T2,Y , T2, ¯Y and T2, ¯Y Y for the cases of the “standard” and the “mild” regulators of the LO potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 39 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' “Standard” regulator For the “standard” regulators of the LO potential, in particular, for the local regulators of the spin-triplet part of the one-pion-exchange potential of power n ≥ 2, the binding functions g and h have the form (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D25)) g(p1) = λ(p1/Λ)2 , h(p) = 1 , if p < Λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (E3) From the bounds on V2 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D31)) and V0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D20)), we obtain |T2,Y (p′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon)| ≤ MV2,0nPW 8π2MYmax ΛV � (|p1|2 + |p′|2)d|p1| (2π)3 ˜flog(p′, p1)λ(p1/Λ)2 = MV2,0nPW 8π2MYmax mNΛ2 V Λ2 b � d|p1| π (|p1|2 + |p′|2)flog(p′, p1)λ(p1/Λ)2 = MV2,0nPW 8π2MYmax mNΛ2 V Λ2 b � � |p′|2Iλ,1a + Iλ,1b � � 1 + θ(|p′| − Mπ) ln |p′| Mπ � + |p′|2Iλ,2a + Iλ,2b � , (E4) where the typical integrals Ii are defined and estimated in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Setting all external momenta on shell, p = p′ = pon, and using pon ≪ Λ, gives |T2,Y (pon)| ≤ 8π2MT2,Y MYmax mNΛ2 V Λ2 b Λ3 ln Λ Mπ , (E5) or, assuming Λ ∼ ΛV , |T2,Y (pon)| ≤ 8π2 ˜ MT2,Y MYmax mNΛV Λ2 Λ2 b ln Λ Mπ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (E6) Symmetrically, the same bound holds for T2, ¯Y (p′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Next, we consider the contribution T2, ¯Y Y : ��T2, ¯Y Y (p′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) �� ≤ MV2,0 �8π2nPWMYmax ΛV �2 × � d|p1| (2π)3 d|p′ 1| (2π)3 (|p1|2 + |p′2 1 |) ˜flog(p′ 1, p1)λ(p1/Λ)2λ(p′ 1/Λ)2 = MV2,0n2 PW 8π2M2 Ymax mNΛ3 V Λ2 b � d|p1|d|p′ 1| π2 (|p1|2 + |p′ 1|2)flog(p′, p1)λ(p1/Λ)2λ(p′ 1/Λ)2 = MV2,0n2 PW 8π2M2 Ymax mNΛ3 V Λ2 b 2 (Iλ,1aIλ,1b + Iλ,2aIλ,1b + Iλ,2bIλ,1a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (E7) Setting all external momenta on shell, p = p′ = pon, and using pon ≪ Λ, we obtain ��T2, ¯Y Y (pon) �� ≤ 8π2MT2, ¯Y Y M2 Ymax mNΛ3 V Λ2 b Λ4 ln Λ Mπ , (E8) or, assuming Λ ∼ ΛV : ��T2, ¯Y Y (pon) �� ≤ 8π2 ˜ MT2, ¯Y Y M2 Ymax mNΛV Λ2 Λ2 b ln Λ Mπ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (E9) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' “Mild” regulator For the “mild” regulator of the LO potential, including the case when the spin-triplet one-pion-exchange contribution is regularized by the local dipole regulator, the binding functions g and h have the form (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D24)) g(p1) = λlog(p1/Λ) , h(p) = 1 , if p < Λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (E10) 40 By analogy with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (E4) from the bounds on the regularized V2 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D32)) and V0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D20)), we obtain |T2,Y (p′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon)| ≤ MV2,0nPW 8π2MYmax mNΛ2 V Λ2 b � d|p1| π (|p1|2 + |p′|2) × flog(p′, p1)λlog(p1/Λ)λlog(p1/ΛNLO) = MV2,0nPW 8π2MYmax mNΛ2 V Λ2 b � � |p′|2Iλlog,1a + Iλlog,1b � � 1 + θ(|p′| − Mπ) ln |p′| Mπ � + |p′|2Iλlog,2a + Iλlog,2b � , (E11) where the typical integrals Ii are defined and estimated in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Setting all external momenta on shell, p = p′ = pon, and using pon ≪ Λ ≪ ΛNLO, yields |T2,Y (pon)| ≤ 8π2MT2,Y MYmax mNΛ2 V Λ2 b Λ2ΛNLO ln ΛNLO Λ ln ΛNLO Mπ , (E12) or, assuming Λ ∼ ΛV : |T2,Y (pon)| ≤ 8π2 ˜ MT2,Y MYmax mNΛV ΛΛNLO Λ2 b ln ΛNLO Λ ln ΛNLO Mπ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (E13) Symmetrically, the same bound holds for T2, ¯Y (p′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Analogously to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (E7), the following bound holds for T2, ¯Y Y : ��T2, ¯Y Y (p′, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon) �� ≤ MV2,0n2 PW 8π2M2 Ymax mNΛ3 V Λ2 b × � d|p1|d|p′ 1| π2 flog(p′ 1, p1)λlog(p1/Λ)λlog(p′ 1/Λ) × � |p1|2λlog(p1/ΛNLO) + |p′ 1|2λlog(p′ 1/ΛNLO) � = MV2,0n2 PW 8π2M2 Ymax mNΛ3 V Λ2 b × 2 � Iλlog,1Iλlog,1b + Iλlog,2Iλlog,1b + Iλlog,2bIλlog,1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (E14) Setting all external momenta on shell, p = p′ = pon, and using pon ≪ Λ ≪ ΛNLO, we obtain ��T2, ¯Y Y (pon) �� ≤ 8π2MT2, ¯Y Y M2 Ymax mNΛ3 V Λ2 b Λ3ΛNLO ln ΛNLO Λ ln ΛNLO Mπ , (E15) or, assuming Λ ∼ ΛV : ��T2, ¯Y Y (pon) �� ≤ 8π2 ˜ MT2, ¯Y Y M2 Ymax mNΛV ΛΛNLO Λ2 b ln ΛNLO Λ ln ΛNLO Mπ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (E16) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Bounds on the function ν(pon) In this subsection we provide bounds on the function νl(pon), defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (118).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' We introduce another function νY,l as follows: νl(pon) = D(pon) [δl,0 + νY,l(pon)] , (E17) which equals (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (105)) νY,l(pon) = � p2 1dp1 (2π)3 Y0,l(p1, pon;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' pon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (E18) 41 Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (61), we derive the following bound for the function nY,l in the case of the “standard” regulator of the LO potential (see Appendix D 2 a): |νY,l(pon)| ≤ MYmax ΛV � d|p1| π g(p1/Λ)h(pon) = MYmax ΛV � d|p1| π λ(p1/Λ)2 = MYmax ΛV Iλ,1a = MYmaxMλ Λ ΛV , (E19) where we have utilized the bounds for typical integrals provided in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Assuming Λ ∼ ΛV yields |νY,l(pon)| ≤ MYmax ˜ Mλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (E20) For the “mild” regulator of the LO potential, one should replace λ(p1/Λ)2 with λlog(p1/Λ) and Iλ,1a with Iλlog,1 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (E19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Since our bounds for Iλlog,1 and Iλ,1a are the same, see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (F2) and (F4), equation (E20) holds also for the “mild” regulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Since the Fredholm determinant D is bounded by a constant of order one (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (51)), the same is true for the function νl(pon): νl(pon) ≤ Mν, (E21) as follows from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (E20) and (E17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Series remainders From the bounds on the matrix elements of the operator Y ( ¯Y ) and its series remainders (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (61) and (65)) as well as the bounds on the Fredholm determinant D and its series remainders (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (51) and (53)), it is straightforward to deduce also the bounds for the series remainders of the quantities T2,Y , T2, ¯Y , T2, ¯Y Y and νY by just replacing MYmax with NδnY = MY δnYmax and M2 Ymax with 2MYmaxNδnY + N 2 δnY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Being proportional to δnYmax or (δnYmax)2, T2,Y , T2, ¯Y , T2, ¯Y Y and νY decrease faster than exponential with any base, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The series remainder of the Fredholm determinant possesses the same property, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Therefore, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (E1) we conclude that N2 also decreases faster than exponential as well as the renormalized quantity R( ˜N2) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (122)), because those are polynomials in T2,Y , T2, ¯Y , T2, ¯Y Y , νY and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' To be specific, the following bound holds: |δn[R( ˜N2)]| = ��� ∞ � k1,k2=0 R( ˜N2)[k1,k2] − n � k1,k2=0 R( ˜N2)[k1,k2]��� ≤ 8π2 mNΛV N ˜ N2e−Mδ ˜ N2n, for n > ˜ Mδ ˜ N2, (E22) where ˜ Mδ ˜ N2 is of order ˜ Mδ ˜ N2 ≳ (eΣ)2 in the general case but is typically much smaller in realistic calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' The prefactors N ˜ N2 follow from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (E6), (E9), (E13), (E16), (E21) and (51): N ˜ N2 = Λ2 Λ2 b ln Λ Mπ (E23) in the case of the “standard” regulators of the LO potential and N ˜ N2 = ΛΛNLO Λ2 b ln ΛNLO Λ ln ΛNLO Mπ (E24) in the case of the “mild” regulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Appendix F: Bounds on typical integrals In this appendix we provide the bounds for typical integrals that appear in the course of evaluation of various amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 42 The integrals Iλlog,1 = � d|p| π λlog(p/Λ), Iλlog,1a = � d|p| π λlog(p/ΛNLO)λlog(p/Λ), Iλlog,2 = � d|p| π λlog(p/Λ)θ(|p| − Mπ) ln |p| Mπ , Iλlog,2a = � d|p| π λlog(p/ΛNLO)λlog(p/Λ)θ(|p| − Mπ) ln |p| Mπ , (F1) with functions λ and λlog defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (D10) can be bounded as follows: Iλlog,1 = Λ � dξ π λlog(ξ) =: MλΛ, Iλlog,1a < Iλlog,1 = MλΛ, Iλlog,2 = 1 π � 2 + Λ + 2Λ ln Λ Mπ � ≤ Mλ,2Λ ln Λ Mπ , Iλlog,2a < Iλlog,2 ≤ Mλ,2Λ ln Λ Mπ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (F2) Analogously,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' for the integrals Iλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='1 = � d|p| π λ(p/Λ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Iλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='1a = � d|p| π λ(p/Λ)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Iλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='1b = � |p|2d|p| π λ(p/Λ)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Iλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='2 = � d|p| π λ(p/Λ)θ(|p| − Mπ) ln |p| Mπ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Iλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='2a = � d|p| π λ(p/Λ)2θ(|p| − Mπ) ln |p| Mπ Iλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='2b = � |p|2d|p| π λ(p/Λ)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' θ(|p| − Mπ) ln |p| Mπ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (F3) we obtain the following bounds: Iλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='1 = Λ � dξ π λ(ξ) < Λ � dξ π λlog(ξ) = MλΛ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Iλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='1a < Iλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='1 ≤ MλΛ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Iλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='1b = Λ3 � ξ2dξ π λ(ξ)2 < Λ3 � dξ π λ(ξ) < Λ3 � dξ π λlog(ξ) = MλΛ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Iλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='2 < Iλlog,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='2 ≤ Mλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='2Λ ln Λ Mπ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Iλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='2a < Iλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='2 ≤ Mλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='2Λ ln Λ Mπ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Iλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='2b < Λ2Iλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='2 ≤ Mλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='2Λ3 ln Λ Mπ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (F4) Next, we estimate the integral Iλlog,1b = � |p|2d|p| π λlog(p/ΛNLO)λlog(p/Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (F5) 43 Direct estimation under the assumption ΛNLO ≫ Λ gives Iλlog,1b = 2 π Λ2ΛNLO ln ΛNLO Λ + O(ΛNLO) ≤ Mλ,1aΛ2ΛNLO ln ΛNLO Λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (F6) Finally, we derive a bound for the integral Iλlog,2b = � |p|2d|p| π λlog(p/ΛNLO)λlog(p/Λ)θ(|p| − Mπ) ln |p| Mπ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (F7) Direct calculation yields Iλlog,2b = 1 π Λ2ΛNLO ln ΛNLO Λ ln ΛNLO Mπ + O(ΛNLO ln ΛNLO/Mπ) ≤ Mλ,1aΛ2ΛNLO ln ΛNLO Λ ln ΛNLO Mπ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' (F8) [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Weinberg, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' B251, 288 (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [2] S.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Reinert, Front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' in Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 8, 98 (2020), 1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='11875.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' [8] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' -G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' Meißner, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content='Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} +page_content=' A747, 362 (2005), nucl-th/0405048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFPT4oBgHgl3EQfODR9/content/2301.13032v1.pdf'} diff --git a/vdE0T4oBgHgl3EQfbwB-/content/tmp_files/2301.02353v1.pdf.txt b/vdE0T4oBgHgl3EQfbwB-/content/tmp_files/2301.02353v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9b719d02aca905d29d171e98a4365df071d11773 --- /dev/null +++ b/vdE0T4oBgHgl3EQfbwB-/content/tmp_files/2301.02353v1.pdf.txt @@ -0,0 +1,673 @@ +Spatio-temporal determinantal point processes +Nafiseh Vafaei +Department of Computer and Statistics Sciences, Faculty of Sciences, University of Mohaghegh +Ardabili, Ardabil, Iran +E-mail: N.Vafaei@uma.ac.ir +Mohammad Ghorbani1 +Department of Engineering Sciences and Mathematics, Luleå University of Technology, Sweden +E-mail: mohammad.ghorbani@ltu.se +Masoud Ganji +Department of Computer and Statistics Sciences, Faculty of Sciences, University of Mohaghegh +Ardabili, Ardabil, Iran +E-mail: mganji@uma.ac.ir +Mari Myllymäki +Natural Resources Institute Finland (Luke), Helsinki, Finland. +E-mail: mari.myllymaki@luke.fi +Abstract +Determinantal point processes are models for regular spatial point patterns, with appealing +probabilistic properties. We present their spatio-temporal counterparts and give examples of +these models, based on spatio-temporal covariance functions which are separable and non-separable +in space and time. +Keywords: Covariance function; point process; regularity; spatio-temporal; spectral density +1 +Introduction +Spatio-temporal point processes are random countable subsets X of R2 ×R+, where a point (u,t) ∈ X +corresponds to an event u ∈ R2 occurring at time t ∈ R+. We assume that the points do not overlap, i.e. +(u1,t1) ̸= (u2,t2). Examples of such events are the occurrence of epidemic diseases (such as corona +or flu), sightings or births of a species, the occurrence of fires, earthquakes, tsunamis, or volcanic +eruptions. We are interested here in spatio-temporal regular point processes, where neighbouring +points in the process tend to repel each other. +In the context of spatial point processes, Gibbs point processes including Markov point processes +and pairwise interaction point process models are generally used to model repulsiveness. Another +class of regular spatial point process models are determinantal point processes (DPPs), which have +1Corresponding Author +1 +arXiv:2301.02353v1 [math.ST] 6 Jan 2023 + +their origin in quantum physics. They were first identified as a class by Macchi (1975), who called +them fermion processes because they reflect the distributions of fermion systems in thermal equi- +librium that exhibit repulsive behaviour. DPPs have been extensively studied in probability theory +and have found applications in random matrix theory, quantum physics, wireless network modelling, +Monte Carlo integration, and machine learning (see e.g. Lavancier et al.; 2015). Recently, Lavancier +et al. (2015) studied the statistical properties of DPPs. +Some regular spatial point process models have already their spatio-temporal counterparts, but +likelihood-based inference or simulation of these models is usually complicated and time-consuming. +To circumvent these challenges, our objective here is to introduce the spatio-temporal determinantal +point processes (STDPPs) and study their properties, which is an open problem according to Lavancier +et al. (2015, pages 875-876). We present the basic properties of these processes and give examples of +them based on separable and non-separable spatio-temporal covariance functions. We derive the +key summary characteristics for these examples that can be used, for example, for model fitting and +evaluation. +2 +Basic concepts and statistical properties +Assume that X is a spatio-temporal point process with nth-order product density ρ(n), n ≥ 1 which +describes the frequency of possible configurations of n points. Suppose that B1,...,Bn are pair- +wise disjoint cylindrical regions having infinitesimal volumes dV1,...,dVn and containing the points +((u1,t1),...,(un,tn)), respectively. Then, ρ(n)� +(u1,t1),...,(un,tn) +� +dV1,...dVn is the probability that X +has a point in each of B1,...,Bn. +Definition 1. Let C be a kernel function from (R2 ×R+)×(R2 ×R+) to R. We say that a spatio-temporal +point process X is a determinantal point process with kernel C and write X ∼ STDPP(C), if its nth-order +product density function is given by +ρ(n)((u1,t1),...,(un,tn)) = det{C((ui,ti),(u j,t j))}1≤i,j≤n, +for (ui,ti) ∈ (R2×R+) and n = 1,2,..., where {C((ui,ti),(u j,t j)}1≤i,j≤n is the n×n matrix withC((ui,ti),(u j,t j)) +as its (i, j)-th entry and det{A} is the determinant of the matrix A. +The point process X is well-defined if, for each n, ρ(n)((u1,t1),··· ,(un,tn)) ≥ 0 for all {(ui,ti)}n +i=1. +This implies that, in Definition 1, C = {C((ui,ti),(u j,t j)}1≤i,j≤n should be a non-negative definite ma- +trix. Then all eigenvalues of the matrix C are non-negative and thus its determinant is also non- +negative (this follows from the fact that the determinant of a matrix is equal to the product of its +eigenvalues) (Radhakrishna Rao and Bhaskara Rao; 1998). Therefore, covariance functions are pos- +sible choices for the kernel C. Denoting the eigenvalues of C by λl (l = 1,2,...), another condition +for the existence of STDPP(C) is that λl ≤ 1 (l = 1,2,...). This is a straightforward extension from the +spatial setting, see details in Lavancier et al. (2015) and the references therein. The Poisson process +with intensity ρ(u,s) results as a special case of STDPP(C) with setting C((u,s),(u,s)) = ρ(u,s) and +C((u,s),(v,t)) = 0 if u ̸= v ors ̸= t. +By Definition 1, the moment characteristics of arbitrary order n (n ≥ 1) can be easily attained for +STDPPs, e.g. the intensity function is given by ρ(u,t) = C((u,t),(u,t)) and the second-order product +2 + +density is given by +ρ(2)((u1,t1),(u2,t2)) = C ((u1,t1),(u1,t1))C((u2,t2),(u2,t2)) +−C((u1,t1),(u2,t2))C((u1,t1),(u2,t2)). +In what follows, we assume that the kernel function is of the form +C((u1,t1),(u2,t2)) = +� +ρ(u1,t1)R +� +(u1,t1)/αs,(u2,t2)/αt +�� +ρ(u2,t2), +where ρ plays the role of the intensity function of the process, αs > 0 and αt > 0 are the spatial and +temporal correlation parameters respectively, and R(·) is the correlation function correspondent to +C. Then, the (inhomogeneous space-time) pair correlation function is given by +g((u1,t1),(u2,t2)) = ρ(2)((u1,t1),(u2,t2)) +ρ(u1,t1)ρ(u2,t2) += 1− +���R +� +(u1,t1)/αs,(u2,t2)/αt +���� +2 +≤ 1. +(1) +Since g ≤ 1, the points of X repel each other, which is a characteristic of regular point patterns. +In general, a STDPP is called second-order intensity reweighted stationary (SOIRS) if its space- +time pair correlation function is a function of the spatial difference u = u2 − u1 and the temporal +difference t = t2 − t1 (see, e.g. Ghorbani; 2013, and the references therein). Thus, a STDPP with ker- +nel function C is SOIRS if the correlation function R is a function of u and r only, i.e. if R takes the +form R((u1,t1),(u2,t2)) = R0(u,t) for some function R0. For a SOIRS STDPP(C), R0(0,0) = 1, and hence +ρ(u,t) = C(0,0). If in addition, the correlation function R0 is invariant under rotation, i.e. isotropic, +then it as well as the pair correlation function (1) depend only on the spatial distance ∥u∥ and the +temporal lag |t|. The pair correlation function is then given by +g(u,t) = 1− +���R0(∥u∥/αs,|t|/αt) +��� +2 +. +(2) +Further, the space-time K -function (see e.g. Gabriel and Diggle; 2009; Møller and Ghorbani; 2012) is +given by +K (u,t) = 2π +�t +0 +�u +0 +g(u′,t′)u′du′dt′ += πu2t − +�u +0 +�t +0 +|R0(∥u′∥/αs,|t′|/αt)|2u′du′dt′. +(3) +If the intensity ρ is constant, i.e. ρ(u,t) = ρ, then the process is stationary and the corresponding +stationary space-time g- and K -functions obtain the same formulas (2) and (3) under the isotropy +assumption. +3 +Examples of spatio-temporal determinantal point processes +Section 3.1 recalls the relationship between the covariance function and its spectral density. Then, +in Sections 3.2 and 3.3, the Fourier transform of positive finite measures, i.e. spectral measures, are +3 + +used to construct stationary spatio-temporal covariance functions, and characteristics of STDPPs with +these kernels are derived. +3.1 +The covariance function and its spectral density +The covariance function of a stationary process can be represented as a Fourier transform of a posi- +tive finite measure. According to the Wiener-Khintchine theorem (see, e.g. Rasmussen and Williams; +2006), if the spectral density function ϕ(·,·) exists, the covariance function C(·,·) and the spectral den- +sity ϕ(·,·) are Fourier duals of each other given by +C(u,t) = +� +Rd+1 e2πi(ωT u+τt)ϕ(ω,τ)dωdτ, +ϕ(ω,τ) = +� +Rd+1 e−2πi(ωT u+τt)C(u,t)dudt, +(4) +where T stands for transpose, ω is the d-dimensional spatial component and τ is the temporal com- +ponent. +A straightforward generalization of Proposition 3.1 in Lavancier et al. (2015) to the spatio-temporal +setting, under continuity and stationarity of C(u,t), when C(u,t) ∈ L2(Rd ×R+), implies that a STDPP +with kernel C exists if the corresponding spectral density satisfies +ϕ(ω,τ) ≤ 1. +(5) +3.2 +Separable spatio-temporal covariance functions +A class of separable spatio-temporal covariance functions is usually given by +C0(u,t) ∝ C s +0(u)C t +0(t), +(6) +which is valid (i.e. a positive definite function) if both the spatial covariance function, C s +0(u), and the +temporal covariance function, C t +0(t), are valid covariance functions. For a separable class of covari- +ance functions, the spectral density ϕ(ω,τ) has also a separable form, namely +ϕ(ω,τ) ∝ +�� +Rd e−2πiωT uC s +0(u)du +� +× +�� +R +e−2πiτtC t +0(t)dt +� += ϕs +0(ω)ϕt +0(τ), +where ϕs +0(ω) and ϕt +0(τ) are the spatial and temporal spectral densities, respectively. According to (5), +the condition ϕs +0(ω)ϕt +0(τ) < 1 must be satisfied for a STDPP with kernel (6) to exist. Further, for this +class, the pair correlation function takes the following simple form +g(u,t) = 1−|Rs +0(∥u∥/αs)|2|Rt +0(|t|/αt)|2, +where Rs +0 and Rt +0 are the correlation functions in space and time corresponding to C s +0 and C t +0, respec- +tively. Therefore, by (3) the corresponding K -function is given by +K (u,t) = πu2t − +�u +0 +u′|Rs +0(u′/αs)|2du′ +�t +0 +|Rt +0(t′/αt)|2dt′. +4 + +There are a large number of classes of valid spatial and valid temporal covariance functions in +the literature, for example the Matérn, power exponential and Gaussian classes, to name a few (see, +e.g. Cressie and Wikle; 2011). As an example, we consider the Gaussian covariance function C s +0(u) = +�ρσ2 +s exp(−∥u∥2/αs), u ∈ R2, with spectral density ϕs +0(ω) = �ρπσ2 +sα2 +s exp(−π2α2 +s∥ω∥2), and the expo- +nential covariance functionC t +0(t) = �ρσ2 +t exp(−|t|/αt), t ∈ R+, with spectral density ϕt +0(τ) = (2�ρσ2 +t αt)/(1+ +4π2α2 +t |τ|2). Here σ2 +s and σ2 +t are the variance parameters of the spatial and time components, respec- +tively, and αs > 0 and αt > 0 are the corresponding range parameters. A stationary STDPP with inten- +sity ρ and the separable covariance function (6) with these components, i.e +C0(u,t) = ρσ2 +sσ2 +t exp +� +− ∥u∥2 +αs +− |t| +αt +� +(7) +will exist if +ϕsep(ω,τ) = 2πρα2 +sαtσ2 +sσ2 +t +(1+4π2α2 +t τ2) +exp +� +−π2α2 +s∥ω∥2� +< 1. +Since the maximum of the spectral density occurs at (0,0), so a STDPP(C0) exists if ϕsep(0,0) < 1, which +implies that ρ < (2πα2 +sαtσ2 +sσ2 +t )−1, and hence the maximal intensity is ρmax = (2πα2 +sαtσ2 +sσ2 +t )−1. For this +process the pair correlation function is simply given by +g(u,t) = 1−exp +� +− 2∥u∥2 +αs +− 2|t| +αt +� +. +(8) +The corresponding K -function also has a closed-form expression, which is given in A. +Figure 1 (top row) shows that the values of the pair correlation function (8) decrease by the in- +crease of the spatial range αs and the temporal delay αt. Thus, these parameters determine the de- +gree of repulsion for the above separable model. +3.3 +Non-separable spatio-temporal covariance function +Following Fuentes et al. (2007), we consider the spatio-temporal spectral density +ϕϵ(ω,τ) = γ(α2 +sα2 +t +α2 +t |ω|2 +α2 +sτ2 +ϵ|ω|2τ2)−ν, +(9) +which is an extension of the commonly used Matérn spectral density (Cressie and Wikle; 2011). Here, +the non-negative parameter α−1 +s +(spatial range) explains the rate of decay of the spatial correlation, +the non-negative parameter α−1 +t +(temporal delay) explains the rate of decay for the temporal correla- +tion. Further, γ > 0 is a scale parameter. The parameter ν measures the degree of smoothness of the +process and it should be larger than (d +1)/2 to have a well-defined spectral density. The parameter +ϵ controls the interaction between the spatial and temporal components. For 0 ≤ ϵ < 1 the spectral +density is non-separable while it is separable when ϵ = 1. The maximum of the spectral density is +γ(α2 +sα2 +t )−ν and accordingly a STDPP with spectral density (9) exists if γ < (α2 +sα2 +t )ν. +In the separable case, i.e. when ϵ = 1, the spectral density is given by +ϕϵ=1(ω,τ) = γ(α2 +s +∥ω∥2)−ν(α2 +t +τ2)−ν = ϕs +0(ω)ϕt +0(τ). +(10) +5 + +0.1 +0.3 +0.5 +0.7 +0.9 +0.1 +0.3 +0.5 +0.7 +0.9 +0.3 +0.5 +0.7 +0.9 +0.1 +0.3 +0.5 +0.7 +0.9 +0.1 +0.3 +0.5 +0.7 +0.9 +0.3 +0.5 +0.7 +0.9 +0.1 +0.3 +0.5 +0.7 +0.9 +0.1 +0.3 +0.5 +0.7 +0.9 +0.3 +0.5 +0.7 +0.9 +0.1 +0.3 +0.5 +0.7 +0.1 +0.3 +0.5 +0.7 +0.9 +0.3 +0.5 +0.7 +0.9 +αs = 1.4, αt = 1.4 +αs = 1.2, αt = 1.2 +αs = 1.1, αt = 1.1 +αs = 1.0, αt = 1.0 +αs = 1.4, αt = 1.4 +αs = 1.2, αt = 1.2 +αs = 1.1, αt = 1.1 +αs = 1.0, αt = 1.0 +αs = 0.05, αt = 0.05 +αs = 0.10, αt = 0.10 +αs = 0.15, αt = 0.15 +αs = 0.25, αt = 0.25 +0.0 +0.1 +0.2 +0.0 +0.1 +0.2 +0.0 +0.1 +0.2 +0.0 +0.1 +0.2 +0.0 +0.1 +0.2 +0.0 +0.1 +0.2 +0.0 +0.1 +0.2 +Spatial distance +Temporal lag +0.25 +0.50 +0.75 +value +Figure 1: Theoretical pair correlation functions (8) (top row), (12) (middle row) and (15) (bottom row) +for different values of the parameters given on top of the plots. +In this case, ϕs +0(ω) and ϕt +0(τ) are Matérn-type spectral densities in space and time, respectively. Conse- +quently, the corresponding separable spatio-temporal covariance function, combining (4), and (10) +and using the equations 6.726.4 and 8.432.5 in Gradshteyn and Ryzhik (2007) and setting d = 2, is +given by +C0(u,t) = +4γπ�π +22ν−1/2(Γ(ν))2 +�2π|t| +αt +�ν−1/2�2π∥u∥ +αs +�ν−1 +×Kν− 1 +2 +� +2παt|t| +� +Kν−1 +� +2παs∥u∥ +� +, +where Kν(·) is the modified Bessel function of the second kind of order ν. C0(u,t) is proportional +to the product of Matérn covariance functions in space and time. Using the special cases, K 1 +2 (r) = +e−r (2r/π)−1/2 and K 3 +2 (r) = e−r (1+r −1)(2r/π)−1/2 (Abramowitz and Stegun; 1992), the above covari- +ance function for ν = 2 can be presented as +C0(u,t) = +γπ2 +4α2 +sα3 +t +(2παt|t|+1)exp(−2παt|t|)(2παs∥u∥)K1(2παs∥u∥). +(11) +For this case, considering the fact that limx→0 xK1(x) = 1 (Yang and Chu; 2017), the intensity of the +6 + +process is ρ = C0(0,0) = (γπ2)/(4α2 +sα3 +t ). Hence, taking into account that γ < α4 +sα4 +t , a STDPP with kernel +(11) exists if 4ρ ≤ π2α2 +sαt. For this separable case with ϵ = 1, considering (2), the pair correlation +function is +g(u,t) = 1−(2παt|t|+1)2 exp(−4παt|t|) +� +(2παs∥u∥)K1(2παs∥u∥) +�2 +. +(12) +For ϵ ∈ (0,1) the spatio-temporal covariance function corresponding to (9) should be computed +numerically as there is no exact closed-form expression. For the case ϵ = 0, the stationary non- +separable spatial-temporal spectral density is given by +ϕϵ=0(ω,τ) = γ(α2 +sα2 +t +α2 +t |ω|2 +α2 +sτ2)−ν. +(13) +Combining (4) and (13), and using the equations 6.726.4 and 8.432.5 in Gradshteyn and Ryzhik (2007), +the covariance function when ϵ = 0 is +C0(u,t) = +γπ +d+1 +2 +2ν− d+1 +2 Γ(ν)α(2ν−d) +s +α(2ν−1) +t +� +2παs +� +(αt +αs +t)2 +∥u∥2�1/2�ν− d+1 +2 +×Kν− d+1 +2 +� +2παs +� +(αt +αs +t)2 +∥u∥2�1/2� +. +According to (5), for d = 2 and ν = 2, there exists a STDPP with kernel +C0(u,t) = +γπ2 +2α2 +sα3 +t +exp +� +−2π +� +α2 +t |t|2 +α2 +s∥u∥2�1/2� +(14) +if and only if γ < α4 +sα4 +t . Further, it holds that ρ = C0(0,0) = (γπ2)/(2α2 +sα3 +t ) for the covariance functions +(14). Thus, under the condition γ < α4 +sα4 +t , it holds that 2ρ ≤ π2α2 +sαt. Therefore, for a STDPP with the +above covariance function, the intensity should be at most ρmax = π2α2 +sαt/2. Further, for this process +the pair correlation function is simply given by +g(u,t) = 1−exp +� +−4π +� +α2 +t |t|2 +α2 +s∥u∥2�1/2� +. +(15) +The expression for the corresponding K -function can be found in A. +Figure 1 (middle and bottom rows) shows that the values of the theoretical pair correlations (12) +and (15) decrease as α−1 +s +and α−1 +t +increase. Thus, for these models, the parameters α−1 +s +and α−1 +t +play the role of spatial range and time delay that determine the degree of repulsion. Moreover, for +fixed range parameters, the separable covariance model with (12) leads to smaller values of the pair +correlation function and thus more repulsive patterns than the non-separable model with (15). While +the separable covariance function controls to repulsiveness of points in space and time separately, in +the non-separable case the points repel each other in the 3D space. This leads to the different small +scale interactions. +7 + +4 +Discussion and conclusion +The different forms of covariance functions presented here allow for STDPPs with different types of +repulsion. While empirical experiments in the spatio-temporal setting are to be conducted in fu- +ture work, model fitting for DPPs is available through the maximum likelihood or minimum contrast +methods (Lavancier et al.; 2015) based on the summary functions such as the pair correlation func- +tion presented here for the given examples, and for model assessment, e.g. the global envelope test +(Myllymäki et al.; 2017) can be employed. +Acknowledgement +The authors are grateful to Frederic Lavancier and Ege Rubak for good discussions. MG was finan- +cially supported by the Kempe Foundations (JCSMK22-0134) and MM by the Academy of Finland +(project numbers 295100 and 327211). +A +K -functions of the proposed models +Here we give the K -functions for two covariance models discussed in Section 3. The K -function of the +model with the separable covariance function model (7), and correspondent to the pair correlation +function (8), has the following closed-form expression: +K (u,t) = πu2t − +�u +0 +�t +0 +exp +�−2u′2 +αs +− 2t′ +αt +� +u′du′dt′ += πu2t −αsαt/ +� +8 +� +1−e(−2u2/αs)�� +1−e(−2t/αt)�� +. +(16) +Employing the general formula of the K -function (3) for the covariance model (14), the space-time +K -function correspondent to the pair correlation function (15) is given by +K (u,t) = 2π +�u +0 +�t +0 +� +1−|R(u′/αs,t′/αt)|2� +u′du′dt′ += πu2t − 2π +ρ2 +�u +0 +�t +0 +|C0(u′,t′)|2u′du′dt′ += πu2t − +γ2π3 +8ρ2α6 +sα6 +t +� +e−4παt t�−2παt t +e4παt t −1 +2παt +� +− +�αsu +αt ++ 4πα2 +su2 +αt +sinhnt +2nn!(n +1−(n +1)!)! +� +J1(4π,t) +� +, +(17) +where J1(4π,t) = �∞ +n=0(−2π)n �n +m=0 +sinh(1+n−2m)t +m!(n−m!)!(1+n−2m) is an incomplete Bessel function (see more de- +tails in Jones (2007)). +References +Abramowitz, M. and Stegun, I. A. (1992). Handbook of mathematical functions with formulas, graphs, +and mathematical tables, Dover Publications Inc., New York. +8 + +Cressie, N. and Wikle, C. K. (2011). Statistics for spatio-temporal data, John Wiley and Sons. +Fuentes, M., Chen, L. and Davis, J. (2007). A class of nonseparable and nonstationary spatial temporal +covariance functions, Environmetrics 9: 487–507. +Gabriel, E. and Diggle, P. J. (2009). Second-order analysis of inhomogeneous spatio-temporal point +process data, Statistica Neerlandica 63: 43–51. +Ghorbani, M. (2013). Testing the weak stationarity of a spatio-temporal point process, Stochastic +Environonmental Research and Risk Assessesment 27: 517–524. +Gradshteyn, I. S. and Ryzhik, I. M. (2007). Table of Integrals, Series, and Products., 7th edn, Academic +Press. +Jones, D. S. (2007). Incomplete Bessel functions. I, Proceedings of the Edinburgh Mathematical Society +50: 173–183. +Lavancier, F., Møller, J. and Rubak, E. (2015). Determinantal point process models and statistical +inference, Journal of the Royal Statistical Society: Series B (Statistical Methodology) 77(4): 853–877. +Macchi, O. (1975). The coincidence approach to stochastic point processes, Advances in Applied Prob- +ability 7: 83–122. +Møller, J. and Ghorbani, M. (2012). Aspects of second-order analysis of structured inhomogeneous +spatio-temporal point processes, Statistica Neerlandica 66(4): 472–491. +Myllymäki, M., Mrkviˇcka, T., Grabarnik, P., Seijo, H. and Hahn, U. (2017). +Global envelope tests +for spatial processes, Journal of the Royal Statistical Society: Series B (Statistical Methodology) +79(2): 381–404. +Radhakrishna Rao, C. and Bhaskara Rao, M. (1998). Matrix Algebra and Its Applications to Statistics +and Econometrics, World Scientific Publishing Co. +Rasmussen, C. E. and Williams, C. K. I. (2006). Gaussian processes for machine learning, The MIT +Press, London. +Yang, Z. H. and Chu, Y. M. (2017). On approximating the modified bessel function of the second kind, +Journal of Inequalities and Applications 41. +9 + diff --git a/vdE0T4oBgHgl3EQfbwB-/content/tmp_files/load_file.txt b/vdE0T4oBgHgl3EQfbwB-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ead7f9e86ffb0d146557c0d716792463473d7770 --- /dev/null +++ b/vdE0T4oBgHgl3EQfbwB-/content/tmp_files/load_file.txt @@ -0,0 +1,379 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf,len=378 +page_content='Spatio-temporal determinantal point processes Nafiseh Vafaei Department of Computer and Statistics Sciences, Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil, Iran E-mail: N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='Vafaei@uma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='ir Mohammad Ghorbani1 Department of Engineering Sciences and Mathematics, Luleå University of Technology, Sweden E-mail: mohammad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='ghorbani@ltu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='se Masoud Ganji Department of Computer and Statistics Sciences, Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil, Iran E-mail: mganji@uma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='ir Mari Myllymäki Natural Resources Institute Finland (Luke), Helsinki, Finland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' E-mail: mari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='myllymaki@luke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='fi Abstract Determinantal point processes are models for regular spatial point patterns, with appealing probabilistic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' We present their spatio-temporal counterparts and give examples of these models, based on spatio-temporal covariance functions which are separable and non-separable in space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Keywords: Covariance function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' point process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' regularity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' spatio-temporal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' spectral density 1 Introduction Spatio-temporal point processes are random countable subsets X of R2 ×R+, where a point (u,t) ∈ X corresponds to an event u ∈ R2 occurring at time t ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' We assume that the points do not overlap, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' (u1,t1) ̸= (u2,t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Examples of such events are the occurrence of epidemic diseases (such as corona or flu), sightings or births of a species, the occurrence of fires, earthquakes, tsunamis, or volcanic eruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' We are interested here in spatio-temporal regular point processes, where neighbouring points in the process tend to repel each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' In the context of spatial point processes, Gibbs point processes including Markov point processes and pairwise interaction point process models are generally used to model repulsiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Another class of regular spatial point process models are determinantal point processes (DPPs), which have 1Corresponding Author 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='02353v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='ST] 6 Jan 2023 their origin in quantum physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' They were first identified as a class by Macchi (1975), who called them fermion processes because they reflect the distributions of fermion systems in thermal equi- librium that exhibit repulsive behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' DPPs have been extensively studied in probability theory and have found applications in random matrix theory, quantum physics, wireless network modelling, Monte Carlo integration, and machine learning (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Lavancier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Recently, Lavancier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' (2015) studied the statistical properties of DPPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Some regular spatial point process models have already their spatio-temporal counterparts, but likelihood-based inference or simulation of these models is usually complicated and time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' To circumvent these challenges, our objective here is to introduce the spatio-temporal determinantal point processes (STDPPs) and study their properties, which is an open problem according to Lavancier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' (2015, pages 875-876).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' We present the basic properties of these processes and give examples of them based on separable and non-separable spatio-temporal covariance functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' We derive the key summary characteristics for these examples that can be used, for example, for model fitting and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' 2 Basic concepts and statistical properties Assume that X is a spatio-temporal point process with nth-order product density ρ(n), n ≥ 1 which describes the frequency of possible configurations of n points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Suppose that B1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=',Bn are pair- wise disjoint cylindrical regions having infinitesimal volumes dV1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=',dVn and containing the points ((u1,t1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=',(un,tn)), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Then, ρ(n)� (u1,t1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=',(un,tn) � dV1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='dVn is the probability that X has a point in each of B1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=',Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Let C be a kernel function from (R2 ×R+)×(R2 ×R+) to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' We say that a spatio-temporal point process X is a determinantal point process with kernel C and write X ∼ STDPP(C), if its nth-order product density function is given by ρ(n)((u1,t1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=',(un,tn)) = det{C((ui,ti),(u j,t j))}1≤i,j≤n, for (ui,ti) ∈ (R2×R+) and n = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=', where {C((ui,ti),(u j,t j)}1≤i,j≤n is the n×n matrix withC((ui,ti),(u j,t j)) as its (i, j)-th entry and det{A} is the determinant of the matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' The point process X is well-defined if, for each n, ρ(n)((u1,t1),··· ,(un,tn)) ≥ 0 for all {(ui,ti)}n i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' This implies that, in Definition 1, C = {C((ui,ti),(u j,t j)}1≤i,j≤n should be a non-negative definite ma- trix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Then all eigenvalues of the matrix C are non-negative and thus its determinant is also non- negative (this follows from the fact that the determinant of a matrix is equal to the product of its eigenvalues) (Radhakrishna Rao and Bhaskara Rao;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Therefore, covariance functions are pos- sible choices for the kernel C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Denoting the eigenvalues of C by λl (l = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='), another condition for the existence of STDPP(C) is that λl ≤ 1 (l = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' This is a straightforward extension from the spatial setting, see details in Lavancier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' (2015) and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' The Poisson process with intensity ρ(u,s) results as a special case of STDPP(C) with setting C((u,s),(u,s)) = ρ(u,s) and C((u,s),(v,t)) = 0 if u ̸= v ors ̸= t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' By Definition 1, the moment characteristics of arbitrary order n (n ≥ 1) can be easily attained for STDPPs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' the intensity function is given by ρ(u,t) = C((u,t),(u,t)) and the second-order product 2 density is given by ρ(2)((u1,t1),(u2,t2)) = C ((u1,t1),(u1,t1))C((u2,t2),(u2,t2)) −C((u1,t1),(u2,t2))C((u1,t1),(u2,t2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' In what follows, we assume that the kernel function is of the form C((u1,t1),(u2,t2)) = � ρ(u1,t1)R � (u1,t1)/αs,(u2,t2)/αt �� ρ(u2,t2), where ρ plays the role of the intensity function of the process, αs > 0 and αt > 0 are the spatial and temporal correlation parameters respectively, and R(·) is the correlation function correspondent to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Then, the (inhomogeneous space-time) pair correlation function is given by g((u1,t1),(u2,t2)) = ρ(2)((u1,t1),(u2,t2)) ρ(u1,t1)ρ(u2,t2) = 1− ���R � (u1,t1)/αs,(u2,t2)/αt ���� 2 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' (1) Since g ≤ 1, the points of X repel each other, which is a characteristic of regular point patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' In general, a STDPP is called second-order intensity reweighted stationary (SOIRS) if its space- time pair correlation function is a function of the spatial difference u = u2 − u1 and the temporal difference t = t2 − t1 (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Ghorbani;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' 2013, and the references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Thus, a STDPP with ker- nel function C is SOIRS if the correlation function R is a function of u and r only, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' if R takes the form R((u1,t1),(u2,t2)) = R0(u,t) for some function R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' For a SOIRS STDPP(C), R0(0,0) = 1, and hence ρ(u,t) = C(0,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' If in addition, the correlation function R0 is invariant under rotation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' isotropic, then it as well as the pair correlation function (1) depend only on the spatial distance ∥u∥ and the temporal lag |t|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' The pair correlation function is then given by g(u,t) = 1− ���R0(∥u∥/αs,|t|/αt) ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' (2) Further, the space-time K -function (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Gabriel and Diggle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Møller and Ghorbani;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' 2012) is given by K (u,t) = 2π �t 0 �u 0 g(u′,t′)u′du′dt′ = πu2t − �u 0 �t 0 |R0(∥u′∥/αs,|t′|/αt)|2u′du′dt′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' (3) If the intensity ρ is constant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' ρ(u,t) = ρ, then the process is stationary and the corresponding stationary space-time g- and K -functions obtain the same formulas (2) and (3) under the isotropy assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' 3 Examples of spatio-temporal determinantal point processes Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='1 recalls the relationship between the covariance function and its spectral density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Then, in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='3, the Fourier transform of positive finite measures, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' spectral measures, are 3 used to construct stationary spatio-temporal covariance functions, and characteristics of STDPPs with these kernels are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='1 The covariance function and its spectral density The covariance function of a stationary process can be represented as a Fourier transform of a posi- tive finite measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' According to the Wiener-Khintchine theorem (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Rasmussen and Williams;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' 2006), if the spectral density function ϕ(·,·) exists, the covariance function C(·,·) and the spectral den- sity ϕ(·,·) are Fourier duals of each other given by C(u,t) = � Rd+1 e2πi(ωT u+τt)ϕ(ω,τ)dωdτ, ϕ(ω,τ) = � Rd+1 e−2πi(ωT u+τt)C(u,t)dudt, (4) where T stands for transpose, ω is the d-dimensional spatial component and τ is the temporal com- ponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' A straightforward generalization of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='1 in Lavancier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' (2015) to the spatio-temporal setting, under continuity and stationarity of C(u,t), when C(u,t) ∈ L2(Rd ×R+), implies that a STDPP with kernel C exists if the corresponding spectral density satisfies ϕ(ω,τ) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' (5) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='2 Separable spatio-temporal covariance functions A class of separable spatio-temporal covariance functions is usually given by C0(u,t) ∝ C s 0(u)C t 0(t), (6) which is valid (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' a positive definite function) if both the spatial covariance function, C s 0(u), and the temporal covariance function, C t 0(t), are valid covariance functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' For a separable class of covari- ance functions, the spectral density ϕ(ω,τ) has also a separable form, namely ϕ(ω,τ) ∝ �� Rd e−2πiωT uC s 0(u)du � × �� R e−2πiτtC t 0(t)dt � = ϕs 0(ω)ϕt 0(τ), where ϕs 0(ω) and ϕt 0(τ) are the spatial and temporal spectral densities, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' According to (5), the condition ϕs 0(ω)ϕt 0(τ) < 1 must be satisfied for a STDPP with kernel (6) to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Further, for this class, the pair correlation function takes the following simple form g(u,t) = 1−|Rs 0(∥u∥/αs)|2|Rt 0(|t|/αt)|2, where Rs 0 and Rt 0 are the correlation functions in space and time corresponding to C s 0 and C t 0, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Therefore, by (3) the corresponding K -function is given by K (u,t) = πu2t − �u 0 u′|Rs 0(u′/αs)|2du′ �t 0 |Rt 0(t′/αt)|2dt′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' 4 There are a large number of classes of valid spatial and valid temporal covariance functions in the literature, for example the Matérn, power exponential and Gaussian classes, to name a few (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Cressie and Wikle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' As an example, we consider the Gaussian covariance function C s 0(u) = �ρσ2 s exp(−∥u∥2/αs), u ∈ R2, with spectral density ϕs 0(ω) = �ρπσ2 sα2 s exp(−π2α2 s∥ω∥2), and the expo- nential covariance functionC t 0(t) = �ρσ2 t exp(−|t|/αt), t ∈ R+, with spectral density ϕt 0(τ) = (2�ρσ2 t αt)/(1+ 4π2α2 t |τ|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Here σ2 s and σ2 t are the variance parameters of the spatial and time components, respec- tively, and αs > 0 and αt > 0 are the corresponding range parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' A stationary STDPP with inten- sity ρ and the separable covariance function (6) with these components, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='e C0(u,t) = ρσ2 sσ2 t exp � − ∥u∥2 αs − |t| αt � (7) will exist if ϕsep(ω,τ) = 2πρα2 sαtσ2 sσ2 t (1+4π2α2 t τ2) exp � −π2α2 s∥ω∥2� < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Since the maximum of the spectral density occurs at (0,0), so a STDPP(C0) exists if ϕsep(0,0) < 1, which implies that ρ < (2πα2 sαtσ2 sσ2 t )−1, and hence the maximal intensity is ρmax = (2πα2 sαtσ2 sσ2 t )−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' For this process the pair correlation function is simply given by g(u,t) = 1−exp � − 2∥u∥2 αs − 2|t| αt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' (8) The corresponding K -function also has a closed-form expression, which is given in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Figure 1 (top row) shows that the values of the pair correlation function (8) decrease by the in- crease of the spatial range αs and the temporal delay αt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Thus, these parameters determine the de- gree of repulsion for the above separable model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='3 Non-separable spatio-temporal covariance function Following Fuentes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' (2007), we consider the spatio-temporal spectral density ϕϵ(ω,τ) = γ(α2 sα2 t +α2 t |ω|2 +α2 sτ2 +ϵ|ω|2τ2)−ν, (9) which is an extension of the commonly used Matérn spectral density (Cressie and Wikle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Here, the non-negative parameter α−1 s (spatial range) explains the rate of decay of the spatial correlation, the non-negative parameter α−1 t (temporal delay) explains the rate of decay for the temporal correla- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Further, γ > 0 is a scale parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' The parameter ν measures the degree of smoothness of the process and it should be larger than (d +1)/2 to have a well-defined spectral density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' The parameter ϵ controls the interaction between the spatial and temporal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' For 0 ≤ ϵ < 1 the spectral density is non-separable while it is separable when ϵ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' The maximum of the spectral density is γ(α2 sα2 t )−ν and accordingly a STDPP with spectral density (9) exists if γ < (α2 sα2 t )ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' In the separable case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' when ϵ = 1, the spectral density is given by ϕϵ=1(ω,τ) = γ(α2 s +∥ω∥2)−ν(α2 t +τ2)−ν = ϕs 0(ω)ϕt 0(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' (10) 5 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='9 αs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='4, αt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='4 αs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='2, αt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='2 αs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='1, αt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='1 αs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='0, αt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='0 αs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='4, αt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='4 αs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='2, αt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='2 αs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='1, αt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='1 αs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='0, αt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='0 αs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='05, αt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='05 αs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='10, αt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='10 αs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='15, αt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='15 αs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='25, αt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='2 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='2 Spatial distance Temporal lag 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='75 value Figure 1: Theoretical pair correlation functions (8) (top row), (12) (middle row) and (15) (bottom row) for different values of the parameters given on top of the plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' In this case, ϕs 0(ω) and ϕt 0(τ) are Matérn-type spectral densities in space and time, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Conse- quently, the corresponding separable spatio-temporal covariance function, combining (4), and (10) and using the equations 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='726.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='4 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='5 in Gradshteyn and Ryzhik (2007) and setting d = 2, is given by C0(u,t) = 4γπ�π 22ν−1/2(Γ(ν))2 �2π|t| αt �ν−1/2�2π∥u∥ αs �ν−1 ×Kν− 1 2 � 2παt|t| � Kν−1 � 2παs∥u∥ � , where Kν(·) is the modified Bessel function of the second kind of order ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' C0(u,t) is proportional to the product of Matérn covariance functions in space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Using the special cases, K 1 2 (r) = e−r (2r/π)−1/2 and K 3 2 (r) = e−r (1+r −1)(2r/π)−1/2 (Abramowitz and Stegun;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' 1992), the above covari- ance function for ν = 2 can be presented as C0(u,t) = γπ2 4α2 sα3 t (2παt|t|+1)exp(−2παt|t|)(2παs∥u∥)K1(2παs∥u∥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' (11) For this case, considering the fact that limx→0 xK1(x) = 1 (Yang and Chu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' 2017), the intensity of the 6 process is ρ = C0(0,0) = (γπ2)/(4α2 sα3 t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Hence, taking into account that γ < α4 sα4 t , a STDPP with kernel (11) exists if 4ρ ≤ π2α2 sαt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' For this separable case with ϵ = 1, considering (2), the pair correlation function is g(u,t) = 1−(2παt|t|+1)2 exp(−4παt|t|) � (2παs∥u∥)K1(2παs∥u∥) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' (12) For ϵ ∈ (0,1) the spatio-temporal covariance function corresponding to (9) should be computed numerically as there is no exact closed-form expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' For the case ϵ = 0, the stationary non- separable spatial-temporal spectral density is given by ϕϵ=0(ω,τ) = γ(α2 sα2 t +α2 t |ω|2 +α2 sτ2)−ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' (13) Combining (4) and (13), and using the equations 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='726.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='4 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='5 in Gradshteyn and Ryzhik (2007), the covariance function when ϵ = 0 is C0(u,t) = γπ d+1 2 2ν− d+1 2 Γ(ν)α(2ν−d) s α(2ν−1) t � 2παs � (αt αs t)2 +∥u∥2�1/2�ν− d+1 2 ×Kν− d+1 2 � 2παs � (αt αs t)2 +∥u∥2�1/2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' According to (5), for d = 2 and ν = 2, there exists a STDPP with kernel C0(u,t) = γπ2 2α2 sα3 t exp � −2π � α2 t |t|2 +α2 s∥u∥2�1/2� (14) if and only if γ < α4 sα4 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Further, it holds that ρ = C0(0,0) = (γπ2)/(2α2 sα3 t ) for the covariance functions (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Thus, under the condition γ < α4 sα4 t , it holds that 2ρ ≤ π2α2 sαt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Therefore, for a STDPP with the above covariance function, the intensity should be at most ρmax = π2α2 sαt/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Further, for this process the pair correlation function is simply given by g(u,t) = 1−exp � −4π � α2 t |t|2 +α2 s∥u∥2�1/2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' (15) The expression for the corresponding K -function can be found in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Figure 1 (middle and bottom rows) shows that the values of the theoretical pair correlations (12) and (15) decrease as α−1 s and α−1 t increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Thus, for these models, the parameters α−1 s and α−1 t play the role of spatial range and time delay that determine the degree of repulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Moreover, for fixed range parameters, the separable covariance model with (12) leads to smaller values of the pair correlation function and thus more repulsive patterns than the non-separable model with (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' While the separable covariance function controls to repulsiveness of points in space and time separately, in the non-separable case the points repel each other in the 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' This leads to the different small scale interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' 7 4 Discussion and conclusion The different forms of covariance functions presented here allow for STDPPs with different types of repulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' While empirical experiments in the spatio-temporal setting are to be conducted in fu- ture work, model fitting for DPPs is available through the maximum likelihood or minimum contrast methods (Lavancier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' 2015) based on the summary functions such as the pair correlation func- tion presented here for the given examples, and for model assessment, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' the global envelope test (Myllymäki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' 2017) can be employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' Acknowledgement The authors are grateful to Frederic Lavancier and Ege Rubak for good discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' MG was finan- cially supported by the Kempe Foundations (JCSMK22-0134) and MM by the Academy of Finland (project numbers 295100 and 327211).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' A K -functions of the proposed models Here we give the K -functions for two covariance models discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' The K -function of the model with the separable covariance function model (7), and correspondent to the pair correlation function (8), has the following closed-form expression: K (u,t) = πu2t − �u 0 �t 0 exp �−2u′2 αs − 2t′ αt � u′du′dt′ = πu2t −αsαt/ � 8 � 1−e(−2u2/αs)�� 1−e(−2t/αt)�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' (16) Employing the general formula of the K -function (3) for the covariance model (14), the space-time K -function correspondent to the pair correlation function (15) is given by K (u,t) = 2π �u 0 �t 0 � 1−|R(u′/αs,t′/αt)|2� u′du′dt′ = πu2t − 2π ρ2 �u 0 �t 0 |C0(u′,t′)|2u′du′dt′ = πu2t − γ2π3 8ρ2α6 sα6 t � e−4παt t�−2παt t +e4παt t −1 2παt � − �αsu αt + 4πα2 su2 αt sinhnt 2nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' (n +1−(n +1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' )!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' � J1(4π,t) � , (17) where J1(4π,t) = �∞ n=0(−2π)n �n m=0 sinh(1+n−2m)t m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content='(n−m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=')!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' (1+n−2m) is an incomplete Bessel function (see more de- tails in Jones (2007)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} +page_content=' References Abramowitz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfbwB-/content/2301.02353v1.pdf'} diff --git a/vtE4T4oBgHgl3EQfXAy5/content/tmp_files/2301.05038v1.pdf.txt b/vtE4T4oBgHgl3EQfXAy5/content/tmp_files/2301.05038v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..86a5bda151fa6c808c14c4d11747e08a675ccd5c --- /dev/null +++ b/vtE4T4oBgHgl3EQfXAy5/content/tmp_files/2301.05038v1.pdf.txt @@ -0,0 +1,1752 @@ +Exchange interaction for Mn acceptor in GaAs: +revealing its strong deformation dependence +I. V. Krainov,1, ∗ K. A. Baryshnikov,1, † A. A. Karpova,1, 2 and N. S. Averkiev1 +1Ioffe Institute, 194021 St. Petersburg, Russia +2Saint-Petersburg Electrotechnical University, 197022 St. Petersburg, Russia +(Dated: January 13, 2023) +In this paper we calculate exchange interaction constant between manganese ion inner electronic d- +shell and GaAs valence band bounded hole using their microscopic multiparticle wave functions. We +reveal its parametric dependence on crystal lattice deformations and find out that it could be about +and even more than dozens percent when the strain tensor reaches values of 10−3÷10−2. This fact is +in accordance with the previous hypothesis of deformation dependence of Mn acceptors in GaAs fine +energy structure obtained from Raman spectroscopy, and we show that this dependence has the same +magnitude. Also, we resolve here the problem of a substantial high temperature mismatch between +well-developed theory and experimental data for the static magnetic susceptibility of Mn ions in +GaAs. We show by numerical estimates and calculations that quite a strong parametric dependence +of the exchange coupling value on GaAs lattice expansion determines the high temperature (above +50 K) magnetic susceptibility reduction as well. +I. +INTRODUCTION +Modern material science is focused on functional mate- +rials combining different properties with maximal func- +tionality. +One of these important kinds of such mate- +rials is magnetic semiconductors mixing electrical, opti- +cal and magnetic properties. Different ways to control +these properties merge into the important directions of +research, including the production of new compounds [1– +6], nanostructure design [1, 7–11] and investigation of the +effects of external forces application [12–16]. One of the +most well-known functional materials is GaMnAs [17–20]. +In this material manganese ion with its inner magnetic +3d-shell containing 5 electrons brings magnetism to GaAs +semiconductor host. This is due to the exchange inter- +actions between manganese inner d-electrons with GaAs +holes. Also, the manganese impurity acts as an accep- +tor increasing hole concentration in GaAs semiconductor +crystal. For an isolated impurity the exchange interac- +tion between Mn half-filled d-shell with the total spin +of electrons 5/2 and localized hole in the Γ8 symmetry +state acting like a 3/2 spin results in initially 24-hold +degenerate state into into 4 sublevels with total angular +momentum F = 1, 2, 3, 4 with F = 1 being the ground +state [16]. Here we stress our attention on this exchange +interaction between isolated manganese ion and a hole +bounded on it. +Exchange interaction constant A for Mn acceptor con- +sist of two parts. The first part includes the exchange +between Mn d-shell orbital electrons and Bloch orbital of +the Γ8 hole, and the second part includes value of hole +envelope at impurity site, i.e., the probability to couple +with the half-filled d-shell as a whole. +In all previous +works [21–23], in which such interaction was discussed, +∗ igor.kraynov@mail.ru +† barysh.1989@gmail.com +only the second part (value of hole envelope at impu- +rity site) was assumed to change in different conditions, +while the first part (exchange between Bloch functions) +was assumed to be unperturbed, and its value was pos- +tulated [16]. The deformation influence on the envelope +part of exchange constant has been investigated in [23], +but it has been found that it changes by less than one +percent at pressures on the limit of GaAs hardness. The +purpose of this work is to calculate exchange interaction +value between Bloch functions of Mn d-shell and Γ8 hole +bounded from the GaAs valence band, and to treat its +dependence on deformation. We demonstrate that this +part of exchange interaction is sensitive to the presence +of crystal strains. +Previously, the assumption of a strong dependence of +exchange constant A on the crystal deformation played a +crucial role in the study of the fine structure of an isolated +Mn acceptor in GaAs. The latter was investigated using +Raman spin-flip scattering and its dependence on mag- +netic fields and external deformations at helium temper- +atures [24]. The theoretical fit of intra and inter transi- +tions between Mn-hole levels based on a standard model +of the Mn acceptor eigenstates was also carried out in +[24], but to make a satisfactory agreement between all +experimental curves and theoretical calculations the de- +formation dependence of exchange interaction constant +was phenomenologically proposed and its value was es- +timated from comparison with the experiment. The ex- +change interaction value changes by 20 % for the pres- +sure 5 kbar, which is about half of GaAs critical value +of hardness, and hence this change is much larger than +previously mentioned nearly one percent dependence on +hole envelope wave function change. +Independently, there are drastically different measure- +ments of static magnetic susceptibility behavior in a wide +temperature range in GaAs samples with low concentra- +tion of Mn ions. The first experiments were made by An- +drianov’s group [25], but their work contains an irrelevant +arXiv:2301.05038v1 [cond-mat.mtrl-sci] 12 Jan 2023 + +2 +theoretical model of the Mn center, which mismatches +with a bunch of low temperature properties of the center. +Other measurements were carried out by Frey’s group +and reported in [26], where the relevant theoretical model +was applied, which, however, has some discrepancies with +the data at the very high temperature edge. The state +of art of these studies is that the theoretical fit based on +that true and now standard Mn-hole interaction model +of experimental data is in a good agreement with the low +temperature region below 50 K. But for the high temper- +ature region, there is a reduction of magnetic suscepti- +bility compared with the theoretical prediction, which is +still puzzling nowadays. A recent paper [16] containing +a deep review of different experimental and theoretical +facts about Mn center in GaAs proposed a hypothesis +that variance mentioned above could be explained by the +Jahn–Teller effect (JTE). +In this paper we also test this hypothesis (see the Sup- +plementary material). It is known from many other ex- +perimental facts [16] that the Mn ground F = 1 state is +unaffected by the static Jahn–Teller distortion, so only +dynamical JTE should be tested [27]. Moreover, one can +show that at high temperatures, there is only one way for +dynamical JTE to occur, which is reduced to the Jahn- +Teller interaction of hole in Γ8 state with local lattice dis- +tortions. As we show (see the Supplementary material, +part 2), the high temperature dependence of magnetic +susceptibility is negligibly dependent on the Jahn–Teller +effect and ceases quite rapidly as temperature increases +that can not explain observed reduction of magnetic sus- +ceptibility discussed above. +Also we test a hypothesis +of the crystal field influence, but it also can not explain +magnetic susceptibility reduction at high temperatures +(see the Supplementary material, part 1). But here we +show that if we link the phenomenological dependence of +exchange interaction value on external deformation from +[24] with the thermal expansion coefficient of the crystal, +the problem of high-temperature magnetic susceptibility +reduction can be elegantly resolved. +In this paper we will calculate Mn-hole exchange- +interaction value part associated with the Bloch wave +functions overlapping. Then we provide an estimate for +this strain dependence. Note that the trace of strain ten- +sor for the pressure about 5 kbar is in the range of 10−3 – +10−2, and it is quite surprising how it can lead to a strong +dependence of the exchange constant ∼ 20%. We elabo- +rate and explain a simple mechanism that could explain +this fact. Further, we show that such purely theoreti- +cal estimates result in the similar variation for Mn-hole +exchange constant on stress as assumed in [24], which +has the same order of value. +Finally, we show by di- +rect calculations that the obtained dependence of A on +the crystal strains ε, which theoretically fits Mn fine en- +ergy structure [24], leads to a better agreement between +high-temperature magnetic-susceptibility calculation re- +sults and experimental data. We also believe that the +developed model could be applied to another magnetic +impurities and hosts with appropriate modifications in +symmetry analysis. +II. +THEORY +A. +Exchange Hamiltonian and representation of +total angular momenta F = 1, 2, 3, and 4. +Eigenstates of Mn acceptor are composed from the six- +fold degenerate state of Mn ion d-shell electrons in the +ground state with total spin S = 5/2 and the fourfold +degenerate state of a localized hole having the Γ8 symme- +try, which corresponds to the total angular momentum +J = 3/2. +Further, to simplify all conclusions, we will +work in the hole basis of the d-shell, which has the same +properties as the electronic one, because the shell is half- +filled, and one-particle states simply have opposite spins. +These eigenstates are split by exchange interaction be- +tween the half-filled d-shell and the localized hole result- +ing in the total angular momentum states F = 1, 2, 3, 4 +with corresponding degeneracy equal to 2F + 1. +So, if we assume that the exchange interaction between +the ion’s d-shell and the hole is described by the only one +constant A, i.e., if we set the corresponding Hamiltonian +as +ˆHex = A( ˆS · ˆ +J) = A +2 +� +ˆF 2 − ˆS2 − ˆJ2� +, +(1) +ˆF = ˆS + ˆJ, +then one can easily find out all its energy eigenvalues, +which are A(F(F +1)−S(S +1)−J(J +1))/2. All other +possible terms proportional to ( ˆS · ˆ +J)2 and ( ˆS · ˆ +J)3 are +connected with the second-order and higher-order per- +turbation terms of Coloumb interaction causing change +of spin projections of d-shell electrons. We will neglect +such terms because the energy of spin-spin interaction be- +tween the d-shell electrons is assumed to be the largest +among all other energies. This assumption allows us to +consider all processes as if no changes in spin states of +the inner shell electrons occur. Note also that there are +no spin-orbit splittings in the d-shell, which is confirmed +by the Raman scattering data of Mn0 centers in GaAs, +which has g-factor strictly equal to 2 [24, 28]. +To calculate the eigenenergies of ˆHex, it is sufficient to +use the subset from the whole basis of acceptor states, +because of the spherical symmetry of the Hamiltonian. +Let us consider such a subset consisting of only 4 wave +functions |F, Fz = 0⟩ (where F = 1, 2, 3, 4), and taking +it from [16] (note that the prefactor coefficient in |2, 0⟩ +function is changed to normalize correctly the wave func- + +3 +tion) one can write it down as +|1, 0⟩ = +1 +√ +5 +� +ΨS +3/2ΨJ +−3/2 − ΨS +−3/2ΨJ +3/2− +− +� +3 +2ΨS +1/2ΨJ +−1/2 + +� +3 +2ΨS +−1/2ΨJ +1/2 +� +, +(2) +|2, 0⟩ = +� +3 +7 +� +ΨS +3/2ΨJ +−3/2 + ΨS +−3/2ΨJ +3/2− +− +� +1 +6ΨS +1/2ΨJ +−1/2 − +� +1 +6ΨS +−1/2ΨJ +1/2 +� +, +(3) +|3, 0⟩ = +1 +√ +5 +�� +3 +2ΨS +3/2ΨJ +−3/2 − +� +3 +2ΨS +−3/2ΨJ +3/2+ ++ΨS +1/2ΨJ +−1/2 − ΨS +−1/2ΨJ +1/2 +� +, +(4) +|4, 0⟩ = +� +3 +7 +�� +1 +6ΨS +3/2ΨJ +−3/2 + +� +1 +6ΨS +−3/2ΨJ +3/2+ ++ΨS +1/2ΨJ +−1/2 + ΨS +−1/2ΨJ +1/2 +� +. +(5) +Then one can calculate all energy differences between +ˆHexch eigenstates as +EF +1 − EF = += ⟨F + 1, 0| ˆHexch|F + 1, 0⟩ − ⟨F, 0| ˆHexch|F, 0⟩ = += 2A, 3A, 4A. +(6) +This result could be obtained by taking subset of 4 wave +functions, which contain only zero projections of the +total momentum on z axis: +� +ΨS +3/2ΨJ +−3/2 ; ΨS +−3/2ΨJ +3/2; +ΨS +1/2ΨJ +−1/2; ΨS +−1/2ΨJ +1/2 +� +, generating |F, 0⟩ states. +By +calculating the exchange Hamiltonian using these wave +functions as bra and ket functions, one can obtain a 4×4 +matrix, which eigenvalues give us the same energy differ- +ences as in Eq. (6). +So, the main idea for microscopic calculation of A via +exchange integrals is to consider the first-order correc- +tion to the energies of d-states and of the hole state due +to the Coulomb interaction calculated using only these +4 wave functions with appropriate symmetrization and +antisymmetrization of all multiparticle orbitals and spin +states. +B. +Microscopic calculation of exchange integrals. +Wave functions of bounded Γ8 hole corresponding to +the total moment J = 3/2 include envelope and Bloch +parts. +Within the framework of the effective mass +method for shallow acceptors in cubic semiconductors in +the spherical approximation, the wave function of this +hole is the sum of the products of the Bloch amplitudes +Xµ and the smooth envelopes R0(r) and R2(r) +ΨJ +3/2 = R0(r)Y00X3/2 + 1 +√ +5R2(r)Y20X3/2 − +− +� +2 +5R2(r)Y21X1/2 + +� +2 +5R2(r)Y22X−1/2, (7) +ΨJ +1/2 = R0(r)Y00X1/2 − 1 +√ +5R2(r)Y20X1/2 + ++ +� +2 +5R2(r)Y2,−1X3/2 + +� +2 +5R2(r)Y22X−3/2, (8) +ΨJ +−1/2 = R0(r)Y00X−1/2 − 1 +√ +5R2(r)Y20X−1/2 + ++ +� +2 +5R2(r)Y2,1X−3/2 + +� +2 +5R2(r)Y2,−2X3/2, (9) +ΨJ +−3/2 = R0(r)Y00X−3/2 + 1 +√ +5R2(r)Y20X−3/2 − +− +� +2 +5R2(r)Y2,−1X−1/2 + +� +2 +5R2(r)Y2,−2X1/2, (10) +where Ylm are the spherical functions corresponding to +the orbital moment l and its projection m. +The ex- +change interaction integral will involve these functions +and the d-shell wave functions, which are located in one +elementary cell at the impurity site. +We can neglect +the effect of R2(r) functions because they tend to zero +limit at the magnetic impurity site, while R0(r) func- +tions take nonzero values (see calculations results in [16]). +Thus, in App. A we calculate all exchange integrals us- +ing only Bloch parts of Γ8 hole wave functions, setting +ΨJ +µ ≈ f(0)Xµ, where f(0) = R0(0)/ +√ +4π. +Basing on spin configurations of wave functions ψi (i = +1, 2, 3, 4) +� +ΨS +3/2ΨJ +−3/2; ΨS +−3/2ΨJ +3/2; ΨS +1/2ΨJ +−1/2; ΨS +−1/2ΨJ +1/2 +� +(11) +one can show that the Hamiltonian of Coulomb interac- +tion has a following 4 × 4 matrix form in this basis +ˆHC = +� +� +� +X Z +0 +0 +Z Y V +0 +0 +V Y Z +0 +0 Z X +� +� +� . +(12) +The details of calculation one can see in App. A, where +the true microscopical multiparticle structure of all wave +functions is taken into account, here we represent the + +4 +results +X = ⟨ΨS +3/2ΨJ +−3/2|U(r1 − r2)|ΨS +3/2ΨJ +−3/2⟩ = += W + æ|f(0)|2, +(13) +Y = ⟨ΨS +1/2ΨJ +−1/2|U(r1 − r2)|ΨS +1/2ΨJ +−1/2⟩ = += W + 7 +3æ|f(0)|2, +(14) +Z = ⟨ΨS +3/2ΨJ +−3/2|U(r1 − r2)|ΨS +1/2ΨJ +−1/2⟩ = += 2 +√ +2 +√ +3 æ|f(0)|2, +(15) +V = ⟨ΨS +1/2ΨJ +−1/2|U(r1 − r2)|ΨS +−1/2ΨJ +1/2⟩ = += 2æ|f(0)|2. +(16) +Here the Coloumb potential is used, which is given by +the expression +U(r1 − r2) = +e2 +|r1 − r2|. +(17) +Note that we treat the Coloumb interaction between the +localized hole and holes in the d-shell (as empty states +in the half-filled shell), and hence we have the positive +sign in Eq. (17). +The W terms in Eqs. (13–14) could +be excluded from the consideration because they result +in equal general energy shift of all 4 states due to the +Coloumb interaction. The main result is the connection +of X, Y, Z and V terms with the exchange integral æ, +which reads as +æ = +5 +� +j=1 +�� +Ω +dr1dr2 +15 +ϕj∗(r1)ϕj(r2)U(r1 − r2)χ∗(r2)χ(r1), +(18) +where integrations goes over directly doubled GaAs- +crystal elementary cell volume Ω, the sum is taken over +all five one-electron orbitals of the 3d-shell of the man- +ganese ion ϕj (the upper index numerates all possible +orbital states j = 1, . . . , 5), and there is an overlapping +with a p-like Bloch part of the localized hole wave func- +tion χ. +The eigenvalues of matrix (12) give us the following +energy differences between eigenstates of this system +E2 − E1 = 4 +3æ|f(0)|2, +E3 − E2 = 2æ|f(0)|2, +E4 − E3 = 8 +3æ|f(0)|2. +(19) +One can see from (6) that they give the same ratio be- +tween energy differences as in the phenomenological ap- +proach using Hamiltonian (1). And these results totally +coincide if one puts +A = 2 +3æ|f(0)|2. +(20) +The latter expression gives us the tool for microscopic +calculations of external forces effects on the exchange +constant A, which is relevant for a lot of measurements. +C. +Exchange constant dependence on deformation. +From the symmetry point of view possible dependence +of exchange constant on deformation reads as +ˆH = A0( ˆS · ˆ +J) + BP Tr(ˆε)( ˆS · ˆ +J) + CP +� +i,j +ˆSi ˆJjεij. (21) +If one consider hydrostatic deformation, constant A de- +pends only on the trace of deformation tensor εij = +δijTr(ˆε)/3 (here δij is the Kronecker delta-symbol) +ˆH = A0( ˆS · ˆ +J) + (BP + CP /3)Tr(ˆε)( ˆS · ˆ +J). +(22) +Further, we will neglect the dependence of envelope wave +functions f(0) on deformation ε, because their change is +too small (it is in the order of 1 % of observed values +[23]). +To understand the microscopic foundations of such +Hamiltonian dependence on deformation, we assume that +the true wave functions of the p-type forming the Bloch +eigenstates of the valley band could be admixed by some +other atomic states, for example, via the pd hybridiza- +tion mechanism keeping the total symmetry of the state +unchanged. Such hybridization can occur due to differ- +ent reasons, for example, due to the lack of inversion +symmetry in the Td group or the action of some inter- +nal potentials. We suggest here to consider the admix- +ing mechanism stemming from the existence of random +electric fields that commonly present near Mn impurity +centers in GaAs [16]. +Such random fields are usually +considered as an additional source of fine structure split- +tings in the Mn acceptor energy spectrum [16, 24], but +they also could affect local wave functions of bounded +holes, i.e., the Bloch wave functions due to the pd hy- +bridization. Thus, in Eq. (18) the χ functions should be +substituted by the hybridized combinations like +˜χ ≈ χ + +� +d +γdϕd, +˜ϕd ≈ ϕd − γ∗ +dχ, +γd = ⟨ϕd| ˆV |χ⟩ +Ep − Ed +. +(23) +Here Ep and Ed represent the pure atomic energies of +pure p- and d-states without hybridization Ep − Ed ∼ +1 eV (we assume here that for Mn-acceptor in GaAs, the +pure d-state is lying not very far from the top of the +valence band, and hence pd interaction is the most large +one), and term ˆV = r · F stands for the hybridization +operator admixing one state to another via electro-dipole +induction mechanism, which is due to some local random +electric force F . The latter could be very sensible to the +change of the elementary cell if the deformation of the +crystal occurs +F ′ +i = Fi + αεijFj, +(24) +Here we introduce dimensionless parameter α that taking +into account deformation dependence of random fields. +We assume that the applied stress is a small parameter + +5 +of the theory, so αε ≪ 1, and further we take into account +only linear terms on stress. +Thus, we can estimate the change of æ under a pressure +or a temperature-affected widening using the following +assumptions about local electric force properties +ˆV 2 ≈ rirk(FiFk + αεkmFiFm + αεijFjFk). +⟨⟨Fi⟩⟩ = 0, +⟨⟨FiFj⟩⟩ = ζδij. +Here double angle brackets represent averaging by pos- +sible realizations of random forces. Of course, the true +averaging should be processed over observable values, al- +though the mean value of an observable depends on de- +formation approximately the same way as the observable +calculated with such averaged value of exchange constant. +Finally, one can conclude that after averaging by ran- +dom forces Eq. (18) could be represented by the following +terms +æ ≈ æppdd + +� +l,i +æll +dddd +⟨χ|ri|ϕl⟩⟨ϕl|ri|χ⟩ +(Ep − Ed)2 +ζ +� +1+ 2 +3αTr(ˆε) +� +, +(25) +where +æppdd = +5 +� +j=1 +�� +Ω +dr1dr2 +15 +ϕj∗(r1)ϕj(r2)U(r1 − r2)× +×χ∗(r2)χ(r1), +æln +dddd = +5 +� +j=1 +�� +Ω +dr1dr2 +15 +ϕj∗(r1)ϕj(r2)U(r1 − r2)× +×ϕl∗(r2)ϕn(r1), +which are exchange integrals with different integrand +functions. We should note that the values of these terms +depend on the functions overlap, and hence the more d- +functions of Mn ion are involved, the larger the value of +the Coulomb term is æppdd ≪ ædddd. +To estimate the magnitude of the effect, we first +take into account that all exchange integrals between d- +functions have the same value in sum in eq. 25. Then +using the hydrogen atom functions χ corresponding to +4p orbitals and ϕd corresponding to 3d orbitals one can +obtain an estimate ædddd/æppdd ≈ 104. Also, we can take +matrix elements of coordinates approximately equal to +the Bohr radius of the atom ⟨χ|ri|ϕl⟩ ≈ ⟨ϕl|ri|χ⟩ ≈ aB ≈ +10−8 cm, and the value of the random forces dispersion +could be estimated as having the order of a typical inter- +atomic interaction term √ζ = F∗ ≈ 106 eV/cm (which is +comparable with typical values of the mean force affect- +ing the nuclear complex of the lattice cell in GaAs in the +case of the Cu ion, for which F ≈ 5·106 eV/cm [29, 30]). +Then we can write an estimate for exchange constant A +change with deformation (AP ≡ BP + CP /3) +A = A0 + AP Tr(ˆε), +(26) +where +AP +A0 +≈ 2 +3α +(15(aBF∗)2/(Ep − Ed)2)ædddd/æppdd +1 + (15(aBF∗)2/(Ep − Ed)2)ædddd/æppdd +. +(27) +From data analysis in [24], we can estimate alpha as +AP /A0 = (900 meV/2.5 meV) ≈ 360, which is equiv- +alent to the relative change of A nearly by δA/A0 ∼ +αTr(ε) ∼ 0.2 at a half of critical strain of GaAs crystal +corresponding to hardness limit at helium temperatures. +III. +CALCULATIONS AND DISCUSSION +We have discussed above the parametric dependence +of exchange constant value on crystal deformation and +its microscopic reasons. +This fact had already played +its role in the explanation of Raman scattering experi- +ment results [24], and now we are going to demonstrate +clearly that the same fact is responsible for high temper- +ature magnetic susceptibility reduction measured inde- +pendently in a completely different experimental setting +[16]. +As GaAs crystal undergoes thermal expansion, we are +going to test our hypothesis of this expansion being re- +sponsible for anomalous reduction of magnetic suscepti- +bility at relatively high temperatures. The temperature +dependence of linear expansion coefficient αT could be +found in literature (see, for example, [31] or [32]). We +show this dependence in Fig. 1. +0 +50 +100 +150 +200 +250 +300 +–1 +0 +1 +2 +3 +4 +5 +6 +FIG. 1. +Temperature dependence of linear expansion coef- +ficient αT . Dark orange circles are experimental results from +[32] (see Table 80 on page 233), black line is our interpolation +for this dependence up to 300 K. +We will use a simple function to interpolate the αT de- +pendence on temperature, which makes the interpolation + +6 +work up to 300 K quite well (see Fig. 1) +˜αT = +� +� +� +0, +T < 50 K, +C tanh +� T − 50 +180 − 50 +� +, 50 K ≤ T ≤ 300 K. +(28) +It is implied that T is measured in kelvins. The coefficient +C = 6 · 10−6 K−1. Note that there is a slight increase +in the αT coefficient above 300 K (at 800 K it reaches +7.4·10−6 K−1, see the full table of its values in [32]), and +hence the approximation in Eq. (28) does not work if +T > 300 K. But for our purposes it is enough to consider +the region of T < 300 K, in which the interpolation in +Eq. (28) describes experimental data quite well. +Note +that a very small decrease in αT values between 25 K and +50 K does not affect the observables in any reasonable +manner, thus, we neglect it. +We are interested in temperature range T = 0÷300 K. +So we can write the dependence of exchange value A on +T taking into account Eq. (26) +A(T) = A0 + AP · 3˜αT · T, +(29) +where A0 = 2.6 meV [24]. We have multiplied ˜αT by a +factor of 3 to get the bulk thermal expansion coefficient +from the linear one, because Tr(ε) = εxx(T) + εyy(T) + +εzz(T) = 3εxx(T). Here we use the same value of AP = +900 meV as in [24]. One can see the calculations results +in Fig. 2. +Note that in [16] the electron-hole basis is +used, hence one should change the sign of the exchange +constant into opposite one compared with the our result +to obtain the same order of energy levels for manganese +acceptor. Thus, substituting Eq. 29 into the formulas in +[16], we need to multiply A(T) by (−1). +As can be seen from Fig. 2 the relative mismatch be- +tween theory and experimental results at T > 100 K re- +duces approximately from 50% to 20%, if we use Eq. 29. +This reduction of the systematic mismatch leads to a bet- +ter agreement between theoretical results and the exper- +imental data in the high-temperature region, which have +the allowable magnitude of the experimental error (see +discussion in [16], experimental data have been first ob- +tained in [25], and the same mismatch has also been inde- +pendently mentioned in [26]). Also we point out that the +sign of changes of exchange interaction constant, which +we use to fit magnetic susceptibility data, is the same as +used in Raman experiments [24]. Note that other possible +factors, such as crystal field effect or reduction of mag- +netic susceptibility caused by the dynamical Jahn-Teller +effect observed by us in Supplementary materials, give no +pronounce effects on magnetic susceptibility. Moreover, +their effects diminish at high temperatures, and they also +ruin the well-established theory predictions at low tem- +peratures below 50 K. +Thus, the effect of exchange constant parametric de- +pendence on lattice deformation is the only effect that +provides reasonable explanation of both high and low +temperature behaviour characteristics of the manganese +acceptor center in GaAs. Note also that A(T) at T = +0.004 0.008 0.012 0.016 0.020 +0 +1 +2 +3 +4 +5 +FIG. 2. +Temperature dependence of static magnetic suscep- +tibility κ of manganese ions in GaAs crystal (concentration of +manganese ions 5.3 · 1018 cm−3). Both axes have logarithmic +scale. Black circles connected by dash lines represent experi- +mental data from [16, 25]. Light orange solid line is a result of +calculation based on the ordinary theory from [16], which im- +plies that A(T) = A0 = 2.6 meV. Violet solid line represents +our calculation result, where the expression for magnetic sus- +ceptibility taken from [16] is modified by taking into account +the exchange constant variation with temperature A(T) due +to thermal expansion effect. +300 K is nearly three times larger than A0, and it could +reach even higher values at higher temperatures accord- +ing to [32] and Eq. 29. Note that at such big changes in A +the nonlinear terms on lattice deformation should be also +taken into account in the pd hybridization mechanism of +exchange constant renormalization via random fields as +soon as the parameter αTr(ε) reaches and exceeds the +limit of 1. But we show in Fig. 2 that even linear terms +give the right trend in temperature dependence of mag- +netic susceptibility. +IV. +CONCLUSION +Exchange constant value between d-electrons of man- +ganese ion impurity in GaAs crystal and the hole, local- +ized from the valence band on the impurity ion, is mi- +croscopically derived. The effect of crystal lattice period +change on the value of the exchange coupling constant +occurring via the hybridization of exchanging orbitals +is shown and estimated. +We also discuss the effect of +the thermal expansion causing the change in magnetic +susceptibility. We show that accounting for this effect +leads to a better agreement between theoretical results +and magnetic susceptibility data measured at high tem- +peratures. The considered thermal widening mechanism +does not influence the low-temperature magnetic suscep- +tibility behaviour. This result is also in agreement with + +7 +another experiment of Raman scattering on Mn accep- +tors in GaAs with applied external strain. We believe +that the approach to the analytical calculation of the ex- +change constant could be generalized to the case of other +magnetic centers in semiconductor structures and semi- +magnetic compounds. +ACKNOWLEDGMENTS +This work has been supported by the Russian Science +Foundation (analytical theory – Project 18-72-10111). +K. A. B. thanks the Theoretical Physics and Mathemat- +ics Advancement Foundation ”BASIS”. We also thank +M. O. Nestoklon, S. A. Tarasenko, and M. M. Glazov for +fruitful discussions. We dedicate this article to the mem- +ory of our colleague and co-author V. F. Sapega (Ioffe +Institute) who passed away in 2022. +Appendix A: Calculation of exchange integrals. +The localized-on-the-ion hole has the Bloch part of +wave function, which describes both spin and orbital de- +grees of freedom in Γ8-state +ΨJ +3/2 = −αX + iY +√ +2 +f(0); +(A1) +ΨJ +1/2 = +�� +2 +3αZ − β X + iY +√ +6 +� +f(0); +(A2) +ΨJ +−1/2 = +�� +2 +3βZ + αX − iY +√ +6 +� +f(0); +(A3) +ΨJ +−3/2 = β X − iY +√ +2 +f(0). +(A4) +Here α and β means spin-up and spin-down states of the +hole captured and localized from the valley band of GaAs +crystal, respectively. Space orbitals X, Y and Z corre- +spond to a p-like orbitals, which form the valley band of +the crystal, and hence one can prove that they are quite +similar from the cubic symmetry point of view. So we +will use more compact notations as χ+ = −(X +iY )/ +√ +2 +and χ− = (X − iY )/ +√ +2. +The half-filled 3d-shell of the Mn ion is described by a +five-hole wave function with the totally symmetrical spin +part. We assume that Hund’s rule is the most powerful +here, and all spin-spin interaction in the shell has already +led to the appearance of co-directed spins of all five d- +holes resulting in the total spin S = 5/2, and hence it +has an antisymmetric orbital part +ΨS +Sz = Φd +1,2,3,4,5|S, Sz⟩; +(A5) +Φd +1,2,3,4,5 = +1 +√ +5! +��������� +ϕ1 +1 ϕ2 +1 ϕ3 +1 ϕ4 +1 ϕ5 +1 +ϕ1 +2 ϕ2 +2 ϕ3 +2 ϕ4 +2 ϕ5 +2 +ϕ1 +3 ϕ2 +3 ϕ3 +3 ϕ4 +3 ϕ5 +3 +ϕ1 +4 ϕ2 +4 ϕ3 +4 ϕ4 +4 ϕ5 +4 +ϕ1 +5 ϕ2 +5 ϕ3 +5 ϕ4 +5 ϕ5 +5 +��������� +; +(A6) +|S, 5/2⟩ = Θ5/2 +1,2,3,4,5 = α1α2α3α4α5; +(A7) +|S, 3/2⟩ = Θ3/2 +1,2,3,4,5 = += +1 +√ +5 (α1α2α3α4β5 + α1α2α3β4α5 + . . . ) ; +(A8) +|S, 1/2⟩ = Θ1/2 +1,2,3,4,5 = +1 +√ +10 (α1α2α3β4β5+ ++α1α2β3α4β5 + · · · + α1α2β3β4α5 + . . . ) . +(A9) +The lower indices of the d-holes orbital coordinates +r1, r2, r3, r4, r5 are the lower indices k = 1, 2, 3, 4, 5 of the +functions in Eq. (A6). The upper indices of ϕj +k functions +list five d-shell different orbitals j = 1, 2, 3, 4, 5. Accord- +ing to Hund’s rule we take all the five possible d-orbitals +for the ground state of the ion, because the states with +identical orbital functions (and hence with opposite di- +rections of spins) correspond to the excited states of the +Mn ion having the excitation energy of electronvolts, and +they are out of consideration. Spin coordinates of differ- +ent d-holes are also indicated by the corresponding in- +dices. The dots in the brackets of (A8) and (A9) mean +that all possible permutations of four α and one β for +(A8) and three α and two β for (A9) over d-hole indices +are taken into account. +Wave functions corresponding +to the negative projections of the total spin on the z +axis Sz = −1/2, −3/2, −5/2 are the same if one changes +all α to β and vice versa. The normalization constants +for those wave functions are equal to one over square +root of the number of permutations of α and β positions +in each case, i.e., 1/ +� +C2 +5 = 1/ +√ +10, 1/ +� +C1 +5 = 1/ +√ +5, +1/ +� +C0 +5 = 1, respectively. +Thus, we have five d-holes in the 3d-shell, where the +strongest Coloumb interaction has already led to realiza- +tion of Hund’s rule, and there is the sixth localized-on- +the-ion hole in Γ8 state, which interacts with all those +five d-holes. Let us introduce the potential energy oper- +ator of remaining weaker Coloumb interactions between +the particles +ˆU = +5 +� +i=1 +U(ri − r6). +(A10) +Let us calculate the first diagonal element of such +Coloumb operator in the basis of zero total-momentum +projection functions. +Using the notation introduced +above, we can write an antisymmetrized form of the wave + +8 +function +ΨS +3/2ΨJ +−3/2 = f(0) +√ +6 +� +Φd +1,2,3,4,5Θ3/2 +1,2,3,4,5 χ− +6 β6− +−Φd +1,2,3,4,6Θ3/2 +1,2,3,4,6 χ− +5 β5− +−Φd +1,2,3,6,5Θ3/2 +1,2,3,6,5 χ− +4 β4 − . . . +� +. +(A11) +This many-particle wave function is formed by the mul- +tiplication of wave functions of the localized hole and +of five d-holes with the fixed order of their coordinates +(i = 1, 2, 3, 4, 5), followed by subtraction of all possible +multiples with consequently interchanged coordinates of +the localized hole (i = 6) and the d-shell holes. One can +prove that this procedure gives us the antisymmetric to- +tal wave function of the system in accordance with the +properties of determinant columns interchange. +Here we illustrate this result with the example of a +three-electron system. If one has an antisymmetric com- +bination of two electron wave functions φ1,2 = ϕ1ψ2 − +ϕ2ψ1 with the fixed order of arguments, then one can +show that the procedure gives us the fully antisymmetric +wave function when adding the third electron +φ1,2χ3 − φ1,3χ2 − φ3,2χ1 = += (ϕ1ψ2 − ϕ2ψ1)χ3 − (ϕ1ψ3 − ϕ3ψ1)χ2− +−(ϕ3ψ2 − ϕ2ψ3)χ1 = += +������ +ϕ1 ϕ2 ϕ3 +ψ1 ψ2 ψ3 +χ1 χ2 χ3 +������ +. +(A12) +Then +X = ⟨ΨS +3/2ΨJ +−3/2|U(r1 − r2)|ΨS +3/2ΨJ +−3/2⟩ = += |f(0)|2 +6 +� +dr1 . . . dr6× +× +� +Φd∗ +1,2,3,4,5 Θ3/2† +1,2,3,4,5 χ∗ +6 β† +6 − (5)† − (4)† − . . . +� +× +× ˆU × +� +Φd +1,2,3,4,5 Θ3/2 +1,2,3,4,5 χ6 β6 − (5) − (4) − . . . +� += += W + æ|f(0)|2, +(A13) +where the notation (5), (4) and etc. are introduced for +the terms, in which the localized hole index 6 (and hence +its coordinate r6), is swapped with the corresponding +intershell d-hole index 5, 4 and etc. +The Coloumb term W is determined by the direct +product of the multiples of the same type as it is shown +on the scheme in Fig 3 below, and it reads as +W = |f(0)|2 +�� +dr1dr2|χ−(r1)|2 +5 +� +j=1 +|ϕj(r2)|2U(r1−r2). +(A14) +And the exchange term is given by Eq. (18), where χ +stands for χ− and the numerical prefactor stems from +the normalization of wave functions and convolution of +spin wave functions with all possible cross-multiples with +FIG. 3. +The scheme of bra and ket direct multiples in Eq. A. +The total number of direct multiples is equal to 6. +permutable indices, which are shown in Figs 4(a) – 4(e). +Note that all multiples give the same contribution but +with different signs. Thus, we carry out the calculations +for the case of multiplication of (6) and (5) terms shown +in Fig. 4(a) and take proper account of the summation +of all terms with positive and negative signs +Θ3/2† +1,2,3,4,5 β† +6 Θ3/2 +1,2,3,4,6 β5 (−5 · 2 + (4 + 3 + 2 + 1) · 2) = 2. +(A15) +Note also that the remaining orbital part of the ex- +change integral (after the summation by the spin indices) +has the following form +æ = 2 +6 +� +dr1 . . . dr6 χ∗ +6 χ5 Φd∗ +1,2,3,4,5 Φd +1,2,3,4,6 × +× (U(|r5 − r6|) + . . . ) = += 1 +3 · 4! +5! +�� +dr5dr6 χ∗ +6 χ5 +5 +� +j=1 +ϕj∗ +5 ϕj +6 U(|r5 − r6|). +(A16) +The integration with all terms denoted by dots in the +first part of Eq. (A) gives us zero due to the orthogo- +nality of all orbital wave functions. The summation over +d-orbital indices in the last part of the equation is car- +ried out considering only one index j = 1, . . . 5 for both +one-particle functions ϕj∗ +5 +and ϕj +6. The latter could be +easily checked by treating the multiples in explicit forms +written one under another +χ∗ +6 Φd∗ +1,2,3,4,5 = χ∗ +6 +√ +5! +� +ϕ1 +1ϕ2 +2ϕ3 +3ϕ4 +4ϕ5 +5 − ϕ1 +1ϕ2 +2ϕ3 +3ϕ4 +5ϕ5 +4 + . . . +�∗, +(A17) +χ5 Φd +1,2,3,4,6 = χ5 +√ +5! +� +ϕ1 +1ϕ2 +2ϕ3 +3ϕ4 +4ϕ5 +6 − ϕ1 +1ϕ2 +2ϕ3 +3ϕ4 +6ϕ5 +4 + . . . +� +. +(A18) +Here the first term is determined by the fixed sequence +of the coordinate indices, and all others are determined +by the coordinate indices swapping accompanied by a +change of the sign. +One can see that the non-zero +multiples are only those which are the products of two +terms written strictly under each other in Eqs. (A17) and +(A18). All other multiples gives us zero due to the mu- +tual orthogonality of all functions. Thus, all terms are +summed up only with positive signs. The number of such +summands with fixed position of 5-th and 6-th particles + ++(6) - (5) - (4) - (3) - (2) - (1 ++(6) - (5) - (4) - (3) - (2) - (19 +(a) +(b) +(c) +(d) +(e) +FIG. 4. +The scheme of bra and ket cross-multiples in Eq. (A). +(a): There are 5·2 = 10 cross-multiples with the minus sign if +one consider five cross-multiples in the inset and their mirror +twins emerging as if the virtual reflection in the horizontal +plane takes place. (b), (c), (d) and (e): The number of each +multiple of the plus sign should be multiplied by 2 due to the +same reason as discussed for inset (a). The total number of +multiples equals to (4 + 3 + 2 + 1) · 2 = 20. +equals to the number of permutation of other four elec- +trons over remaining four orbitals, and hence it is equal +to 4! = 24. +Let us explain now, in brief, the calculation details for +the Y, Z and V terms. The Y term also involves bra and +ket functions of the same type +ΨS +1/2ΨJ +−1/2 = += f(0) +√ +6 +� +Φd +1,2,3,4,5Θ1/2 +1,2,3,4,5 +�� +2 +3Z6β6 + +� +1 +3χ− +6 α6 +� +− +−(5) − (4) − . . . } . +(A19) +Here the same notation ((5), (4), etc.) is introduced as +for the Eq. (A). One can see that the same scheme which +we use when calculating the matrix elements of direct +Coulomb interaction terms gives us, as denoted in Fig. 3, +the same value W (due to the symmetry properties of +χ− and Z functions of the localized hole), and the lat- +ter could be excluded from consideration. The exchange +terms calculation requires consideration of two possible +results of wave-functions spin parts convolution. The first +is +Θ1/2† +1,2,3,4,5 β† +6 Θ1/2 +1,2,3,4,6 β5 = += 1 +10 +� +α† +1α† +2α† +3β† +4β† +5 + α† +1α† +2β† +3α† +4β† +5 + . . . +� +β† +6× +× (α1α2α3β4β6 + α1α2β3α4β6 + . . . ) β5 = += 1 +10 +� +α† +1α† +2α† +3β† +4 + α† +1α† +2β† +3α† +4 + . . . +� +× +× (α1α2α3β4 + α1α2β3α4 + . . . ) = 4 +10. +(A20) +and the second is +Θ1/2† +1,2,3,4,5 α† +6 Θ1/2 +1,2,3,4,6 α5 = += 1 +10 +� +α† +1α† +2α† +3β† +4β† +5 + α† +1α† +2β† +3α† +4β† +5 + . . . +� +α† +6× +× (α1α2α3β4β6 + α1α2β3α4β6 + . . . ) α5 = += 1 +10 +� +α† +1α† +2β† +3β† +4 + α† +1β† +2α† +3β† +4 + . . . +� +× +× (α1α2β3β4 + α1β2α3β4 + . . . ) = 6 +10. +(A21) +Then, after the summation over the spin indices and tak- +ing into account possible cross-multiples, as in Figs. 4(a) +– 4(e), we obtain an additional multiplier (−5 · 2 + (4 + +3 + 2 + 1) · 2) = 10 as in Eq. (A15), and then we get the +exchange part of Y equal to +|f(0)|2 10 +6 +� +dr1 . . . dr6 +�2 +3 +4 +10Z∗ +6 Z5 + 1 +3 +6 +10χ∗ +6 χ5 +� +× +×Φd∗ +1,2,3,4,5 Φd +1,2,3,4,6 (U(|r5 − r6|) + . . . ) = += |f(0)|2 7 +9 · 4! +5! +�� +dr5dr6 χ∗ +6 χ5× +× +5 +� +j=1 +ϕj∗ +5 ϕj +6 U(|r5 − r6|) = 7 +3æ|f(0)|2. +(A22) +Here we used the symmetry equivalence of Z and χ func- +tions when calculating such type of integrals. +When calculating Z, the off-diagonal matrix element +between quantum states from Eq. (A) and Eq. (A19) is +taken. Thus, there is no Coloumb term, and the exchange +integral in this case reads as +Z = |f(0)|2 10 +6 +� +dr1 . . . dr6 Φd∗ +1,2,3,4,5 Θ3/2† +1,2,3,4,5 χ−∗ +6 +β† +6× +× ˆU Φd +1,2,3,4,6 Θ1/2 +1,2,3,4,6 +�� +2 +3Z5β5 + +� +1 +3χ− +5 α5 +� += += |f(0)|2 10 +6 +� +dr1 . . . dr6 Φd∗ +1,2,3,4,5 Θ3/2† +1,2,3,4,5 χ−∗ +6 +β† +6× +× ˆU Φd +1,2,3,4,6 Θ1/2 +1,2,3,4,6 +� +1 +3χ− +5 α5. +(A23) + ++(6) - (5) - (4) - (3) - (2) - (1 +5 ++(6) - (5) - (4) - (3) - (2) - (1+(6) - (5) - (4) - (3) - (2) - (1 +4 ++(6) - (5) - (4) - (3) - (2) - (1+(6) - (5) - (4) - (3) - (2) - (1 ++(6) - (5) - (4) - (3) - (2) - (1)+(6) - (5) - (4) - (3) - (2) - (1 +2 ++(6) - (5) - (4) - (3) - (2) - (1)+(6) - (5) - (4) - (3) - (2) - (1 ++(6) - (5) - (4) - (3) - (2) - (1)10 +The multiplier 10 arises as a result of the usage of the +introduced scheme, which implies taking into account all +exchange integrals (see Figs. 4(a) – 4(e)). There are no +multiples with Z functions because the convolution of +spin functions gives zero, as the considered summands +have 4α and 1β parts of Θ3/2† +1,2,3,4,5 spin function, and 3α +and 2β enter Θ1/2 +1,2,3,4,6 spin function. The spin convolu- +tion of the latter part is equal to +Θ3/2† +1,2,3,4,5β† +6Θ1/2 +1,2,3,4,6α5 = += +1 +√ +5 +� +α† +1α† +2α† +3α† +4β† +5 + α† +1α† +2α† +3β† +4α† +5 + . . . +� +β† +6× +× 1 +√ +10 (α1α2α3β4β6 + α1α2β3α4β6 + . . . ) α5 = += +1 +√ +50 +� +α† +1α† +2α† +3β† +4 + α† +1α† +2β† +3α† +4 + . . . +� +× +× (α1α2α3β4 + α1α2β3α4 + . . . ) = +4 +√ +50. +(A24) +Then finally we get +Z = |f(0)|2 10 +6 +4 +√ +50 +1 +√ +3 +4! +5! +�� +dr5dr6 χ∗ +6 χ5 × +× +5 +� +j=1 +ϕj∗ +5 ϕj +6 U(|r5 − r6|) = 2 +√ +2 +√ +3 æ|f(0)|2. +(A25) +The calculation of another off-diagonal matrix element +V requires usage of the following ket wave function ac- +cording to Eq. (16) +ΨS +−1/2ΨJ +1/2 = += f(0) +√ +6 +� +Φd +1,2,3,4,5Θ−1/2 +1,2,3,4,5 +�� +2 +3Z6α6 + +� +1 +3χ+ +6 β6 +� +− +−(5) − (4) − . . . } . +(A26) +As the spin function Θ1/2 +1,2,3,4,5 contains 3α and 2β in +each summation term, while the function Θ−1/2 +1,2,3,4,5, on +the contrary, contains 2α and 3β in each summand, one +can see that the exchange matrix element of interaction +between quantum states given by Eq. (A19) and Eq. (A) +will reduce to the following expression, having non-zero +contributions from only Z-orbital terms. +V = |f(0)|2 10 +6 +� +dr1 . . . dr6 Φd∗ +1,2,3,4,5 Θ1/2† +1,2,3,4,5× +× +� +2 +3Z∗ +6 β† +6 ˆU Φd +1,2,3,4,6 Θ−1/2 +1,2,3,4,6 +� +2 +3Z5α5 = += |f(0)|2 10 +6 +2 +3 +6 +10 +4! +5! +�� +dr5dr6 Z∗ +6 Z5× +× +5 +� +j=1 +ϕj∗ +5 ϕj +6 U(|r5 − r6|) = 2æ|f(0)|2, +(A27) +in which the result of spin functions convolution is in- +cluded +Θ1/2† +1,2,3,4,5β† +6Θ−1/2 +1,2,3,4,6β5 = += 1 +10 +� +α† +1α† +2α† +3β† +4β† +5 + α† +1α† +2β† +3α† +4β† +5 + . . . +� +β† +6× +× (β1β2β3α4α6 + β1β2α3β4α6 + . . . ) α5 = += 1 +10 +� +α† +1α† +2β† +3β† +4 + . . . +� +(α1α2β3β4 + . . . ) = 6 +10. +(A28) +[1] D. D. Awschalom and M. E. Flatt´e, Nature physics 3, +153 (2007). +[2] B. R. Ortiz, L. C. Gomes, J. R. Morey, M. Winiarski, +M. Bordelon, J. S. Mangum, I. W. Oswald, J. A. +Rodriguez-Rivera, J. R. Neilson, S. D. Wilson, et al., +Physical Review Materials 3, 094407 (2019). +[3] M. Otrokov, I. P. Rusinov, M. 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Novikova, (Izdatel Nauka, Moscow, 1974). + diff --git a/vtE4T4oBgHgl3EQfXAy5/content/tmp_files/load_file.txt b/vtE4T4oBgHgl3EQfXAy5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6c22f1d2f4a574bb0e02b5da001a559095675f34 --- /dev/null +++ b/vtE4T4oBgHgl3EQfXAy5/content/tmp_files/load_file.txt @@ -0,0 +1,726 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf,len=725 +page_content='Exchange interaction for Mn acceptor in GaAs: revealing its strong deformation dependence I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Krainov,1, ∗ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Baryshnikov,1, † A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Karpova,1, 2 and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Averkiev1 1Ioffe Institute, 194021 St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Petersburg, Russia 2Saint-Petersburg Electrotechnical University, 197022 St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Petersburg, Russia (Dated: January 13, 2023) In this paper we calculate exchange interaction constant between manganese ion inner electronic d- shell and GaAs valence band bounded hole using their microscopic multiparticle wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' We reveal its parametric dependence on crystal lattice deformations and find out that it could be about and even more than dozens percent when the strain tensor reaches values of 10−3÷10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' This fact is in accordance with the previous hypothesis of deformation dependence of Mn acceptors in GaAs fine energy structure obtained from Raman spectroscopy, and we show that this dependence has the same magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Also, we resolve here the problem of a substantial high temperature mismatch between well-developed theory and experimental data for the static magnetic susceptibility of Mn ions in GaAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' We show by numerical estimates and calculations that quite a strong parametric dependence of the exchange coupling value on GaAs lattice expansion determines the high temperature (above 50 K) magnetic susceptibility reduction as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' INTRODUCTION Modern material science is focused on functional mate- rials combining different properties with maximal func- tionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' One of these important kinds of such mate- rials is magnetic semiconductors mixing electrical, opti- cal and magnetic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Different ways to control these properties merge into the important directions of research, including the production of new compounds [1– 6], nanostructure design [1, 7–11] and investigation of the effects of external forces application [12–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' One of the most well-known functional materials is GaMnAs [17–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' In this material manganese ion with its inner magnetic 3d-shell containing 5 electrons brings magnetism to GaAs semiconductor host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' This is due to the exchange inter- actions between manganese inner d-electrons with GaAs holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Also, the manganese impurity acts as an accep- tor increasing hole concentration in GaAs semiconductor crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' For an isolated impurity the exchange interac- tion between Mn half-filled d-shell with the total spin of electrons 5/2 and localized hole in the Γ8 symmetry state acting like a 3/2 spin results in initially 24-hold degenerate state into into 4 sublevels with total angular momentum F = 1, 2, 3, 4 with F = 1 being the ground state [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Here we stress our attention on this exchange interaction between isolated manganese ion and a hole bounded on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Exchange interaction constant A for Mn acceptor con- sist of two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The first part includes the exchange between Mn d-shell orbital electrons and Bloch orbital of the Γ8 hole, and the second part includes value of hole envelope at impurity site, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=', the probability to couple with the half-filled d-shell as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' In all previous works [21–23], in which such interaction was discussed, ∗ igor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content='kraynov@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content='ru † barysh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content='1989@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content='com only the second part (value of hole envelope at impu- rity site) was assumed to change in different conditions, while the first part (exchange between Bloch functions) was assumed to be unperturbed, and its value was pos- tulated [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The deformation influence on the envelope part of exchange constant has been investigated in [23], but it has been found that it changes by less than one percent at pressures on the limit of GaAs hardness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The purpose of this work is to calculate exchange interaction value between Bloch functions of Mn d-shell and Γ8 hole bounded from the GaAs valence band, and to treat its dependence on deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' We demonstrate that this part of exchange interaction is sensitive to the presence of crystal strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Previously, the assumption of a strong dependence of exchange constant A on the crystal deformation played a crucial role in the study of the fine structure of an isolated Mn acceptor in GaAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The latter was investigated using Raman spin-flip scattering and its dependence on mag- netic fields and external deformations at helium temper- atures [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The theoretical fit of intra and inter transi- tions between Mn-hole levels based on a standard model of the Mn acceptor eigenstates was also carried out in [24], but to make a satisfactory agreement between all experimental curves and theoretical calculations the de- formation dependence of exchange interaction constant was phenomenologically proposed and its value was es- timated from comparison with the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The ex- change interaction value changes by 20 % for the pres- sure 5 kbar, which is about half of GaAs critical value of hardness, and hence this change is much larger than previously mentioned nearly one percent dependence on hole envelope wave function change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Independently, there are drastically different measure- ments of static magnetic susceptibility behavior in a wide temperature range in GaAs samples with low concentra- tion of Mn ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The first experiments were made by An- drianov’s group [25], but their work contains an irrelevant arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content='05038v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content='mtrl-sci] 12 Jan 2023 2 theoretical model of the Mn center, which mismatches with a bunch of low temperature properties of the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Other measurements were carried out by Frey’s group and reported in [26], where the relevant theoretical model was applied, which, however, has some discrepancies with the data at the very high temperature edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The state of art of these studies is that the theoretical fit based on that true and now standard Mn-hole interaction model of experimental data is in a good agreement with the low temperature region below 50 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' But for the high temper- ature region, there is a reduction of magnetic suscepti- bility compared with the theoretical prediction, which is still puzzling nowadays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' A recent paper [16] containing a deep review of different experimental and theoretical facts about Mn center in GaAs proposed a hypothesis that variance mentioned above could be explained by the Jahn–Teller effect (JTE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' In this paper we also test this hypothesis (see the Sup- plementary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' It is known from many other ex- perimental facts [16] that the Mn ground F = 1 state is unaffected by the static Jahn–Teller distortion, so only dynamical JTE should be tested [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Moreover, one can show that at high temperatures, there is only one way for dynamical JTE to occur, which is reduced to the Jahn- Teller interaction of hole in Γ8 state with local lattice dis- tortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' As we show (see the Supplementary material, part 2), the high temperature dependence of magnetic susceptibility is negligibly dependent on the Jahn–Teller effect and ceases quite rapidly as temperature increases that can not explain observed reduction of magnetic sus- ceptibility discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Also we test a hypothesis of the crystal field influence, but it also can not explain magnetic susceptibility reduction at high temperatures (see the Supplementary material, part 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' But here we show that if we link the phenomenological dependence of exchange interaction value on external deformation from [24] with the thermal expansion coefficient of the crystal, the problem of high-temperature magnetic susceptibility reduction can be elegantly resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' In this paper we will calculate Mn-hole exchange- interaction value part associated with the Bloch wave functions overlapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Then we provide an estimate for this strain dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Note that the trace of strain ten- sor for the pressure about 5 kbar is in the range of 10−3 – 10−2, and it is quite surprising how it can lead to a strong dependence of the exchange constant ∼ 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' We elabo- rate and explain a simple mechanism that could explain this fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Further, we show that such purely theoreti- cal estimates result in the similar variation for Mn-hole exchange constant on stress as assumed in [24], which has the same order of value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Finally, we show by di- rect calculations that the obtained dependence of A on the crystal strains ε, which theoretically fits Mn fine en- ergy structure [24], leads to a better agreement between high-temperature magnetic-susceptibility calculation re- sults and experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' We also believe that the developed model could be applied to another magnetic impurities and hosts with appropriate modifications in symmetry analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' THEORY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Exchange Hamiltonian and representation of total angular momenta F = 1, 2, 3, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Eigenstates of Mn acceptor are composed from the six- fold degenerate state of Mn ion d-shell electrons in the ground state with total spin S = 5/2 and the fourfold degenerate state of a localized hole having the Γ8 symme- try, which corresponds to the total angular momentum J = 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Further, to simplify all conclusions, we will work in the hole basis of the d-shell, which has the same properties as the electronic one, because the shell is half- filled, and one-particle states simply have opposite spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' These eigenstates are split by exchange interaction be- tween the half-filled d-shell and the localized hole result- ing in the total angular momentum states F = 1, 2, 3, 4 with corresponding degeneracy equal to 2F + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' So, if we assume that the exchange interaction between the ion’s d-shell and the hole is described by the only one constant A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=', if we set the corresponding Hamiltonian as ˆHex = A( ˆS · ˆ J) = A 2 � ˆF 2 − ˆS2 − ˆJ2� , (1) ˆF = ˆS + ˆJ, then one can easily find out all its energy eigenvalues, which are A(F(F +1)−S(S +1)−J(J +1))/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' All other possible terms proportional to ( ˆS · ˆ J)2 and ( ˆS · ˆ J)3 are connected with the second-order and higher-order per- turbation terms of Coloumb interaction causing change of spin projections of d-shell electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' We will neglect such terms because the energy of spin-spin interaction be- tween the d-shell electrons is assumed to be the largest among all other energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' This assumption allows us to consider all processes as if no changes in spin states of the inner shell electrons occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Note also that there are no spin-orbit splittings in the d-shell, which is confirmed by the Raman scattering data of Mn0 centers in GaAs, which has g-factor strictly equal to 2 [24, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' To calculate the eigenenergies of ˆHex, it is sufficient to use the subset from the whole basis of acceptor states, because of the spherical symmetry of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Let us consider such a subset consisting of only 4 wave functions |F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Fz = 0⟩ (where F = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' and taking it from [16] (note that the prefactor coefficient in |2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 0⟩ function is changed to normalize correctly the wave func- 3 tion) one can write it down as |1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 0⟩ = 1 √ 5 � ΨS 3/2ΨJ −3/2 − ΨS −3/2ΨJ 3/2− − � 3 2ΨS 1/2ΨJ −1/2 + � 3 2ΨS −1/2ΨJ 1/2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (2) |2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 0⟩ = � 3 7 � ΨS 3/2ΨJ −3/2 + ΨS −3/2ΨJ 3/2− − � 1 6ΨS 1/2ΨJ −1/2 − � 1 6ΨS −1/2ΨJ 1/2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (3) |3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 0⟩ = 1 √ 5 �� 3 2ΨS 3/2ΨJ −3/2 − � 3 2ΨS −3/2ΨJ 3/2+ +ΨS 1/2ΨJ −1/2 − ΨS −1/2ΨJ 1/2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (4) |4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 0⟩ = � 3 7 �� 1 6ΨS 3/2ΨJ −3/2 + � 1 6ΨS −3/2ΨJ 3/2+ +ΨS 1/2ΨJ −1/2 + ΨS −1/2ΨJ 1/2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (5) Then one can calculate all energy differences between ˆHexch eigenstates as EF +1 − EF = = ⟨F + 1, 0| ˆHexch|F + 1, 0⟩ − ⟨F, 0| ˆHexch|F, 0⟩ = = 2A, 3A, 4A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (6) This result could be obtained by taking subset of 4 wave functions, which contain only zero projections of the total momentum on z axis: � ΨS 3/2ΨJ −3/2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' ΨS −3/2ΨJ 3/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' ΨS 1/2ΨJ −1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' ΨS −1/2ΨJ 1/2 � , generating |F, 0⟩ states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' By calculating the exchange Hamiltonian using these wave functions as bra and ket functions, one can obtain a 4×4 matrix, which eigenvalues give us the same energy differ- ences as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' So, the main idea for microscopic calculation of A via exchange integrals is to consider the first-order correc- tion to the energies of d-states and of the hole state due to the Coulomb interaction calculated using only these 4 wave functions with appropriate symmetrization and antisymmetrization of all multiparticle orbitals and spin states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Microscopic calculation of exchange integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Wave functions of bounded Γ8 hole corresponding to the total moment J = 3/2 include envelope and Bloch parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Within the framework of the effective mass method for shallow acceptors in cubic semiconductors in the spherical approximation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' the wave function of this hole is the sum of the products of the Bloch amplitudes Xµ and the smooth envelopes R0(r) and R2(r) ΨJ 3/2 = R0(r)Y00X3/2 + 1 √ 5R2(r)Y20X3/2 − − � 2 5R2(r)Y21X1/2 + � 2 5R2(r)Y22X−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (7) ΨJ 1/2 = R0(r)Y00X1/2 − 1 √ 5R2(r)Y20X1/2 + + � 2 5R2(r)Y2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content='−1X3/2 + � 2 5R2(r)Y22X−3/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (8) ΨJ −1/2 = R0(r)Y00X−1/2 − 1 √ 5R2(r)Y20X−1/2 + + � 2 5R2(r)Y2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content='1X−3/2 + � 2 5R2(r)Y2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content='−2X3/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (9) ΨJ −3/2 = R0(r)Y00X−3/2 + 1 √ 5R2(r)Y20X−3/2 − − � 2 5R2(r)Y2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content='−1X−1/2 + � 2 5R2(r)Y2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content='−2X1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (10) where Ylm are the spherical functions corresponding to the orbital moment l and its projection m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The ex- change interaction integral will involve these functions and the d-shell wave functions, which are located in one elementary cell at the impurity site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' We can neglect the effect of R2(r) functions because they tend to zero limit at the magnetic impurity site, while R0(r) func- tions take nonzero values (see calculations results in [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Thus, in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' A we calculate all exchange integrals us- ing only Bloch parts of Γ8 hole wave functions, setting ΨJ µ ≈ f(0)Xµ, where f(0) = R0(0)/ √ 4π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Basing on spin configurations of wave functions ψi (i = 1, 2, 3, 4) � ΨS 3/2ΨJ −3/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' ΨS −3/2ΨJ 3/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' ΨS 1/2ΨJ −1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' ΨS −1/2ΨJ 1/2 � (11) one can show that the Hamiltonian of Coulomb interac- tion has a following 4 × 4 matrix form in this basis ˆHC = � � � X Z 0 0 Z Y V 0 0 V Y Z 0 0 Z X � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (12) The details of calculation one can see in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' A, where the true microscopical multiparticle structure of all wave functions is taken into account, here we represent the 4 results X = ⟨ΨS 3/2ΨJ −3/2|U(r1 − r2)|ΨS 3/2ΨJ −3/2⟩ = = W + æ|f(0)|2, (13) Y = ⟨ΨS 1/2ΨJ −1/2|U(r1 − r2)|ΨS 1/2ΨJ −1/2⟩ = = W + 7 3æ|f(0)|2, (14) Z = ⟨ΨS 3/2ΨJ −3/2|U(r1 − r2)|ΨS 1/2ΨJ −1/2⟩ = = 2 √ 2 √ 3 æ|f(0)|2, (15) V = ⟨ΨS 1/2ΨJ −1/2|U(r1 − r2)|ΨS −1/2ΨJ 1/2⟩ = = 2æ|f(0)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (16) Here the Coloumb potential is used, which is given by the expression U(r1 − r2) = e2 |r1 − r2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (17) Note that we treat the Coloumb interaction between the localized hole and holes in the d-shell (as empty states in the half-filled shell), and hence we have the positive sign in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The W terms in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (13–14) could be excluded from the consideration because they result in equal general energy shift of all 4 states due to the Coloumb interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The main result is the connection of X, Y, Z and V terms with the exchange integral æ, which reads as æ = 5 � j=1 �� Ω dr1dr2 15 ϕj∗(r1)ϕj(r2)U(r1 − r2)χ∗(r2)χ(r1), (18) where integrations goes over directly doubled GaAs- crystal elementary cell volume Ω, the sum is taken over all five one-electron orbitals of the 3d-shell of the man- ganese ion ϕj (the upper index numerates all possible orbital states j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' , 5), and there is an overlapping with a p-like Bloch part of the localized hole wave func- tion χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The eigenvalues of matrix (12) give us the following energy differences between eigenstates of this system E2 − E1 = 4 3æ|f(0)|2, E3 − E2 = 2æ|f(0)|2, E4 − E3 = 8 3æ|f(0)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (19) One can see from (6) that they give the same ratio be- tween energy differences as in the phenomenological ap- proach using Hamiltonian (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' And these results totally coincide if one puts A = 2 3æ|f(0)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (20) The latter expression gives us the tool for microscopic calculations of external forces effects on the exchange constant A, which is relevant for a lot of measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Exchange constant dependence on deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' From the symmetry point of view possible dependence of exchange constant on deformation reads as ˆH = A0( ˆS · ˆ J) + BP Tr(ˆε)( ˆS · ˆ J) + CP � i,j ˆSi ˆJjεij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (21) If one consider hydrostatic deformation, constant A de- pends only on the trace of deformation tensor εij = δijTr(ˆε)/3 (here δij is the Kronecker delta-symbol) ˆH = A0( ˆS · ˆ J) + (BP + CP /3)Tr(ˆε)( ˆS · ˆ J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (22) Further, we will neglect the dependence of envelope wave functions f(0) on deformation ε, because their change is too small (it is in the order of 1 % of observed values [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' To understand the microscopic foundations of such Hamiltonian dependence on deformation, we assume that the true wave functions of the p-type forming the Bloch eigenstates of the valley band could be admixed by some other atomic states, for example, via the pd hybridiza- tion mechanism keeping the total symmetry of the state unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Such hybridization can occur due to differ- ent reasons, for example, due to the lack of inversion symmetry in the Td group or the action of some inter- nal potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' We suggest here to consider the admix- ing mechanism stemming from the existence of random electric fields that commonly present near Mn impurity centers in GaAs [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Such random fields are usually considered as an additional source of fine structure split- tings in the Mn acceptor energy spectrum [16, 24], but they also could affect local wave functions of bounded holes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=', the Bloch wave functions due to the pd hy- bridization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Thus, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (18) the χ functions should be substituted by the hybridized combinations like ˜χ ≈ χ + � d γdϕd, ˜ϕd ≈ ϕd − γ∗ dχ, γd = ⟨ϕd| ˆV |χ⟩ Ep − Ed .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (23) Here Ep and Ed represent the pure atomic energies of pure p- and d-states without hybridization Ep − Ed ∼ 1 eV (we assume here that for Mn-acceptor in GaAs, the pure d-state is lying not very far from the top of the valence band, and hence pd interaction is the most large one), and term ˆV = r · F stands for the hybridization operator admixing one state to another via electro-dipole induction mechanism, which is due to some local random electric force F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The latter could be very sensible to the change of the elementary cell if the deformation of the crystal occurs F ′ i = Fi + αεijFj, (24) Here we introduce dimensionless parameter α that taking into account deformation dependence of random fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' We assume that the applied stress is a small parameter 5 of the theory, so αε ≪ 1, and further we take into account only linear terms on stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Thus, we can estimate the change of æ under a pressure or a temperature-affected widening using the following assumptions about local electric force properties ˆV 2 ≈ rirk(FiFk + αεkmFiFm + αεijFjFk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' ⟨⟨Fi⟩⟩ = 0, ⟨⟨FiFj⟩⟩ = ζδij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Here double angle brackets represent averaging by pos- sible realizations of random forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Of course, the true averaging should be processed over observable values, al- though the mean value of an observable depends on de- formation approximately the same way as the observable calculated with such averaged value of exchange constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Finally, one can conclude that after averaging by ran- dom forces Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (18) could be represented by the following terms æ ≈ æppdd + � l,i æll dddd ⟨χ|ri|ϕl⟩⟨ϕl|ri|χ⟩ (Ep − Ed)2 ζ � 1+ 2 3αTr(ˆε) � , (25) where æppdd = 5 � j=1 �� Ω dr1dr2 15 ϕj∗(r1)ϕj(r2)U(r1 − r2)× ×χ∗(r2)χ(r1), æln dddd = 5 � j=1 �� Ω dr1dr2 15 ϕj∗(r1)ϕj(r2)U(r1 − r2)× ×ϕl∗(r2)ϕn(r1), which are exchange integrals with different integrand functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' We should note that the values of these terms depend on the functions overlap, and hence the more d- functions of Mn ion are involved, the larger the value of the Coulomb term is æppdd ≪ ædddd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' To estimate the magnitude of the effect, we first take into account that all exchange integrals between d- functions have the same value in sum in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Then using the hydrogen atom functions χ corresponding to 4p orbitals and ϕd corresponding to 3d orbitals one can obtain an estimate ædddd/æppdd ≈ 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Also, we can take matrix elements of coordinates approximately equal to the Bohr radius of the atom ⟨χ|ri|ϕl⟩ ≈ ⟨ϕl|ri|χ⟩ ≈ aB ≈ 10−8 cm, and the value of the random forces dispersion could be estimated as having the order of a typical inter- atomic interaction term √ζ = F∗ ≈ 106 eV/cm (which is comparable with typical values of the mean force affect- ing the nuclear complex of the lattice cell in GaAs in the case of the Cu ion, for which F ≈ 5·106 eV/cm [29, 30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Then we can write an estimate for exchange constant A change with deformation (AP ≡ BP + CP /3) A = A0 + AP Tr(ˆε), (26) where AP A0 ≈ 2 3α (15(aBF∗)2/(Ep − Ed)2)ædddd/æppdd 1 + (15(aBF∗)2/(Ep − Ed)2)ædddd/æppdd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (27) From data analysis in [24], we can estimate alpha as AP /A0 = (900 meV/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content='5 meV) ≈ 360, which is equiv- alent to the relative change of A nearly by δA/A0 ∼ αTr(ε) ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content='2 at a half of critical strain of GaAs crystal corresponding to hardness limit at helium temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' CALCULATIONS AND DISCUSSION We have discussed above the parametric dependence of exchange constant value on crystal deformation and its microscopic reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' This fact had already played its role in the explanation of Raman scattering experi- ment results [24], and now we are going to demonstrate clearly that the same fact is responsible for high temper- ature magnetic susceptibility reduction measured inde- pendently in a completely different experimental setting [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' As GaAs crystal undergoes thermal expansion, we are going to test our hypothesis of this expansion being re- sponsible for anomalous reduction of magnetic suscepti- bility at relatively high temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The temperature dependence of linear expansion coefficient αT could be found in literature (see, for example, [31] or [32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' We show this dependence in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 0 50 100 150 200 250 300 –1 0 1 2 3 4 5 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Temperature dependence of linear expansion coef- ficient αT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Dark orange circles are experimental results from [32] (see Table 80 on page 233), black line is our interpolation for this dependence up to 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' We will use a simple function to interpolate the αT de- pendence on temperature, which makes the interpolation 6 work up to 300 K quite well (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 1) ˜αT = � � � 0, T < 50 K, C tanh � T − 50 180 − 50 � , 50 K ≤ T ≤ 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (28) It is implied that T is measured in kelvins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The coefficient C = 6 · 10−6 K−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Note that there is a slight increase in the αT coefficient above 300 K (at 800 K it reaches 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content='4·10−6 K−1, see the full table of its values in [32]), and hence the approximation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (28) does not work if T > 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' But for our purposes it is enough to consider the region of T < 300 K, in which the interpolation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (28) describes experimental data quite well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Note that a very small decrease in αT values between 25 K and 50 K does not affect the observables in any reasonable manner, thus, we neglect it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' We are interested in temperature range T = 0÷300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' So we can write the dependence of exchange value A on T taking into account Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (26) A(T) = A0 + AP · 3˜αT · T, (29) where A0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content='6 meV [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' We have multiplied ˜αT by a factor of 3 to get the bulk thermal expansion coefficient from the linear one, because Tr(ε) = εxx(T) + εyy(T) + εzz(T) = 3εxx(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Here we use the same value of AP = 900 meV as in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' One can see the calculations results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Note that in [16] the electron-hole basis is used, hence one should change the sign of the exchange constant into opposite one compared with the our result to obtain the same order of energy levels for manganese acceptor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Thus, substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 29 into the formulas in [16], we need to multiply A(T) by (−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' As can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 2 the relative mismatch be- tween theory and experimental results at T > 100 K re- duces approximately from 50% to 20%, if we use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' This reduction of the systematic mismatch leads to a bet- ter agreement between theoretical results and the exper- imental data in the high-temperature region, which have the allowable magnitude of the experimental error (see discussion in [16], experimental data have been first ob- tained in [25], and the same mismatch has also been inde- pendently mentioned in [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Also we point out that the sign of changes of exchange interaction constant, which we use to fit magnetic susceptibility data, is the same as used in Raman experiments [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Note that other possible factors, such as crystal field effect or reduction of mag- netic susceptibility caused by the dynamical Jahn-Teller effect observed by us in Supplementary materials, give no pronounce effects on magnetic susceptibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Moreover, their effects diminish at high temperatures, and they also ruin the well-established theory predictions at low tem- peratures below 50 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Thus, the effect of exchange constant parametric de- pendence on lattice deformation is the only effect that provides reasonable explanation of both high and low temperature behaviour characteristics of the manganese acceptor center in GaAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Note also that A(T) at T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content='020 0 1 2 3 4 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Temperature dependence of static magnetic suscep- tibility κ of manganese ions in GaAs crystal (concentration of manganese ions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content='3 · 1018 cm−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Both axes have logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Black circles connected by dash lines represent experi- mental data from [16, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Light orange solid line is a result of calculation based on the ordinary theory from [16], which im- plies that A(T) = A0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content='6 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Violet solid line represents our calculation result, where the expression for magnetic sus- ceptibility taken from [16] is modified by taking into account the exchange constant variation with temperature A(T) due to thermal expansion effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 300 K is nearly three times larger than A0, and it could reach even higher values at higher temperatures accord- ing to [32] and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Note that at such big changes in A the nonlinear terms on lattice deformation should be also taken into account in the pd hybridization mechanism of exchange constant renormalization via random fields as soon as the parameter αTr(ε) reaches and exceeds the limit of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' But we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 2 that even linear terms give the right trend in temperature dependence of mag- netic susceptibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' CONCLUSION Exchange constant value between d-electrons of man- ganese ion impurity in GaAs crystal and the hole, local- ized from the valence band on the impurity ion, is mi- croscopically derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The effect of crystal lattice period change on the value of the exchange coupling constant occurring via the hybridization of exchanging orbitals is shown and estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' We also discuss the effect of the thermal expansion causing the change in magnetic susceptibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' We show that accounting for this effect leads to a better agreement between theoretical results and magnetic susceptibility data measured at high tem- peratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The considered thermal widening mechanism does not influence the low-temperature magnetic suscep- tibility behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' This result is also in agreement with 7 another experiment of Raman scattering on Mn accep- tors in GaAs with applied external strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' We believe that the approach to the analytical calculation of the ex- change constant could be generalized to the case of other magnetic centers in semiconductor structures and semi- magnetic compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work has been supported by the Russian Science Foundation (analytical theory – Project 18-72-10111).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' thanks the Theoretical Physics and Mathemat- ics Advancement Foundation ”BASIS”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' We also thank M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Nestoklon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Tarasenko, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Glazov for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' We dedicate this article to the mem- ory of our colleague and co-author V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Sapega (Ioffe Institute) who passed away in 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Appendix A: Calculation of exchange integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The localized-on-the-ion hole has the Bloch part of wave function, which describes both spin and orbital de- grees of freedom in Γ8-state ΨJ 3/2 = −αX + iY √ 2 f(0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A1) ΨJ 1/2 = �� 2 3αZ − β X + iY √ 6 � f(0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A2) ΨJ −1/2 = �� 2 3βZ + αX − iY √ 6 � f(0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A3) ΨJ −3/2 = β X − iY √ 2 f(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A4) Here α and β means spin-up and spin-down states of the hole captured and localized from the valley band of GaAs crystal, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Space orbitals X, Y and Z corre- spond to a p-like orbitals, which form the valley band of the crystal, and hence one can prove that they are quite similar from the cubic symmetry point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' So we will use more compact notations as χ+ = −(X +iY )/ √ 2 and χ− = (X − iY )/ √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The half-filled 3d-shell of the Mn ion is described by a five-hole wave function with the totally symmetrical spin part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' We assume that Hund’s rule is the most powerful here, and all spin-spin interaction in the shell has already led to the appearance of co-directed spins of all five d- holes resulting in the total spin S = 5/2, and hence it has an antisymmetric orbital part ΨS Sz = Φd 1,2,3,4,5|S, Sz⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A5) Φd 1,2,3,4,5 = 1 √ 5!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' ��������� ϕ1 1 ϕ2 1 ϕ3 1 ϕ4 1 ϕ5 1 ϕ1 2 ϕ2 2 ϕ3 2 ϕ4 2 ϕ5 2 ϕ1 3 ϕ2 3 ϕ3 3 ϕ4 3 ϕ5 3 ϕ1 4 ϕ2 4 ϕ3 4 ϕ4 4 ϕ5 4 ϕ1 5 ϕ2 5 ϕ3 5 ϕ4 5 ϕ5 5 ��������� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A6) |S, 5/2⟩ = Θ5/2 1,2,3,4,5 = α1α2α3α4α5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A7) |S, 3/2⟩ = Θ3/2 1,2,3,4,5 = = 1 √ 5 (α1α2α3α4β5 + α1α2α3β4α5 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A8) |S, 1/2⟩ = Θ1/2 1,2,3,4,5 = 1 √ 10 (α1α2α3β4β5+ +α1α2β3α4β5 + · · · + α1α2β3β4α5 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A9) The lower indices of the d-holes orbital coordinates r1, r2, r3, r4, r5 are the lower indices k = 1, 2, 3, 4, 5 of the functions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The upper indices of ϕj k functions list five d-shell different orbitals j = 1, 2, 3, 4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Accord- ing to Hund’s rule we take all the five possible d-orbitals for the ground state of the ion, because the states with identical orbital functions (and hence with opposite di- rections of spins) correspond to the excited states of the Mn ion having the excitation energy of electronvolts, and they are out of consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Spin coordinates of differ- ent d-holes are also indicated by the corresponding in- dices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The dots in the brackets of (A8) and (A9) mean that all possible permutations of four α and one β for (A8) and three α and two β for (A9) over d-hole indices are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Wave functions corresponding to the negative projections of the total spin on the z axis Sz = −1/2, −3/2, −5/2 are the same if one changes all α to β and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The normalization constants for those wave functions are equal to one over square root of the number of permutations of α and β positions in each case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=', 1/ � C2 5 = 1/ √ 10, 1/ � C1 5 = 1/ √ 5, 1/ � C0 5 = 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Thus, we have five d-holes in the 3d-shell, where the strongest Coloumb interaction has already led to realiza- tion of Hund’s rule, and there is the sixth localized-on- the-ion hole in Γ8 state, which interacts with all those five d-holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Let us introduce the potential energy oper- ator of remaining weaker Coloumb interactions between the particles ˆU = 5 � i=1 U(ri − r6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A10) Let us calculate the first diagonal element of such Coloumb operator in the basis of zero total-momentum projection functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Using the notation introduced above, we can write an antisymmetrized form of the wave 8 function ΨS 3/2ΨJ −3/2 = f(0) √ 6 � Φd 1,2,3,4,5Θ3/2 1,2,3,4,5 χ− 6 β6− −Φd 1,2,3,4,6Θ3/2 1,2,3,4,6 χ− 5 β5− −Φd 1,2,3,6,5Θ3/2 1,2,3,6,5 χ− 4 β4 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A11) This many-particle wave function is formed by the mul- tiplication of wave functions of the localized hole and of five d-holes with the fixed order of their coordinates (i = 1, 2, 3, 4, 5), followed by subtraction of all possible multiples with consequently interchanged coordinates of the localized hole (i = 6) and the d-shell holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' One can prove that this procedure gives us the antisymmetric to- tal wave function of the system in accordance with the properties of determinant columns interchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Here we illustrate this result with the example of a three-electron system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' If one has an antisymmetric com- bination of two electron wave functions φ1,2 = ϕ1ψ2 − ϕ2ψ1 with the fixed order of arguments, then one can show that the procedure gives us the fully antisymmetric wave function when adding the third electron φ1,2χ3 − φ1,3χ2 − φ3,2χ1 = = (ϕ1ψ2 − ϕ2ψ1)χ3 − (ϕ1ψ3 − ϕ3ψ1)χ2− −(ϕ3ψ2 − ϕ2ψ3)χ1 = = ������ ϕ1 ϕ2 ϕ3 ψ1 ψ2 ψ3 χ1 χ2 χ3 ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A12) Then X = ⟨ΨS 3/2ΨJ −3/2|U(r1 − r2)|ΨS 3/2ΨJ −3/2⟩ = = |f(0)|2 6 � dr1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' dr6× × � Φd∗ 1,2,3,4,5 Θ3/2† 1,2,3,4,5 χ∗ 6 β† 6 − (5)† − (4)† − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' � × × ˆU × � Φd 1,2,3,4,5 Θ3/2 1,2,3,4,5 χ6 β6 − (5) − (4) − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' � = = W + æ|f(0)|2, (A13) where the notation (5), (4) and etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' are introduced for the terms, in which the localized hole index 6 (and hence its coordinate r6), is swapped with the corresponding intershell d-hole index 5, 4 and etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The Coloumb term W is determined by the direct product of the multiples of the same type as it is shown on the scheme in Fig 3 below, and it reads as W = |f(0)|2 �� dr1dr2|χ−(r1)|2 5 � j=1 |ϕj(r2)|2U(r1−r2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A14) And the exchange term is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (18), where χ stands for χ− and the numerical prefactor stems from the normalization of wave functions and convolution of spin wave functions with all possible cross-multiples with FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The scheme of bra and ket direct multiples in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The total number of direct multiples is equal to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' permutable indices, which are shown in Figs 4(a) – 4(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Note that all multiples give the same contribution but with different signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Thus, we carry out the calculations for the case of multiplication of (6) and (5) terms shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 4(a) and take proper account of the summation of all terms with positive and negative signs Θ3/2† 1,2,3,4,5 β† 6 Θ3/2 1,2,3,4,6 β5 (−5 · 2 + (4 + 3 + 2 + 1) · 2) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A15) Note also that the remaining orbital part of the ex- change integral (after the summation by the spin indices) has the following form æ = 2 6 � dr1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' dr6 χ∗ 6 χ5 Φd∗ 1,2,3,4,5 Φd 1,2,3,4,6 × × (U(|r5 − r6|) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' ) = = 1 3 · 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 5!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' �� dr5dr6 χ∗ 6 χ5 5 � j=1 ϕj∗ 5 ϕj 6 U(|r5 − r6|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A16) The integration with all terms denoted by dots in the first part of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A) gives us zero due to the orthogo- nality of all orbital wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The summation over d-orbital indices in the last part of the equation is car- ried out considering only one index j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 5 for both one-particle functions ϕj∗ 5 and ϕj 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The latter could be easily checked by treating the multiples in explicit forms written one under another χ∗ 6 Φd∗ 1,2,3,4,5 = χ∗ 6 √ 5!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' � ϕ1 1ϕ2 2ϕ3 3ϕ4 4ϕ5 5 − ϕ1 1ϕ2 2ϕ3 3ϕ4 5ϕ5 4 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' �∗, (A17) χ5 Φd 1,2,3,4,6 = χ5 √ 5!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' � ϕ1 1ϕ2 2ϕ3 3ϕ4 4ϕ5 6 − ϕ1 1ϕ2 2ϕ3 3ϕ4 6ϕ5 4 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A18) Here the first term is determined by the fixed sequence of the coordinate indices, and all others are determined by the coordinate indices swapping accompanied by a change of the sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' One can see that the non-zero multiples are only those which are the products of two terms written strictly under each other in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A17) and (A18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' All other multiples gives us zero due to the mu- tual orthogonality of all functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Thus, all terms are summed up only with positive signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The number of such summands with fixed position of 5-th and 6-th particles +(6) - (5) - (4) - (3) - (2) - (1 +(6) - (5) - (4) - (3) - (2) - (19 (a) (b) (c) (d) (e) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The scheme of bra and ket cross-multiples in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (a): There are 5·2 = 10 cross-multiples with the minus sign if one consider five cross-multiples in the inset and their mirror twins emerging as if the virtual reflection in the horizontal plane takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (b), (c), (d) and (e): The number of each multiple of the plus sign should be multiplied by 2 due to the same reason as discussed for inset (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The total number of multiples equals to (4 + 3 + 2 + 1) · 2 = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' equals to the number of permutation of other four elec- trons over remaining four orbitals, and hence it is equal to 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Let us explain now, in brief, the calculation details for the Y, Z and V terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The Y term also involves bra and ket functions of the same type ΨS 1/2ΨJ −1/2 = = f(0) √ 6 � Φd 1,2,3,4,5Θ1/2 1,2,3,4,5 �� 2 3Z6β6 + � 1 3χ− 6 α6 � − −(5) − (4) − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A19) Here the same notation ((5), (4), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=') is introduced as for the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' One can see that the same scheme which we use when calculating the matrix elements of direct Coulomb interaction terms gives us, as denoted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 3, the same value W (due to the symmetry properties of χ− and Z functions of the localized hole), and the lat- ter could be excluded from consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The exchange terms calculation requires consideration of two possible results of wave-functions spin parts convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The first is Θ1/2† 1,2,3,4,5 β† 6 Θ1/2 1,2,3,4,6 β5 = = 1 10 � α† 1α† 2α† 3β† 4β† 5 + α† 1α† 2β† 3α† 4β† 5 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' � β† 6× × (α1α2α3β4β6 + α1α2β3α4β6 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' ) β5 = = 1 10 � α† 1α† 2α† 3β† 4 + α† 1α† 2β† 3α† 4 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' � × × (α1α2α3β4 + α1α2β3α4 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' ) = 4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A20) and the second is Θ1/2† 1,2,3,4,5 α† 6 Θ1/2 1,2,3,4,6 α5 = = 1 10 � α† 1α† 2α† 3β† 4β† 5 + α† 1α† 2β† 3α† 4β† 5 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' � α† 6× × (α1α2α3β4β6 + α1α2β3α4β6 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' ) α5 = = 1 10 � α† 1α† 2β† 3β† 4 + α† 1β† 2α† 3β† 4 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' � × × (α1α2β3β4 + α1β2α3β4 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' ) = 6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A21) Then, after the summation over the spin indices and tak- ing into account possible cross-multiples, as in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 4(a) – 4(e), we obtain an additional multiplier (−5 · 2 + (4 + 3 + 2 + 1) · 2) = 10 as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A15), and then we get the exchange part of Y equal to |f(0)|2 10 6 � dr1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' dr6 �2 3 4 10Z∗ 6 Z5 + 1 3 6 10χ∗ 6 χ5 � × ×Φd∗ 1,2,3,4,5 Φd 1,2,3,4,6 (U(|r5 − r6|) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' ) = = |f(0)|2 7 9 · 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 5!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' �� dr5dr6 χ∗ 6 χ5× × 5 � j=1 ϕj∗ 5 ϕj 6 U(|r5 − r6|) = 7 3æ|f(0)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A22) Here we used the symmetry equivalence of Z and χ func- tions when calculating such type of integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' When calculating Z, the off-diagonal matrix element between quantum states from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A19) is taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Thus, there is no Coloumb term, and the exchange integral in this case reads as Z = |f(0)|2 10 6 � dr1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' dr6 Φd∗ 1,2,3,4,5 Θ3/2† 1,2,3,4,5 χ−∗ 6 β† 6× × ˆU Φd 1,2,3,4,6 Θ1/2 1,2,3,4,6 �� 2 3Z5β5 + � 1 3χ− 5 α5 � = = |f(0)|2 10 6 � dr1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' dr6 Φd∗ 1,2,3,4,5 Θ3/2† 1,2,3,4,5 χ−∗ 6 β† 6× × ˆU Φd 1,2,3,4,6 Θ1/2 1,2,3,4,6 � 1 3χ− 5 α5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A23) +(6) - (5) - (4) - (3) - (2) - (1 5 +(6) - (5) - (4) - (3) - (2) - (1+(6) - (5) - (4) - (3) - (2) - (1 4 +(6) - (5) - (4) - (3) - (2) - (1+(6) - (5) - (4) - (3) - (2) - (1 +(6) - (5) - (4) - (3) - (2) - (1)+(6) - (5) - (4) - (3) - (2) - (1 2 +(6) - (5) - (4) - (3) - (2) - (1)+(6) - (5) - (4) - (3) - (2) - (1 +(6) - (5) - (4) - (3) - (2) - (1)10 The multiplier 10 arises as a result of the usage of the introduced scheme,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' which implies taking into account all exchange integrals (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 4(a) – 4(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' There are no multiples with Z functions because the convolution of spin functions gives zero, as the considered summands have 4α and 1β parts of Θ3/2† 1,2,3,4,5 spin function, and 3α and 2β enter Θ1/2 1,2,3,4,6 spin function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' The spin convolu- tion of the latter part is equal to Θ3/2† 1,2,3,4,5β† 6Θ1/2 1,2,3,4,6α5 = = 1 √ 5 � α† 1α† 2α† 3α† 4β† 5 + α† 1α† 2α† 3β† 4α† 5 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' � β† 6× × 1 √ 10 (α1α2α3β4β6 + α1α2β3α4β6 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' ) α5 = = 1 √ 50 � α† 1α† 2α† 3β† 4 + α† 1α† 2β† 3α† 4 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' � × × (α1α2α3β4 + α1α2β3α4 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' ) = 4 √ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A24) Then finally we get Z = |f(0)|2 10 6 4 √ 50 1 √ 3 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 5!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' �� dr5dr6 χ∗ 6 χ5 × × 5 � j=1 ϕj∗ 5 ϕj 6 U(|r5 − r6|) = 2 √ 2 √ 3 æ|f(0)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A25) The calculation of another off-diagonal matrix element V requires usage of the following ket wave function ac- cording to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (16) ΨS −1/2ΨJ 1/2 = = f(0) √ 6 � Φd 1,2,3,4,5Θ−1/2 1,2,3,4,5 �� 2 3Z6α6 + � 1 3χ+ 6 β6 � − −(5) − (4) − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A26) As the spin function Θ1/2 1,2,3,4,5 contains 3α and 2β in each summation term, while the function Θ−1/2 1,2,3,4,5, on the contrary, contains 2α and 3β in each summand, one can see that the exchange matrix element of interaction between quantum states given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A19) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A) will reduce to the following expression, having non-zero contributions from only Z-orbital terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' V = |f(0)|2 10 6 � dr1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' dr6 Φd∗ 1,2,3,4,5 Θ1/2† 1,2,3,4,5× × � 2 3Z∗ 6 β† 6 ˆU Φd 1,2,3,4,6 Θ−1/2 1,2,3,4,6 � 2 3Z5α5 = = |f(0)|2 10 6 2 3 6 10 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 5!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' �� dr5dr6 Z∗ 6 Z5× × 5 � j=1 ϕj∗ 5 ϕj 6 U(|r5 − r6|) = 2æ|f(0)|2, (A27) in which the result of spin functions convolution is in- cluded Θ1/2† 1,2,3,4,5β† 6Θ−1/2 1,2,3,4,6β5 = = 1 10 � α† 1α† 2α† 3β† 4β† 5 + α† 1α† 2β† 3α† 4β† 5 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' � β† 6× × (β1β2β3α4α6 + β1β2α3β4α6 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' ) α5 = = 1 10 � α† 1α† 2β† 3β† 4 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' � (α1α2β3β4 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' ) = 6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' (A28) [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Awschalom and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Flatt´e, Nature physics 3, 153 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' [2] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Ortiz, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Gomes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' Morey, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE4T4oBgHgl3EQfXAy5/content/2301.05038v1.pdf'} +page_content=' 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b/wNAzT4oBgHgl3EQf7f4e/content/tmp_files/2301.01889v1.pdf.txt @@ -0,0 +1,640 @@ +FPGA Implementation of SIMON-128 +Cryptographic Algorithm Using Artix-7 +Ridha Ghayoula1,2, Jaouhar Fattahi3, Amor Smida4, Issam El Gmati5, Emil Pricop6 and Marwa Ziadia3 +1Unit of Research in High Frequency Electronic Circuits and Systems, University of Tunis El Manar- 2092, Tunis, Tunisia. +2Department of Electrical and Computer Engineering, Laval University, Quebec, Canada. +3Department of Computer Science and Software Engineering, Laval University, Quebec, Canada. +4Department of Medical Equipment Technology, College of Applied Medical Sciences, Majmaah University, +Majmaah, 11952, KSA. +5College of Engineering at al Gunfudha Umm Al Qura University, KSA. +6Automatic Control, Computers and Electronics Department. Petroleum-Gas University of Ploiesti, Romania. +Abstract—FPGA is a hardware architecture based on a matrix +of programmable and configurable logic circuits thanks to which +a large number of functionalities inside the device can be modified +using a hardware description language. These functionalities +must often be secured especially when the context is sensitive +(military, banking, medical, legal, etc.). In this paper, we put +forward an efficient implementation of SIMON’s block cipher +algorithm using Xilinx Vivado 2018.2. The proposed design is +analyzed through simulation on Xilinx Artix-7. A prototype of +our design is implemented using the xc7a35tcsg324-1 FPGA chip. +Performance and results are discussed. +Index Terms—Artix-7, SIMON-128, FPGA, Security, Cryptog- +raphy. +I. INTRODUCTION +Cryptography [1] has been used for thousands of years. +Nowadays, it is more and more present in our daily life. +Contemporary cryptography is mainly interested in, but not +limited to, the six following properties: +1) Secrecy (or confidentiality): the property that ensures +that secret or sensitive information is not discovered by +an unauthorized party; +2) Authentication: the property that ensures that a stake- +holder or service requester is who they claim to be, by +presenting something they have, something they are, or +something they know. +3) Integrity: the property that ensures that the information +has not been modified in a malicious, accidental, or +intentional way by a third party; +4) Authenticity: authentication and integrity usually results +in authenticity, which is the guarantee that the informa- +tion is authentic and comes really from its purported +source as it is sent; +5) Non-repudiation: the property that ensures that a stake- +holder cannot deny his action, or his partial participation +in an action. This results in the fact that one can always +establish irrefutable proof of an stakeholder’s action; +6) Availability [2]: the property that ensures that the infor- +mation or service has not ceased to exist in a malicious +way. +The ability to keep an encrypted message secret is based +not on the encryption algorithm but on a secret piece of +information called a key that must be used with the algorithm +to produce the encrypted message. The size of the key, +expressed in number of bits, is a key element and plays a +crucial role in the security of the cryptographic algorithm. +Depending on whether the key used for encryption and +decryption is the same or not, we speak of a symmetric or +asymmetric cryptographic system. Symmetric cryptography, +also known as secret key cryptography, uses a unique key +to encrypt and decrypt data. This key must be shared with +the recipient. The advantage of symmetric cryptography is +that it is easy to implement. Its disadvantage is that the +secret key must be shared with the recipient, which adds a +key management burden. Unlike symmetric cryptography, +asymmetric cryptography requires two keys for its operation: +first, a so-called public key that must be made public to +recipients; second, a private key that must be kept secret. +The public key and the private key are two totally different +things, nevertheless, they are linked by mathematical bonds. +The advantage of asymmetric cryptography is that one does +not manage the security of key sharing, but its disadvantage +is that it takes a lot of time. Also, encrypted messages are +much larger than those encrypted using symmetric keys. +Cryptographic protocols, using symmetric or asymmetric +cryptographic algorithms, [3], [4], [5], [6] are rules of +exchange between network points whose role is precisely +to secure communications. They are used for example in +e-commerce, when a customer enters his credit card number +to pay for a purchase. But they are also used in a multitude of +other situations, such as when connecting to a computer in a +secure manner, when sending e-mails if one wishes to prevent +an eavesdropper from reading them, or when checking one’s +bank account balance. They are used for any use of the bank +card, such as withdrawing money from an ATM or paying +in a restaurant. They have also been used for a long time +in the decoders of pay TV channels to allow the customer +to have access to the channels to which he has subscribed +and to prevent him from accessing other channels, while +arXiv:2301.01889v1 [cs.CR] 5 Jan 2023 + +allowing possible changes in the subscription. Nowadays, +it is possible to implement a cryptographic algorithm in a +software or hardware way. The hardware implementation of +a cryprographic algorithm consists of the use of computer +hardware (e.g. processors, dedicated chips, etc.) in the data +encryption process. In general, this implementation is put +integrated in the instruction set of the processor, which means +that a part of the processor is dedicated to the cryptographic +mission. This also means that a significant increase in speed +will be observed. By the same stream of ideas, parallel +architectures of modern processors are capable of executing +other instructions at the same time. A secure cryptoprocessor +is a processor optimized for cryptographic tasks (modular +exponentiation, +DES +encryption, +etc.) +incorporated +with +multiple physical security measures, giving it some resistance +to tampering. It can be realized in various ways depending +on the profile of the use: FPGA, ASIC or microcontroller. +The reconfiguration capabilities of FPGAs allow considerable +optimization of operations and correction of implementations +if necessary. Some encryption algorithms are less suitable +than others for modern hardware. DES, for example, is based +on permutations between bits, which may not be suitable for +some types of hardware. +In this paper, we propose a hardware implementation of the +SIMON-128 cryptographic algorithm[7], [8], [9], [10]. The +architecture we are putting forward is designed using Xilinx +Vivado 2018.2. It is implemented on the xc7a35tcsg324-1 +Artix-7 FPGA board. Artix-7[11], [12] is a development +platform designed around the Xilinx in-situ programmable +gate array. It is designed entirely for use as a MicroBlaze +softcore processor system. The Artix-7 FPGA is optimized +for high-performance logic and offers better performance as +well as more capacity than old designs. +The methodology including the SIOMON algorithm de- +scription is presented in section II. Design, synthesis, imple- +mentation, and experimental results are presented in section +III. Discussion is made in section IV. Finally, section V makes +conclusions. +II. METHODOLOGY +The algorithm we implement in this paper is the SIMON +encryption algorithm. It was proposed by researchers in cryp- +tography at the NSA. One of SIMON’s security goals was +to keep a reasonable level of security in an environment +where power, memory and processors are severely limited. +The detailed description of the algorithm is freely available +on the web. SIMON is a symmetric block cipher algorithm. +The computation scheme used is a Feistel network[13], [14]. +Feistel’s process 1 hinges on the idea that repeating judiciously +chosen simple operations enough times allows good security. +Encryption is a succession of similar steps (called rounds) each +using a subkey. This process is characterized by: +1) An iterative and modular construction; +2) Subkeys are derived from the secret key; +3) The functions used by a lathe must be optimized and +are generally simple operations. +These simple operations are generally: +1) Permutation: the symbols of the plain text are exchanged +between them. Permutation adds diffusion; +2) Substitution: a symbol is replaced by another symbol. +Substitution adds confusion. +Figure 1. +Feistel round. +The encryption is decomposed into several rounds. In each +round, two blocks are exchanged and one block is combined +with a transformed version of the second block and a key. +The transformation function is a non-linear bijection while the +combination function is usually the exclusive function (XOR). +The rotating key is generated from an initial secret key. The +key generation mechanism is usually called a key program. +This scheme thus provides the two properties of diffusion and +confusion necessary in an encryption algorithm. SIMON is +an algorithm intended to be physically implemented in highly +constrained embedded systems. For this the following design +choices have been made [15], [16], [17]: +• The size of the block and the key are configurable; +• The number of laps required for encryption then varies; +• The nonlinear function used at each turn is very simple; +• The complexity of decryption grows exponentially with +the number of rounds. +SIMON respects the symmetry operation regarding the +circular shift operation on n-bit words. The key schedule uses +a succession of 1-bit round constants devised for the sliding +properties and circular shift symmetry. +Table I sums up all configurations of SIMON-128. +Let Sj denote a j-bit-left circular shift. The key schedule +is mathematically described as: +The key schedule structure can be either balanced or +unbalanced. The number of keywords m is used to establish +the key expansion structure, yielding a total bit width of + +Li +Ri +Li+1=Ri +Ri1=F(K) @ LiTABLE I +SIMON-128 PARAMETERS +Bloc size (bits) +key size +key word (m) +Round constant +Rounds +124 +2 +z2 +68 +128 +192 +3 +z3 +69 +256 +4 +z4 +72 +� +� +� +� +� +� +� +� +� +c ⊕ zji ⊕ ki ⊕ � +i ⊕ S−1�� +S−3Ki+1 +� +, with m = 2 +c ⊕ zji ⊕ ki ⊕ � +i ⊕ S−1�� +S−3Ki+2 +� +, with m = 3 +c ⊕ zji ⊕ ki ⊕ � +i ⊕ S−1�� +S−3Ki+3 ⊕ Ki+1 +� +, with m = 4 +m ∗ n. The keyword expansion consists of a right shift, an +XOR and a constant sequence, zx. The zx bit operates on the +lowest bit of the keyword once every round [18], [19]. +The SIMON-128 encryption is expressed by Equation 1: +R (l, r, k) = +�� +S1 (l) &S8 (l) +� +⊕ S2 (l) ⊕ r ⊕ k, l +� +(1) +The SIMON-128 decryption is expressed by Equation 2: +R−1 (l, r, k) = +� +r, +� +S1 (r) &S8 (r) +� +⊕ S2 (r) ⊕ l ⊕ k +� +(2) +Where l is the left-most word of a block, r the right-most +word and k the corresponding round key [20]. +III. IMPLEMENTATION AND EXPERIMENTAL RESULTS +We assume that the block of data to be encrypted as well +as the key are presented at the same time. A pulse of the +”start” signal indicates that the encryption can start. Once the +encryption is finished, the ”done” signal goes to one indicating +that the value presented on the output ”ciphertext” is valid. In +addition to these signals, we also have an input for the clock +signal ”clk” and reset ”nrst”. The module is divided into three +parts: +• SIMONdp: for encrypting data blocks; +• SIMONks: for generating the turn key; +• SIMONctrl: for generating control signals. +The architecture of the SIMON cipher block consists of a +parallel cipher that uses round functions and key generation +blocks. +The SIMON-128 architecture is implemented on a Xilinx +Arty board which is a Artix-7 Pro-based embedded develop- +ment platform. The xc7a35tcsg324−1 FPGA contains 20800 +LUT, 4600 Flip Flip and 9600 LUTRAM modules. We used +Xilinx Vivado 2018.2 Softwares to implement our architecture +on the board. All VHDL modules are extensively simulated +Figure 2. +SIMON Design. +using Vivado 2018.2 and synthesized using Xilinx synthesis +technologies. Figure 3 shows the experimental setup for the +SIMON-128 architecture. Table II presents the implementation +resources (Post-synthesis and post-implementation). +Figure 3. +SIMON-Code application using Artix-7. +Summary of on-Chip static and dynamic power are shown +in Table III using xc7a35tcsg324 − 1 device of Artix-7 +family. Table III and Table IV present the thermal and power +characteristics of this implementation. +IV. DISCUSSION +Table V gives the performance of our implementation and +compares it with the results obtained with Zynq-7000 and +Virtex-7 presented in [21]. Our implementation presents sig- +nificant improvement over both of them regarding all metrics. + +plaintext +ciphertext +Datapath +Key +Key Schedule +compute +start +done +Control +clk +nrstCode Application +Simon +Free +TCP/IP +RTOS +Stack +Device DriversFigure 4. +SIMON-128 XDC file. +TABLE II +IMPLEMENTATION RESOURCES (POST-SYNTHESIS AND +POST-IMPLEMENTATION) +Resource +Utilization +Available +Utilization (%) +LUT +45 +20800 +0.22 +LUTRAM +12 +9600 +0.13 +FF +27 +4600 +0.06 +IO +5 +210 +2.38 +BUFG +1 +32 +3.13 +TABLE III +ON-CHIP DYNAMIC AND STATIC POWER +Dynamic +Static +Power (W) +Percentage +Power (W) +Percentage +Signals +< 0.001 +10% +Logic +< 0.001 +13% +PL Static +0.070 +97% +I/O +0.001 +46% +Clocks +0.001 +31% +TABLE IV +THERMAL AND POWER CHARACTERISTICS +Power +Total On-Chip Power +0.072 w +Junction Temperature +25.3◦ +Thermal margin +59.7◦ (12.) w +Effective JA +4.8◦ c/w +Power Supplied to off-hip devices +0 w +Confidence level +Medium +We note a gain in power consumption of 69.87% compared to +Zynq-7000 and 70.56% compared to Virtex-7, a gain in delay +of 26.69% compared to Zynq-7000 and 8.95% compared to +Virtex-7. SIMON Artix-7 also uses less area of lookup table +(e.g. 45 LUT) compared to both Zynq-7000 and Virtex-7 (73 +LUT). Artix-7 confirms its reputation of being quicker and +low-cost than other models. +Other +recent +work +comparable +to +our +present +implementation is worthy of mention. For instance, Rashidi in +TABLE V +PERFORMANCE AND COMPARISON OF OUR IMPLEMENTATION WITH +ZYNQ-7000 AND SIMON VIRTEX-7 +Metric +SIMON Zynq-7000 +[21] +SIMON Virtex-7 +[21] +SIMON Artix-7 +Power (mW) +239 +248 +72 +Delay (ns) +5.448 +4.415 +4.020 +Area (LUT) +73 +73 +45 +[22] presented an ASIC implementation of several sizes of the +SIMON algorithm using Sklansky adder. Encouraging results +were observed regarding critical path delay. In the same vein, +Sheikhpour et al. in [23] presented a flexible implementation +of SIOMON with various key sizes, which has capabilities +for various kinds of attacks. In [24], Abed et al. presented +several types of SIMON implementations (pipelined, scalar, +etc.) and showed a comparison between these families in +terms of throughput and drew up an implementation guideline +depending on the technological need. +Despite the good performance of our implementation, it +requires very sharp knowledge of the hardware and tools and +the design is sometimes tedious to set up. +V. CONCLUSION +In this paper, we have presented an implementation of +SIMON-128 algorithm using Artix-7of with low-cost FPGA +platform. The Feistel feature of SIMON is chosen to reduce +the hardware impact of encryption without sacrificing software +performance. Such a structure has the advantage that the +encryption and decryption operations are very similar, which +is enough to reverse the operation of the key manager to obtain +the decryption operation. We used circular offsets (just hard- +ware cabling) and bit-to-bit operations. The implementation +led to a good performance that we discussed in this paper +compared to other implementations in the state of the art. + +clock signal +set property-dictPACKAGEPIN E3 +IOSTANDARD LVCMOS33}[getports (cIk 1]:IOL12PT1 MRCC35Sch=gc1k[100] +create clock -add -name sys cik pin -period 1o.o0 -waveform (0 5] [get ports (cik)l; +-Switches +set property-dict PAcKAGEPIN A8 +IOSTANDARDLVCMOS33][getports(data_in]:#I0L12NT1 MRCC 16Sch=sw[0] +setproperty-dictf PACKAGE PINCi1 +IOSTANDARDLVCMOS33[getports1input[0]J:#IO L13PT2MRCC16 Sch=3W[1] +set property-dict f PACKAGEPIN Cio +IOSTANDARDLVCMOS33 +lget ports +input[1]9]:#I0L13NT2MRCc16Sch=sw[2] +LED3 +set property-dict (PACKAGE_PIN F6 +IOSTANDARDLVCMOS33][getports(cipher_out[7]1]:IO_L19N_T3_VREF_35Sch=1ed0_gREFERENCES +[1] H. C. v. Tilborg, Encyclopedia of Cryptography and Security. +Berlin, +Heidelberg: Springer-Verlag, 2005. +[2] E. Pricop, S. F. Mihalache, N. Paraschiv, J. Fattahi, and F. 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Comput. +Syst., +vol. +16, +no. +4, +may +2017. +[Online]. +Available: +https: +//doi.org/10.1145/3055514 +[22] B. Rashidi, “High-throughput and flexible ASIC implementations of +SIMON and SPECK lightweight block ciphers,” Int. J. Circuit Theory +Appl., vol. 47, no. 8, pp. 1254–1268, 2019. [Online]. Available: +https://doi.org/10.1002/cta.2645 +[23] S. +Sheikhpour, +M. +H. +Sadi, +and +A. +Mahani, +“High-throughput +configurable SIMON architecture for flexible security,” Microelectron. +J., vol. 113, p. 105085, 2021. [Online]. Available: https://doi.org/10. +1016/j.mejo.2021.105085 +[24] S. +Abed, +R. +Jaffal, +B. +J. +Mohd, +and +M. +Alshayeji, +“FPGA +modeling and optimization of a SIMON lightweight block cipher,” +Sensors, vol. 19, no. 4, p. 913, 2019. [Online]. Available: https: +//doi.org/10.3390/s19040913 +NOTICE +©2022 IEEE. Personal use of this material is permitted. +Permission from IEEE must be obtained for all other uses, in +any current or future media, including reprinting/republishing +this material for advertising or promotional purposes, creating +new collective works, for resale or redistribution to servers or +lists, or reuse of any copyrighted component of this work in +other works. + diff --git a/wNAzT4oBgHgl3EQf7f4e/content/tmp_files/load_file.txt b/wNAzT4oBgHgl3EQf7f4e/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a3bcc4126ad2a256ffd52c87e387f6eba1505870 --- /dev/null +++ b/wNAzT4oBgHgl3EQf7f4e/content/tmp_files/load_file.txt @@ -0,0 +1,467 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf,len=466 +page_content='FPGA Implementation of SIMON-128 Cryptographic Algorithm Using Artix-7 Ridha Ghayoula1,2, Jaouhar Fattahi3, Amor Smida4, Issam El Gmati5, Emil Pricop6 and Marwa Ziadia3 1Unit of Research in High Frequency Electronic Circuits and Systems, University of Tunis El Manar- 2092, Tunis, Tunisia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' 2Department of Electrical and Computer Engineering, Laval University, Quebec, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' 3Department of Computer Science and Software Engineering, Laval University, Quebec, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' 4Department of Medical Equipment Technology, College of Applied Medical Sciences, Majmaah University, Majmaah, 11952, KSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' 5College of Engineering at al Gunfudha Umm Al Qura University, KSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' 6Automatic Control, Computers and Electronics Department.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Petroleum-Gas University of Ploiesti, Romania.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Abstract—FPGA is a hardware architecture based on a matrix of programmable and configurable logic circuits thanks to which a large number of functionalities inside the device can be modified using a hardware description language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' These functionalities must often be secured especially when the context is sensitive (military, banking, medical, legal, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' In this paper, we put forward an efficient implementation of SIMON’s block cipher algorithm using Xilinx Vivado 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' The proposed design is analyzed through simulation on Xilinx Artix-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' A prototype of our design is implemented using the xc7a35tcsg324-1 FPGA chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Performance and results are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Index Terms—Artix-7, SIMON-128, FPGA, Security, Cryptog- raphy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' INTRODUCTION Cryptography [1] has been used for thousands of years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Nowadays, it is more and more present in our daily life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Contemporary cryptography is mainly interested in, but not limited to, the six following properties: 1) Secrecy (or confidentiality): the property that ensures that secret or sensitive information is not discovered by an unauthorized party;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' 2) Authentication: the property that ensures that a stake- holder or service requester is who they claim to be, by presenting something they have, something they are, or something they know.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' 3) Integrity: the property that ensures that the information has not been modified in a malicious, accidental, or intentional way by a third party;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' 4) Authenticity: authentication and integrity usually results in authenticity, which is the guarantee that the informa- tion is authentic and comes really from its purported source as it is sent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' 5) Non-repudiation: the property that ensures that a stake- holder cannot deny his action, or his partial participation in an action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' This results in the fact that one can always establish irrefutable proof of an stakeholder’s action;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' 6) Availability [2]: the property that ensures that the infor- mation or service has not ceased to exist in a malicious way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' The ability to keep an encrypted message secret is based not on the encryption algorithm but on a secret piece of information called a key that must be used with the algorithm to produce the encrypted message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' The size of the key, expressed in number of bits, is a key element and plays a crucial role in the security of the cryptographic algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Depending on whether the key used for encryption and decryption is the same or not, we speak of a symmetric or asymmetric cryptographic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Symmetric cryptography, also known as secret key cryptography, uses a unique key to encrypt and decrypt data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' This key must be shared with the recipient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' The advantage of symmetric cryptography is that it is easy to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Its disadvantage is that the secret key must be shared with the recipient, which adds a key management burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Unlike symmetric cryptography, asymmetric cryptography requires two keys for its operation: first, a so-called public key that must be made public to recipients;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' second, a private key that must be kept secret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' The public key and the private key are two totally different things, nevertheless, they are linked by mathematical bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' The advantage of asymmetric cryptography is that one does not manage the security of key sharing, but its disadvantage is that it takes a lot of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Also, encrypted messages are much larger than those encrypted using symmetric keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Cryptographic protocols, using symmetric or asymmetric cryptographic algorithms, [3], [4], [5], [6] are rules of exchange between network points whose role is precisely to secure communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' They are used for example in e-commerce, when a customer enters his credit card number to pay for a purchase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' But they are also used in a multitude of other situations, such as when connecting to a computer in a secure manner, when sending e-mails if one wishes to prevent an eavesdropper from reading them, or when checking one’s bank account balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' They are used for any use of the bank card, such as withdrawing money from an ATM or paying in a restaurant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' They have also been used for a long time in the decoders of pay TV channels to allow the customer to have access to the channels to which he has subscribed and to prevent him from accessing other channels, while arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content='01889v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content='CR] 5 Jan 2023 allowing possible changes in the subscription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Nowadays, it is possible to implement a cryptographic algorithm in a software or hardware way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' The hardware implementation of a cryprographic algorithm consists of the use of computer hardware (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' processors, dedicated chips, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=') in the data encryption process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' In general, this implementation is put integrated in the instruction set of the processor, which means that a part of the processor is dedicated to the cryptographic mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' This also means that a significant increase in speed will be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' By the same stream of ideas, parallel architectures of modern processors are capable of executing other instructions at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' A secure cryptoprocessor is a processor optimized for cryptographic tasks (modular exponentiation, DES encryption, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=') incorporated with multiple physical security measures, giving it some resistance to tampering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' It can be realized in various ways depending on the profile of the use: FPGA, ASIC or microcontroller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' The reconfiguration capabilities of FPGAs allow considerable optimization of operations and correction of implementations if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Some encryption algorithms are less suitable than others for modern hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' DES, for example, is based on permutations between bits, which may not be suitable for some types of hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' In this paper, we propose a hardware implementation of the SIMON-128 cryptographic algorithm[7], [8], [9], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' The architecture we are putting forward is designed using Xilinx Vivado 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' It is implemented on the xc7a35tcsg324-1 Artix-7 FPGA board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Artix-7[11], [12] is a development platform designed around the Xilinx in-situ programmable gate array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' It is designed entirely for use as a MicroBlaze softcore processor system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' The Artix-7 FPGA is optimized for high-performance logic and offers better performance as well as more capacity than old designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' The methodology including the SIOMON algorithm de- scription is presented in section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Design, synthesis, imple- mentation, and experimental results are presented in section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Discussion is made in section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Finally, section V makes conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' METHODOLOGY The algorithm we implement in this paper is the SIMON encryption algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' It was proposed by researchers in cryp- tography at the NSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' One of SIMON’s security goals was to keep a reasonable level of security in an environment where power, memory and processors are severely limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' The detailed description of the algorithm is freely available on the web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' SIMON is a symmetric block cipher algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' The computation scheme used is a Feistel network[13], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Feistel’s process 1 hinges on the idea that repeating judiciously chosen simple operations enough times allows good security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Encryption is a succession of similar steps (called rounds) each using a subkey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' This process is characterized by: 1) An iterative and modular construction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' 2) Subkeys are derived from the secret key;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' 3) The functions used by a lathe must be optimized and are generally simple operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' These simple operations are generally: 1) Permutation: the symbols of the plain text are exchanged between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Permutation adds diffusion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' 2) Substitution: a symbol is replaced by another symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Substitution adds confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Feistel round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' The encryption is decomposed into several rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' In each round, two blocks are exchanged and one block is combined with a transformed version of the second block and a key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' The transformation function is a non-linear bijection while the combination function is usually the exclusive function (XOR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' The rotating key is generated from an initial secret key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' The key generation mechanism is usually called a key program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' This scheme thus provides the two properties of diffusion and confusion necessary in an encryption algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' SIMON is an algorithm intended to be physically implemented in highly constrained embedded systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' For this the following design choices have been made [15], [16], [17]: The size of the block and the key are configurable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' The number of laps required for encryption then varies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' The nonlinear function used at each turn is very simple;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' The complexity of decryption grows exponentially with the number of rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' SIMON respects the symmetry operation regarding the circular shift operation on n-bit words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' The key schedule uses a succession of 1-bit round constants devised for the sliding properties and circular shift symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Table I sums up all configurations of SIMON-128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Let Sj denote a j-bit-left circular shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' The key schedule is mathematically described as: The key schedule structure can be either balanced or unbalanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' The number of keywords m is used to establish the key expansion structure, yielding a total bit width of Li Ri Li+1=Ri Ri1=F(K) @ LiTABLE I SIMON-128 PARAMETERS Bloc size (bits) key size key word (m) Round constant Rounds 124 2 z2 68 128 192 3 z3 69 256 4 z4 72 � � � � � � � � � c ⊕ zji ⊕ ki ⊕ � i ⊕ S−1�� S−3Ki+1 � , with m = 2 c ⊕ zji ⊕ ki ⊕ � i ⊕ S−1�� S−3Ki+2 � , with m = 3 c ⊕ zji ⊕ ki ⊕ � i ⊕ S−1�� S−3Ki+3 ⊕ Ki+1 � , with m = 4 m ∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' The keyword expansion consists of a right shift, an XOR and a constant sequence, zx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' The zx bit operates on the lowest bit of the keyword once every round [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' The SIMON-128 encryption is expressed by Equation 1: R (l, r, k) = �� S1 (l) &S8 (l) � ⊕ S2 (l) ⊕ r ⊕ k, l � (1) The SIMON-128 decryption is expressed by Equation 2: R−1 (l, r, k) = � r, � S1 (r) &S8 (r) � ⊕ S2 (r) ⊕ l ⊕ k � (2) Where l is the left-most word of a block, r the right-most word and k the corresponding round key [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' IMPLEMENTATION AND EXPERIMENTAL RESULTS We assume that the block of data to be encrypted as well as the key are presented at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' A pulse of the ”start” signal indicates that the encryption can start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Once the encryption is finished, the ”done” signal goes to one indicating that the value presented on the output ”ciphertext” is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' In addition to these signals, we also have an input for the clock signal ”clk” and reset ”nrst”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' The module is divided into three parts: SIMONdp: for encrypting data blocks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' SIMONks: for generating the turn key;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' SIMONctrl: for generating control signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' The architecture of the SIMON cipher block consists of a parallel cipher that uses round functions and key generation blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' The SIMON-128 architecture is implemented on a Xilinx Arty board which is a Artix-7 Pro-based embedded develop- ment platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' The xc7a35tcsg324−1 FPGA contains 20800 LUT, 4600 Flip Flip and 9600 LUTRAM modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' We used Xilinx Vivado 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content='2 Softwares to implement our architecture on the board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' All VHDL modules are extensively simulated Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' SIMON Design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' using Vivado 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content='2 and synthesized using Xilinx synthesis technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Figure 3 shows the experimental setup for the SIMON-128 architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Table II presents the implementation resources (Post-synthesis and post-implementation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' SIMON-Code application using Artix-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Summary of on-Chip static and dynamic power are shown in Table III using xc7a35tcsg324 − 1 device of Artix-7 family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Table III and Table IV present the thermal and power characteristics of this implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' DISCUSSION Table V gives the performance of our implementation and compares it with the results obtained with Zynq-7000 and Virtex-7 presented in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Our implementation presents sig- nificant improvement over both of them regarding all metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' plaintext ciphertext Datapath Key Key Schedule compute start done Control clk nrstCode Application Simon Free TCP/IP RTOS Stack Device DriversFigure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' SIMON-128 XDC file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' TABLE II IMPLEMENTATION RESOURCES (POST-SYNTHESIS AND POST-IMPLEMENTATION) Resource Utilization Available Utilization (%) LUT 45 20800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content='22 LUTRAM 12 9600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content='13 FF 27 4600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content='06 IO 5 210 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content='38 BUFG 1 32 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content='13 TABLE III ON-CHIP DYNAMIC AND STATIC POWER Dynamic Static Power (W) Percentage Power (W) Percentage Signals < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content='001 10% Logic < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content='001 13% PL Static 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content='070 97% I/O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content='001 46% Clocks 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content='001 31% TABLE IV THERMAL AND POWER CHARACTERISTICS Power Total On-Chip Power 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content='072 w Junction Temperature 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content='3◦ Thermal margin 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content='7◦ (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=') w Effective JA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content='8◦ c/w Power Supplied to off-hip devices 0 w Confidence level Medium We note a gain in power consumption of 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content='87% compared to Zynq-7000 and 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content='56% compared to Virtex-7, a gain in delay of 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content='69% compared to Zynq-7000 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content='95% compared to Virtex-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' SIMON Artix-7 also uses less area of lookup table (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' 45 LUT) compared to both Zynq-7000 and Virtex-7 (73 LUT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Artix-7 confirms its reputation of being quicker and low-cost than other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Other recent work comparable to our present implementation is worthy of mention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' For instance, Rashidi in TABLE V PERFORMANCE AND COMPARISON OF OUR IMPLEMENTATION WITH ZYNQ-7000 AND SIMON VIRTEX-7 Metric SIMON Zynq-7000 [21] SIMON Virtex-7 [21] SIMON Artix-7 Power (mW) 239 248 72 Delay (ns) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content='448 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content='415 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content='020 Area (LUT) 73 73 45 [22] presented an ASIC implementation of several sizes of the SIMON algorithm using Sklansky adder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Encouraging results were observed regarding critical path delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' In the same vein, Sheikhpour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' in [23] presented a flexible implementation of SIOMON with various key sizes, which has capabilities for various kinds of attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' In [24], Abed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' presented several types of SIMON implementations (pipelined, scalar, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=') and showed a comparison between these families in terms of throughput and drew up an implementation guideline depending on the technological need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Despite the good performance of our implementation, it requires very sharp knowledge of the hardware and tools and the design is sometimes tedious to set up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' CONCLUSION In this paper, we have presented an implementation of SIMON-128 algorithm using Artix-7of with low-cost FPGA platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' The Feistel feature of SIMON is chosen to reduce the hardware impact of encryption without sacrificing software performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Such a structure has the advantage that the encryption and decryption operations are very similar, which is enough to reverse the operation of the key manager to obtain the decryption operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' We used circular offsets (just hard- ware cabling) and bit-to-bit operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' The implementation led to a good performance that we discussed in this paper compared to other implementations in the state of the art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' clock signal set property-dictPACKAGEPIN E3 IOSTANDARD LVCMOS33}[getports (cIk 1]:IOL12PT1 MRCC35Sch=gc1k[100] create clock -add -name sys cik pin -period 1o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content='o0 -waveform (0 5] [get ports (cik)l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Switches set property-dict PAcKAGEPIN A8 IOSTANDARDLVCMOS33][getports(data_in]:#I0L12NT1 MRCC 16Sch=sw[0] setproperty-dictf PACKAGE PINCi1 IOSTANDARDLVCMOS33[getports1input[0]J:#IO L13PT2MRCC16 Sch=3W[1] set property-dict f PACKAGEPIN Cio IOSTANDARDLVCMOS33 lget ports input[1]9]:#I0L13NT2MRCc16Sch=sw[2] LED3 set property-dict (PACKAGE_PIN F6 IOSTANDARDLVCMOS33][getports(cipher_out[7]1]:IO_L19N_T3_VREF_35Sch=1ed0_gREFERENCES [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' C.' metadata={'source': 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Alshayeji, “FPGA modeling and optimization of a SIMON lightweight block cipher,” Sensors, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' 19, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' 4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' 913, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content=' Available: https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQf7f4e/content/2301.01889v1.pdf'} +page_content='org/10.' metadata={'source': 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Hannukainen3, Quentin Marsal4, Daniel Mu˜noz-Segovia5 and Adolfo G. +Grushin4 +(a) (b) +1 Department of Materials Science, University of California, Berkeley, California 94720, USA +2 Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA +3 Department of Physics, KTH Royal Institute of Technology, 106 91 Stockholm, Sweden +4 Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut N´eel, 38000 Grenoble, France +5 Donostia International Physics Center, 20018 Donostia-San Sebastian, Spain +PACS nn.mm.xx – First pacs description +PACS nn.mm.xx – Second pacs description +Abstract – Topological phases of matter are ubiquitous in crystals, but less is known about +their existence in amorphous systems, that lack long-range order. In this perspective, we review +the recent progress made on theoretically defining amorphous topological phases and the new +phenomenology that they can open. We revisit key experiments suggesting that amorphous topo- +logical phases exist in both solid-state and synthetic amorphous systems. We finish by discussing +the open questions in the field, that promises to significantly enlarge the set of materials and +synthetic systems benefiting from the robustness of topological matter. +Introduction. – +The quantum Hall state of a two di- +mensional electron gas, the first topological phase ever ob- +served, was discovered in a non-crystalline system [1].Here +free electrons are confined to move in two dimensions un- +der a perpendicular magnetic field, display Landau lev- +els, and an associated quantized metallic topological edge +states caused by a confining potential, with no assumption +of an underlying crystalline lattice. +The quantum Hall +state does, however, display a continuous translational in- +variance, since the electron’s momentum p is well defined. +The momentum enters the parabolic dispersion relation +p2/2m, with m being the electron’s mass. By promoting +m to the effective mass of the electron within a medium, +the parabolic dispersion and its corresponding Landau lev- +els can be thought of as arising from the long-wavelength +limit of a lattice tight-binding crystalline model [2]. With +this notion of translational invariance in place, the con- +densed matter community discovered how to dispose of +magnetic fields to define topological states in crystalline +systems [3], establishing topological phases in crystals of +any dimension, irrespective of their insulating, conducting +or superconducting nature [4–6]. +Topological phases do exist in the absence of long-range +periodicity, as we are not forced to regularize a continuum +(a)adolfo.grushin@neel.cnrs.fr +(b)All authors contributed equally to the writing of the manuscript. +theory using a periodic lattice. This observation is at the +heart of this perspective article. Our goal is to summarize +the recent progress made to understand how topological +phases emerge on the largest class of non-crystalline sys- +tems, amorphous systems [7–10]. +Characterizing topol- +ogy in amorphous matter, without the convenience of +Bloch’s theorem, has lead to the emergence of new phe- +nomenology, unique to amorphous matter. Topology re- +mains largely unexplored in this class of solids, which may +offer different functionalities compared to crystals. +We start by discussing the main properties of amor- +phous and topological matter, followed by a review of the +progress made in combining these two fields. We finish +by summarizing the experimental status and offering some +perspectives on the main open questions. For a more tech- +nical review we refer the reader to Ref. [11]. +Basic properties of amorphous matter. +Amorphous +materials are defined by their lack of long-range order [12]. +However, they display short- and even medium-range or- +der, as well defined nearest and next-to-nearest neighbour +distances, respectively (Fig. 1 (a-c)). The short range or- +der manifests itself as preferred bond lengths and angles, +peaked around the values of its crystalline counterpart. +Due to the short range order, amorphous materials have +a well defined coordination environment with a distinct +number of nearest neighbours. In solid state systems this +p-1 +arXiv:2301.04176v1 [cond-mat.mes-hall] 10 Jan 2023 + +P. Corbae et al. +is a result of the electronic configuration of the atoms in- +volved in bonding. Hence, amorphous solids remain lo- +cally ordered [12,13]. +Elucidating the atomic structure of amorphous solids is +necessary to understand most of their electronic proper- +ties [12,14]. The disordered atomic positions in amorphous +solids result in diffuse rings in the diffraction pattern and +a lack of sharp Bragg peaks characteristic of crystalline +materials [12]. The absence of discrete crystalline symme- +try, in favour of local short range order and well defined +diffraction rings demonstrates that amorphous systems are +isotropic on average (see Fig. 1) [15–19]. Fluctuations of +the bond lengths account for the broadening of the rings. +The radii of the diffraction rings and its weight can be used +to determine the structure factor and estimate an aver- +age bond distance and coordination number of the amor- +phous structure [20, 21]. The absence of Bragg peaks in +the diffraction pattern, and thus the absence of long-range +order, determines which solids are amorphous. +Amorphous materials are commonplace in science and +technology [12]. Their applications range from common +objects such as window glass to technological devices like +computer memories or solar cells [22,23]. Amorphous ma- +terials are advantageous for technological applications as +they can be grown under less stringent conditions than +single crystals require. In solids state systems, they can +be grown in a range of compositions unlike typical crys- +talline compounds. Transitioning between the amorphous +to the crystalline state in a controlled and reversible man- +ner, for example using current or laser pulses [24], is a +useful and defining property of phase-change materials. +These are commonly used in computer memory-storage +devices [22, 23]. Additionally, amorphous materials play +a major role in fundamental science e.g. as coatings in +gravitational waves detectors at LIGO [25]. +Similarly to crystals, amorphous materials can be insu- +lators, semiconductors, metals, and superconductors [12]. +Amorphous oxides used in glassware, such as silicon oxide +or lead glass, are century-old insulators. Amorphous semi- +conductors, such as silicon or germanium, have also been +extensively studied, due to their possible use in electronic +devices [13]. +Amorphous metals are exceptionally hard +and can display unique magnetic properties [26]. Amor- +phous superconductors can also be synthesised, as conven- +tional superconductivity is robust to disorder, an observa- +tion known as Anderson’s theorem [27]. Remarkably, the +critical superconducting temperature has been observed +to be higher in several amorphous materials compared to +their crystalline counterpart (Fig. 1(d)). +The existence and robustness of topological phases +poses the natural question of whether they can be realized +in amorphous systems. Before reviewing how topological +amorphous phases were first achieved [7–10] and extended, +we revisit the main properties of topological phases. +Basic properties of topological matter. +The discovery +of the quantum Hall effect and its quantized Hall conduc- +tance [28–30] introduced the field of topological matter; +phases of matter characterized by their metallic boundary +states and quantized responses to external fields, which +are robust against impurities and local perturbations [31]. +The quantum Hall effect is an example of a strong topo- +logical insulator [4,5], phases of matter where the bound- +ary states are protected by local symmetries. These sym- +metries are the time reversal-, particle hole-, and chiral +symmetry, the latter being the product of the other two. +These three symmetries can be combined in ten different +ways, defining the Altland-Zirnbauer classification of first +quantized free fermion Hamiltonians [32,33], leading to the +full classification of strong topological insulators and su- +perconductors [34–38]. There are five non-trivial Altland- +Zirnbauer classes, classes that can host topological phases, +in every dimension, each defined by a topological invari- +ant, either integer valued Chern or winding numbers or +a Z2 invariant [36], characterizing the phase. Two states +are defined to be in the same topological phase if they +can be adiabatically perturbed into one another smoothly +without closing the conduction gap and not breaking the +underlying symmetries, while keeping the number of or- +bitals fixed during the process. +Translational invariance in crystal lattices allows use of +Bloch’s theorem to define crystal momentum, simplifying +the characterization of the topological phases and yielding +closed-form momentum-space expressions of the topologi- +cal invariants. For example, the Chern number [39] char- +acterizing the quantum Hall phase in two dimensions (2D) +is evaluated as the integral of the Berry curvature [40] over +the first Brillouin zone. +Electronic topological phases extend beyond strong +topological insulators and superconductors, +including +weak [41–43]- and crystalline [44]- topological insulators, +and topological metals [6]. +Weak topological insulators +can be constructed by stacking strong topological insula- +tors, giving rise to symmetry protected surface states per- +pendicular to the stacking direction. +Crystalline (point +group) symmetries, including rotations and reflections, +can protect topological states called higher order topo- +logical insulators which host symmetry protected states +on those surfaces which are invariant under the crystalline +symmetry. Crystalline symmetries simplify how to iden- +tify topological phases, through to the concept of symme- +try indicators [45–48]— eigenvalues of point group opera- +tors whose products determine topological invariants. +Although translation invariance simplifies describing +and classifying topological phases, it is not necessary for +their existence. For example, strong topology is protected +by local symmetries, irrespective of the lattice details. +Non-trivial topology only requires the existence of a mo- +bility gap, and not a spectral gap. However, characterizing +topological phases of matter far from the crystalline limit, +notably for non-crystalline lattices, requires new tools, as +the known momentum space expressions for topological in- +variants are no longer applicable. We describe these tools +and the models introduced to study amorphous topologi- +p-2 + +Amorphous topological matter: theory and experiment +(a) +(b) +(c) +(d) +(e) +(f) +Bi +Be +Al +5 mK +6 K +26 mK +10 K +1.18 K +6 K +Amorphous +Crystal +Supercond. Tc +Element +amorphous +crystal +quasicrystal +(a) +(b) +(c) +(d) +(e) +(f) +Figure 1: (a) Crystalline lattice and corresponding structure +factor with sharp Bragg peaks. (b) Amorphous lattices with +local order result in broad diffraction rings. (c) Quasicrystals +break translational invariance but retain long range order, re- +sulting in sharp Bragg peaks. (d) Local Chern marker of the 2D +threefold-coordinated Weaire-Thorpe model [49]. The bulk av- +erage equals −1, indicating a nontrivial Chern insulator phase. +(e) Configuration-averaged spectral gap, Eg, of a 2D quantum +spin Hall model on a structurally disordered trigonal lattice +as a function of disorder strength σ and spin-orbit coupling +λ. The quantum spin Hall (QSH) and normal insulator (NI) +phases are labelled according to the value of the spin Bott +index. Adapted from Ref. [50]. (f) Superconducting critical +temperature of different amorphous and crystalline solids [51]. +cal matter next. +Theory of amorphous topological matter. – +Overview. +There are a variety of amorphous models +displaying topological phases, ranging from strong topo- +logical states to spatial-symmetry-protected topological +phases. Amorphous strong topological states include 2D +Chern insulators in class A [7–10, 49, 52–55], 2D and 3D +time-reversal invariant topological insulators in class AII +[7,50,54,56–58], and 2D time-reversal breaking topological +superconductors in class D [59,60]. Amorphous structures +also support phases a priori protected by crystalline sym- +metries, such as 2D reflection-symmetry-protected topo- +logical insulators [61], 2D and 3D higher-order topological +insulators [62–64], 2D and 3D obstructed insulators [65], +and 3D topological metals [66]. While structural disorder +is detrimental to some of these states, it can also induce +nontrivial phases when starting from a trivial crystalline +state [50,63,66], and it can give rise to new phenomenology +intrinsically associated with amorphous topological mat- +ter and phase transitions [54,55,61,65,66]. +A common starting point is a crystalline tight bind- +ing Hamiltonian known to host a topologically nontriv- +ial phase. +The hopping terms are generalized to ac- +count for arbitrary angles and distances between sites. +For example the angular dependence can be modelled us- +ing the Slater-Koster parametrization [67], and the rea- +dial dependence can be accounted for by an exponen- +tial [7, 54, 55, 60–63, 66, 68] or polynomial [50] decay with +the radial distance. There are several ways to introduce +structural disorder, including lattices with uncorrelated +random sites [7,54,55,59–63,66,68], more realistic models +which preserve the local coordination number [8,49,52,65], +and lattices with controllable deviations from the crys- +talline limit [9,10,50,63]. +Characterizing topology without translational symmetry. +Among the different methods to characterize topological +phases far from a translationally invariant limits topolog- +ical markers are a wide-spread tool. Topological marker +is a unifying term that includes the local markers [69–76], +the spectral localizers [77–80], the nonlocal (spin) Bott in- +dices [66,81–89], and similar generalizations of the winding +of the quadrupole and octupole moment [62,63]. Markers +characterizing the two-dimensional quantum Hall phase +are especially well explored, including the local Chern +marker [69, 70] and the nonlocal Bott index [83]. +The +local Chern marker [69, 70] is the Fourier transform of +the Chern character. +For a crystalline lattice it quan- +tizes to the Chern number at each lattice point. For non- +crystalline lattices quantization requires averaging over a +large enough region, where the size of the region is model +dependent [70,73], (see Fig. 1(d)). The Chiral- and Chern- +Simons markers [73] are local markers analogous to the +Chern marker in odd dimensions. The chiral marker char- +acterizes the Z invariant topological phases with chiral +symmetry, whilst the Chern-Simons marker characterizes +Z2 invariant phases with either time reversal or particle- +hole symmetry, depending on the dimension. +Topological states often display a characteristic trans- +port or electromagnetic response, such as quantized longi- +tudinal conductance, the Hall conductivity, and the Wit- +ten effect [68,90–92], which can also be used to character- +ize the topological phase. The local marker in Ref. [75] is +for example based on the local Hall conductivity measured +in the bulk of the system. Alternatively, the scattering +matrix can determine topological indices without relying +on the Hamiltonian eigenstates [93]. +Topological phases can be detected by the presence of +anomalous boundary states in the local density of states +calculated with open boundary conditions [4, 5]. Neural +networks can also detect non-trivial topology, by efficiently +learning features associated with topology from for exam- +ple the wavefunctions [56], and the flow of the entangle- +p-3 + +(a) +(b) +(c) +2π +2π +2π +ky +0 +0 +0 +-2π +-2π +-2π +-2元 +0 +2π +2元 +0 +2元 +2元 +0 +2元 +一 +ka +kaSCBA +QSH +1.4 +2.1 +1.2 +1.8 +1.0 +1.5 +(a)(6) +入(eV) +0.8 +1.2 +0.9 +0.6 +0.6 +0.4 +NI +0.3 +0.2 +0.0 +0.0 0.06 0.13 0.19 0.25 0.31 0.38 0.44P. Corbae et al. +ment spectrum [94]. Other approaches include the effec- +tive Hamiltonian [49, 95], symmetry indicators [49], and +the structural spillage [96], which take advantage of the +gap closing and band inversion in a topological phase tran- +sition. The effective Hamiltonian Heff is defined as the in- +verse of the Green’s function of the system projected into +plane waves [49,95]. If the spectral gap of the total Hamil- +tonian closes, so does the spectral gap of Heff, allowing the +detection of topological phase transitions. Therefore, one +can construct topological invariants defined in terms of +Heff, which only change when the full Hamiltonian under- +goes a phase transition. Some amorphous models display +average local symmetries which are used to construct sym- +metry indicators based on the symmetry properties of the +filled states [49]. The structural spillage is a topological +indicator that measures the amount of band inversion be- +tween an amorphous system and a crystal [96], where the +knowledge of the topological state of the crystal is used to +determine the topology of the amorphous system. +Amorphous models with strong topology. +The Chern +insulator was the first amorphous topological phase to be +characterized [7, 9, 10, 49, 52, 53]. +Ref. [7] introduced a +random lattice implementation of a model that displays +a Chern insulator phase on a square lattice. +The ran- +dom lattice exhibits a gapped topological phase charac- +terized by a nontrivial Bott index, edge states, and a +quantized longitudinal conductance, which are all hall- +marks of a Chern insulator, where the nontrivial phase +is separated from trivial atomic insulators by bulk gap +closings. There exists a similar random lattice implemen- +tation of a quantum Hall state, but in the presence of a +magnetic field [97]. The three- and fourfold-coordinated +Weaire-Thorpe amorphous lattices [13] with complex in- +trasite hoppings [49], provide a more realistic model for +covalently-bonded amorphous solids. +The local symme- +tries of these models makes it possible to compute sym- +metry indicators analogous to the ones defined for crystals +[98]. +These symmetry indicators predict a Chern insu- +lator phase, which is confirmed by the presence of edge +states, the nontrivial local Chern marker, (see Fig. 1(d)), +and the effective Hamiltonian [49]. +Amorphous Chern +insulators are also present in artificial systems, such as +mechanical metamaterials [8], gyromagnetic photonic lat- +tices [9,10,99,100], and magnetic impurities on the surface +of topological insulators [52]. The Chern insulator phase +also survives in an atomic liquid, defined via tight-binding +molecular dynamics, which not only lacks long-range or- +der, but has thermally moving atoms [53]. +Amorphous quantum spin Hall insulators [7, 50, 57, 58, +94, 101] are characterized by a nonzero spin Bott index +and edge states carrying a quantized 2e2/h conductance. +Ref. [7] realized a quantum spin Hall phase by placing +the Bernevig-Hughes-Zhang model [102] on a random lat- +tice. Ref. [94] studied a similar model and calculated its +topological phase diagram using a neural network algo- +rithm that learns the flow of the entanglement spectrum. +Refs. [57, 58] performed a realistic modelling of amor- +phous monolayer Bismuth using density functional theory, +showing that the topology of the crystal survives in the +amorphous structure. +Based on both tight-binding and +density functional theory calculations, Ref. [96] showed +that the amorphous Bismuth bilayer remains topologi- +cal, as indicated by the structural spillage and the con- +ductance. +Ref. [50] demonstrated a structural-disorder- +induced quantum spin Hall phase, constructing a phase +diagram as a function of spin-orbit coupling and disor- +der strength, by modelling the disorder by Gaussian de- +viations from an initial triangular lattice. +For a range +of parameters where the initial crystal is a trivial insula- +tor, the disorder decreases the bulk gap and favours the +topological phase (see Fig. 1(e)), as happens in the onsite- +disorder-driven topological Anderson insulators [103,104]. +Amorphous structures also display 3D time-reversal- +invariant topological insulators +[7, 56, 68]. +Ref. [7] de- +scribed a 3D random lattice model with exponentially- +decaying hoppings that, for appropriate onsite energy +M and range of the hopping r0, displays surface states. +Ref. [68] further characterized the r0 − M phase diagram +of the same model, and found that the phase with surface +states features the Witten effect—due to the axion electro- +magnetic term in the action, a magnetic monopole binds +a half-odd integer electric charge, forming a dyon [90–92]. +Moreover, Ref. [56] studied a discrete random lattice typi- +cal of quantum percolation theory: a 3D cubic lattice with +nearest neighbor hoppings whose sites are occupied with +a given probability p, which controls the number and size +of vacancies. The analysis of the zero-energy wavefunc- +tions with a convolutional neural network show that the +topological insulator survives until p ∼ 0.5. +Finally, Refs. +[59, 60] have reported gapped time- +reversal-breaking 2D amorphous topological superconduc- +tors in class D. Ref. [59] studied a Shiba glass, an en- +semble of randomly distributed magnetic moments on a +gapped superconducting surface with Rashba spin-orbit +coupling. Analogously to the topological superconducting +phases induced by the subgap Yu-Shiba-Rusinov states in +periodic arrays of magnetic atoms [105–107], the random +Shiba glass effectively realizes a 2D px + ipy chiral topo- +logical superconductor with nontrivial Chern number and +quantized thermal conductance [59]. +In contrast to the +long-range pairing in this system, Ref. [60] has realised +this topological superconductor in 2D Dirac models with +local pairing when implemented not only in random lat- +tices, but also in quasicrystalline and fractal lattices. +Spatial-symmetry-protected topological amorphous mod- +els. +Amorphous systems support and induce topological +phases beyond strong topological states, including systems +protected by spatial symmetries. [61–63, 65, 66]. The ap- +pearance of these phases is related to the concept of sta- +tistical topological insulators [108–111], which are spectral +insulators protected by an average symmetry. They dis- +play gapless boundary states pinned to the critical point +p-4 + +Amorphous topological matter: theory and experiment +of a topological phase transition, and protected from lo- +calization by the average symmetry. +Based on this idea, Ref. [61] has classified all 2D amor- +phous statistical topological insulators protected by the +average continuous rotation and reflection symmetries +present in amorphous matter. Unlike in crystals, where +reflection-symmetry-protected topological insulators dis- +play edge states only on the boundaries respecting the +symmetry, their amorphous counterparts show delocalized +boundary states at all edge terminations. Furthermore, +they are characterized by a bulk Z2 topological invariant +that can be defined from the effective Hamiltonian. +Higher order topological insulators are another ex- +ample of topological insulators protected by combina- +tions of crystalline and discrete onsite symmetries, whose +amorphous counterparts have also been reported [62–64]. +First, Ref. [62] showed that a 2D (3D) chiral-symmetry- +protected higher order topological insulator with 0D cor- +ner states is robust against bulk structural disorder as +long as the boundaries remain crystalline, as indicated by +the quantized quadrupolar (octupolar) moment. +Then, +Ref. [63] realized a structural-disorder-induced 3D higher +order topological insulator with chiral hinge modes, char- +acterized by a quantized longitudinal conductance 2e2/h +and a quantized winding number of the quadrupole mo- +ment with respect to an applied magnetic flux. +Obstructed atomic insulators are a class of insulators +that are topologically trivial, in the sense of being de- +scribed by exponentially localized and symmetric wave- +functions, but are not adiabatically connected to the triv- +ial atomic limit [112–118]. The simplest example is the +half-filled inversion-symmetric Su-Schrieffer-Heeger chain +[119]. +These examples suggest that an average Peierls- +like dimerization can give rise to obstructed phases, which +has been exploited by Ref. [65] to realize amorphous ob- +structed insulators. Ref. [65] suggested that phase-change +materials, whose amorphous form can exhibit an aver- +age dimerization characterized by a double-peak struc- +ture in the three-particle correlation function [120], can +controllably realize an obstructed amorphous phase. The +main experimental signature of amorphous obstructed in- +sulators, which differentiates them from their crystalline +counterparts, is the appearance of a flatband of fractional +charges at all terminations, not only at the corners. +Finally, there are amorphous generalizations of Weyl +semimetals, dubbed a topological amorphous metal [66]. +In crystals their topological charge can be measured by +the Chern number of a surface enclosing the node in mo- +mentum space [6]. Ref. [66] defined the amorphous coun- +terpart based on a known time-reversal-breaking two-band +Weyl semimetal model defined on a random lattice. The +topological amorphous metal is signaled by the nonzero +Bott index and Hall conductivity in the planes perpendic- +ular to the Weyl node separation in the crystal, and by the +boundary states at these planes. Furthermore, in contrast +to its crystalline version, the topological amorphous metal +displays diffusive metallic behaviour. +Amorphous topological phase transitions. +The critical +theory of topological quantum phase transitions has been +extensively studied for disordered systems, especially for +the quantum Hall plateau transitions [121,122]. The stan- +dard theory postulates that the transition is of the Ander- +son localization type, characterized by a two-parameter +scaling and a diverging localization length with univer- +sal critical exponent ν. However, recent theoretical and +numerical works point to a marginal scaling with non- +universal effective critical exponents, which could explain +the model-dependence of ν [123,124]. +Motivated by the different nature of the disorder and +of the driving parameter of the transition, Refs. [54, 55] +numerically analyzed amorphous systems. In particular, +they considered both continuum 2D random geometries as +well as discrete (square and triangular) 2D lattices with +randomly occupied sites, as studied in percolation the- +ory. In their models, a Chern insulator in class D appears +above a critical density, dependent on the parameters of +the Hamiltonian. They examined the critical scaling of +both the Chern number and the conductance, as well as +the conductance distribution curves. While their analy- +sis is compatible with the standard two-parameter scaling +form, the localization length critical exponent ν is highly +non-universal. The exponents interpolate between a ge- +ometric classical percolation transition [125] and a stan- +dard Anderson localization transition [122]. While these +differences with standard theory of disordered systems re- +main to be fully understood, it is possible that changing +the density of sites introduces a variable length scale that +modifies the range of the geometric correlations in the sys- +tem, which are believed to affect the critical exponents of +the transition [126–130]. +Strongly interacting amorphous topological models. +All the above phases concern amorphous but non- +interacting systems. +The first step towards topologi- +cal amorphous many-body systems was taken by Pro- +dan [131], who defined toric code models, which display +topological order with anyonic excitations, in random tri- +angulations. The groundstate degeneracy and anyonic ex- +citation survive amorphization, even if some commutation +relations of the Hamiltonian terms are modified. +Electron-electron interactions could also lead to many- +body amorphous topological phases. However, identifying +these phases is challenging due to the lack of local topo- +logical markers for interacting systems. Ref. [132] circum- +vented this issue by solving an amorphous Chern insulator +model [7] with strong Hubbard interactions using a par- +ton construction. Fractionalizing the electron into a neu- +tral fermion f and a charged boson b lead to a mean-field +phase diagram with a phase displaying protected electri- +cally neutral chiral edge modes of f, dubbed the fraction- +alized amorphous Chern insulator. Recently, it was shown +that the Kitaev spin-liquid is exactly solvable in a three- +fold coordinated amorphous lattice [133], which survives +even if the lattice is not fully amorphous [134]. Contrary +p-5 + +P. Corbae et al. +Mechanical +Photonic +10% +-0.0 +-0.1 +-0.2 +-0.3 +-0.4 +-0.5 +-0.6 +-0.7 +-10 +10 +5 +-5 +0 +E – EF (eV) +ф (°) +10% +0.0 +-0.2 +-0.4 +-0.6 +-0.8 +-1.0 +-10 +10 +5 +-5 +0 +ф (°) +E – EF (eV) +(a) +(b) +(c) +(d) +Electronic +⇡ +⇡/2 +3⇡/4 +2⇡ +phase +c +Figure 2: (a) A mechanical amorphous Chern insulator. Cou- +pled gyroscopes excited at the edge result in a chiral edge mode. +Adapted from [136]. (b) A photonic amorphous Chern insula- +tor with excited chiral edge modes. Adapted from [137]. (c) +ARPES spectrum of amorphous Bi2Se3 showing well-defined +dispersive features crossing the bulk gap. These midgap states +(−0.2 > E − EF +> −0.6) are two-dimensional and spin- +polarized (d). The switch in polarization at E − EF < −0.6 +and E − EF > −0.2 is consistent with bulk states. Adapted +from [138]. +to the Kitaev Honeycomb model [135], which realizes a +gapless spin-liquid, the amorphous model groundstate is a +gapped chiral spin-liquid, featuring chiral majorana edge +modes. This opens the tantalising possibility to engineer- +ing disorder, for example by focused-ion beam irradiation, +to induce a chiral quantum spin-liquid without magnetic +fields [134]. +Experimental status of amorphous topological +matter. – +Amorphous topological matter has been ex- +perimentally studied in both synthetic and solid-state sys- +tems. The first experimental observation was reported in +a mechanical system of coupled gyroscopes [8]. Later on, +the observation of spin-momentum locked surface states +was reported in an amorphous electronic system [138], as +well as topological edge states in an amorphous photonic +lattices [137,139,140]. +Despite the few experimental observations of topological +states in amorphous matter, amorphous phases of topo- +logical matter have been frequently studied. In solid state +systems, amorphous phases of topological materials have +been studied both before and after the discovery of the +quantum spin Hall effect [141]. +However experimental +studies of amorphous materials did not address the sur- +vival of topological properties. For example, phase-change +materials have been studied extensively, with GeSb2Te4 +being one of the most widely studied representative [24]. +Interestingly, GeSb2Te4is also a topological insulator in its +crystalline phase [142]. Amorphous Bi2Se3 has also been +studied long before it was predicted to be a 3D topologi- +cal insulator in its crystalline form [143]. Amorphous and +structurally disordered counterparts of crystalline topolog- +ical materials have provided materials systems that show +large spin-orbit torque efficiencies, but the existence and +role of topological surface states have not been explored +[144, 145]. Skyrmions, which have topologically distinct +spin textures, have also been observed in amorphous sys- +tems [146]. +Using fixed-coordination amorphous structures of cou- +pled gyroscopes, generated from different point sets such +as hyperuniform or jammed, Mitchel, et al [8,147] showed +the existence of a mechanical amorphous Chern insula- +tor with chiral, propogating edge modes, Fig. 2(a). The +authors used d.c. motors that interacted via a magnetic +interaction, findidng that the local connectivity, which is +predictive of the global density of states, is crucial for the +existence of topological states in amorphous systems. Sim- +ilar findings were reported in photonic systems [137,139]. +By placing an amorphous arrangement of gyromagnetic +rods into a waveguide and biasing them with a magnetic +field, the authors observe photonic topological edge states +Fig. 2(b). Interestingly, topological states exist while the +system has short range order, and disappear at the glass- +to-liquid transition [137]. Moreover, lattice disorder [140] +enhances light confinement increasing the generation rate +of correlated photon pairs by an order of magnitude com- +pared to periodic topological platforms. +Regarding electronic materials, physical vapor deposi- +tion (PVD) is a particularly useful growth technique for +amorphous materials and has been found to make amor- +phous materials which are not available by liquid quench- +ing. PVD has several advantages since it allows to con- +trol a variety of different properties, such as the substrate +temperature, growth rate (which affects the time absorbed +atoms have to diffuse to ideal positions), irradiation, and +chemical dopants to frustrate crystallization. Modifying +the substrate temperature enables the growth of amor- +phous films with different local ordering and produces +what is called an ”ideal glass” [148,149]. +Growth conditions are critical for achieving high quality +amorphous films, especially amorphous topological mate- +rials. Growing the amorphous phase of a known topolog- +ical crystal does not always preserve topological proper- +ties. Several groups have grown amorphous counterparts +of known crystalline topological insulators finding no ev- +idence for topological surface states, but rather a highly +insulating, localized state [150,151]. First-principles calcu- +lations indicate that the local environment plays an impor- +tant role in the electronic structure in three dimensional +solids [152]. If the disorder associated with the new atomic +p-6 + +Experiment @ 11.2 GHz +C +0 +Max. +. +O +- +Excitehere00 +00 +00 +00 +00 +: +. +0000 +0.0 +00 +00 +. +: +: +. +0000 +00 +0 +00 +. +. +0000 +00 +00 +00 +: +. +. +. +0000 +00 +o +06 +00 +. +. +00Amorphous topological matter: theory and experiment +positions (new atomic environment) closes the mobility +gap, the system can be trivial. These subtleties might ex- +plain why some amorphous versions of known crystalline +topological insulators do not display evidence for a topo- +logical bulk. Controlling the growth conditions may en- +able tuning of the local amorphous structure. For exam- +ple, by controlling the growth rate, atoms can be deposited +with enough time to diffuse to their preferred neighbour +(based on the chemistry of elements involved) before the +next monolayer is deposited. This leads to a well defined +local environment and subsequent electronic structure. +Focusing on electronic systems, the first demonstration +of topological properties in an amorphous solid-state sys- +tem was inspired by a known crystalline topological in- +sulator. Bi2Se3 is a textbook topological insulator with +quintuple layers separated by a van der Waals gap. Us- +ing PVD, Corbae, et al [138] grew amorphous Bi2Se3 thin +films with short and medium range order (next nearest +neighbours) as well as no van der Waals gap. In transport +measurements, the films showed an increased bulk resis- +tance that was largely temperature independent, and the +weak-antilocalization effect resulting from quantum inter- +ference in two dimensions in the presence of spin orbit +coupling. +Using, ARPES/SARPES the authors showed +that two dimensional surface states cross the bulk elec- +tronic gap and are spin polarized. +The spin polariza- +tion switches multiple times as a function of binding en- +ergy matching the spin resolved spectral function from an +amorphous topological model. These results contrast data +taken on nanocrystalline samples which show a lack of dis- +person in ARPES and an insulating resistivity, consistent +with earlier works [153]. Amorphous Bi2Se3 in this study +possesses a local environment similar to the crystal, as +seen in Raman measurements, suggesting that by preserv- +ing a similar local environment to that of the crystal the +topological bulk mobility gap is not closed preserving the +topological nature. In contrast, the atomic environemnt +at grain boundaries in nanocrystalline systems is quite dis- +ordered, providing a possible explanation for the absence +of topological features. +Looking forward, developing a workflow from growth +to measurement will help the experimental observation +of amorphous topological states in the solid state. +In- +situ measurement capabilities greatly enable spectroscopic +meausurements, as they do not require thin film capping. +Scanning tunneling microscopy (STM) can directly mea- +sure the electronic and real space structure. +Combined +with ARPES and transport, STM would be invaluable to +discover amorphous topological materials and shed light +on the nature of grain boundaries in nanocrystalline sys- +tems. Nano-ARPES is also promising as beam sizes scale +down to the order of hundreds of nanometers. +Perspective and open questions. – +The growing +field of topological phases in amorphous matter is an op- +portunity to establish a deeper understanding of topolog- +ical phases and the systems that host them. In particu- +lar, the quest to define real-space topological markers and +invariants to characterize topological phases is an ongo- +ing quest. Defining topological indicators that signal non- +crystalline topological metals remains an open question. +Specifically, generalizations of Weyl semimetals to amor- +phous systems that respect time-reversal symmetry can- +not be described by the Bott index or the Chern marker, +and thus require the development of new tools. +An important open question is the lack of experimental +evidence for solids that are both amorphous and topolog- +ical, relating to the theoretical challenge of how to effi- +ciently find them. The field would benefit from a textbook +amorphous topological material, where topology is unam- +biguously confirmed by combining different experiments. +However, we lack a criterion with which to establish a +hierarchy of amorphous materials where to find topolog- +ical phases. Currently, we draw from criteria applicable +to crystals, such as large spin-orbit coupling. However, +this methodology precludes reaching the major milestone +of finding materials that are only topological when grown +amorphous, and that are otherwise trivial crystals. Ma- +terial candidates include amorphous Sb2Se3, hosting rich +electronic properties as pressure changes the local envi- +ronment [154], and BiTeI, predicted to host a structural +topological phase transition [155]. A promising possibility +is to integrate methods such as the structural spillage and +symmetry indicators, with realistic molecular dynamics +predictions based on first principle calculations. Develop- +ing these may establish a pipeline to manufacture candi- +date material databases which can guide experiments. +A related open problem is the prediction of an amor- +phous topological superconductor beyond toy models. +Such an achievement could widen the search for platforms +useful for topological quantum computing. While topo- +logical superconductivity has been found by assuming a +finite pairing [59, 60] its appearance in a self-consistent +calculation is yet to be demonstrated. +Engineering the +interactions to obtain a self-consistent topological ground +state is a nontrivial problem since Anderson’s theorem [27] +is strictly applicable only to conventional s-wave pairing. +The search for novel topological phases and phe- +nomenology should also incorporate phenomena familiar +from crystals. +Amorphous topological states have for +example not been fully explored in amorphous interact- +ing [11,131–133], driven, or non-hermitian systems [156]. +Our focus on amorphous systems has necessarily left +aside other non-crystalline solids. Disordered crystals, al- +loys, quasicrystals, fractals, and moir´e heterostructures all +present exciting opportunities to apply and extend numer- +ous concepts presented here. +In summary, amorphous solids are central to fundamen- +tal science and technology. If we aim to establish a full +theory of topological matter that is technologically useful +it seems unavoidable to consider amorphous materials as +the largest subset of non-crystalline solids. We are con- +fident that research in this direction will bring a deeper +understanding of condensed matter, as well as novel and +p-7 + +P. Corbae et al. +interesting phenomenology. +Acknowledgements. – +We thank S. Franca and J. +Schirmann for discussions and related collaborations. We +thank F. Hellman for her invaluable help in understand- +ing amorphous solids. A.G.G. and Q. Marsal acknowledge +financial support from the European Union Horizon 2020 +research and innovation program under grant agreement +No. 829044 (SCHINES). A. G. G. is also supported by +the European Research Council (ERC) Consolidator grant +under grant agreement No. 101042707 (TOPOMORPH). +P.C. was primarily funded by the US Department of En- +ergy, Office of Science, Office of Basic Energy Sciences, +Materials Sciences and Engineering Division under Con- +tract No. DE-AC02-05-CH11231 (NEMM program MS- +MAG). D.M.S. is supported by an FPU predoctoral con- +tract from Spanish MCIU No. FPU19/03195. J.D.H is +supported by the Swedish Research Council (VR) through +grant numbers 2019-04736 and 2020-00214. +References +[1] von Klitzing K., Rev. Mod. Phys., 58 (1986) 519. +https://link.aps.org/doi/10.1103/RevModPhys.58. +519 +[2] Hofstadter D. R., Phys. Rev. B, 14 (1976) 2239. +https://link.aps.org/doi/10.1103/PhysRevB.14. +2239 +[3] Haldane F. D. M., Phys. Rev. Lett., 61 (1988) 2015. +https://link.aps.org/doi/10.1103/PhysRevLett. +61.2015 +[4] Hasan M. Z. and Kane C. 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Grushin4 (a) (b) 1 Department of Materials Science, University of California, Berkeley, California 94720, USA 2 Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 3 Department of Physics, KTH Royal Institute of Technology, 106 91 Stockholm, Sweden 4 Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Grenoble Alpes, CNRS, Grenoble INP, Institut N´eel, 38000 Grenoble, France 5 Donostia International Physics Center, 20018 Donostia-San Sebastian, Spain PACS nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='xx – First pacs description PACS nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='xx – Second pacs description Abstract – Topological phases of matter are ubiquitous in crystals, but less is known about their existence in amorphous systems, that lack long-range order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' In this perspective, we review the recent progress made on theoretically defining amorphous topological phases and the new phenomenology that they can open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' We revisit key experiments suggesting that amorphous topo- logical phases exist in both solid-state and synthetic amorphous systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' We finish by discussing the open questions in the field, that promises to significantly enlarge the set of materials and synthetic systems benefiting from the robustness of topological matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' – The quantum Hall state of a two di- mensional electron gas, the first topological phase ever ob- served, was discovered in a non-crystalline system [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='Here free electrons are confined to move in two dimensions un- der a perpendicular magnetic field, display Landau lev- els, and an associated quantized metallic topological edge states caused by a confining potential, with no assumption of an underlying crystalline lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The quantum Hall state does, however, display a continuous translational in- variance, since the electron’s momentum p is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The momentum enters the parabolic dispersion relation p2/2m, with m being the electron’s mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' By promoting m to the effective mass of the electron within a medium, the parabolic dispersion and its corresponding Landau lev- els can be thought of as arising from the long-wavelength limit of a lattice tight-binding crystalline model [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' With this notion of translational invariance in place, the con- densed matter community discovered how to dispose of magnetic fields to define topological states in crystalline systems [3], establishing topological phases in crystals of any dimension, irrespective of their insulating, conducting or superconducting nature [4–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Topological phases do exist in the absence of long-range periodicity, as we are not forced to regularize a continuum (a)adolfo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='grushin@neel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='cnrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='fr (b)All authors contributed equally to the writing of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' theory using a periodic lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' This observation is at the heart of this perspective article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Our goal is to summarize the recent progress made to understand how topological phases emerge on the largest class of non-crystalline sys- tems, amorphous systems [7–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Characterizing topol- ogy in amorphous matter, without the convenience of Bloch’s theorem, has lead to the emergence of new phe- nomenology, unique to amorphous matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Topology re- mains largely unexplored in this class of solids, which may offer different functionalities compared to crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' We start by discussing the main properties of amor- phous and topological matter, followed by a review of the progress made in combining these two fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' We finish by summarizing the experimental status and offering some perspectives on the main open questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' For a more tech- nical review we refer the reader to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Basic properties of amorphous matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Amorphous materials are defined by their lack of long-range order [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' However, they display short- and even medium-range or- der, as well defined nearest and next-to-nearest neighbour distances, respectively (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' 1 (a-c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The short range or- der manifests itself as preferred bond lengths and angles, peaked around the values of its crystalline counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Due to the short range order, amorphous materials have a well defined coordination environment with a distinct number of nearest neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' In solid state systems this p-1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='04176v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='mes-hall] 10 Jan 2023 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Corbae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' is a result of the electronic configuration of the atoms in- volved in bonding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Hence, amorphous solids remain lo- cally ordered [12,13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Elucidating the atomic structure of amorphous solids is necessary to understand most of their electronic proper- ties [12,14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The disordered atomic positions in amorphous solids result in diffuse rings in the diffraction pattern and a lack of sharp Bragg peaks characteristic of crystalline materials [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The absence of discrete crystalline symme- try, in favour of local short range order and well defined diffraction rings demonstrates that amorphous systems are isotropic on average (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' 1) [15–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Fluctuations of the bond lengths account for the broadening of the rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The radii of the diffraction rings and its weight can be used to determine the structure factor and estimate an aver- age bond distance and coordination number of the amor- phous structure [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The absence of Bragg peaks in the diffraction pattern, and thus the absence of long-range order, determines which solids are amorphous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Amorphous materials are commonplace in science and technology [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Their applications range from common objects such as window glass to technological devices like computer memories or solar cells [22,23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Amorphous ma- terials are advantageous for technological applications as they can be grown under less stringent conditions than single crystals require.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' In solids state systems, they can be grown in a range of compositions unlike typical crys- talline compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Transitioning between the amorphous to the crystalline state in a controlled and reversible man- ner, for example using current or laser pulses [24], is a useful and defining property of phase-change materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' These are commonly used in computer memory-storage devices [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Additionally, amorphous materials play a major role in fundamental science e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' as coatings in gravitational waves detectors at LIGO [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Similarly to crystals, amorphous materials can be insu- lators, semiconductors, metals, and superconductors [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Amorphous oxides used in glassware, such as silicon oxide or lead glass, are century-old insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Amorphous semi- conductors, such as silicon or germanium, have also been extensively studied, due to their possible use in electronic devices [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Amorphous metals are exceptionally hard and can display unique magnetic properties [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Amor- phous superconductors can also be synthesised, as conven- tional superconductivity is robust to disorder, an observa- tion known as Anderson’s theorem [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Remarkably, the critical superconducting temperature has been observed to be higher in several amorphous materials compared to their crystalline counterpart (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' 1(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The existence and robustness of topological phases poses the natural question of whether they can be realized in amorphous systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Before reviewing how topological amorphous phases were first achieved [7–10] and extended, we revisit the main properties of topological phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Basic properties of topological matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The discovery of the quantum Hall effect and its quantized Hall conduc- tance [28–30] introduced the field of topological matter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' phases of matter characterized by their metallic boundary states and quantized responses to external fields, which are robust against impurities and local perturbations [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The quantum Hall effect is an example of a strong topo- logical insulator [4,5], phases of matter where the bound- ary states are protected by local symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' These sym- metries are the time reversal-, particle hole-, and chiral symmetry, the latter being the product of the other two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' These three symmetries can be combined in ten different ways, defining the Altland-Zirnbauer classification of first quantized free fermion Hamiltonians [32,33], leading to the full classification of strong topological insulators and su- perconductors [34–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' There are five non-trivial Altland- Zirnbauer classes, classes that can host topological phases, in every dimension, each defined by a topological invari- ant, either integer valued Chern or winding numbers or a Z2 invariant [36], characterizing the phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Two states are defined to be in the same topological phase if they can be adiabatically perturbed into one another smoothly without closing the conduction gap and not breaking the underlying symmetries, while keeping the number of or- bitals fixed during the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Translational invariance in crystal lattices allows use of Bloch’s theorem to define crystal momentum, simplifying the characterization of the topological phases and yielding closed-form momentum-space expressions of the topologi- cal invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' For example, the Chern number [39] char- acterizing the quantum Hall phase in two dimensions (2D) is evaluated as the integral of the Berry curvature [40] over the first Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Electronic topological phases extend beyond strong topological insulators and superconductors, including weak [41–43]- and crystalline [44]- topological insulators, and topological metals [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Weak topological insulators can be constructed by stacking strong topological insula- tors, giving rise to symmetry protected surface states per- pendicular to the stacking direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Crystalline (point group) symmetries, including rotations and reflections, can protect topological states called higher order topo- logical insulators which host symmetry protected states on those surfaces which are invariant under the crystalline symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Crystalline symmetries simplify how to iden- tify topological phases, through to the concept of symme- try indicators [45–48]— eigenvalues of point group opera- tors whose products determine topological invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Although translation invariance simplifies describing and classifying topological phases, it is not necessary for their existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' For example, strong topology is protected by local symmetries, irrespective of the lattice details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Non-trivial topology only requires the existence of a mo- bility gap, and not a spectral gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' However, characterizing topological phases of matter far from the crystalline limit, notably for non-crystalline lattices, requires new tools, as the known momentum space expressions for topological in- variants are no longer applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' We describe these tools and the models introduced to study amorphous topologi- p-2 Amorphous topological matter: theory and experiment (a) (b) (c) (d) (e) (f) Bi Be Al 5 mK 6 K 26 mK 10 K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='18 K 6 K Amorphous Crystal Supercond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Tc Element amorphous crystal quasicrystal (a) (b) (c) (d) (e) (f) Figure 1: (a) Crystalline lattice and corresponding structure factor with sharp Bragg peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' (b) Amorphous lattices with local order result in broad diffraction rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' (c) Quasicrystals break translational invariance but retain long range order, re- sulting in sharp Bragg peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' (d) Local Chern marker of the 2D threefold-coordinated Weaire-Thorpe model [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The bulk av- erage equals −1, indicating a nontrivial Chern insulator phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' (e) Configuration-averaged spectral gap, Eg, of a 2D quantum spin Hall model on a structurally disordered trigonal lattice as a function of disorder strength σ and spin-orbit coupling λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The quantum spin Hall (QSH) and normal insulator (NI) phases are labelled according to the value of the spin Bott index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Adapted from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' (f) Superconducting critical temperature of different amorphous and crystalline solids [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' cal matter next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Theory of amorphous topological matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' – Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' There are a variety of amorphous models displaying topological phases, ranging from strong topo- logical states to spatial-symmetry-protected topological phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Amorphous strong topological states include 2D Chern insulators in class A [7–10, 49, 52–55], 2D and 3D time-reversal invariant topological insulators in class AII [7,50,54,56–58], and 2D time-reversal breaking topological superconductors in class D [59,60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Amorphous structures also support phases a priori protected by crystalline sym- metries, such as 2D reflection-symmetry-protected topo- logical insulators [61], 2D and 3D higher-order topological insulators [62–64], 2D and 3D obstructed insulators [65], and 3D topological metals [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' While structural disorder is detrimental to some of these states, it can also induce nontrivial phases when starting from a trivial crystalline state [50,63,66], and it can give rise to new phenomenology intrinsically associated with amorphous topological mat- ter and phase transitions [54,55,61,65,66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' A common starting point is a crystalline tight bind- ing Hamiltonian known to host a topologically nontriv- ial phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The hopping terms are generalized to ac- count for arbitrary angles and distances between sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' For example the angular dependence can be modelled us- ing the Slater-Koster parametrization [67], and the rea- dial dependence can be accounted for by an exponen- tial [7, 54, 55, 60–63, 66, 68] or polynomial [50] decay with the radial distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' There are several ways to introduce structural disorder, including lattices with uncorrelated random sites [7,54,55,59–63,66,68], more realistic models which preserve the local coordination number [8,49,52,65], and lattices with controllable deviations from the crys- talline limit [9,10,50,63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Characterizing topology without translational symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Among the different methods to characterize topological phases far from a translationally invariant limits topolog- ical markers are a wide-spread tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Topological marker is a unifying term that includes the local markers [69–76], the spectral localizers [77–80], the nonlocal (spin) Bott in- dices [66,81–89], and similar generalizations of the winding of the quadrupole and octupole moment [62,63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Markers characterizing the two-dimensional quantum Hall phase are especially well explored, including the local Chern marker [69, 70] and the nonlocal Bott index [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The local Chern marker [69, 70] is the Fourier transform of the Chern character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' For a crystalline lattice it quan- tizes to the Chern number at each lattice point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' For non- crystalline lattices quantization requires averaging over a large enough region, where the size of the region is model dependent [70,73], (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' 1(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The Chiral- and Chern- Simons markers [73] are local markers analogous to the Chern marker in odd dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The chiral marker char- acterizes the Z invariant topological phases with chiral symmetry, whilst the Chern-Simons marker characterizes Z2 invariant phases with either time reversal or particle- hole symmetry, depending on the dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Topological states often display a characteristic trans- port or electromagnetic response, such as quantized longi- tudinal conductance, the Hall conductivity, and the Wit- ten effect [68,90–92], which can also be used to character- ize the topological phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The local marker in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' [75] is for example based on the local Hall conductivity measured in the bulk of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Alternatively, the scattering matrix can determine topological indices without relying on the Hamiltonian eigenstates [93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Topological phases can be detected by the presence of anomalous boundary states in the local density of states calculated with open boundary conditions [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Neural networks can also detect non-trivial topology, by efficiently learning features associated with topology from for exam- ple the wavefunctions [56], and the flow of the entangle- p-3 (a) (b) (c) 2π 2π 2π ky 0 0 0 2π 2π 2π 2元 0 2π 2元 0 2元 2元 0 2元 一 ka kaSCBA QSH 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='5 (a)(6) 入(eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='4 NI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='44P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Corbae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' ment spectrum [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Other approaches include the effec- tive Hamiltonian [49, 95], symmetry indicators [49], and the structural spillage [96], which take advantage of the gap closing and band inversion in a topological phase tran- sition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The effective Hamiltonian Heff is defined as the in- verse of the Green’s function of the system projected into plane waves [49,95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' If the spectral gap of the total Hamil- tonian closes, so does the spectral gap of Heff, allowing the detection of topological phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Therefore, one can construct topological invariants defined in terms of Heff, which only change when the full Hamiltonian under- goes a phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Some amorphous models display average local symmetries which are used to construct sym- metry indicators based on the symmetry properties of the filled states [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The structural spillage is a topological indicator that measures the amount of band inversion be- tween an amorphous system and a crystal [96], where the knowledge of the topological state of the crystal is used to determine the topology of the amorphous system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Amorphous models with strong topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The Chern insulator was the first amorphous topological phase to be characterized [7, 9, 10, 49, 52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' [7] introduced a random lattice implementation of a model that displays a Chern insulator phase on a square lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The ran- dom lattice exhibits a gapped topological phase charac- terized by a nontrivial Bott index, edge states, and a quantized longitudinal conductance, which are all hall- marks of a Chern insulator, where the nontrivial phase is separated from trivial atomic insulators by bulk gap closings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' There exists a similar random lattice implemen- tation of a quantum Hall state, but in the presence of a magnetic field [97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The three- and fourfold-coordinated Weaire-Thorpe amorphous lattices [13] with complex in- trasite hoppings [49], provide a more realistic model for covalently-bonded amorphous solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The local symme- tries of these models makes it possible to compute sym- metry indicators analogous to the ones defined for crystals [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' These symmetry indicators predict a Chern insu- lator phase, which is confirmed by the presence of edge states, the nontrivial local Chern marker, (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' 1(d)), and the effective Hamiltonian [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Amorphous Chern insulators are also present in artificial systems, such as mechanical metamaterials [8], gyromagnetic photonic lat- tices [9,10,99,100], and magnetic impurities on the surface of topological insulators [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The Chern insulator phase also survives in an atomic liquid, defined via tight-binding molecular dynamics, which not only lacks long-range or- der, but has thermally moving atoms [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Amorphous quantum spin Hall insulators [7, 50, 57, 58, 94, 101] are characterized by a nonzero spin Bott index and edge states carrying a quantized 2e2/h conductance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' [7] realized a quantum spin Hall phase by placing the Bernevig-Hughes-Zhang model [102] on a random lat- tice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' [94] studied a similar model and calculated its topological phase diagram using a neural network algo- rithm that learns the flow of the entanglement spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' [57, 58] performed a realistic modelling of amor- phous monolayer Bismuth using density functional theory, showing that the topology of the crystal survives in the amorphous structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Based on both tight-binding and density functional theory calculations, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' [96] showed that the amorphous Bismuth bilayer remains topologi- cal, as indicated by the structural spillage and the con- ductance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' [50] demonstrated a structural-disorder- induced quantum spin Hall phase, constructing a phase diagram as a function of spin-orbit coupling and disor- der strength, by modelling the disorder by Gaussian de- viations from an initial triangular lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' For a range of parameters where the initial crystal is a trivial insula- tor, the disorder decreases the bulk gap and favours the topological phase (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' 1(e)), as happens in the onsite- disorder-driven topological Anderson insulators [103,104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Amorphous structures also display 3D time-reversal- invariant topological insulators [7, 56, 68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' [7] de- scribed a 3D random lattice model with exponentially- decaying hoppings that, for appropriate onsite energy M and range of the hopping r0, displays surface states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' [68] further characterized the r0 − M phase diagram of the same model, and found that the phase with surface states features the Witten effect—due to the axion electro- magnetic term in the action, a magnetic monopole binds a half-odd integer electric charge, forming a dyon [90–92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Moreover, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' [56] studied a discrete random lattice typi- cal of quantum percolation theory: a 3D cubic lattice with nearest neighbor hoppings whose sites are occupied with a given probability p, which controls the number and size of vacancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The analysis of the zero-energy wavefunc- tions with a convolutional neural network show that the topological insulator survives until p ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Finally, Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' [59, 60] have reported gapped time- reversal-breaking 2D amorphous topological superconduc- tors in class D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' [59] studied a Shiba glass, an en- semble of randomly distributed magnetic moments on a gapped superconducting surface with Rashba spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Analogously to the topological superconducting phases induced by the subgap Yu-Shiba-Rusinov states in periodic arrays of magnetic atoms [105–107], the random Shiba glass effectively realizes a 2D px + ipy chiral topo- logical superconductor with nontrivial Chern number and quantized thermal conductance [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' In contrast to the long-range pairing in this system, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' [60] has realised this topological superconductor in 2D Dirac models with local pairing when implemented not only in random lat- tices, but also in quasicrystalline and fractal lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Spatial-symmetry-protected topological amorphous mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Amorphous systems support and induce topological phases beyond strong topological states, including systems protected by spatial symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' [61–63, 65, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The ap- pearance of these phases is related to the concept of sta- tistical topological insulators [108–111], which are spectral insulators protected by an average symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' They dis- play gapless boundary states pinned to the critical point p-4 Amorphous topological matter: theory and experiment of a topological phase transition, and protected from lo- calization by the average symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Based on this idea, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' [61] has classified all 2D amor- phous statistical topological insulators protected by the average continuous rotation and reflection symmetries present in amorphous matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Unlike in crystals, where reflection-symmetry-protected topological insulators dis- play edge states only on the boundaries respecting the symmetry, their amorphous counterparts show delocalized boundary states at all edge terminations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Furthermore, they are characterized by a bulk Z2 topological invariant that can be defined from the effective Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Higher order topological insulators are another ex- ample of topological insulators protected by combina- tions of crystalline and discrete onsite symmetries, whose amorphous counterparts have also been reported [62–64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' First, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' [62] showed that a 2D (3D) chiral-symmetry- protected higher order topological insulator with 0D cor- ner states is robust against bulk structural disorder as long as the boundaries remain crystalline, as indicated by the quantized quadrupolar (octupolar) moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Then, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' [63] realized a structural-disorder-induced 3D higher order topological insulator with chiral hinge modes, char- acterized by a quantized longitudinal conductance 2e2/h and a quantized winding number of the quadrupole mo- ment with respect to an applied magnetic flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Obstructed atomic insulators are a class of insulators that are topologically trivial, in the sense of being de- scribed by exponentially localized and symmetric wave- functions, but are not adiabatically connected to the triv- ial atomic limit [112–118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The simplest example is the half-filled inversion-symmetric Su-Schrieffer-Heeger chain [119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' These examples suggest that an average Peierls- like dimerization can give rise to obstructed phases, which has been exploited by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' [65] to realize amorphous ob- structed insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' [65] suggested that phase-change materials, whose amorphous form can exhibit an aver- age dimerization characterized by a double-peak struc- ture in the three-particle correlation function [120], can controllably realize an obstructed amorphous phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The main experimental signature of amorphous obstructed in- sulators, which differentiates them from their crystalline counterparts, is the appearance of a flatband of fractional charges at all terminations, not only at the corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Finally, there are amorphous generalizations of Weyl semimetals, dubbed a topological amorphous metal [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' In crystals their topological charge can be measured by the Chern number of a surface enclosing the node in mo- mentum space [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' [66] defined the amorphous coun- terpart based on a known time-reversal-breaking two-band Weyl semimetal model defined on a random lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The topological amorphous metal is signaled by the nonzero Bott index and Hall conductivity in the planes perpendic- ular to the Weyl node separation in the crystal, and by the boundary states at these planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Furthermore, in contrast to its crystalline version, the topological amorphous metal displays diffusive metallic behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Amorphous topological phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The critical theory of topological quantum phase transitions has been extensively studied for disordered systems, especially for the quantum Hall plateau transitions [121,122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The stan- dard theory postulates that the transition is of the Ander- son localization type, characterized by a two-parameter scaling and a diverging localization length with univer- sal critical exponent ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' However, recent theoretical and numerical works point to a marginal scaling with non- universal effective critical exponents, which could explain the model-dependence of ν [123,124].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Motivated by the different nature of the disorder and of the driving parameter of the transition, Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' [54, 55] numerically analyzed amorphous systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' In particular, they considered both continuum 2D random geometries as well as discrete (square and triangular) 2D lattices with randomly occupied sites, as studied in percolation the- ory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' In their models, a Chern insulator in class D appears above a critical density, dependent on the parameters of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' They examined the critical scaling of both the Chern number and the conductance, as well as the conductance distribution curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' While their analy- sis is compatible with the standard two-parameter scaling form, the localization length critical exponent ν is highly non-universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The exponents interpolate between a ge- ometric classical percolation transition [125] and a stan- dard Anderson localization transition [122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' While these differences with standard theory of disordered systems re- main to be fully understood, it is possible that changing the density of sites introduces a variable length scale that modifies the range of the geometric correlations in the sys- tem, which are believed to affect the critical exponents of the transition [126–130].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Strongly interacting amorphous topological models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' All the above phases concern amorphous but non- interacting systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The first step towards topologi- cal amorphous many-body systems was taken by Pro- dan [131], who defined toric code models, which display topological order with anyonic excitations, in random tri- angulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The groundstate degeneracy and anyonic ex- citation survive amorphization, even if some commutation relations of the Hamiltonian terms are modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Electron-electron interactions could also lead to many- body amorphous topological phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' However, identifying these phases is challenging due to the lack of local topo- logical markers for interacting systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' [132] circum- vented this issue by solving an amorphous Chern insulator model [7] with strong Hubbard interactions using a par- ton construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Fractionalizing the electron into a neu- tral fermion f and a charged boson b lead to a mean-field phase diagram with a phase displaying protected electri- cally neutral chiral edge modes of f, dubbed the fraction- alized amorphous Chern insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Recently, it was shown that the Kitaev spin-liquid is exactly solvable in a three- fold coordinated amorphous lattice [133], which survives even if the lattice is not fully amorphous [134].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Contrary p-5 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Corbae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Mechanical Photonic 10% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='7 10 10 5 5 0 E – EF (eV) ф (°) 10% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='0 10 10 5 5 0 ф (°) E – EF (eV) (a) (b) (c) (d) Electronic ⇡ ⇡/2 3⇡/4 2⇡ phase c Figure 2: (a) A mechanical amorphous Chern insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Cou- pled gyroscopes excited at the edge result in a chiral edge mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Adapted from [136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' (b) A photonic amorphous Chern insula- tor with excited chiral edge modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Adapted from [137].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' (c) ARPES spectrum of amorphous Bi2Se3 showing well-defined dispersive features crossing the bulk gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' These midgap states (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='2 > E − EF > −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='6) are two-dimensional and spin- polarized (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The switch in polarization at E − EF < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='6 and E − EF > −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='2 is consistent with bulk states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Adapted from [138].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' to the Kitaev Honeycomb model [135], which realizes a gapless spin-liquid, the amorphous model groundstate is a gapped chiral spin-liquid, featuring chiral majorana edge modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' This opens the tantalising possibility to engineer- ing disorder, for example by focused-ion beam irradiation, to induce a chiral quantum spin-liquid without magnetic fields [134].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Experimental status of amorphous topological matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' – Amorphous topological matter has been ex- perimentally studied in both synthetic and solid-state sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The first experimental observation was reported in a mechanical system of coupled gyroscopes [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Later on, the observation of spin-momentum locked surface states was reported in an amorphous electronic system [138], as well as topological edge states in an amorphous photonic lattices [137,139,140].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Despite the few experimental observations of topological states in amorphous matter, amorphous phases of topo- logical matter have been frequently studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' In solid state systems, amorphous phases of topological materials have been studied both before and after the discovery of the quantum spin Hall effect [141].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' However experimental studies of amorphous materials did not address the sur- vival of topological properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' For example, phase-change materials have been studied extensively, with GeSb2Te4 being one of the most widely studied representative [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Interestingly, GeSb2Te4is also a topological insulator in its crystalline phase [142].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Amorphous Bi2Se3 has also been studied long before it was predicted to be a 3D topologi- cal insulator in its crystalline form [143].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Amorphous and structurally disordered counterparts of crystalline topolog- ical materials have provided materials systems that show large spin-orbit torque efficiencies, but the existence and role of topological surface states have not been explored [144, 145].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Skyrmions, which have topologically distinct spin textures, have also been observed in amorphous sys- tems [146].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Using fixed-coordination amorphous structures of cou- pled gyroscopes, generated from different point sets such as hyperuniform or jammed, Mitchel, et al [8,147] showed the existence of a mechanical amorphous Chern insula- tor with chiral, propogating edge modes, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The authors used d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' motors that interacted via a magnetic interaction, findidng that the local connectivity, which is predictive of the global density of states, is crucial for the existence of topological states in amorphous systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Sim- ilar findings were reported in photonic systems [137,139].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' By placing an amorphous arrangement of gyromagnetic rods into a waveguide and biasing them with a magnetic field, the authors observe photonic topological edge states Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Interestingly, topological states exist while the system has short range order, and disappear at the glass- to-liquid transition [137].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Moreover, lattice disorder [140] enhances light confinement increasing the generation rate of correlated photon pairs by an order of magnitude com- pared to periodic topological platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Regarding electronic materials, physical vapor deposi- tion (PVD) is a particularly useful growth technique for amorphous materials and has been found to make amor- phous materials which are not available by liquid quench- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' PVD has several advantages since it allows to con- trol a variety of different properties, such as the substrate temperature, growth rate (which affects the time absorbed atoms have to diffuse to ideal positions), irradiation, and chemical dopants to frustrate crystallization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Modifying the substrate temperature enables the growth of amor- phous films with different local ordering and produces what is called an ”ideal glass” [148,149].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Growth conditions are critical for achieving high quality amorphous films, especially amorphous topological mate- rials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Growing the amorphous phase of a known topolog- ical crystal does not always preserve topological proper- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Several groups have grown amorphous counterparts of known crystalline topological insulators finding no ev- idence for topological surface states, but rather a highly insulating, localized state [150,151].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' First-principles calcu- lations indicate that the local environment plays an impor- tant role in the electronic structure in three dimensional solids [152].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' If the disorder associated with the new atomic p-6 Experiment @ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='2 GHz C 0 Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' O Excitehere00 00 00 00 00 : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' 0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='0 00 00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' : : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' 0000 00 0 00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' 0000 00 00 00 : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' 0000 00 o 06 00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' 00Amorphous topological matter: theory and experiment positions (new atomic environment) closes the mobility gap, the system can be trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' These subtleties might ex- plain why some amorphous versions of known crystalline topological insulators do not display evidence for a topo- logical bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Controlling the growth conditions may en- able tuning of the local amorphous structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' For exam- ple, by controlling the growth rate, atoms can be deposited with enough time to diffuse to their preferred neighbour (based on the chemistry of elements involved) before the next monolayer is deposited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' This leads to a well defined local environment and subsequent electronic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Focusing on electronic systems, the first demonstration of topological properties in an amorphous solid-state sys- tem was inspired by a known crystalline topological in- sulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Bi2Se3 is a textbook topological insulator with quintuple layers separated by a van der Waals gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Us- ing PVD, Corbae, et al [138] grew amorphous Bi2Se3 thin films with short and medium range order (next nearest neighbours) as well as no van der Waals gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' In transport measurements, the films showed an increased bulk resis- tance that was largely temperature independent, and the weak-antilocalization effect resulting from quantum inter- ference in two dimensions in the presence of spin orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Using, ARPES/SARPES the authors showed that two dimensional surface states cross the bulk elec- tronic gap and are spin polarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The spin polariza- tion switches multiple times as a function of binding en- ergy matching the spin resolved spectral function from an amorphous topological model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' These results contrast data taken on nanocrystalline samples which show a lack of dis- person in ARPES and an insulating resistivity, consistent with earlier works [153].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Amorphous Bi2Se3 in this study possesses a local environment similar to the crystal, as seen in Raman measurements, suggesting that by preserv- ing a similar local environment to that of the crystal the topological bulk mobility gap is not closed preserving the topological nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' In contrast, the atomic environemnt at grain boundaries in nanocrystalline systems is quite dis- ordered, providing a possible explanation for the absence of topological features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Looking forward, developing a workflow from growth to measurement will help the experimental observation of amorphous topological states in the solid state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' In- situ measurement capabilities greatly enable spectroscopic meausurements, as they do not require thin film capping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Scanning tunneling microscopy (STM) can directly mea- sure the electronic and real space structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Combined with ARPES and transport, STM would be invaluable to discover amorphous topological materials and shed light on the nature of grain boundaries in nanocrystalline sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Nano-ARPES is also promising as beam sizes scale down to the order of hundreds of nanometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Perspective and open questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' – The growing field of topological phases in amorphous matter is an op- portunity to establish a deeper understanding of topolog- ical phases and the systems that host them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' In particu- lar, the quest to define real-space topological markers and invariants to characterize topological phases is an ongo- ing quest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Defining topological indicators that signal non- crystalline topological metals remains an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Specifically, generalizations of Weyl semimetals to amor- phous systems that respect time-reversal symmetry can- not be described by the Bott index or the Chern marker, and thus require the development of new tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' An important open question is the lack of experimental evidence for solids that are both amorphous and topolog- ical, relating to the theoretical challenge of how to effi- ciently find them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The field would benefit from a textbook amorphous topological material, where topology is unam- biguously confirmed by combining different experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' However, we lack a criterion with which to establish a hierarchy of amorphous materials where to find topolog- ical phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Currently, we draw from criteria applicable to crystals, such as large spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' However, this methodology precludes reaching the major milestone of finding materials that are only topological when grown amorphous, and that are otherwise trivial crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Ma- terial candidates include amorphous Sb2Se3, hosting rich electronic properties as pressure changes the local envi- ronment [154], and BiTeI, predicted to host a structural topological phase transition [155].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' A promising possibility is to integrate methods such as the structural spillage and symmetry indicators, with realistic molecular dynamics predictions based on first principle calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Develop- ing these may establish a pipeline to manufacture candi- date material databases which can guide experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' A related open problem is the prediction of an amor- phous topological superconductor beyond toy models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Such an achievement could widen the search for platforms useful for topological quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' While topo- logical superconductivity has been found by assuming a finite pairing [59, 60] its appearance in a self-consistent calculation is yet to be demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Engineering the interactions to obtain a self-consistent topological ground state is a nontrivial problem since Anderson’s theorem [27] is strictly applicable only to conventional s-wave pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' The search for novel topological phases and phe- nomenology should also incorporate phenomena familiar from crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Amorphous topological states have for example not been fully explored in amorphous interact- ing [11,131–133], driven, or non-hermitian systems [156].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Our focus on amorphous systems has necessarily left aside other non-crystalline solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Disordered crystals, al- loys, quasicrystals, fractals, and moir´e heterostructures all present exciting opportunities to apply and extend numer- ous concepts presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' In summary, amorphous solids are central to fundamen- tal science and technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' If we aim to establish a full theory of topological matter that is technologically useful it seems unavoidable to consider amorphous materials as the largest subset of non-crystalline solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' We are con- fident that research in this direction will bring a deeper understanding of condensed matter, as well as novel and p-7 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Corbae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' interesting phenomenology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' – We thank S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Franca and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Schirmann for discussions and related collaborations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' We thank F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Hellman for her invaluable help in understand- ing amorphous solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Marsal acknowledge financial support from the European Union Horizon 2020 research and innovation program under grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' 829044 (SCHINES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' is also supported by the European Research Council (ERC) Consolidator grant under grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' 101042707 (TOPOMORPH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' was primarily funded by the US Department of En- ergy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division under Con- tract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' DE-AC02-05-CH11231 (NEMM program MS- MAG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' is supported by an FPU predoctoral con- tract from Spanish MCIU No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' FPU19/03195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content='H is supported by the Swedish Research Council (VR) through grant numbers 2019-04736 and 2020-00214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' References [1] von Klitzing K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=', Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytE2T4oBgHgl3EQf4AhB/content/2301.04176v1.pdf'} +page_content=' Mod.' metadata={'source': 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This is the author’s version which has not been fully edited and content may change +prior to final publication. Citation information: DOI 10.1007/s00371-022-02703-y +Screen Space Indirect Lighting with Visibility Bitmask +Olivier Therrien1† Yannick Levesque2‡ Guillaume Gilet3§ +1CDRIN, QC, Canada +2Cégep de Matane, QC, Canada +3 University of Sherbrooke +Figure 1: Left: Direct illumination of the scene. Middle: Indirect lighting produced by our method (without texture). Right: Final frame +rendered with our method, exhibiting directionally occluded ambient lighting, and a GI bounce that avoids typical thin surface artifacts. +Abstract +Horizon-based indirect illumination efficiently estimates a diffuse light bounce in screen space by analytically integrat- +ing the horizon angle difference between samples along a given direction. Like other horizon-based methods, this technique +cannot properly simulate light passing behind thin surfaces. We propose the concept of a visibility bitmask that re- +places the two horizon angles by a bit field representing the binary state (occluded / un-occluded) of N sectors uniformly +distributed around the hemisphere slice. It allows light to pass behind surfaces of constant thickness while keeping the ef- +ficiency of horizon-based methods. It can also do more accurate ambient lighting than bent normal by sampling more than +one visibility cone. This technique improves the visual quality of ambient occlusion, indirect diffuse, and ambient light com- +pared to previous screen space methods while minimizing noise and keeping a low performance overhead. +Keywords: Real-Time Rendering, Indirect Lighting, Ambient Occlusion, Visibility +1. Introduction +Indirect diffuse lighting is challenging to compute in real-time. +Screen space approximations can be attractive as they reduce the di- +mensionality of the problem and make the execution cost constant +regardless of the scene’s geometric complexity. Modern Screen +Space Global Illumination (SSGI) implementations often gather +indirect light by doing ray marching on screen pixels similar to +† therrien.olivier@cdrin.com +‡ levesqueyannick@cgmatane.qc.ca +§ guillaume.gilet@usherbrooke.ca +Screen Space Reflections (SSR) [SKS11]. This approach tends to +generate a lot of noise because it implies the numerical integra- +tion of irradiance over the entire hemisphere around the surface. +Horizon-Based Indirect Illumination (HBIL) [May18], based on +Horizon-Based Ambient Occlusion (HBAO) [BSD08, Bav11] and +Ground Truth Ambient Occlusion (GTAO) [JWPJ16], improve the +efficiency by numerically integrating over a set of directions around +the view vector v (Figure 2) while doing analytic integration of the +horizon angle difference between samples. +Fundamentaly, the core principle of these methods lies in the es- +timation of the scene local geometry around each shading sample +arXiv:2301.11376v1 [cs.GR] 26 Jan 2023 + +2 +O. Therrien, Y. Levesque, G. Gilet / Screen Space Indirect Lighting with Visibility Bitmask +Figure 2: +3D view of the scene, centered on the shaded pixel. +GTAO/HBIL generates a set of hemisphere slices (in green) in var- +ious directions around the view vector v. Occluders (in black) in- +tersect some of the slices, producing occlusion cones (in red). +by relying on readily-available screen-space information, such as +the discrete depth buffer. However, such information is by essence +discrete and incomplete, and must be reconstructed. All those tech- +niques evaluate Ambient Occlusion, the modulation of indirect ir- +radiance due to local geometry, in screen space from a single layer +depth buffer, and assume infinite surface thickness by treating it as a +height-field (see Figure 3). While this is a valid assumption in some +cases, not knowing what the real geometry looks like, it causes ha- +los and over-darkening around thin surfaces (see figure 4). Falloff +heuristics are used to mitigate those artifacts but fail when using a +large sampling radius. +Figure 3: Side view of one slice centered on the view vector v. Left: +Ground truth scene with multiple occluders (in black) producing +multiple visibility cones (in green). Right: GTAO takes a fixed num- +ber of samples (red dots) in the depth buffer on both sides of the +hemisphere to find highest elevation angles θ1 and θ2. +Our proposed method rejects the assumption that the depth +buffer is strictly a height-field and models the behavior of light +passing behind surfaces. Additionally, to preserve performance, we +want to avoid explicitly tracing new rays from scratch to adequately +sample multiple elevation angles. To do so, we introduce the con- +cept of visibility bitmask, which is essentially a discretization of +the hemisphere slice in Nb sectors, that allows us to approximate +the tracing of Nb rays at the same performance cost as one hori- +zon search. +Figure 4: Left: GTAO produces halos around the poles. Right: Our +method is able let light pass behind the poles without introducing +halos artifacts. +A common limitation of single layer depth buffers is that oc- +cluded surfaces are not represented, which can cause missing oc- +clusion or lighting. Approaches like Deep G-Buffer [MMNL16] or +Multi-layer SSRT [HBSS17] alleviate this issue by storing mul- +tiple layers or sample per pixel, to provide more information on +background surfaces. As rendering multiple layers is very expen- +sive, we chose to limit ourselves to only one layer. This means that +some artifacts caused by inaccurate background surface estimation +will remain, but using a constant thickness t at each depth sam- +ple with a visibility bitmask greatly improves quality around thin +surfaces compared to horizon-based techniques. In this study, we +show that visibility bitmasks can tremendously reduce noise in the +image compared to SSR-like tracing, while handling the light pass- +ing behind surfaces much more accurately than horizon-based tech- +niques. We also show that using a visibility bitmask to sample am- +bient light gives a more precise ambient estimation than traditional +methods of sampling along the surface normal or even bent nor- +mal. +The main contributions of this paper revolve around the introduc- +tion of a visibility bitmask in traditional SSGI methods, retaining +the efficiency and noise reduction qualities of horizon sampling, +while handling light passing behind surfaces of constant thick- +ness. We demonstrate the capabilities of our method through sev- +eral SSGI applications, such as ambient occlusion, directional am- +bient occlusion and indirect diffuse lighting. +2. Previous Work +Estimating ambient occlusion and indirect diffuse lighting from +screen space information is a well-known idea. It mainly stems +from the seminal Screen Space Ambient Occlusion method +[Mit07], approximating ambient occlusion by sampling random +points in the depth buffer in a circle around each pixel and has been +thoroughly extended over the years [Mit07, MOBH11, MML12]. +The keypoint of these methods is to estimate the local geometry +around a sample using the incomplete information contained in the +various buffers (geometrical normals, depth...) while maintaining +high rendering performance. +Reconstruction of local geometry can be improved by gather- +ing more information from the scene during the rendering passes. +Reflective Shadow Maps (RSM) [DS05] approximate indirect dif- +fuse lighting coming from a point light source using essentially + +nO. Therrien, Y. Levesque, G. Gilet / Screen Space Indirect Lighting with Visibility Bitmask +3 +a G-Buffer generated from the light’s view point. This tech- +nique is costly so in practice it’s usage is limited to less than +a handful of light simultaneously, and doesn’t take into ac- +count indirect light occlusion. Deep Screen Space [NRS14] +adaptively tessellate scene geometry into an unstructured sur- +fel cloud used for rendering different effects like AO, GI and +more. It bypasses major screen-space limitations like hidden sur- +faces and under-sampling of oblique geometry, but is expensive to +compute and cannot handle indirect light occlusion between sur- +fels. +More recently, Stochastic-Depth Ambient Occlusion (SDAO) +[VSE21] introduced the notion of stochastic depth map, captur- +ing multiple scene layers per pixel at random. This technique is +effective at detecting hidden surfaces, but, since it’s used in con- +junction with HBAO [BSD08], it doesn’t prevent over-darkening +around thin objects. However, while these methods improve re- +construction of local geometry, they are more computationally ex- +pensive than single-layer approaches, both during sample capture +(by forgoing early-z optimization) and reconstruction (by hav- +ing to evaluate multiple layers). +Improving the quality of the reconstruction from a single layer +is a difficult problem that has been widely studied. Alchemy am- +bient obscurance [MOBH11] is based on a similar approach than +SSAO and improves robustness and artistic control, with the follow +up Scalable Ambient Obscurance (SAO) [MML12] that also im- +proves performance. Horizon-Based techniques [BSD08, Bav11] +generates high quality results with low amount of noise by sam- +pling elevation along a set of directions but causes over-darkening +around thin surface. Several methods focus on improving perfor- +mance, such as Line Sweep Ambient Obscurance (LSAO) [Tim13], +which pre-caches sample information along azimuthal lines in +GPU shared memory and reuses the same samples to shade +multiple pixels. More recently, Ground Truth Ambient Occlu- +sion [JWPJ16] improved the accuracy of HBAO by making it +match a path-traced reference, and support a multi-bounce occlu- +sion approximation. +These techniques has been derived to propose more advanced in- +direct illumination features, taking advantage of the local geometry +reconstruction. Silvennoinen et al. [ST15] added support for indi- +rect lighting in real-time via an SSGI implementation using LSAO +as a basis. This method is approximative because only one color +sample is taken per horizon highpoint, and it doesn’t take into ac- +count partial occlusion that could have occurred along the way. +Other SSGI variants like Screen Space Ray Tracing Global Illu- +mination (SSRTGI) [SVF17] use an SSR-like technique to sam- +ple GI at the ray hit location. However, this approach introduces a +lot of noise which is difficult to remove without over-blurring. An- +other technique known as HBIL (which is based on HBAO and +GTAO) showed how to compute GI accurately from multiple sam- +ples by weighting the sample contribution by the angle difference +relative to the previous sample. While this method gives accu- +rate results within the visibility cone, it’s based on the assumption +that the depth buffer is a height field, and it cannot take into ac- +count light bounces that would pass behind surfaces. +Finally, Bitmask Soft Shadows (BMSS) [SS07] determine the +visibility of an area light source with a bit field where each bit +tracks the visibility of a sample point on the light source. Our +method solves a slightly different problem but nonetheless shares +many similarities regarding surface shape estimation from a depth +map and addresses overlapping sample visibility in the same way +using a bitmask. +3. Proposed Algorithm +In this section, we present how our method propose to improve the +quality of reconstruction of local geometry, especially in the case +of thin surfaces, by treating the depth buffer as a set of unconnected +samples each associated with an arbitrary thickness. +3.1. Ambient Occlusion +Ambient Occlusion (AO) [ZIK98] is a non-physically based light- +ing approximation of global illumination that assumes that the +scene is lit by uniform ambient lighting and that all objects are oc- +cluders. It’s a very common effect in real-time applications because +it can be computed efficiently in screen space and adds a lot of per- +ceived realism to the scenes. It can be expressed as a an estimation +of the visibility function V in the hemisphere around each sample : +AO = 1− 1 +π +� +Ω +V(p,ω)(np.ω)dω +(1) +By using a parameterization of the hemisphere, it can be decom- +posed into : +AO = 1− 1 +π +� π +0 +AO2(φ)dφ +(2) +and +AO2(φ) = +� π +0 +V(p,θ,φ)cosθsinθdθ +(3) +In practice, the integral of equation 2 is computed using Monte +Carlo integration over a few slices. Method such as GTAO and +HBAO propose an analytic solution of equation 3 by estimating, +through depth sampling, two horizons θ1 and θ2. +In our proposed algorithm, the two horizon angles θ1 and θ2 of +GTAO are replaced by a bit field of size Nb, representing the binary +state (occluded/un-occluded) of Nb visibility sectors uniformly dis- +tributed around the hemisphere slice. Samples are still taken on +each side of the view vector v, but the bit field is centered on pro- +jected normal n : +AO2(φi) ≈ 1 +Nb +Nb +∑ +j=1 +V(φi,θj) +(4) +Each sample taken along the hemisphere slice will determine a +potential occluder and impact the visibility function V of the given +sector. To determine the occluded state of a sector, we consider each +sample as a local thin geometry having a thickness t, acting as an +occluding geometry between angles θf and θb. The activation of a +sector depends on the hit criterion which ensures sufficient overlap +of these angles with the sector to get registered. θ f is equivalent + +4 +O. Therrien, Y. Levesque, G. Gilet / Screen Space Indirect Lighting with Visibility Bitmask +to θ used in GTAO (directly at the sample), and θb depends on +sample thickness t, (figure 3). For the proposed algorithm, we used +the round criterion which requires the sector to be half covered by +the sample. The pseudo-code at line 15 to 17 in Algorithm 1 shows +how (θf ,θb) are inferred from sample position and thickness. All +occluded sectors are set at once, making the algorithm perform in +O(1) for any sector count. Note that we must convert the angles +from cosine space to angular space for the samples to be properly +distributed around the hemisphere. +This enables fast directional occlusion and partial integration, at +the cost of precision. In the depth buffer, we take a fixed number of +azimuthal directions around each pixel and sample along these di- +rections to find (θf ,θb) pairs that can be integrated into the hemi- +sphere slice (see Figure 5). Like GTAO we distribute the occlusion +integral spatially and temporally to increase the number of effec- +tive samples. +Figure 5: The hemisphere is divided into Nb uniform sectors that +can be either occluded (in red) or un-occluded (in green). θb is de- +rived from θf and thickness t. Sectors that are at least half covered +by the (θ f ,θb) pair get occluded by the sample. We assume v and n +are aligned in this diagram for simplicity. +The choice of t has a big impact on visual fidelity. GTAO with- +out falloff is equivalent to using an infinite value for t. Ideally we +would want to use the per-pixel surface thickness of objects as +value for t, unfortunately this is more expensive to compute in real- +time and impossible to know precisely using a single layer depth +buffer. We propose using a small constant value since artifacts of +over-occlusion around thin objects are much more noticeable than +light leaks behind thick objects. This is because bigger objects usu- +ally occlude completely what is behind them so a light leak can +still look plausible, whereas over-occlusion around thin objects is +Figure 6: Left: GTAO using horizon angles without falloff term +exhibits no light leaks. Middle: our method using visibility bit- +masks with fixed thickness causes light leaks at depth disconti- +nuities. Right: GTAO using horizon angles with falloff term also +causes some light leaks at depth discontinuities. +directly visible (see Figures 6 and 7). Using a fixed world-space +thickness can cause an over-attenuation of occlusion for objects far +away from the camera, so we give the option to increase t linearly +over the distance to counter this effect. This causes a slight change +in occlusion when the camera moves, but is barely noticeable and +fixes the problem effectively. Finding an efficient heuristic to esti- +mate an accurate thickness for each sample would further improve +the accuracy of the method but remains a difficult problem we leave +for future work. +Algorithm 1 Generate AO and GI using visibility bitmasks +1: t ← constant thickness +2: Nb ← bitmask size +3: p ← view space fragment position +4: np ← view space fragment normal +5: r ← projected radius onto image plane +6: Determine stepsize as r/(Ns +1) +7: Determine directions with random offset +8: AO,GI ← 0 +9: for direction di where i = 0 to Nd do +10: +tp ← slice plane tangent vector in direction di +11: +tθ ← angle of tp with XY-plane +12: +Bitmask bi ← 0 +13: +for step s j where j = 0 to Ns do +14: +Front sample s f ← view-space position at step j +15: +Back sample sb ← s f − +p +∥p∥t +16: +θf , θb ← angles of s f and sb on XY-plane +17: +θmin, θmax ← min(θ f ,θb),max(θf ,θb) +18: +a, b ← ⌊ θmin+ π +2 +π +Nb⌋, ⌈ θmax−θmin+ π +2 +π +Nb⌉ +19: +bj ← 2b −1 ≪ a +20: +cj ← direct lighting at step j (from GBuffer) +21: +nj ← normal at step j (from GBuffer) +22: +lj ← +sf −p +∥sf −p∥ +23: +GI ← GI + COUNTBITS(bj&∼bi) +Nb +cj(np ·l j)(n j ·−l j) +24: +bi ← bi |b j +25: +end for +26: +AO ← AO+1− COUNTBITS(bi)/Nb +27: end for +28: return AO/Nd, GI/Nd +The following sections explain the usage of the above-described +core algorithm in implementing ambient occlusion, directionally +occluded ambient lighting, and indirect diffuse bounce. +3.2. Directionally Occluded Ambient Lighting +Ambient lighting in real-time applications is usually sampled using +the surface normal, but tends to give poor results because it doesn’t +take into account the directional occlusion of lighting. Screen space +bent normal [KRES11] addresses this problem by sampling to- +wards the largest non-occluded direction, but is limited to a sin- +gle ambient direction per pixel and does not handle thickness prop- +erly. Visibility bitmasks can improve this by weighting the ambient +lighting in a given direction by the directional occlusion while al- +lowing light to pass behind surfaces. To this end, we divide the + +v=n +0.O. Therrien, Y. Levesque, G. Gilet / Screen Space Indirect Lighting with Visibility Bitmask +5 +Figure 7: From left to right: Ray tracing reference, GTAO without falloff, GTAO with falloff, visibiliy bitmask using a thickness of 1, 0.5, +0.25, 0.1. All methods use a radius of 2. +hemisphere into as many subregions as the number of ambient sam- +ples, generating a sampling direction vector at the center of each +subregion. The ambient source is then sampled with this vector and +multiplied the lighting intensity by the amount of un-occluded sec- +tors over the total sector count (see Figure 8). It made the ambient +light color vary smoothly according to changes in occlusion direc- +tionality. +Figure 8: In this example, 4 ambient samples per slice per pixel +are taken so the hemisphere is divided into 4 subregions. Vectors a1, +a2, a3, a4 are generated at the center of their respective subregion. +3.3. Indirect Diffuse +Indirect diffuse lighting is the bouncing of light on nearby surfaces. +It’s traditionally expensive to compute accurately, even in screen +space. HBIL can do it efficiently but fails to account for light pass- +ing behind surfaces. Figure 9 shows how we computed this effect +with better thickness handling using visibility bitmasks. Samples +were taken along the slice direction and detected (in O(1)) how +many un-occluded sectors are covered to estimate lighting contri- +bution. These sectors were then set to an occluded state (also in +O(1)) to handle partial or total occlusion of light coming from sub- +sequent samples. The more the visibility sectors, the more precise +the estimation of lighting and occlusion. It was observed that 32 +sectors gave good quality and makes the bit field fit nicely within a +single unsigned integer. +Figure 9: Left: The yellow sample intersects one un-occluded sec- +tor and can contribute lighting equivalent to one over the total +number of visibility sectors. The sector is set to an occluded state +for subsequent samples. Right: Sampling continues and a new ob- +ject on the right is found, but it intersects an already occluded sec- +tor, so it cannot contribute lighting. The yellow sample on the left +crosses an un-occluded sector and can contribute. +The pseudo-code at line 23 in Algorithm 1 shows how the light- +ing contribution of a sample is implemented, using the number +of occluded zones by the current sample. If one or more sectors +are covered, the sample contributed lighting and the light buffer +is sampled at the sample location. Then n · l and nl · l are com- +puted and used for weighting light intensity. The light is further- +more weighted by the occluded sector count over the total sector +count. +4. Results and evaluation +4.1. Ambient Occlusion +In this subsection, the core algorithm will be demonstrated con- +jointly with AO, as it’s much easier to discern the properties of vis- +ibility bitmasks in this mode than using indirect diffuse. The ren- +ders in Figure 10 are produced by our extension of Unity’s GTAO +implementation, where the two horizon angles θ1 and θ2 have been +replaced by a single visibility bitmask. + +Reference +GTAO (no falloff) +Ours (t = 1) +Ours (t = 0.5) +Ours (t = 0.25) +Ours (t = 0.1) +GTAO (falloff)v=n +a2 +a3 +a4 +a1v=n +v=n6 +O. Therrien, Y. Levesque, G. Gilet / Screen Space Indirect Lighting with Visibility Bitmask +Figure 10: Comparison of GTAO, our method based on visibility bitmasks, and a ray-traced AO reference at different radius and sample +count values. +It highlights how easy it is to implement our method on top of an +existing horizon-based technique, and how this single modification +can dramatically enhance visual quality compared to a ray-traced +reference. The Lumberyard Bistro scene has been chosen because it +contains a lot of thin and shallow surfaces that are typically a prob- +lem with horizon-based techniques, but are improved by visibility +bitmasks. All benchmarks use one hemisphere slices per pixel jit- +tered over multiple frames. +The radius parameter (Figure 10) is the radius of the hemisphere +aligned to the screen in world units. Wider hemispheres will cover +wider regions of the screen, casting farther-reaching occlusion. A +wide radius is problematic for GTAO because the single cone ap- +proximation tends to cast too much occlusion in regions enclosed +by thin objects. Even around not-so-thin objects, a wide radius +tends to produce a blurry occlusion blob that does not capture fine +detail. In contrast, our method (with the exact same samples) is able +to let light pass behind surfaces, avoiding over-occlusion and cap- +turing a lot of small details. +Figure 11 demonstrates an even more difficult case for GTAO +where most of the objects are behind a fence. The light gets +Figure 11: Left: GTAO exhibit too much occlusion behind the bars. +Right: Our method is able let light pass behind the bars, minimizing +the over-occlusion artifact. +trapped behind the bars instead of passing by, causing a lot of over- +occlusion. Our method is able to handle this situation much better. +When using a wide radius, GTAO has a tendency to produce ha- +los around objects (Figure 4). Our method doesn’t have this prob- +lem, and capture more geometric and normal details. +The number of samples (Figure 10) indicates the number of + +GTAO +Our Method +Ray Traced Reference +Radius 0.8 +8 samples +Radius 1.0 +12 samples +Radius 1.0 +16 samples +Radius 2.0 +16 samples +Radius 3.0 +16 samplesO. Therrien, Y. Levesque, G. Gilet / Screen Space Indirect Lighting with Visibility Bitmask +7 +fetches taken in the depth buffer along one horizon side. There- +fore, for one slice, the actual number of samples taken is twice that +number. A wider radius causes the samples to be sparser on-screen, +so it’s typical to increase sample count for a wider radius to main- +tain the same sampling density. A low sampling density increases +the likelihood of missing potential occluders, especially if they are +thin on-screen. Increasing the number of samples has a big impact +on performance, so it’s a tradeoff. +Performance depends primarily on the number of samples taken +and the radius of the effect. A wider radius increases the proba- +bility of occurrence of a cache miss (sample not being present in +the cache) and consequently can lower performance. The execu- +tion time of the technique tends to scale linearly with the number +of samples. Table 1 compares the performance of the horizon-based +GTAO implementation versus our method using visibility bitmasks. +Both techniques are composed of a sampling pass and a denoising +pass. Only the results of the sampling pass are included in the ta- +ble since it’s the only one impacted by our method. The denoising +pass has a constant cost of 0.3 ms in 1080p. It can be observed that +our method has a modest impact on performance around 0.01-0.02 +milliseconds, with a fixed ALU overhead of about 15 GPU instruc- +tions per sample. Increasing the radius masks this overhead as the +execution becomes bandwidth-limited. +Radius +Sample Count +GTAO +Our Method +0.8 +8 +0.49 ms +0.51 ms +1 +12 +0.75 ms +0.77 ms +1 +16 +0.95 ms +0.97 ms +2 +16 +1.12 ms +1.13 ms +3 +16 +1.12 ms +1.13 ms +Table 1: Render time of the sampling pass for GTAO and our +method with various radius and sample parameters, at 1920x1080, +with 32 visibility sectors per hemisphere slice. Benchmarks are +done on an Nvidia RTX 2080 GPU. +Another parameter that impacts image quality is the number of +visibility sectors. When the sector count is too low, banding arti- +facts can appear, particularly around thin objects. The proposed im- +plementation used 32 visibility sectors because it just crossed the +threshold where the artifacts became almost invisible. Additionally, +it nicely fits into a single unsigned integer which gives good per- +formance on the GPU. By contrast, a 128 bits version require four +unsigned integers and the use of vector instructions, which lim- +its the amount of instruction packing that the compiler could do, +negatively impacting the performance. A performance overhead of +around 5-10% was observed with 128 visibility sectors compared +to 32. +4.2. Directionally Occluded Ambient Lighting +In this subsection, we compare different ambient sampling strate- +gies and show how they can dramatically influence the resulting +lighting and occlusion. Figure 12 shows the average normal on the +left, and the resulting ambient lighting on the right for each strat- +egy. In most 3D applications, ambient lighting is sampled in the di- +rection of the surface normal. This approach does not take into ac- +count the fact that some light could be occluded in some direction +and tend to make ambient lighting change sharply with the scene +geometry. A better approach is to use a screen space bent normal +per pixel that is modulated according to nearby occlusion. It points +towards the direction of incoming light and gives a smoother re- +sult. However, it cannot handle multiple light directions and will +misrepresent the ambient lighting of surfaces enclosed by thin ob- +jects. +Figure 12: Comparison of ambient lighting using the surface nor- +mal, bent normals, and visibility bitmasks. The average sampling +normal direction is shown on the left to make the difference more +visible. Top row: G-Buffer normals. Middle row: Bent normals. +Bottom row: visibility bitmask. +Our approach used visibility bitmasks to take multiple samples +along each hemisphere slice in the directions that were not oc- +cluded. Doing so allow ambient light to pass behind surfaces, giv- +ing smooth lighting from multiple directions. It’s also worth noting +that each ambient sample is weighted according to the occlusion +in that specific direction. If the ambient color had been simply av- +eraged out and then multiplied by ambient occlusion, a lot of the +color variation in the lighting would have been lost. +4.3. Indirect Diffuse +In this subsection, we look at the visual quality and performance +of the indirect diffuse portion of the algorithm. In addition to sam- +pling the depth buffer, we also sample the HDR light buffer and +the screen space normal buffer for every sample taken. The light +buffer contains only the direct lighting (with shadows), as the am- +bient light is computed by our method. Figure 13 compares our +result with a path tracing reference for single and multi-bounce in- +direct diffuse lighting. The direct lighting coming from the sun on +the left wall bounces on the brick wall, illuminating it and cast- +ing indirect shadows. With multiple bounces, the light is even able +to bounce back on the left wall, illuminating a shadowed region of + +1 +EL8 +O. Therrien, Y. Levesque, G. Gilet / Screen Space Indirect Lighting with Visibility Bitmask +Figure 13: Comparison of indirect diffuse algorithm and path tracing reference with single and multi-bounce indirect diffuse. +the wall. Multiple bounces are implemented by injecting the indi- +rect illumination into the light buffer to be used as input for the +next frame. Light intensity needs to be properly balanced when +using multi-bounce, or it can cause a feedback loop resulting in +lighting accumulation over time. The result cannot match perfectly +the path-traced reference since the algorithm operates only on the +screen pixels as opposed to the entire scene geometry. One ma- +jor drawback of our technique, along with screen-space methods, +is that if direct light leaves the screen, the indirect lighting disap- +pears. +Figure 14: When samples are evenly spaced along the sampling di- +rection, a banding pattern can appear. Jittering the samples along +the sampling direction helps mask the banding artifact. +The number of samples taken has a big impact on quality. Hav- +ing a too low sampling density can introduce banding artifacts. To +mitigate the issue, samples are jittered along the sampling direc- +tion, as shown in Figure 14. +Low sample density can also cause a loss of detail around small +objects. To improve this, we distribute the samples exponentially +around the shaded pixel, as nearby surfaces usually have more in- +fluence on the result than farther ones. +Figure 16 compares the amount of noise incurred when sampling +the scene with SSR-like tracing and our method based on visibil- +ity bitmasks. Both techniques are taking the same maximum num- +ber of samples but the visibility bitmask approach has a lot less +noise. Rays in SSR are doing at most one accumulation operation +(when a hit is found). In contrast, the visibility bitmask approach is +accumulating each sample that is visible from the current pixel. +This algorithm is bandwidth-intensive because we need to sam- +ple the HDR light buffer and the screen space normal buffer for +every sample taken. Those sample locations are not correlated and +impair caching. Table 2 compares the performances of the indirect +diffuse part of the algorithm with different sampling parameters. +The corresponding renders are shown in Figure 15. Full resolution +means that the shader is executed for every pixel of the final ren- +der resolution, whereas half-resolution executes it on a screen that +is half the size (a quarter the number of pixels). The image is then +upscaled with a classic bilateral upsampler to avoid aliasing. Ren- +dering in half resolution is much more efficient (around 4x), but +can introduce more flickering in the image and a blurrier result. +5. Conclusion and future work +Previous screen space GI methods that rely on HBAO cannot han- +dle thickness because of the assumption that the depth buffer is +strictly a height field. Other techniques based on SSR tracing +squander a lot of rays that end up passing behind and over sur- +faces, exiting the screen without hitting anything, thereby introduc- + +Visibility Sector Gl +Path Tracing Reference +Single- +Bounce +Multi- +BounceNon-Jittered +JitteredO. Therrien, Y. Levesque, G. Gilet / Screen Space Indirect Lighting with Visibility Bitmask +9 +Figure 15: Resulting renders using different sample, radius, stepping, and resolution parameters for the indirect diffuse lighting. The param- +eters of each figure are indicated in Table 2. +Figure 16: Left: 2 rays per pixel SSR tracing, with 16 steps per +ray. There is a lot of noise, as a lot of rays exit the screen or pass +behind surfaces without hitting anything. Right: Horizon search on +both sides of the pixel using a visibility bitmask with 16 steps per +horizon is a lot less noisy. +ing a lot of noise. In contrast, our method combined the sampling +efficiency of horizon-based methods, while retaining the capabil- +ity of handling thickness properly along the way. The proposed al- +gorithm is easy to understand and implement on modern GPUs and +can be integrated into any horizon-based technique with only a mild +performance overhead. Moreover, this method can also be used to +improve ambient light sampling by taking into account the direc- +tionality of occlusion when integrating ambient light. +Even though the ray-tracing part of the algorithm handles thick- +ness properly, since we operate on a single layer of the depth buffer, +the actual object thickness is unknown. We are forced to rely on a +Configuration +Sampling +Denoising +Total +a) 8 samples, radius 1, +const. steps, full res. +0.9 ms +0.33 ms +1.23 ms +b) 8 samples, radius 4, +const. steps, full res. +1.7 ms +0.33 ms +2.03 ms +c) 16 samples, radius 4, +const. steps, full res. +2.3 ms +0.33 ms +2.63 ms +d) 16 samples, radius 4, +exp. steps, full res. +2.6 ms +0.33 ms +2.93 ms +e) 32 samples, radius 4, +exp. steps, full res. +4.0 ms +0.33 ms +4.33 ms +f) 16 samples, radius 4, +exp. steps, half res. +0.97 ms +0.1 ms +1.07 ms +Table 2: Render time at 1920x1080, with a constant thickness value +of 0.2. +constant thickness value that optionally increases linearly over the +distance. A thickness heuristic that would give a plausible thick- +ness per pixel would help improve the occlusion leaks around some +very thin objects or light leaks behind very thick ones and would +be interesting to explore in future work. At the performance level, +SSGI can be an expensive algorithm because it is very demanding +on GPU bandwidth and utilizes the cache poorly. Using a caching +method similar to LSAO could potentially improve this. Finally, + +a) +b) +c) +d) +eSSR-based.Gl +Our Method10 +O. Therrien, Y. Levesque, G. Gilet / Screen Space Indirect Lighting with Visibility Bitmask +the common ambient light sources are either static (constant color, +light probes) or expensive to update at runtime. We might investi- +gate in the future ways to approximate low-frequency ambient irra- +diance dynamically. +6. Acknowledgments and data statements +Special thanks to Deepti Joshi (CDRIN) and Peter Shirley +(NVIDIA) for their guidance in the redaction of this paper. We +also want to thank our other colleagues namely Antoine Fortin +(CDRIN), Olivier Leclerc (CDRIN), Steven Pigeon (UQAR), and +Vahe Vardanyan (CDRIN) who have actively supported the current +body of work. Most of the models are from the Amazon Lumber- +yard Bistro scene. This research is financed in part by the province +of Quebec (Canada) via the grant "Programme d’aide à la recherche +et au transfert (PART)". Data sharing not applicable to this article +as no datasets were generated or analysed during the current study. +References +[Bav11] +BAVOIL L.: Horizon-based ambient occlusion using compute +shaders. Nvidia DirectX 11 (2011). +[BSD08] +BAVOIL L., SAINZ M., DIMITROV R.: Image-space horizon- +based ambient occlusion. In ACM SIGGRAPH 2008 talks. 2008, pp. 1–1. +[DS05] +DACHSBACHER C., STAMMINGER M.: Reflective shadow maps. +In Proceedings of the 2005 symposium on Interactive 3D graphics and +games (2005), pp. 203–231. +[HBSS17] +HOFMANN N., BOGENDÖRFER P., STAMMINGER M., SEL- +GRAD K.: Hierarchical multi-layer screen-space ray tracing. In Proceed- +ings of High Performance Graphics. 2017, pp. 1–10. +[JWPJ16] +JIMÉNEZ J., WU X., PESCE A., JARABO A.: +Practical +real-time strategies for accurate indirect occlusion. SIGGRAPH 2016 +Courses: Physically Based Shading in Theory and Practice (2016). +[KRES11] +KLEHM O., RITSCHEL T., EISEMANN E., SEIDEL H.-P.: +Bent normals and cones in screen-space. +In VMV (2011), Citeseer, +pp. 177–182. +[May18] +MAYAUX B.: Horizon-based indirect lighting. 2018. +[Mit07] +MITTRING M.: Finding next gen: Cryengine 2. In ACM SIG- +GRAPH 2007 courses. 2007, pp. 97–121. +[MML12] +MCGUIRE M., MARA M., LUEBKE D. P.: Scalable ambient +obscurance. In High Performance Graphics (2012), Citeseer, pp. 97– +103. +[MMNL16] +MARA M., MCGUIRE M., NOWROUZEZAHRAI D., LUE- +BKE D. P.: Deep g-buffers for stable global illumination approximation. +In High Performance Graphics (2016), pp. 87–98. +[MOBH11] +MCGUIRE M., OSMAN B., BUKOWSKI M., HENNESSY P.: +The alchemy screen-space ambient obscurance algorithm. In Proceed- +ings of the ACM SIGGRAPH Symposium on High Performance Graphics +(2011), pp. 25–32. +[NRS14] +NALBACH O., RITSCHEL T., SEIDEL H.-P.: +Deep screen +space. In Proceedings of the 18th meeting of the ACM SIGGRAPH Sym- +posium on Interactive 3D Graphics and Games (2014), pp. 79–86. +[SKS11] +SOUSA T., KASYAN N., SCHULZ N.: Secrets of cryengine 3 +graphics technology. In ACM SIGGRAPH (2011), vol. 1. +[SS07] +SCHWARZ M., STAMMINGER M.: +Bitmask soft shadows. +In +Computer Graphics Forum (2007), vol. 26, Wiley Online Library, +pp. 515–524. +[ST15] +SILVENNOINEN A., TIMONEN V.: Multi-scale global illumina- +tion in quantum break. In ACM SIGGRAPH (2015). +[SVF17] +SHERGIN D., VIDIGER D., FOFANOVA A.: +Superposition +benchmark: innovative ssrtgi lighting in real time. In ACM SIGGRAPH +2017 Real Time Live! 2017, pp. 25–25. +[Tim13] +TIMONEN V.: Line-sweep ambient obscurance. In Computer +graphics forum (2013), vol. 32, Wiley Online Library, pp. 97–105. +[VSE21] +VERMEER J., SCANDOLO L., EISEMANN E.: Stochastic-depth +ambient occlusion. Proceedings of the ACM on Computer Graphics and +Interactive Techniques 4, 1 (2021), 1–15. +[ZIK98] +ZHUKOV S., IONES A., KRONIN G.: An ambient light illumina- +tion model. In Eurographics Workshop on Rendering Techniques (1998), +Springer, pp. 45–55. + diff --git a/ztFIT4oBgHgl3EQf2Ssk/content/tmp_files/load_file.txt b/ztFIT4oBgHgl3EQf2Ssk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5ee8f2f5f6707361d028c40401d4d77dd55aa845 --- /dev/null +++ b/ztFIT4oBgHgl3EQf2Ssk/content/tmp_files/load_file.txt @@ -0,0 +1,473 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf,len=472 +page_content='This article has been accepted for publication in The Visual Computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' This is the author’s version which has not been fully edited and content may change prior to final publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Citation information: DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='1007/s00371-022-02703-y Screen Space Indirect Lighting with Visibility Bitmask Olivier Therrien1† Yannick Levesque2‡ Guillaume Gilet3§ 1CDRIN, QC, Canada 2Cégep de Matane, QC, Canada 3 University of Sherbrooke Figure 1: Left: Direct illumination of the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Middle: Indirect lighting produced by our method (without texture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Right: Final frame rendered with our method, exhibiting directionally occluded ambient lighting, and a GI bounce that avoids typical thin surface artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Abstract Horizon-based indirect illumination efficiently estimates a diffuse light bounce in screen space by analytically integrat- ing the horizon angle difference between samples along a given direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Like other horizon-based methods, this technique cannot properly simulate light passing behind thin surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' We propose the concept of a visibility bitmask that re- places the two horizon angles by a bit field representing the binary state (occluded / un-occluded) of N sectors uniformly distributed around the hemisphere slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' It allows light to pass behind surfaces of constant thickness while keeping the ef- ficiency of horizon-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' It can also do more accurate ambient lighting than bent normal by sampling more than one visibility cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' This technique improves the visual quality of ambient occlusion, indirect diffuse, and ambient light com- pared to previous screen space methods while minimizing noise and keeping a low performance overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Keywords: Real-Time Rendering, Indirect Lighting, Ambient Occlusion, Visibility 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Introduction Indirect diffuse lighting is challenging to compute in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Screen space approximations can be attractive as they reduce the di- mensionality of the problem and make the execution cost constant regardless of the scene’s geometric complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Modern Screen Space Global Illumination (SSGI) implementations often gather indirect light by doing ray marching on screen pixels similar to † therrien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='olivier@cdrin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='com ‡ levesqueyannick@cgmatane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='qc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='ca § guillaume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='gilet@usherbrooke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='ca Screen Space Reflections (SSR) [SKS11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' This approach tends to generate a lot of noise because it implies the numerical integra- tion of irradiance over the entire hemisphere around the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Horizon-Based Indirect Illumination (HBIL) [May18], based on Horizon-Based Ambient Occlusion (HBAO) [BSD08, Bav11] and Ground Truth Ambient Occlusion (GTAO) [JWPJ16], improve the efficiency by numerically integrating over a set of directions around the view vector v (Figure 2) while doing analytic integration of the horizon angle difference between samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Fundamentaly, the core principle of these methods lies in the es- timation of the scene local geometry around each shading sample arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='11376v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='GR] 26 Jan 2023 2 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Therrien, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Levesque, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Gilet / Screen Space Indirect Lighting with Visibility Bitmask Figure 2: 3D view of the scene, centered on the shaded pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' GTAO/HBIL generates a set of hemisphere slices (in green) in var- ious directions around the view vector v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Occluders (in black) in- tersect some of the slices, producing occlusion cones (in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' by relying on readily-available screen-space information, such as the discrete depth buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' However, such information is by essence discrete and incomplete, and must be reconstructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' All those tech- niques evaluate Ambient Occlusion, the modulation of indirect ir- radiance due to local geometry, in screen space from a single layer depth buffer, and assume infinite surface thickness by treating it as a height-field (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' While this is a valid assumption in some cases, not knowing what the real geometry looks like, it causes ha- los and over-darkening around thin surfaces (see figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Falloff heuristics are used to mitigate those artifacts but fail when using a large sampling radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Figure 3: Side view of one slice centered on the view vector v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Left: Ground truth scene with multiple occluders (in black) producing multiple visibility cones (in green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Right: GTAO takes a fixed num- ber of samples (red dots) in the depth buffer on both sides of the hemisphere to find highest elevation angles θ1 and θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Our proposed method rejects the assumption that the depth buffer is strictly a height-field and models the behavior of light passing behind surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Additionally, to preserve performance, we want to avoid explicitly tracing new rays from scratch to adequately sample multiple elevation angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' To do so, we introduce the con- cept of visibility bitmask, which is essentially a discretization of the hemisphere slice in Nb sectors, that allows us to approximate the tracing of Nb rays at the same performance cost as one hori- zon search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Figure 4: Left: GTAO produces halos around the poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Right: Our method is able let light pass behind the poles without introducing halos artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' A common limitation of single layer depth buffers is that oc- cluded surfaces are not represented, which can cause missing oc- clusion or lighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Approaches like Deep G-Buffer [MMNL16] or Multi-layer SSRT [HBSS17] alleviate this issue by storing mul- tiple layers or sample per pixel, to provide more information on background surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' As rendering multiple layers is very expen- sive, we chose to limit ourselves to only one layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' This means that some artifacts caused by inaccurate background surface estimation will remain, but using a constant thickness t at each depth sam- ple with a visibility bitmask greatly improves quality around thin surfaces compared to horizon-based techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' In this study, we show that visibility bitmasks can tremendously reduce noise in the image compared to SSR-like tracing, while handling the light pass- ing behind surfaces much more accurately than horizon-based tech- niques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' We also show that using a visibility bitmask to sample am- bient light gives a more precise ambient estimation than traditional methods of sampling along the surface normal or even bent nor- mal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' The main contributions of this paper revolve around the introduc- tion of a visibility bitmask in traditional SSGI methods, retaining the efficiency and noise reduction qualities of horizon sampling, while handling light passing behind surfaces of constant thick- ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' We demonstrate the capabilities of our method through sev- eral SSGI applications, such as ambient occlusion, directional am- bient occlusion and indirect diffuse lighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Previous Work Estimating ambient occlusion and indirect diffuse lighting from screen space information is a well-known idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' It mainly stems from the seminal Screen Space Ambient Occlusion method [Mit07], approximating ambient occlusion by sampling random points in the depth buffer in a circle around each pixel and has been thoroughly extended over the years [Mit07, MOBH11, MML12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' The keypoint of these methods is to estimate the local geometry around a sample using the incomplete information contained in the various buffers (geometrical normals, depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=') while maintaining high rendering performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Reconstruction of local geometry can be improved by gather- ing more information from the scene during the rendering passes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Reflective Shadow Maps (RSM) [DS05] approximate indirect dif- fuse lighting coming from a point light source using essentially nO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Therrien, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Levesque, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Gilet / Screen Space Indirect Lighting with Visibility Bitmask 3 a G-Buffer generated from the light’s view point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' This tech- nique is costly so in practice it’s usage is limited to less than a handful of light simultaneously, and doesn’t take into ac- count indirect light occlusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Deep Screen Space [NRS14] adaptively tessellate scene geometry into an unstructured sur- fel cloud used for rendering different effects like AO, GI and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' It bypasses major screen-space limitations like hidden sur- faces and under-sampling of oblique geometry, but is expensive to compute and cannot handle indirect light occlusion between sur- fels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' More recently, Stochastic-Depth Ambient Occlusion (SDAO) [VSE21] introduced the notion of stochastic depth map, captur- ing multiple scene layers per pixel at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' This technique is effective at detecting hidden surfaces, but, since it’s used in con- junction with HBAO [BSD08], it doesn’t prevent over-darkening around thin objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' However, while these methods improve re- construction of local geometry, they are more computationally ex- pensive than single-layer approaches, both during sample capture (by forgoing early-z optimization) and reconstruction (by hav- ing to evaluate multiple layers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Improving the quality of the reconstruction from a single layer is a difficult problem that has been widely studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Alchemy am- bient obscurance [MOBH11] is based on a similar approach than SSAO and improves robustness and artistic control, with the follow up Scalable Ambient Obscurance (SAO) [MML12] that also im- proves performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Horizon-Based techniques [BSD08, Bav11] generates high quality results with low amount of noise by sam- pling elevation along a set of directions but causes over-darkening around thin surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Several methods focus on improving perfor- mance, such as Line Sweep Ambient Obscurance (LSAO) [Tim13], which pre-caches sample information along azimuthal lines in GPU shared memory and reuses the same samples to shade multiple pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' More recently, Ground Truth Ambient Occlu- sion [JWPJ16] improved the accuracy of HBAO by making it match a path-traced reference, and support a multi-bounce occlu- sion approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' These techniques has been derived to propose more advanced in- direct illumination features, taking advantage of the local geometry reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Silvennoinen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' [ST15] added support for indi- rect lighting in real-time via an SSGI implementation using LSAO as a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' This method is approximative because only one color sample is taken per horizon highpoint, and it doesn’t take into ac- count partial occlusion that could have occurred along the way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Other SSGI variants like Screen Space Ray Tracing Global Illu- mination (SSRTGI) [SVF17] use an SSR-like technique to sam- ple GI at the ray hit location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' However, this approach introduces a lot of noise which is difficult to remove without over-blurring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' An- other technique known as HBIL (which is based on HBAO and GTAO) showed how to compute GI accurately from multiple sam- ples by weighting the sample contribution by the angle difference relative to the previous sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' While this method gives accu- rate results within the visibility cone, it’s based on the assumption that the depth buffer is a height field, and it cannot take into ac- count light bounces that would pass behind surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Finally, Bitmask Soft Shadows (BMSS) [SS07] determine the visibility of an area light source with a bit field where each bit tracks the visibility of a sample point on the light source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Our method solves a slightly different problem but nonetheless shares many similarities regarding surface shape estimation from a depth map and addresses overlapping sample visibility in the same way using a bitmask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Proposed Algorithm In this section, we present how our method propose to improve the quality of reconstruction of local geometry, especially in the case of thin surfaces, by treating the depth buffer as a set of unconnected samples each associated with an arbitrary thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Ambient Occlusion Ambient Occlusion (AO) [ZIK98] is a non-physically based light- ing approximation of global illumination that assumes that the scene is lit by uniform ambient lighting and that all objects are oc- cluders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' It’s a very common effect in real-time applications because it can be computed efficiently in screen space and adds a lot of per- ceived realism to the scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' It can be expressed as a an estimation of the visibility function V in the hemisphere around each sample : AO = 1− 1 π � Ω V(p,ω)(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='ω)dω (1) By using a parameterization of the hemisphere, it can be decom- posed into : AO = 1− 1 π � π 0 AO2(φ)dφ (2) and AO2(φ) = � π 0 V(p,θ,φ)cosθsinθdθ (3) In practice, the integral of equation 2 is computed using Monte Carlo integration over a few slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Method such as GTAO and HBAO propose an analytic solution of equation 3 by estimating, through depth sampling, two horizons θ1 and θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' In our proposed algorithm, the two horizon angles θ1 and θ2 of GTAO are replaced by a bit field of size Nb, representing the binary state (occluded/un-occluded) of Nb visibility sectors uniformly dis- tributed around the hemisphere slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Samples are still taken on each side of the view vector v, but the bit field is centered on pro- jected normal n : AO2(φi) ≈ 1 Nb Nb ∑ j=1 V(φi,θj) (4) Each sample taken along the hemisphere slice will determine a potential occluder and impact the visibility function V of the given sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' To determine the occluded state of a sector, we consider each sample as a local thin geometry having a thickness t, acting as an occluding geometry between angles θf and θb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' The activation of a sector depends on the hit criterion which ensures sufficient overlap of these angles with the sector to get registered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' θ f is equivalent 4 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Therrien, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Levesque, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Gilet / Screen Space Indirect Lighting with Visibility Bitmask to θ used in GTAO (directly at the sample), and θb depends on sample thickness t, (figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' For the proposed algorithm, we used the round criterion which requires the sector to be half covered by the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' The pseudo-code at line 15 to 17 in Algorithm 1 shows how (θf ,θb) are inferred from sample position and thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' All occluded sectors are set at once, making the algorithm perform in O(1) for any sector count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Note that we must convert the angles from cosine space to angular space for the samples to be properly distributed around the hemisphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' This enables fast directional occlusion and partial integration, at the cost of precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' In the depth buffer, we take a fixed number of azimuthal directions around each pixel and sample along these di- rections to find (θf ,θb) pairs that can be integrated into the hemi- sphere slice (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Like GTAO we distribute the occlusion integral spatially and temporally to increase the number of effec- tive samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Figure 5: The hemisphere is divided into Nb uniform sectors that can be either occluded (in red) or un-occluded (in green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' θb is de- rived from θf and thickness t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Sectors that are at least half covered by the (θ f ,θb) pair get occluded by the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' We assume v and n are aligned in this diagram for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' The choice of t has a big impact on visual fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' GTAO with- out falloff is equivalent to using an infinite value for t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Ideally we would want to use the per-pixel surface thickness of objects as value for t, unfortunately this is more expensive to compute in real- time and impossible to know precisely using a single layer depth buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' We propose using a small constant value since artifacts of over-occlusion around thin objects are much more noticeable than light leaks behind thick objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' This is because bigger objects usu- ally occlude completely what is behind them so a light leak can still look plausible, whereas over-occlusion around thin objects is Figure 6: Left: GTAO using horizon angles without falloff term exhibits no light leaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Middle: our method using visibility bit- masks with fixed thickness causes light leaks at depth disconti- nuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Right: GTAO using horizon angles with falloff term also causes some light leaks at depth discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' directly visible (see Figures 6 and 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Using a fixed world-space thickness can cause an over-attenuation of occlusion for objects far away from the camera, so we give the option to increase t linearly over the distance to counter this effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' This causes a slight change in occlusion when the camera moves, but is barely noticeable and fixes the problem effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Finding an efficient heuristic to esti- mate an accurate thickness for each sample would further improve the accuracy of the method but remains a difficult problem we leave for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Algorithm 1 Generate AO and GI using visibility bitmasks 1: t ← constant thickness 2: Nb ← bitmask size 3: p ← view space fragment position 4: np ← view space fragment normal 5: r ← projected radius onto image plane 6: Determine stepsize as r/(Ns +1) 7: Determine directions with random offset 8: AO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='GI ← 0 9: for direction di where i = 0 to Nd do 10: tp ← slice plane tangent vector in direction di 11: tθ ← angle of tp with XY-plane 12: Bitmask bi ← 0 13: for step s j where j = 0 to Ns do 14: Front sample s f ← view-space position at step j 15: Back sample sb ← s f − p ∥p∥t 16: θf ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' θb ← angles of s f and sb on XY-plane 17: θmin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' θmax ← min(θ f ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='θb),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='max(θf ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='θb) 18: a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' b ← ⌊ θmin+ π 2 π Nb⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' ⌈ θmax−θmin+ π 2 π Nb⌉ 19: bj ← 2b −1 ≪ a 20: cj ← direct lighting at step j (from GBuffer) 21: nj ← normal at step j (from GBuffer) 22: lj ← sf −p ∥sf −p∥ 23: GI ← GI + COUNTBITS(bj&∼bi) Nb cj(np ·l j)(n j ·−l j) 24: bi ← bi |b j 25: end for 26: AO ← AO+1− COUNTBITS(bi)/Nb 27: end for 28: return AO/Nd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' GI/Nd The following sections explain the usage of the above-described core algorithm in implementing ambient occlusion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' directionally occluded ambient lighting,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' and indirect diffuse bounce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Directionally Occluded Ambient Lighting Ambient lighting in real-time applications is usually sampled using the surface normal, but tends to give poor results because it doesn’t take into account the directional occlusion of lighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Screen space bent normal [KRES11] addresses this problem by sampling to- wards the largest non-occluded direction, but is limited to a sin- gle ambient direction per pixel and does not handle thickness prop- erly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Visibility bitmasks can improve this by weighting the ambient lighting in a given direction by the directional occlusion while al- lowing light to pass behind surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' To this end, we divide the v=n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Therrien, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Levesque, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Gilet / Screen Space Indirect Lighting with Visibility Bitmask 5 Figure 7: From left to right: Ray tracing reference, GTAO without falloff, GTAO with falloff, visibiliy bitmask using a thickness of 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' All methods use a radius of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' hemisphere into as many subregions as the number of ambient sam- ples, generating a sampling direction vector at the center of each subregion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' The ambient source is then sampled with this vector and multiplied the lighting intensity by the amount of un-occluded sec- tors over the total sector count (see Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' It made the ambient light color vary smoothly according to changes in occlusion direc- tionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Figure 8: In this example, 4 ambient samples per slice per pixel are taken so the hemisphere is divided into 4 subregions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Vectors a1, a2, a3, a4 are generated at the center of their respective subregion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Indirect Diffuse Indirect diffuse lighting is the bouncing of light on nearby surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' It’s traditionally expensive to compute accurately, even in screen space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' HBIL can do it efficiently but fails to account for light pass- ing behind surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Figure 9 shows how we computed this effect with better thickness handling using visibility bitmasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Samples were taken along the slice direction and detected (in O(1)) how many un-occluded sectors are covered to estimate lighting contri- bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' These sectors were then set to an occluded state (also in O(1)) to handle partial or total occlusion of light coming from sub- sequent samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' The more the visibility sectors, the more precise the estimation of lighting and occlusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' It was observed that 32 sectors gave good quality and makes the bit field fit nicely within a single unsigned integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Figure 9: Left: The yellow sample intersects one un-occluded sec- tor and can contribute lighting equivalent to one over the total number of visibility sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' The sector is set to an occluded state for subsequent samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Right: Sampling continues and a new ob- ject on the right is found, but it intersects an already occluded sec- tor, so it cannot contribute lighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' The yellow sample on the left crosses an un-occluded sector and can contribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' The pseudo-code at line 23 in Algorithm 1 shows how the light- ing contribution of a sample is implemented, using the number of occluded zones by the current sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' If one or more sectors are covered, the sample contributed lighting and the light buffer is sampled at the sample location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Then n · l and nl · l are com- puted and used for weighting light intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' The light is further- more weighted by the occluded sector count over the total sector count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Results and evaluation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Ambient Occlusion In this subsection, the core algorithm will be demonstrated con- jointly with AO, as it’s much easier to discern the properties of vis- ibility bitmasks in this mode than using indirect diffuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' The ren- ders in Figure 10 are produced by our extension of Unity’s GTAO implementation, where the two horizon angles θ1 and θ2 have been replaced by a single visibility bitmask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Reference GTAO (no falloff) Ours (t = 1) Ours (t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='5) Ours (t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='25) Ours (t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='1) GTAO (falloff)v=n a2 a3 a4 a1v=n v=n6 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Therrien, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Levesque, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Gilet / Screen Space Indirect Lighting with Visibility Bitmask Figure 10: Comparison of GTAO, our method based on visibility bitmasks, and a ray-traced AO reference at different radius and sample count values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' It highlights how easy it is to implement our method on top of an existing horizon-based technique, and how this single modification can dramatically enhance visual quality compared to a ray-traced reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' The Lumberyard Bistro scene has been chosen because it contains a lot of thin and shallow surfaces that are typically a prob- lem with horizon-based techniques, but are improved by visibility bitmasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' All benchmarks use one hemisphere slices per pixel jit- tered over multiple frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' The radius parameter (Figure 10) is the radius of the hemisphere aligned to the screen in world units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Wider hemispheres will cover wider regions of the screen, casting farther-reaching occlusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' A wide radius is problematic for GTAO because the single cone ap- proximation tends to cast too much occlusion in regions enclosed by thin objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Even around not-so-thin objects, a wide radius tends to produce a blurry occlusion blob that does not capture fine detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' In contrast, our method (with the exact same samples) is able to let light pass behind surfaces, avoiding over-occlusion and cap- turing a lot of small details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Figure 11 demonstrates an even more difficult case for GTAO where most of the objects are behind a fence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' The light gets Figure 11: Left: GTAO exhibit too much occlusion behind the bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Right: Our method is able let light pass behind the bars, minimizing the over-occlusion artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' trapped behind the bars instead of passing by, causing a lot of over- occlusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Our method is able to handle this situation much better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' When using a wide radius, GTAO has a tendency to produce ha- los around objects (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Our method doesn’t have this prob- lem, and capture more geometric and normal details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' The number of samples (Figure 10) indicates the number of GTAO Our Method Ray Traced Reference Radius 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='8 8 samples Radius 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='0 12 samples Radius 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='0 16 samples Radius 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='0 16 samples Radius 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='0 16 samplesO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Therrien, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Levesque, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Gilet / Screen Space Indirect Lighting with Visibility Bitmask 7 fetches taken in the depth buffer along one horizon side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' There- fore, for one slice, the actual number of samples taken is twice that number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' A wider radius causes the samples to be sparser on-screen, so it’s typical to increase sample count for a wider radius to main- tain the same sampling density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' A low sampling density increases the likelihood of missing potential occluders, especially if they are thin on-screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Increasing the number of samples has a big impact on performance, so it’s a tradeoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Performance depends primarily on the number of samples taken and the radius of the effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' A wider radius increases the proba- bility of occurrence of a cache miss (sample not being present in the cache) and consequently can lower performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' The execu- tion time of the technique tends to scale linearly with the number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Table 1 compares the performance of the horizon-based GTAO implementation versus our method using visibility bitmasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Both techniques are composed of a sampling pass and a denoising pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Only the results of the sampling pass are included in the ta- ble since it’s the only one impacted by our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' The denoising pass has a constant cost of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='3 ms in 1080p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' It can be observed that our method has a modest impact on performance around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='01-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='02 milliseconds, with a fixed ALU overhead of about 15 GPU instruc- tions per sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Increasing the radius masks this overhead as the execution becomes bandwidth-limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Radius Sample Count GTAO Our Method 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='8 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='49 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='51 ms 1 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='75 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='77 ms 1 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='95 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='97 ms 2 16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='12 ms 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='13 ms 3 16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='12 ms 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='13 ms Table 1: Render time of the sampling pass for GTAO and our method with various radius and sample parameters, at 1920x1080, with 32 visibility sectors per hemisphere slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Benchmarks are done on an Nvidia RTX 2080 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Another parameter that impacts image quality is the number of visibility sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' When the sector count is too low, banding arti- facts can appear, particularly around thin objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' The proposed im- plementation used 32 visibility sectors because it just crossed the threshold where the artifacts became almost invisible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Additionally, it nicely fits into a single unsigned integer which gives good per- formance on the GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' By contrast, a 128 bits version require four unsigned integers and the use of vector instructions, which lim- its the amount of instruction packing that the compiler could do, negatively impacting the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' A performance overhead of around 5-10% was observed with 128 visibility sectors compared to 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Directionally Occluded Ambient Lighting In this subsection, we compare different ambient sampling strate- gies and show how they can dramatically influence the resulting lighting and occlusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Figure 12 shows the average normal on the left, and the resulting ambient lighting on the right for each strat- egy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' In most 3D applications, ambient lighting is sampled in the di- rection of the surface normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' This approach does not take into ac- count the fact that some light could be occluded in some direction and tend to make ambient lighting change sharply with the scene geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' A better approach is to use a screen space bent normal per pixel that is modulated according to nearby occlusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' It points towards the direction of incoming light and gives a smoother re- sult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' However, it cannot handle multiple light directions and will misrepresent the ambient lighting of surfaces enclosed by thin ob- jects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Figure 12: Comparison of ambient lighting using the surface nor- mal, bent normals, and visibility bitmasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' The average sampling normal direction is shown on the left to make the difference more visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Top row: G-Buffer normals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Middle row: Bent normals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Bottom row: visibility bitmask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Our approach used visibility bitmasks to take multiple samples along each hemisphere slice in the directions that were not oc- cluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Doing so allow ambient light to pass behind surfaces, giv- ing smooth lighting from multiple directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' It’s also worth noting that each ambient sample is weighted according to the occlusion in that specific direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' If the ambient color had been simply av- eraged out and then multiplied by ambient occlusion, a lot of the color variation in the lighting would have been lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Indirect Diffuse In this subsection, we look at the visual quality and performance of the indirect diffuse portion of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' In addition to sam- pling the depth buffer, we also sample the HDR light buffer and the screen space normal buffer for every sample taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' The light buffer contains only the direct lighting (with shadows), as the am- bient light is computed by our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Figure 13 compares our result with a path tracing reference for single and multi-bounce in- direct diffuse lighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' The direct lighting coming from the sun on the left wall bounces on the brick wall, illuminating it and cast- ing indirect shadows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' With multiple bounces, the light is even able to bounce back on the left wall, illuminating a shadowed region of 1 EL8 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Therrien, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Levesque, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Gilet / Screen Space Indirect Lighting with Visibility Bitmask Figure 13: Comparison of indirect diffuse algorithm and path tracing reference with single and multi-bounce indirect diffuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Multiple bounces are implemented by injecting the indi- rect illumination into the light buffer to be used as input for the next frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Light intensity needs to be properly balanced when using multi-bounce, or it can cause a feedback loop resulting in lighting accumulation over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' The result cannot match perfectly the path-traced reference since the algorithm operates only on the screen pixels as opposed to the entire scene geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' One ma- jor drawback of our technique, along with screen-space methods, is that if direct light leaves the screen, the indirect lighting disap- pears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Figure 14: When samples are evenly spaced along the sampling di- rection, a banding pattern can appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Jittering the samples along the sampling direction helps mask the banding artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' The number of samples taken has a big impact on quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Hav- ing a too low sampling density can introduce banding artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' To mitigate the issue, samples are jittered along the sampling direc- tion, as shown in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Low sample density can also cause a loss of detail around small objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' To improve this, we distribute the samples exponentially around the shaded pixel, as nearby surfaces usually have more in- fluence on the result than farther ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Figure 16 compares the amount of noise incurred when sampling the scene with SSR-like tracing and our method based on visibil- ity bitmasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Both techniques are taking the same maximum num- ber of samples but the visibility bitmask approach has a lot less noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Rays in SSR are doing at most one accumulation operation (when a hit is found).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' In contrast, the visibility bitmask approach is accumulating each sample that is visible from the current pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' This algorithm is bandwidth-intensive because we need to sam- ple the HDR light buffer and the screen space normal buffer for every sample taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Those sample locations are not correlated and impair caching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Table 2 compares the performances of the indirect diffuse part of the algorithm with different sampling parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' The corresponding renders are shown in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Full resolution means that the shader is executed for every pixel of the final ren- der resolution, whereas half-resolution executes it on a screen that is half the size (a quarter the number of pixels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' The image is then upscaled with a classic bilateral upsampler to avoid aliasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Ren- dering in half resolution is much more efficient (around 4x), but can introduce more flickering in the image and a blurrier result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Conclusion and future work Previous screen space GI methods that rely on HBAO cannot han- dle thickness because of the assumption that the depth buffer is strictly a height field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Other techniques based on SSR tracing squander a lot of rays that end up passing behind and over sur- faces, exiting the screen without hitting anything, thereby introduc- Visibility Sector Gl Path Tracing Reference Single- Bounce Multi- BounceNon-Jittered JitteredO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Therrien, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Levesque, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Gilet / Screen Space Indirect Lighting with Visibility Bitmask 9 Figure 15: Resulting renders using different sample, radius, stepping, and resolution parameters for the indirect diffuse lighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' The param- eters of each figure are indicated in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Figure 16: Left: 2 rays per pixel SSR tracing, with 16 steps per ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' There is a lot of noise, as a lot of rays exit the screen or pass behind surfaces without hitting anything.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Right: Horizon search on both sides of the pixel using a visibility bitmask with 16 steps per horizon is a lot less noisy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' ing a lot of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' In contrast, our method combined the sampling efficiency of horizon-based methods, while retaining the capabil- ity of handling thickness properly along the way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' The proposed al- gorithm is easy to understand and implement on modern GPUs and can be integrated into any horizon-based technique with only a mild performance overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Moreover, this method can also be used to improve ambient light sampling by taking into account the direc- tionality of occlusion when integrating ambient light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Even though the ray-tracing part of the algorithm handles thick- ness properly, since we operate on a single layer of the depth buffer, the actual object thickness is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' We are forced to rely on a Configuration Sampling Denoising Total a) 8 samples, radius 1, const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' steps, full res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='9 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='33 ms 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='23 ms b) 8 samples, radius 4, const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' steps, full res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='7 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='33 ms 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='03 ms c) 16 samples, radius 4, const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' steps, full res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='3 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='33 ms 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='63 ms d) 16 samples, radius 4, exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' steps, full res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='6 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='33 ms 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='93 ms e) 32 samples, radius 4, exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' steps, full res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='0 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='33 ms 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='33 ms f) 16 samples, radius 4, exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' steps, half res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='97 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='1 ms 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='07 ms Table 2: Render time at 1920x1080, with a constant thickness value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' constant thickness value that optionally increases linearly over the distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' A thickness heuristic that would give a plausible thick- ness per pixel would help improve the occlusion leaks around some very thin objects or light leaks behind very thick ones and would be interesting to explore in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' At the performance level, SSGI can be an expensive algorithm because it is very demanding on GPU bandwidth and utilizes the cache poorly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Using a caching method similar to LSAO could potentially improve this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Finally, a) b) c) d) eSSR-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content='Gl Our Method10 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Therrien, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Levesque, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Gilet / Screen Space Indirect Lighting with Visibility Bitmask the common ambient light sources are either static (constant color, light probes) or expensive to update at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' We might investi- gate in the future ways to approximate low-frequency ambient irra- diance dynamically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Acknowledgments and data statements Special thanks to Deepti Joshi (CDRIN) and Peter Shirley (NVIDIA) for their guidance in the redaction of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' We also want to thank our other colleagues namely Antoine Fortin (CDRIN), Olivier Leclerc (CDRIN), Steven Pigeon (UQAR), and Vahe Vardanyan (CDRIN) who have actively supported the current body of work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Most of the models are from the Amazon Lumber- yard Bistro scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' This research is financed in part by the province of Quebec (Canada) via the grant "Programme d’aide à la recherche et au transfert (PART)".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Data sharing not applicable to this article as no datasets were generated or analysed during the current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' References [Bav11] BAVOIL L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=': Horizon-based ambient occlusion using compute shaders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Nvidia DirectX 11 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=': Multi-scale global illumina- tion in quantum break.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' In ACM SIGGRAPH (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' [SVF17] SHERGIN D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=', VIDIGER D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=', FOFANOVA A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=': Superposition benchmark: innovative ssrtgi lighting in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' In ACM SIGGRAPH 2017 Real Time Live!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' 25–25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' [Tim13] TIMONEN V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=': Line-sweep ambient obscurance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' In Computer graphics forum (2013), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' 32, Wiley Online Library, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' 97–105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' [VSE21] VERMEER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=', SCANDOLO L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=', EISEMANN E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=': Stochastic-depth ambient occlusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' Proceedings of the ACM on Computer Graphics and Interactive Techniques 4, 1 (2021), 1–15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' [ZIK98] ZHUKOV S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=', IONES A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=', KRONIN G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=': An ambient light illumina- tion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' In Eurographics Workshop on Rendering Techniques (1998), Springer, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'} +page_content=' 45–55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFIT4oBgHgl3EQf2Ssk/content/2301.11376v1.pdf'}