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+MNRAS 000, 1–15 (2022)
+Preprint 13 January 2023
+Compiled using MNRAS LATEX style file v3.0
+Shear-driven magnetic buoyancy in the solar tachocline: The mean
+electromotive force due to rotation
+Craig. D. Duguid,★ Paul J. Bushby, and Toby S. Wood
+School of Mathematics, Statistics and Physics, Newcastle University, Newcastle Upon Tyne, NE1 7RU, UK
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+The leading theoretical paradigm for the Sun’s magnetic cycle is an 𝛼𝜔-dynamo process, in
+which a combination of differential rotation and turbulent, helical flows produces a large-scale
+magnetic field that reverses every 11 years. Most 𝛼𝜔 solar dynamo models rely on differential
+rotation in the solar tachocline to generate a strong toroidal field. The most problematic part of
+such models is then the production of the large-scale poloidal field, via a process known as the
+𝛼-effect. Whilst this is usually attributed to small-scale convective motions under the influence
+of rotation, the efficiency of this regenerative process has been called into question by some
+numerical simulations. Motivated by likely conditions within the tachocline, the aim of this
+paper is to investigate an alternative mechanism for the poloidal field regeneration, namely the
+magnetic buoyancy instability in a shear-generated, rotating magnetic layer. We use a local,
+fully compressible model in which an imposed vertical shear winds up an initially vertical
+magnetic field. The field ultimately becomes buoyantly unstable, and we measure the resulting
+mean electromotive force (EMF). For sufficiently rapid rotation, we find that a significant
+component of the mean EMF is aligned with the direction of the mean magnetic field, which
+is the characteristic feature of the classical 𝛼𝜔-dynamo model. Our results therefore suggest
+that magnetic buoyancy could contribute directly to the generation of large-scale poloidal field
+in the Sun.
+Key words: MHD – Sun: magnetic fields – dynamo – Sun: rotation – hydrodynamics –
+instabilities
+1
+INTRODUCTION
+The key properties of the solar magnetic cycle, which waxes and
+wanes with a period of approximately 11 years, are well known.
+However, the nature of the dynamo mechanism that is responsi-
+ble for producing this cyclic behaviour is still not fully understood
+(Charbonneau 2020). One of the most plausible explanations for the
+observed magnetic activity is that of an 𝛼𝜔-dynamo (first proposed
+by Parker 1955b). A key component of this dynamo mechanism is
+differential rotation (the 𝜔-effect), which stretches magnetic field
+lines in the direction of the flow; around the base of the solar convec-
+tion zone, the strong radial shear in the solar tachocline (Thompson
+et al. 2003) would tend to produce a magnetic field with a dominant
+azimuthal (toroidal) component. For a successful dynamo, there
+must also be some mechanism that regenerates the poloidal (radial
+and latitudinal) components of the large-scale magnetic field. In
+Parker’s scenario (Parker 1955b, 1993), the dynamo loop is com-
+pleted by the action of cyclonic convection upon toroidal field lines
+(a process now referred to as the 𝛼-effect). Due to the action of the
+Coriolis force, convective upwellings tend to produce rising, twisted
+★ E-mail: craig.duguid@newcastle.ac.uk
+magnetic loops. The cumulative effect of many such loops could,
+in principle, regenerate a large-scale poloidal field. Using a model
+in which the 𝛼-effect was implemented via a parametrised source
+term, Parker established that an 𝛼𝜔-dynamo of this type should
+be capable of producing oscillatory magnetic behaviour, similar to
+that seen in the Sun, provided that the 𝛼- and 𝜔-effects are efficient
+enough to overcome ohmic dissipation.
+While there is a strong consensus, driven by the observations,
+that the 𝜔-effect is at work in the solar interior, the 𝛼-effect re-
+mains a controversial area. It is possible to place Parker’s heuristic
+arguments on a firmer mathematical footing using the techniques of
+mean-field electrodynamics (Steenbeck et al. 1966; Moffatt 1978).
+Under this approach, an averaging process is introduced, and the
+equation for the mean magnetic field then contains an additional
+source term (a mean electromotive force, EMF) that arises as a
+result of the correlations between the fluctuating parts of the mag-
+netic field and the flow. For systems that lack reflectional symmetry
+(e.g. rotating systems), Parker’s 𝛼-effect then emerges (under certain
+assumptions) as part of this mean EMF. By making the first-order
+smoothing approximation (see, e.g., Moffatt 1978, for more details),
+it is possible to derive an expression for the 𝛼-effect that is directly
+proportional to the kinetic helicity of the flow. However, the valid-
+© 2022 The Authors
+arXiv:2301.05067v1  [astro-ph.SR]  12 Jan 2023
+
+2
+C. D. Duguid et al.
+ity of this expression is questionable under the highly-conducting,
+turbulent conditions present in the solar interior.
+Even in (non-solar-like) situations where there is no differential
+rotation, mean-field theory suggests that rapidly rotating convection
+should be capable of sustaining a magnetic field with a significant
+large-scale component (i.e. a magnetic field that is structured on
+the scale of the system). There are certainly examples of large-scale
+(𝛼2) dynamos in near-onset, rapidly-rotating convection (e.g. Chil-
+dress & Soward 1972) and in moderately supercritical, rotationally-
+dominated convection (e.g. Masada & Sano 2016; Bushby et al.
+2018). However, dynamos driven by turbulent rotating convection
+tend to produce disordered, small-scale magnetic fields (e.g. Catta-
+neo & Hughes 2006; Favier & Bushby 2013). These results are often
+interpreted in terms of the 𝛼-effect, and it is certainly possible to
+measure 𝛼 in numerical simulations of convection in a rotating do-
+main (see, e.g. Cattaneo & Hughes 2006; Käpylä et al. 2009; Favier
+& Bushby 2013). However, the outcome of such measurements is
+dependent upon the techniques used, and this remains an area of
+some disagreement in the literature (a full discussion of which lies
+beyond the scope of the present paper). Nonetheless, these studies
+cast doubt on whether cyclonic convection can produce the required
+𝛼-effect in the turbulent conditions of the solar convection zone.
+Given the above considerations, it is natural to ask whether
+there are any other physical mechanisms that could give rise to
+a similar “rise and twist” effect that Parker originally envisaged
+as being due to cyclonic convection (thus completing the dynamo
+cycle). One candidate for this is magnetic buoyancy. It is well estab-
+lished that isolated magnetic flux tubes tend to be less dense than
+their non-magnetic surroundings, and are therefore buoyant (Parker
+1955a; Jensen 1955), and this has long been cited as an explanation
+for the emergence of magnetically active regions at the solar surface.
+Whilst it has been shown that instabilities in individual flux tubes
+can (in the presence of rotation) give rise to a mean EMF that is
+analogous to the 𝛼-effect (Ferriz-Mas et al. 1994), it is unlikely that
+the magnetic field distribution in the solar interior takes the form of
+discrete, isolated magnetic flux tubes. It is more plausible that there
+is a continuous layer of predominantly azimuthal magnetic flux that
+is (partly as a consequence of flux pumping, e.g. Tobias et al. 2001)
+largely confined to the sub-adiabatically stratified region just below
+the base of the solar convection zone, where poloidal field can be
+stretched out by the shear in the tachocline.
+The evolution of an imposed magnetic layer under the action
+of magnetic buoyancy has been well studied (e.g. Gilman 1970;
+Cattaneo & Hughes 1988; Matthews et al. 1995b; Wissink et al.
+2000; Kersalé et al. 2007; Mizerski et al. 2013; Gilman 2018). The
+preferred mode of instability is usually three-dimensional, produc-
+ing undular motions with a long wavelength parallel to the magnetic
+field lines and a short wavelength perpendicular to them. As noted
+by Hughes (2007), for example, a simple comparison of the typi-
+cal growth rate of the instability compared to the rotation period
+of the Sun suggests that rotation should be playing a role in the
+evolution of this instability at the base of the solar convection zone.
+The effects of rotation upon the magnetic buoyancy instability are
+non-trivial (see, e.g. Hughes 2007, for a review). Here we focus
+upon those studies of magnetic buoyancy in the presence of rotation
+that are directly relevant to the solar dynamo problem. Building
+on the analysis of Gilman (1970), Moffatt (1978) considered the
+magnetic buoyancy instability in a unidirectional layer under the
+magnetostrophic approximation; he was able to show that the re-
+sultant instability produces a systematic mean EMF from which it
+is possible to derive an 𝛼-effect. The mean electromotive force due
+to magnetic buoyancy in the presence of rotation (and its possible
+influence on the solar dynamo) has been explored in a number of
+subsequent studies (Schmitt 1984; Brandenburg & Schmitt 1998;
+Thelen 2000b,a; Chatterjee et al. 2011; Davies & Hughes 2011). It
+should be noted that Davies & Hughes (2011) suggest that attention
+should be focused upon the mean EMF in such cases, rather than the
+𝛼-effect itself, arguing that the decomposition of this quantity into
+standard mean-field coefficients is too simplistic to be meaningful
+in the context of this magnetically-driven instability.
+All of the magnetic buoyancy studies listed above consider the
+evolution of an imposed magnetic layer under the action of this insta-
+bility. Inevitably, the evolution of this layer will depend to some ex-
+tent on the initial magnetic field distribution. In the solar tachocline,
+it is believed that this layer is generated by shearing motions, so it
+is clearly of importance (as a step towards the full dynamo prob-
+lem) to consider the corresponding magnetic buoyancy instability
+in a shear-generated magnetic layer. In a series of papers, Brummell
+et al. (2002a), Cline et al. (2003a) and Cline et al. (2003b) consid-
+ered the evolution of buoyant magnetic structures, generated by an
+idealised shear flow with a strong horizontal (latitudinal) gradient.
+Although the shear flow was not particularly tachocline-like, they
+did manage to generate dynamo action (Cline et al. 2003b), even in
+the absence of rotation, via the combination of magnetic buoyancy
+and Kelvin-Helmholtz instabilities. The more solar-like problem of
+a magnetic layer generated by a vertical shear has also been consid-
+ered (Vasil & Brummell 2008, 2009; Silvers et al. 2009a,b; Barker
+et al. 2012). In their initial study, Vasil & Brummell (2008) found
+that it was only possible to excite a magnetic buoyancy instability in
+this configuration with an unrealistically strong (hydrodynamically
+unstable) shear. A resolution to this problem was later proposed
+by Silvers et al. (2009b), who observed that it was in fact possible
+for such systems to produce magnetic buoyancy instabilities, even
+with a comparatively weak shear, if the ratio of the magnetic to
+thermal diffusivities is sufficiently small. Fortunately, this is exactly
+the expected parameter regime for the solar tachocline.
+In terms of modelling the solar dynamo, the key ingredient
+that is missing from previous studies that have considered magnetic
+buoyancy in a shear-generated layer is rotation. We know from
+the imposed field calculations that magnetic buoyancy instabilities
+can produce a mean EMF that should be conducive to dynamo
+action. What we do not yet know is whether or not a suitable mean
+EMF can be obtained from the shear-generated magnetic buoyancy
+instability. This is the subject of the present paper and this represents
+a crucial step towards building a magnetic buoyancy-driven version
+of Parker’s solar dynamo model.
+The plan for this paper is as follows. In Section 2 we will
+describe the model and governing equations. In Section 3 we will
+report on the results of the hydrodynamic problem on the stability
+of the shear for non-rotating and rotating systems. In section 4 we
+analyse results of simulations with and without rotation for the full
+MHD problem. We will then present the most important results
+of this work in Section 5, namely, the mean EMF driven by this
+magnetic buoyancy instability. We will then discuss the importance
+of our results in Section 6, focusing particularly upon the potential
+for this system to act as a dynamo.
+2
+MODEL
+2.1
+Model set-up
+A schematic of the model geometry is shown in Fig. 1. We consider
+a Cartesian domain, of depth 𝑑, with the coordinate system oriented
+MNRAS 000, 1–15 (2022)
+
+Shear-driven magnetic buoyancy
+3
+so that the 𝑧-axis points vertically downwards, parallel to the con-
+stant gravitational acceleration, 𝑔e𝑧. This domain rotates uniformly
+about the 𝑧-axis, with a constant angular velocity, −Ωe𝑧. The do-
+main is filled with a compressible, electrically-conducting fluid, of
+constant thermal conductivity 𝐾, constant magnetic diffusivity 𝜂,
+and constant dynamic viscosity 𝜇. The (constant) magnetic perme-
+ability of this fluid is denoted by 𝜇0. This fluid is assumed to be an
+ideal gas, with constant specific heats 𝑐𝑝 and 𝑐𝑣 (ℜ = 𝑐𝑝 −𝑐𝑣 is the
+gas constant). In its initial state, this fluid is horizontally-uniform,
+with temperature, 𝑇0, and density, 𝜌0, at the upper surface (𝑧 = 0).
+A fixed heat flux is applied through the lower boundary (𝑧 = 𝑑),
+leading to an initial temperature difference across the domain of
+Δ𝑇. The initial state is that of a polytrope, with
+𝑇(𝑧) = 𝑇0
+�
+1 + 𝜃𝑧
+𝑑
+�
+,
+and
+𝜌(𝑧) = 𝜌0
+�
+1 + 𝜃𝑧
+𝑑
+�𝑚
+,
+(1)
+where 𝜃 = Δ𝑇/𝑇0 is a measure of the thermal stratification of the
+domain, and 𝑚 = (𝑔𝑑/ℜΔ𝑇) − 1 is the polytropic index.
+Following previous studies (Vasil & Brummell 2008; Silvers
+et al. 2009b), we include an additional forcing term in the momen-
+tum equation. The aim is to mimic the key aspects of the differential
+rotation in a local region of the solar tachocline. Identifying the
+𝑥-axis of the domain with the local azimuthal (toroidal) direction,
+this shear flow takes the form u = 𝑈0(𝑧)e𝑥, where 𝑈0(𝑧) will be
+specified below. The forcing is chosen so as to balance this flow (in
+the absence of a magnetic field), and this flow is set as an initial
+condition across the polytropic layer.
+Having set up the polytropic state with a shear flow, we then
+introduce a seed magnetic field into the system. This field is initially
+uniform and vertical, taking the form B = 𝐵0e𝑧. This imposed
+field will be continually stretched in the 𝑥-direction by the shear,
+producing a horizontal magnetic layer (the peak strength of which
+initially increases linearly with time). Eventually, the field in this
+layer will be amplified to a level at which it becomes dynamically
+significant, resisting the shearing motions and ultimately driving a
+magnetic buoyancy instability in the system.
+Given the expected geometry of the magnetic layer, we an-
+ticipate that the ensuing magnetic buoyancy instability will have
+a long length-scale in the 𝑥-direction and a short length-scale in
+the 𝑦 direction. Following Silvers et al. (2009b), we choose to re-
+flect the anisotropic nature of the instability in the domain geom-
+etry. Defining the horizontal dimensions to be 0 ⩽ 𝑥 ⩽ 𝐿𝑥𝑑 and
+0 ⩽ 𝑦 ⩽ 𝐿𝑦𝑑, we set 𝐿𝑥 = 2 and 𝐿𝑦 = 0.5. This narrow domain
+enables us to study the key features of the system without expending
+unnecessary computational effort.
+It is worth emphasising at this stage that it is the inclusion of
+rotation that represents the novel aspect of the present study. To the
+best of our knowledge, this is the first time that the effects of rotation
+have been included in a study of magnetic buoyancy in a shear-
+generated layer. As we shall see later, this raises some interesting
+issues in terms of the shear forcing, and also leads to the excitation
+of mean flows once the magnetic field becomes dynamically sig-
+nificant. Given the additional complexity that this introduces into
+the system, we chose to focus exclusively upon the simplest case
+of a vertical rotation vector in this initial study (which places this
+domain within the polar regions of the solar tachocline). We will
+explore the case of an inclined rotation vector, allowing us to probe
+the latitudinal dependence of the system, in a future paper.
+Figure 1. Schematic of the model domain. The arrows in the 𝑥-direction
+show the shear flow, 𝑈0, whilst the arrows in the 𝑧-direction show the initial
+magnetic field, B0. The shading indicates the initial temperature distribution,
+with red (yellow) shading corresponding to warmer (cooler) fluid.
+2.2
+Governing equations
+In order to make the governing equations dimensionless, we adopt
+similar scalings to those set out in several previous studies (e.g.
+Matthews et al. 1995a; Favier & Bushby 2013). The characteristic
+length-scale is assumed to be the depth of the layer, 𝑑, whilst the
+density and temperature are scaled in terms of their initial values
+at the upper surface of the domain (𝜌0 and 𝑇0 respectively). Our
+adopted time-scale is the (isothermal) acoustic travel time at the top
+of the domain (𝑧 = 0) defined by 𝑑/
+√︁
+ℜ𝑇0. A natural scaling for
+the velocity of the fluid is then the corresponding isothermal sound
+speed,
+√︁
+ℜ𝑇0, whilst the magnetic field is scaled in terms of 𝐵0.
+In these units, the equations describing the temporal evolution
+of the density 𝜌, temperature 𝑇, velocity u and magnetic field B
+become
+𝜌 𝜕u
+𝜕𝑡 + 𝜌(u · ∇)u = −Ta01/2 𝜎𝜅𝜌𝛀 × u − ∇𝑃 + 𝜃(𝑚 + 1)𝜌e𝑧
++ 𝐹(∇ × B) × B + 𝜎𝜅∇ · 𝜏 + F𝑠 ,
+(2a)
+𝜌 𝜕𝑇
+𝜕𝑡 + 𝜌(u · ∇)𝑇 = −(𝛾 − 1)𝑃∇ · u + 𝛾𝜅∇2𝑇
++ 𝐹(𝛾 − 1)𝜁0𝜅|∇ × B|2 + (𝛾 − 1)𝜎𝜅
+2
+𝜏2 ,
+(2b)
+𝜕B
+𝜕𝑡 = ∇ × (u × B − 𝜁0𝜅∇ × B) ,
+(2c)
+𝜕𝜌
+𝜕𝑡 = −∇ · (𝜌u) ,
+(2d)
+∇ · B = 0 .
+(2e)
+We have defined the viscous stress tensor as
+𝜏𝑖 𝑗 = 𝜕𝑢𝑖
+𝜕𝑥 𝑗
++ 𝜕𝑢 𝑗
+𝜕𝑥𝑖
+− 2
+3𝛿𝑖 𝑗
+𝜕𝑢𝑘
+𝜕𝑥𝑘
+(3)
+(where 𝛿𝑖 𝑗 denotes the Kronecker delta), whilst the pressure 𝑃
+satisfies the equation of state for an ideal gas,
+𝑃 = 𝜌𝑇 .
+(4)
+This system contains a number of non-dimensional parameters,
+which are summarised in Table 1 along with their typical values.
+These are discussed in more detail below.
+In Eq. (2a), the forcing term, F𝑠, is chosen to balance the
+viscous and Coriolis forces associated with the imposed shear flow,
+𝑈0(𝑧)e𝑥. In the absence of magnetic field or any hydrodynamic
+MNRAS 000, 1–15 (2022)
+
+12
+y
+Bo
+三
+Uo
+*g
+三(4
+C. D. Duguid et al.
+instabilities, we would therefore expect the fluid to remain in a
+steady state with u = 𝑈0(𝑧)e𝑥. Following a similar approach to
+Vasil & Brummell (2008) and Silvers et al. (2009b) we choose a
+shear flow of the form
+𝑈0(𝑧) = 𝐴 tanh[10(𝑧 − 0.5)] ,
+(5)
+where the constant 𝐴 determines the shear amplitude. The form of
+the shear flow is illustrated schematically in Fig. 1. This hyperbolic
+tangent profile localises the shearing region at the mid-plane of the
+domain. We note that, for sufficiently large values of 𝐴, the shear
+flow would be unstable even in the absence of a magnetic field; the
+hydrodynamic stability of this system (and the consequent choice
+of 𝐴) will be discussed in § 3. In order to maintain this shear flow,
+we take
+F𝑠 =
+������
+−𝜎𝜅𝜕𝑧𝑧𝑈0
+√Ta0 𝜎𝜅𝜌𝑈0
+0
+������
+.
+(6)
+In the 𝑥-direction, this forcing balances the viscous term. Note that
+the presence of rotation requires F𝑠 to have a 𝑦 component in order to
+balance the Coriolis contribution from the imposed shear. Once the
+magnetic field is introduced into the fluid, the Lorentz tension in the
+field lines will eventually act to inhibit the shear, and may also drive
+other kinds of mean flow. However, by making the initial magnetic
+field sufficiently weak, it is possible minimise the influence of some
+of these dynamical effects upon the evolution of the system over the
+timescales of interest.
+2.3
+Boundary and initial conditions
+All variables are assumed to be periodic in both of the horizontal
+directions. The upper and lower bounding surfaces are assumed
+to be stress-free and impermeable. The upper boundary is held at
+fixed temperature, whilst (as already noted) a fixed heat flux is
+imposed through the lower boundary. On both bounding surfaces,
+the horizontal magnetic field is assumed to vanish. Therefore the
+tangential viscous, Reynolds and Maxwell stresses all vanish on the
+upper and lower boundaries. In this dimensionless system, these
+conditions correspond to
+𝑢𝑧 = 𝜕𝑢𝑥
+𝜕𝑧 = 𝜕𝑢𝑦
+𝜕𝑧 = 0
+for
+𝑧 ∈ {0, 1} ,
+(7a)
+𝑇(𝑧 = 0) = 1
+and
+𝜕𝑇
+𝜕𝑧
+���𝑧=1 = 𝜃 ,
+(7b)
+𝐵𝑥 = 𝐵𝑦 = 𝜕𝐵𝑧
+𝜕𝑧 = 0
+for
+𝑧 ∈ {0, 1} .
+(7c)
+Note that the shear is sufficiently localised about the mid-plane that
+there is no contradiction here in terms of the stress-free condition on
+the velocity field. It is straightforward to confirm that these bound-
+ary conditions are consistent with the initial polytropic solution,
+which (in the absence of a magnetic field) is then an equilibrium
+solution of the governing equations. This equilibrium is eventually
+perturbed by the magnetic field as it is amplified by the shear. How-
+ever, a further perturbation is needed in order to seed the magnetic
+buoyancy instability. Therefore, a small thermal perturbation is also
+added to the initial state for each simulation. This takes the form of a
+small, pseudo-random perturbation to the temperature distribution
+(additive noise, localised around the mid-plane, of peak amplitude
+0.05 in dimensionless units).
+Symbol
+Description
+Definition
+Values
+𝐹
+Magnetic field strength
+𝐵2
+0
+ℜ𝑇0𝜌0𝜇0
+Variable
+𝜎
+Prandtl number
+𝜇𝑐𝑝
+𝐾
+0.00025
+𝜃
+Temperature gradient
+Δ𝑇
+𝑇0
+5
+𝜅
+Thermal diffusivity
+𝐾
+𝑑𝜌0𝑐𝑝
+√︁
+ℜ𝑇0
+0.01
+𝜁0
+Inverse Roberts number
+𝜂𝑐𝑝𝜌0
+𝐾
+0.0005
+𝛾
+Ratio of specific heats
+𝑐𝑝
+𝑐𝑣
+5/3
+𝑚
+Polytropic index
+𝑔𝑑
+ℜΔ𝑇 − 1
+1.6
+Ta0
+Taylor number
+4𝜌2
+0Ω2𝑑4
+𝜇2
+Variable
+Re
+Reynolds number
+𝑈0𝜌0
+𝜎𝜅
+8000
+Rm
+Magnetic Reynolds number
+𝑈0
+𝜁0𝜅
+4000
+𝜎𝑚
+Magnetic Prandtl number
+𝜎
+𝜁0
+0.5
+𝜏visc
+Viscous timescale
+1
+𝜎𝜅
+4 × 105
+𝜏ohmic
+Ohmic timescale
+1
+𝜁0𝜅
+2 × 105
+𝜏therm
+Thermal timescale
+1
+𝜅
+100
+Table 1. Non-dimensional parameters in the system including a text de-
+scription/name of the quantity, the definition, and the value the parameter
+takes (where applicable). Also reported are the (shear) fluid and magnetic
+Reynolds numbers (assuming a shear amplitude of 𝐴 = 0.02), the magnetic
+Prandtl number, and the diffusive timescales (viscous, Ohmic, and thermal).
+These Reynolds numbers and timescales are defined using the total layer
+depth as the characteristic length-scale (which is approximately one order
+of magnitude larger than the initial width of the shear layer).
+2.4
+Non-dimensional parameters
+In Table 1 we list the non dimensional parameters with their typical
+values (which largely follows Silvers et al. 2009b). Most of the
+parameters are fixed. In particular, we choose the Prandtl number,
+the thermal diffusivity and the inverse Roberts number so that the
+viscous, Ohmic and thermal diffusion timescales have the correct
+ordering for the tachocline, with the thermal timescale much shorter
+than the other two, and the magnetic Prandtl number less than unity
+(although because of computational limitations these simulations
+are necessarily much more dissipative than the real tachocline).
+The comparatively short thermal diffusion time tends to promote
+magnetic buoyancy in this system (Silvers et al. 2009b). With 𝛾 =
+5/3, a polytropic index of 𝑚 = 1.6 ensures that the layer is sub-
+adiabatically stratified. With this choice of parameters, our domain
+covers 𝑁𝑝 = (𝑚 + 1) ln(1 + 𝜃) ≈ 4.66 pressure scale heights. In
+terms of the parameters to be varied, the quantity 𝐹 gives the ratio
+of the squared Alfvén speed to the squared (isothermal) sound speed
+at 𝑧 = 0 at the start of the calculation. Note that a multiplicative
+factor of
+√
+𝐹 must be included in front of the magnetic field in
+order to make quantitative comparisons between the magnetic field
+strength and the flow. With all other parameters fixed, varying 𝐹
+is equivalent to varying the strength of the initial vertical magnetic
+field. The other parameter to be varied is the Taylor number, Ta0,
+MNRAS 000, 1–15 (2022)
+
+Shear-driven magnetic buoyancy
+5
+which is a dimensionless measure of the rotation rate of the system
+relative to viscous dissipation. Due to the dependence of the Coriolis
+term upon 𝜌, it is arguably the mid-layer value of this ratio that is
+of direct relevance to the shear (and hence the magnetic layer); we
+therefore quote the mid-layer Taylor number in what follows, i.e.
+Ta = Ta0
+�
+1 + 𝜃
+2
+�2𝑚
+=
+4𝜌2
+0Ω2𝑑4
+𝜇2
+�
+1 + 𝜃
+2
+�2𝑚
+.
+(8)
+With this choice of parameters, Ta ≈ 55Ta0.
+2.5
+Numerical method
+We solve the system of governing equations (Eq. 2) numerically.
+In practice, we use a poloidal-toroidal decomposition for the mag-
+netic field, which ensures that it remains solenoidal at all times.
+Horizontal derivatives are computed in Fourier space, whereas a
+4th-order finite difference scheme is applied to the vertical deriva-
+tives (in order to improve stability, an upwind scheme is used
+for the advective terms, where appropriate). All variables are de-
+aliased in Fourier space using the standard 2/3 rule. In order to
+time-step the equations, we use an explicit third-order Adams-
+Bashforth scheme with a variable time-step. We use a resolution
+of (𝑁𝑥, 𝑁𝑦, 𝑁𝑧) = (192, 96, 192) in all cases presented here. We
+have also performed a more extensive low resolution parameter sur-
+vey with a resolution of (𝑁𝑥, 𝑁𝑦, 𝑁𝑧) = (128, 64, 128), the results
+of which are not presented in the main body of the paper. How-
+ever, it should be noted that the differences between the lower and
+higher resolutions are not significant. A full list of all simulations
+performed in this work can be found in Appendix A.
+3
+HYDRODYNAMIC CONSIDERATIONS
+We first consider the stability of the shear flow in the absence of a
+magnetic field. The purpose of this is to ensure that the dynamics
+that we observe in the subsequent magnetic calculations are en-
+tirely driven by magnetic buoyancy, as opposed to an underlying
+hydrodynamic instability.
+Focusing initially upon the hydrodynamic system in the ab-
+sence of rotation, we note that the Richardson number is defined as
+follows:
+Ri =
+𝑁2
+(d𝑈0/d𝑧)2
+= 𝜃2(𝑚 + 1)
+100𝐴2
+�
+𝑚 − 1
+𝛾 [𝑚 + 1]
+� cosh4 �
+10(𝑧 − 0.5)
+�
+1 + 𝜃𝑧
+,
+(9)
+where 𝑁 is the Brunt-Väisälä frequency, which measures the
+strength of stratification, and d𝑈0/d𝑧 is the local shearing rate.
+At fixed 𝜃, 𝑚 and 𝛾, the key point to note is that this quantity is
+inversely proportional to the square of the shear flow amplitude, 𝐴.
+In the absence of diffusion, a necessary condition for stability is
+that Ri > 1/4 is satisfied everywhere in the domain (Miles 1961;
+Howard 1961). This condition was not satisfied in the strong shear
+regime considered by Vasil & Brummell (2008), who found a vig-
+orous hydrodynamic instability in the absence of a magnetic field.
+However, this condition was satisfied in the calculations of Silvers
+et al. (2009b), who quote a value of Ri ≈ 2.96. When diffusive ef-
+fects are included, some care is needed in the limit of rapid thermal
+diffusion, which can have a destabilising influence upon stratified
+Figure 2. Two metrics for the estimated stability of the shear for the shear pa-
+rameters of Silvers et al. (2009b) (𝐴 = 0.05) and the shear considered in this
+work (𝐴 = 0.02), see legend for colours. Left: the normalised (horizontally-
+averaged) vertical shear profile taken at 𝑡 = 500 time units in each case. The
+dotted line represents the normalised target shear profile. Right: the volume
+averaged deviation between the normalised shear flow and the normalised
+target profile.
+shear flows (Zahn 1974; Garaud et al. 2017; Cope 2021). In this
+case, the stability of the system depends not only upon the Richard-
+son number, but also the Prandtl number. Building on the work of
+Zahn (1974), Garaud et al. (2017) find a condition for instability
+that Ri 𝜎 ≲ 0.007. Whilst this threshold is empirically determined,
+probably exhibiting some degree of dependence upon the details of
+the model, it is notable that the system considered by Silvers et al.
+(2009b) yields a value of Ri 𝜎 ≈ 0.00074, which is well below the
+stability threshold suggested by Garaud et al. (2017). In other words,
+their underlying shear is potentially (hydrodynamically) unstable in
+this parameter regime.
+In order to investigate this issue, we consider here two shear
+amplitudes (initially in the absence of rotation). Following Silvers
+et al. (2009b), the stronger shear corresponds to 𝐴 = 0.05. We also
+consider a weaker shear of 𝐴 = 0.02. The choice of 𝐴 = 0.02
+is something of a compromise. Whilst Ri ≈ 18.5 ≫ 1/4 for the
+case of 𝐴 = 0.02, Ri 𝜎 ≈ 0.004, which is still just below the
+empirically-determined stability threshold of Garaud et al. (2017).
+However, the weaker the shear, the longer the time taken for the
+formation of the horizontal magnetic layer (during the early stages
+of evolution, the peak value of 𝐵𝑥 is proportional to 𝐴𝑡), so longer
+(computationally expensive) integrations are needed before the layer
+becomes buoyantly unstable.
+The effects of varying the shear amplitude are illustrated in
+Fig. 2. This figure shows the evolution of the horizontally-averaged
+𝑥-component of the velocity field,
+⟨𝑢𝑥⟩(𝑧, 𝑡) =
+1
+𝐿𝑥𝐿𝑦
+∫ 𝐿𝑦
+0
+∫ 𝐿𝑦
+0
+𝑢𝑥 d𝑥 d𝑦
+(10)
+(angled brackets will be used throughout this paper to denote a
+horizontal average of this form). In particular, this figure shows the
+deviation from the initial shear profile for the two shear amplitudes
+considered, alongside the integrated deviation from the initial pro-
+file. If even a weak instability is present then one would expect the
+shear profile to deviate from the target profile as energy is extracted
+from the shear. After 500 time units (which will be the typical period
+of time over which we evolve the magnetic buoyancy calculations),
+we see that the stronger shear (𝐴 = 0.05) has departed significantly
+from the target shear and the deviations are still increasing with time.
+This suggests that this shear flow is in fact (weakly) unstable. On the
+other hand, the weaker shear profile still shows excellent agreement
+MNRAS 000, 1–15 (2022)
+
+1
+0.08
+S
+0.5
+A
+0
+0.04
+(m)
+-0.5
+A
+0
+0
+0.5
+1
+0
+500
+A = 0.02 A = 0.05
+t
+26
+C. D. Duguid et al.
+Figure 3. The evolution of the mean flows (for 𝐴 = 0.02) for three different
+rotation rates, with Ta ∈ {0, 1, 5}(×108) as denoted by the legend. Top
+left: snapshots of the horizontally averaged vertical shear profile normalised
+by the shear amplitude for the three rotation rates taken at 𝑡 = 500. The
+initial shear profile, 𝑈0(𝑧), is overplotted as a dotted line, and is practically
+indistinguishable. Top right: the volume averaged deviation between the nor-
+malised shear flow and the normalised target profile. Bottom left: snapshots
+of the ⟨𝑢𝑦 ⟩ profile for each rotation rate taken at 𝑡 = 500. Bottom right: the
+volume integrated deviation of ⟨𝑢𝑦 ⟩ from zero.
+with the target shear, with an integrated error that appears to be
+reaching a steady state of approximately 1%. Over the timescales of
+interest the shear is essentially unchanged, so we choose 𝐴 = 0.02
+throughout in what follows; this choice should ensure that any de-
+partures from the target shear are definitely magnetically-driven. It
+is worth noting here that the vertical magnetic field we subsequently
+introduce will (at least initially) tend to inhibit any shear-driven in-
+stabilities (e.g. Silvers et al. 2009a), further reducing the possible
+impact of any underlying hydrodynamical instability.
+The discussion of shear stability has so far been in the context
+of non-rotating systems, and we need to confirm that our conclusions
+are unchanged if the system is rotating. It is also important to check
+that our imposed forcing correctly balances the initial shear when
+Coriolis effects are present. Fixing the shear amplitude to 𝐴 = 0.02,
+Fig. 3 shows a comparison between the non-rotating case presented
+above and rotating cases with Ta ∈ {1, 5}(×108). We see that the
+addition of rotation has no significant influence upon the evolution
+of ⟨𝑢𝑥⟩. In fact, we see that rotation tends to reduce the (already
+insignificant) integrated deviation from the target profile; rotation
+therefore seems to have a stabilising influence upon the system. As
+indicated by the lower part of Fig. 3, due to the presence of Coriolis
+effects, we see that weak mean flows emerge in the 𝑦-direction in
+the rotating cases. These could either be a transient response to
+the small-scale thermal perturbations that are added to the initial
+polytrope or the result of a very weak hydrodynamic instability.
+Either way, it should be stressed that these flows are rather low
+amplitude and are unlikely to have any significant impact upon the
+system over the timescales of interest. The inclusion of rotation has
+no significant (adverse) impact on the hydrodynamic stability of the
+system.
+4
+MAGNETIC BUOYANCY WITH AND WITHOUT
+ROTATION
+Having chosen a shear amplitude of 𝐴 = 0.02, we now impose a
+vertical background magnetic field (as described in § 2). As indi-
+cated in Table 1, where all of the other parameters are specified, we
+still need to choose the values of 𝐹 (which determines the magnetic
+field strength) and the Taylor number (recall that we quote the mid-
+layer value, Ta). The choices for these parameters were guided by
+a preliminary low-resolution parameter survey, the details of which
+can be found in Appendix A.
+As well as using a somewhat weaker shear flow than Silvers
+et al. (2009b), we also impose a weaker initial field, defined by
+𝐹 = 2.5 × 10−6 (compared to 𝐹 ⩾ 1.25 × 10−5 in Silvers et al.
+2009b). This might seem to be a somewhat counter-intuitive deci-
+sion, as this will increase the time taken for the formation of the
+horizontal magnetic layer, necessitating longer numerical calcula-
+tions. However, this also delays the point at which the Lorentz force
+feedback starts to perturb the mean flow (the details of which will
+be described below). Across the parameters surveyed, this reduced
+value of 𝐹 has a beneficial impact upon the magnetic buoyancy
+instability, because it allows steeper magnetic field gradients to de-
+velop. In what follows, we will present results for three different
+values of the Taylor number, corresponding to a non-rotating case
+(Ta = 0) and two rotating cases, Ta = 108 and Ta = 5 × 108.
+4.1
+Non-rotating case
+Before considering the effects of rotation, we first give a detailed
+overview of the evolution in the non-rotating case, the key fea-
+tures of which are illustrated in Fig. 4. In the early stages of the
+simulation, for 𝑡 ≲ 100, the magnetic field is too weak to play a
+significant dynamical role. During this stage there are only small-
+amplitude departures from the forced shear flow, which we attribute
+to acoustic and internal gravity waves that are excited by the initial
+temperature perturbation. These oscillate with periods ≲ 10 (di-
+mensionless units), and dissipate on the thermal diffusion timescale,
+𝜏therm = 100. At the same time, the imposed vertical magnetic field
+is continually stretched out by the shear flow, forming a magnetic
+layer around the mid-plane whose strength grows linearly in time.
+This stretching process is analogous to the 𝜔-effect at the base of
+the solar convection zone.
+In Fig. 4 we also show the state of the system at 𝑡 = 200. By
+this time, the magnetic pressure from the mean toroidal field, ⟨𝐵𝑥⟩,
+has become strong enough to noticeably change the overall mass
+distribution, moving fluid out of the shear region and into the layers
+above, as evident from the plots of Δ⟨𝜌⟩ = ⟨𝜌⟩ − 𝜌init(𝑧), where
+𝜌init is the initial density profile. This upward redistribution of mass
+provides the energy source for the magnetic buoyancy instability. At
+the onset of magnetic buoyancy, we see tube-like structures in the
+isovolumes that are aligned with the shear direction, evident in both
+the magnetic field and the flow. We illustrate these in Fig. 4 through
+the vertical flow component and the toroidal field perturbation. It
+is clear that 𝐵′𝑥, which is defined as 𝐵′𝑥 = 𝐵𝑥 − ⟨𝐵𝑥⟩, and 𝑢𝑧 have
+structures that are localised around the upper regions of the induced
+magnetic layer, where the magnetic field gradient is conducive to
+magnetic buoyancy. As expected, we see that the magnetic buoyancy
+instability is characterised by a long length-scale parallel to the
+MNRAS 000, 1–15 (2022)
+
+1
+2
+d
+S
+0.008
+0.5
+A
+0
+(cm)
+0.004
+-0.5
+-1
+0
+I-A
+0
+0.5
+1
+0
+500
+t
+0.05
+0.007
+V/ <hn)
+1<rn)I
+0
+-0.05
+0
+0
+0.5
+1
+0
+500
+t
+Ta = 0
+-Ta = 108Shear-driven magnetic buoyancy
+7
+Figure 4. The evolution of the magnetic buoyancy instability for the non-rotating case. Top section (a): Vertical profiles of the horizontally averaged density
+perturbation (left), defined as Δ⟨𝜌⟩ = ⟨𝜌⟩ − 𝜌init(𝑧) (where 𝜌init(𝑧) is the initial, dimensionless density distribution of the polytropic layer), and vertical
+profile of the horizontally-averaged toroidal field
+√
+𝐹 ⟨𝐵𝑥 ⟩ (right). These are shown for three times in the evolution denoted by the legend. The bottom section
+consists of 3D renderings at three times (see text labels) of pseudocoloured isovolumes (with associated legends to the left). The backdrop for these isovolumes
+shows 2D slices (with associated legend to the top) of the toroidal field 𝐵𝑥 at the associated times. Middle row (b): the vertical component of velocity 𝑢𝑧 with
+isovolumes defined by the regions |𝑢𝑧 | ⩾ 10−4. Bottom row (c): the toroidal field perturbation defined as
+√
+𝐹 𝐵′𝑥 =
+√
+𝐹 (𝐵𝑥 − ⟨𝐵𝑥 ⟩) in isovolumes defined
+by the regions
+√
+𝐹 |𝐵′𝑥 | ⩾ 1.5 × 10−3.
+field and a short length-scale perpendicular to it. In this case, this
+effect is perhaps exaggerated by the presence of the imposed shear,
+which will tend to promote “near-interchange” modes over those
+with a more undular profile (Tobias & Hughes 2004). These non-
+rotating results are qualitatively consistent with those of Silvers et al.
+(2009b), albeit for a slightly different parameter regime.
+As the simulation evolves, the buoyancy-driven motions be-
+come more pronounced and the non-trivial field/flow structures
+migrate towards the upper boundary of the computational domain.
+This is clearly apparent at 𝑡 = 500 in Fig. 4. It is worth noting that
+there is no appreciable arching of the magnetic field structures as
+the instability evolves. Eventually, the upper boundary condition
+starts to play a significant role in the dynamics of the system. As
+we discuss in more detail below, the system also starts to exhibit
+MNRAS 000, 1–15 (2022)
+
+×10-4
+0.06
+2
+<d>
+0.04
+0.02
+-2 
+0
+0
+0.2
+0.4
+0.6
+0.8
+0.2
+1
+0
+0.4
+0.6
+0.8
+a
+2
+009～00～001～
+t = 100
+t = 200
+t = 500
+VFBa
+0.55
+O1X
+3
+b
+.5 VFB'
+-01 ×
+9
+y
+28
+C. D. Duguid et al.
+Figure 5. Magnetic buoyancy and mean magnetic field evolution for Ta = 0, Ta = 108 and Ta = 5 × 108 (top to bottom). Left: volume renderings consisting of
+a pseudocoloured isovolume of
+√
+𝐹 𝐵𝑥, defined by the region where |𝐵𝑥 | > 30, where the magnitude is denoted by the colour (see legend). Superimposed on
+this are isosurfaces of the vertical component of velocity 𝑢𝑧 defined at values ±2.5 × 10−4 (dark indicates positive and light indicates negative). Each snapshot
+is taken at 𝑡 ≈ 400 non-dimensional time units. Right: time snapshots of the horizontally averaged vertical profiles of the induced toroidal field
+√
+𝐹 𝐵𝑥. Each
+case contains a number of snapshots taken at times denoted by the legend.
+significant (magnetically-driven) deviations away from the initial
+imposed shear flow. Motivated by these considerations, we there-
+fore limit our integration times to 𝑡 ≈ 500 in all simulations. As
+shown in Table 1, the characteristic viscous and Ohmic timescales
+(based on the layer depth) are of the order of 105 time units. Given
+that these timescales are significantly longer than the evolution time
+for these simulations, it is clear that the observed migration of the
+field/flow structures is not a diffusively driven process.
+4.2
+The effects of rotation
+Having set out the key features of this model in the non-rotating case,
+we now turn our attention to the effects of rotation. Throughout this
+subsection, quantitative comparisons are made between the non-
+rotating case (Ta = 0) and two rotating cases with Ta = 108 and
+Ta = 5 × 108.
+Fig. 5 illustrates some of the effects of rotation upon this sys-
+tem, focusing upon the evolution of the vertical velocity, 𝑢𝑧, and the
+mean toroidal magnetic field, ⟨𝐵𝑥⟩. Regardless of the rotation rate
+of the domain, there are many qualitative similarities between the
+rotating cases and the non-rotating case that is illustrated in Fig. 4.
+In particular the shear is always able to produce a magnetic layer
+that is susceptible to magnetic buoyancy. However, there are some
+significant differences that are attributable to the effects of rotation.
+In particular, as the rotation rate increases, we see that the develop-
+ment of the magnetic buoyancy instability is delayed. The left-hand
+panel of Fig. 5 shows the vertical velocity distribution at 𝑡 ≈ 400
+for the three cases; whilst each exhibits clear indications of mag-
+netic buoyancy, the associated flow structures (at this fixed instant
+MNRAS 000, 1–15 (2022)
+
+0.08
+non-rotating
+0.053
+0.06
+0.052
+0.04
+B
+0.02
+0.050
+0
+0.049
+0
+0.2
+0.4
+0.6
+0.8
+1
+2
+0.047
+0.08
+0.056
+:01
+ 0.06
+B
+0.054
+0.04
+二
+B
+0.02
+0.052
+a
+0
+0.050
+0
+0.2
+0.4
+0.6
+0.8
+1
+2
+0.047
+0.08
+0.067
+ 0.06
+B
+0.062
+X
+0.04
+B
+5
+0.02
+0.057
+11
+0
+0.052
+0
+0.2
+0.4
+0.6
+0.8
+1
+a
+0.047
+y
+t ~ 56
+t ～ 119 t ～ 183 t ～ 246
+t ～ 310 t ～ 373 -
+t ～ 437 
+-t ～ 500Shear-driven magnetic buoyancy
+9
+Figure 6. Vertical profiles of the correlation, Corr𝑥 (𝑧), between 𝑢𝑧 and
+𝐵𝑥, as defined by Eq. (11) for Ta = 0, Ta = 108 and Ta = 5 × 108 (top
+to bottom), where each plot contains a number of snapshots taken at times
+denoted by the legend.
+in time) become progressively less well-developed as the rotation
+rate increases. One of the other notable features illustrated by this
+plot is that stronger toroidal fields are generated at the mid-plane
+in the more rapidly-rotating cases. This is another indication of the
+delayed onset of the magnetic buoyancy instability, which would
+otherwise tend to disrupt this layer. In cases where the magnetic
+buoyancy instability is more vigorous, we see a significant redis-
+tribution of the mean toroidal field, with a flattening of the peak
+around the mid-plane.
+To investigate the rotational-dependence of the onset of the
+magnetic buoyancy instability in a more quantitative manner, we
+compute the Pearson correlation between the vertical flows and the
+induced horizontal magnetic field. Following Silvers et al. (2009b),
+we define
+Corr𝑥(𝑧, 𝑡) =
+⟨𝑢𝑧𝐵𝑥⟩ − ⟨𝑢𝑧⟩⟨𝐵𝑥⟩
+max𝑧
+�√︃
+⟨𝑢2𝑧⟩ − ⟨𝑢𝑧⟩2
+√︃
+⟨𝐵2𝑥⟩ − ⟨𝐵𝑥⟩2
+� ,
+(11)
+and then plot this quantity as a function of depth and time in Fig. 6
+(for the non-rotating and the two rotating cases). We emphasise that
+this quantity does not have a straightforward physical interpretation
+(although, as we shall describe later, it is related to one of the terms
+that generate the 𝑦-component of the mean EMF). Nevertheless,
+it can be regarded as an indicator of magnetic buoyancy, because
+Figure 7. Vertical profiles of the horizontally-averaged 𝑢𝑥 component of
+velocity for Ta = 0, Ta = 108 and Ta = 5 × 108 (top, middle, and bottom
+respectively), and all other parameters defined in Table 1. Each case has
+snapshots of the shear profile taken at various times denoted by the legend.
+The dashed black line highlights the target shear profile.
+regions of stronger than average horizontal magnetic field are likely
+to be associated with buoyant upflows. In the presence of magnetic
+buoyancy, we would therefore expect Corr𝑥 to be negative, and
+this is borne out by Fig. 6. In all cases shown, after an initial
+transient phase, we observe significant negative correlations in this
+quantity above the mid-plane of the domain. The depth at which this
+correlation is maximal moves towards the surface as 𝑡 increases.
+However, the rate at which this peak moves is strongly dependent
+upon the rotation rate. In the non-rotating case, this peak has reached
+𝑧 ≈ 0.2 by 𝑡 ≈ 500; at Ta = 5 × 108, this peak is closer to 𝑧 ≈ 0.4
+at the same time. This is generally consistent with the delayed
+development of magnetic buoyancy-induced motions in the rotating
+cases.
+The effects of rotation are also evident in the evolution of
+the shear flow. Fig. 7 shows the evolution of ⟨𝑢𝑥⟩ for the three
+different cases. In all cases, we see a gradual deviation away from
+the initial shear. This is due to the effects of the Lorentz force
+associated with the growing mean toroidal magnetic field, which
+will tend to resist the stretching due to the shear. Initially this Lorentz
+force feedback simply leads to a weakening and broadening of the
+shear profile, which is still largely concentrated about the mid-
+plane. Once magnetic buoyancy develops, the shear profile flattens
+MNRAS 000, 1–15 (2022)
+
+1
+0
+-1
+0
+0.2
+0.4
+0.6
+0.8
+1
+2
+1
+8
+0
+0
+11
+a
+-1 
+0
+0.2
+0.4
+0.6
+0.8
+1
+2
+1
+X
+0
+5
+-1
+0
+0.2
+0.4
+0.6
+0.8
+1
+a
+2
+L
+t ～ 56
+t ～ 119 t ～ 183 t ～ 246
+t. ～ 310
++？.373
+？.437
+t ～ 5000.02
+αm
+0
+-0.02
+0
+0.2
+0.4
+0.6
+0.8
+1
+0.02
+8
+0
+2
+0
+a
+-0.02
+0
+0.2
+0.4
+0.6
+0.8
+1
+801
+0.02
+X
+0
+5
+-0.02
+0
+0.2
+0.4
+0.6
+0.8
+1
+a
+2
+t ～ 56
+t ～ 119
+ t ～ 183
+-t ～ 246
+t ～ 437
+-t ～ 50010
+C. D. Duguid et al.
+about the mid-plane and the resultant momentum redistribution
+leads to a significant perturbation to the flow structure, particularly
+at the upper boundary (where the density is relatively low). Longer
+integrations lead to further departures from the initial shear profile.
+As the rotation rate increases, the deviation of the flow profile away
+from the initial state becomes more gradual. Indeed, in the most
+rapidly rotating case the shear flow still remains localised around
+the mid-plane even at 𝑡 ≈ 500.
+Another important feature of the rotating cases is the emer-
+gence of mean flows in the 𝑦-direction (which are not a feature of
+the non-rotating case). This is illustrated in the left-hand panels of
+Fig. 8, which plot the time and depth dependence of ⟨𝑢𝑦⟩ for the
+three cases under consideration. In the rotating cases, such flows
+arise because the Lorentz force from the growing magnetic field
+disturbs the balance between the imposed forcing, F𝑠, and the Cori-
+olis and viscous forces. Because the density is lower in the upper
+part of the domain, the strongest mean flows in the 𝑦-direction tend
+to occur in this region, where (towards the end of the calculation)
+they reach values that are of a similar order of magnitude to the
+peak mean flow in the 𝑥-direction. We recall that our boundary
+conditions impose zero tangential stresses on the upper and lower
+boundaries, and so there is no flux of horizontal momentum through
+these boundaries. Therefore, the mean flows arise from an internal
+redistribution of momentum within the domain, and the total mo-
+mentum is conserved.
+The right-hand panels of Fig. 8 show the time and depth depen-
+dence of ⟨𝐵𝑦⟩. As expected, this quantity is practically zero in the
+non-rotating case. However, both rotating cases show the appear-
+ance of a systematic ⟨𝐵𝑦⟩, which is an inevitable consequence of
+the behaviour of ⟨𝑢𝑦⟩ that was discussed in the previous paragraph:
+the strong gradient in this quantity at the mid-plane stretches the
+vertical magnetic field into the 𝑦-direction. The peak value of ⟨𝐵𝑦⟩
+remains less than half of the peak value of ⟨𝐵𝑥⟩, so the mean mag-
+netic field is still predominantly in the 𝑥-direction. Nonetheless, the
+rotation of the mean magnetic field away from the 𝑥-axis affects the
+structure of the magnetic buoyancy instability. This is illustrated by
+Fig. 9, which shows snapshots of the vertical flow, at the mid-plane
+of the domain; the tilting of these structures due to the effects of
+rotation is clearly apparent.
+5
+THE MEAN EMF
+We now turn our attention to the most important result of this
+work, which is the analysis of the mean electromotive force (mean
+EMF) that is produced by the magnetic buoyancy instability in the
+presence of rotation. According to the ideas originally put forward
+by Parker (1955b), if the instability can produce a mean EMF with
+a significant component that is parallel to the mean magnetic field,
+then this system might be able to drive an 𝛼𝜔-type dynamo. Before
+discussing the results from these simulations, we first review some
+of the key theoretical ideas.
+5.1
+Theoretical background
+Following the standard procedures of mean-field theory, we can de-
+compose the magnetic field and the velocity field into their mean and
+fluctuating parts. To be specific, we can write B = ⟨B⟩ + B′, where
+(as before) ⟨B⟩ corresponds to the horizontally-averaged magnetic
+field, which is then a function of 𝑧 and 𝑡 only, whilst B′ represents
+the fluctuating component. By construction, ⟨B′⟩ = 0. Similarly, we
+can express the velocity field in the form, u = ⟨u⟩ + u′. Applying
+the averaging operator to the induction equation (Eq. 2c), we see
+that
+𝜕⟨B⟩
+𝜕𝑡
+= ∇ ×
+�
+⟨u⟩ × ⟨B⟩ + E
+�
++ 𝜁0𝜅 𝜕2
+𝜕𝑧2 ⟨B⟩ ,
+(12)
+where we have defined the mean EMF
+E = ⟨u′ × B′⟩ .
+(13)
+The ∇ × E term in Eq. (12) is the key ingredient of any mean-field
+dynamo model; under certain conditions it acts as a source term
+for the mean magnetic field. Due to the fact that E is a function of
+𝑧 and 𝑡 only, it is only the 𝑥 and 𝑦 components of the mean EMF
+that contribute to the evolution of the mean magnetic field. In what
+follows, we therefore restrict our analysis to these two components.
+In order to understand the potential influence of this mean
+EMF upon the dynamo, it is useful to review some of the key
+ideas from mean-field dynamo theory (see, e.g., Moffatt 1978, for
+more details). For the moment, we make the simplifying assumption
+that there is an imposed (but potentially time-dependent) flow that
+evolves independently of the magnetic field, in order to consider
+the induction equation in isolation. It should be stressed that this
+assumption is not applicable for magnetic buoyancy, with the subse-
+quent flow inextricably linked to the magnetic field; nevertheless, it
+is an assumption that allows insights to be gained into the properties
+of the mean EMF that might be conducive to dynamo action. We
+shall also assume that the mean quantities vary on a much longer
+length-scale than the fluctuations.
+Unless the imposed flow has the ability to generate a small
+scale dynamo (allowing the fluctuating magnetic field to grow in
+the absence of a mean field), there should be a linear relationship
+between the mean EMF and the mean magnetic field (and its deriva-
+tives). If the underlying flow were to be homogeneous and isotropic,
+then we could posit an expansion of the form
+E = 𝛼⟨B⟩ − 𝛽∇ × ⟨B⟩ + . . . ,
+(14)
+where 𝛼 and 𝛽 are scalars (under more general conditions these
+would be tensorial quantities). Under these simplifying assump-
+tions, the 𝛼⟨B⟩ term corresponds to the 𝛼-effect and is non-zero
+only if the flow lacks reflectional symmetry (which is typically
+the case in the presence of rotation). In fact, under the first-order
+smoothing approximation, it is possible to derive an expression for
+𝛼 that is directly proportional to the kinetic helicity of the flow1.
+We stress again that some of the simplifications made here are not
+really applicable in the context of magnetic buoyancy, and any in-
+terpretation of the mean EMF in terms of 𝛼 and 𝛽 should be treated
+with a degree of caution (Hughes 2018; Davies & Hughes 2011).
+In the following discussion, we shall therefore focus upon the prop-
+erties of the mean EMF itself. Nevertheless, these insights from
+mean-field theory suggest that the component of the mean EMF in
+the direction of the mean magnetic field has a significant bearing
+upon the potential for the system to act as a dynamo. Based upon
+these insights, we would also expect to see a strong dependence of
+this part of the mean EMF upon the rotation rate.
+Before moving on to analyse the mean EMF in these simu-
+lations, it is useful to relate this to previous work. Two studies of
+particular relevance are those of Davies & Hughes (2011) and Chat-
+terjee et al. (2011). In their linear analysis, Davies & Hughes (2011)
+1 We have not found an obvious relationship between the kinetic helicity
+of the turbulence and the mean EMF in our simulations and hence have not
+presented these results for brevity.
+MNRAS 000, 1–15 (2022)
+
+Shear-driven magnetic buoyancy
+11
+Figure 8. Time snapshots of the horizontally-averaged vertical profiles of the 𝑢𝑦 component of velocity (left) and the 𝐵𝑦 component of the magnetic field
+(right) for Ta = 0, Ta = 108 and Ta = 5 × 108 (top to bottom). Each case contains a number of snapshots taken at times denoted by the legend.
+considered the mean EMF produced by magnetic buoyancy in an
+imposed, rotating, horizontal magnetic layer. Most of their analysis
+focused upon the case of an inclined rotation vector, but there are
+some general conclusions that are applicable to the vertical rotation
+case that is considered in the present paper. Chatterjee et al. (2011)
+carried out numerical simulations that are somewhat related to those
+of the present paper, although they considered the evolution of an
+imposed magnetic layer rather than a shear-generated one, carrying
+out an extended analysis of the latitudinal dependence of the system
+(largely at a fixed rotation rate).
+Considering a unidirectional horizontal field, aligned with the
+𝑥 direction, Davies & Hughes (2011) found that the magnitude of
+the component of the mean EMF in the direction of the mean field
+(E𝑥) does indeed increase with increasing rotation rate. Even in the
+absence of rotation, they also found a substantial component of the
+EMF in the horizontal direction that is perpendicular to the imposed
+magnetic field (i.e. E𝑦), the amplitude of which decreased slightly
+as the rotation rate increased. For the cases considered by Davies
+& Hughes (2011), they typically found that |E𝑦| ≫ |E𝑥|. Allowing
+for differences in the coordinate geometry of the two systems, the
+simulations of Chatterjee et al. (2011) exhibited similar behaviour,
+with the magnitude of the horizontal component of the mean EMF
+that is perpendicular to the imposed magnetic field exceeding that of
+the field-parallel component. In both cases, this result can be traced
+back to the fact that the instability tends to be characterised by
+magnetohydrodynamical perturbations with small 𝑦-components.
+These imposed magnetic field studies provide us with some
+clear points of comparison, and we might expect to see some similar
+behaviour (in terms of the properties of the mean EMF) in the shear-
+generated case. However, given the greater complexity of our model,
+we should not be surprised if there are also some notable differences.
+5.2
+Numerical results
+Fig. 10 shows the depth- and time-dependence of the mean EMFs for
+the three cases considered. We focus initially upon the E𝑥 compo-
+nent. In the cases with Ta = 0 and Ta = 108, E𝑥 takes both positive
+and negative values of similar magnitude, with rapid variations in 𝑧
+and 𝑡, and displays no systematic trend. In the most rapidly-rotating
+case (Ta = 5 × 108), by contrast, E𝑥 is positive for almost all val-
+ues of 𝑧 and 𝑡, with an amplitude that grows superlinearly in time.
+Moreover, the location of the peak value of E𝑥 migrates upward in
+time, roughly mirroring the migration of the buoyancy instability,
+as illustrated in Fig. 6. By 𝑡 ≈ 500, the peak value of E𝑥 is an order
+of magnitude larger than the the peak value of |E𝑥| in either of the
+other two cases. Whilst this is clearly a more complicated model,
+this is qualitatively consistent with the results of Davies & Hughes
+(2011), who (as noted above) also found that this component of the
+mean EMF tended to increase with increasing rotation rate.
+MNRAS 000, 1–15 (2022)
+
+0.01
+0.04
+X10-6
+X10-6
+non-rotatin
+10
+505
+B
+0
+0.02
+0
+-10
+-20
+0
+0
+0.5
+0.5
+0
+-0.01
+-0.02
+0
+0.2
+0.4
+0.6
+0.8
+1
+0.2
+0
+0.4
+0.6
+0.8
+1
+2
+0.01
+0.04
+B
+0.02
+0
+0
+-0.01
+a
+-0.02
+0
+0.2
+0.6
+0.4
+0.8
+0.2
+0.4
+0.6
+0.8
+0
+2
+2
+801
+0.01
+0.04
+9
+B
+0.02
+0
+X
+5
+0
+-0.01
+二
+-0.02
+0
+0.2
+0.4
+0.6
+0.8
+1
+0.2
+0
+0.4
+0.6
+0.8
+1
+a
+2
+7
+t ～ 56
+t ～ 119
+t ～ 183
+t ～ 246
+t ～ 310 
+t ～ 373
+t ～ 437
+t ～ 50012
+C. D. Duguid et al.
+Figure 9. Plots of the 𝑢𝑧 component of velocity, with magnitude denoted by
+the colour bar, for Ta = 0, Ta = 108 and Ta = 5 × 108 (top to bottom). Each
+plot is taken at the mid-plane, 𝑧 = 0.5, from a snapshot at 𝑡 ≈ 400 where
+the tilting effects, or lack thereof, of rotation can be most clearly seen.
+Turning our attention to E𝑦, we find (after an initial transient
+phase) that in each case there is a significant component of the
+mean EMF in the negative 𝑦 direction, whose magnitude is roughly
+an order of magnitude larger than that of E𝑥 (this is qualitatively
+consistent with the results of Davies & Hughes 2011 and Chatterjee
+et al. 2011). Interestingly, the Ta = 108 case has slightly smaller
+values of E𝑦 than the non-rotating case. However, in the more
+rapidly-rotating case (with Ta = 5×108), E𝑦 is initially smaller, but
+eventually grows to be significantly larger, roughly in proportion
+with E𝑥. This non-monotonic dependence of the E𝑦 on Ta probably
+reflects the fact that the geometry of the mean magnetic field changes
+with rotation rate. In the rotating cases, the mean magnetic field has
+significant components in both the 𝑥 and 𝑦 directions, and so the
+distinction between E𝑥 and E𝑦 (in terms of their orientation with
+respect to the mean field) also becomes less meaningful.
+In Fig. 11 we carry out a more detailed analysis of the com-
+ponents of the mean EMF at 𝑡 ≈ 500, for each of the three cases.
+To be specific, we have considered separately the contributions of
+⟨𝑢′𝑦𝐵′𝑧⟩ and −⟨𝑢′𝑧𝐵′𝑦⟩ to E𝑥 (and have carried out a similar de-
+composition for E𝑦). In the Ta = 0 case, there is little systematic
+behaviour evident in these plots; all quantities are of low ampli-
+tude, leading to a negligible overall E𝑥. For Ta = 108, both ⟨𝑢′𝑦𝐵′𝑧⟩
+and −⟨𝑢′𝑧𝐵′𝑦⟩ seem to have a well-defined depth dependence, with
+(approximately) equal and opposite magnitudes, leading to signifi-
+cant levels of cancellation when these quantities are added together.
+Whilst there is no significant net E𝑥 in this case, the coherent depth
+dependence of these quantities does indicate that rotation is already
+playing an important dynamical role. In the most rapidly rotating
+case, although the two contributing quantities still have opposite
+signs, they have very different magnitudes; a stronger contribution
+from −⟨𝑢′𝑧𝐵′𝑦⟩ leads to a net E𝑥 in this case. Given the nature of the
+magnetic buoyancy instability, it is perhaps unsurprising that the
+contribution from 𝑢′𝑧 should be playing a dominant role. Both the
+Ta = 0 and Ta = 108 cases have a well-defined E𝑦 in which ⟨𝑢′𝑧𝐵′𝑥⟩
+and −⟨𝑢′𝑥𝐵′𝑧⟩ both play an important role. (This means that the
+correlation defined in Eq. (11) is generally not a reliable proxy for
+E𝑦.) In the most rapidly rotating case, the ⟨𝑢′𝑧𝐵′𝑥⟩ term is dominant,
+leading to an increase in the magnitude of E𝑦.
+An obvious question to ask at this stage is whether or not
+these mean EMFs are large enough to have a significant regener-
+ative effect on the magnetic field. Some insight can be gained by
+considering the relative magnitude of the mean EMF relative to the
+mean magnetic field, focusing particularly upon the 𝑥-component
+of this quantity (for the reasons presented above). However, this
+process is complicated by the fact that the relevant quantities are
+all depth-dependent. To produce an indicative value of this nor-
+malised magnitude, we therefore define 𝑧𝑚(𝑡) to be the value of
+𝑧 at which |E𝑥| takes its maximum value at that particular time.
+We then consider the quantity E𝑥(𝑧𝑚(𝑡), 𝑡)/⟨𝐵𝑥⟩(𝑧𝑚(𝑡), 𝑡). Whilst
+the definition of this normalised quantity is certainly motivated by
+the definition of 𝛼 in Eq. (14), we stress again that the validity of
+this expansion is questionable in this context, which explains why
+we choose to work with the mean EMF rather than describing this
+system in terms of 𝛼.
+In Fig. 12, we plot |E𝑥(𝑧𝑚(𝑡), 𝑡)|/⟨𝐵𝑥⟩(𝑧𝑚(𝑡), 𝑡), as a function
+of time, for the three cases (Ta = 0, Ta = 108 and Ta = 5 × 108).
+After initial transients have subsided, this normalised magnitude
+is comparable for each of the three cases, until the curves start to
+diverge at 𝑡 ≈ 200. For both Ta = 0 and Ta = 108, there is no
+significant increase in this quantity after this point. In the more
+rapidly-rotating case, the normalised magnitude of the mean EMF
+increases rapidly, reaching a value of approximately 2 × 10−6 at the
+end of the integration time. In dimensional terms, this normalised
+magnitude has the units of velocity, so it is natural to compare this
+quantity with the fluctuating component of the velocity field. As is
+evident from Fig. 9, this peak value for the normalised mean EMF
+is approximately one order of magnitude smaller than the typical
+vertical velocity at the mid-plane of the domain. This is, therefore,
+still relatively small, but it should be noted that this quantity is still
+growing at the end of the simulation.
+In a classical 𝛼𝜔-dynamo model, it is perfectly possible for
+a dynamo to operate with a weak 𝛼-effect provided that the shear
+is sufficiently strong. A full dynamo calculation would be needed
+in order to assess whether or not this EMF (in tandem with the
+shear) is capable of sustaining a Parker-like dynamo. Moreover,
+such a calculation would require a much larger domain in the lat-
+itudinal direction than the model presented here, in order to allow
+for sufficient separation between the small-scale instability and the
+large-scale magnetic field.
+6
+DISCUSSION AND CONCLUSIONS
+Motivated by the original 𝛼𝜔-dynamo model that was first proposed
+by Parker (1955b), the aim of this paper was to assess whether the
+interaction between magnetic buoyancy and rotation, in a shear-
+generated magnetic layer, can produce a regenerative effect for the
+mean magnetic field analogous to that of a convectively-driven 𝛼-
+effect. Our numerical model builds on that of Vasil & Brummell
+MNRAS 000, 1–15 (2022)
+
+8.8
+- 4.4
+0
+4.4
+8.8
+Wz
+X 10-5
+-1.3
+- 0.6
+0.6
+1.3
+0
+8
+X 10-5
+二
+a
+-2.2
+0
+1.1
+2.2
+1.1
+X 10-5
+X
+5
+9
+aShear-driven magnetic buoyancy
+13
+Figure 10. Plots of the mean EMF components E𝑥 (left) and E𝑦 (right), as a function of 𝑧, for Ta = 0 (top row), Ta = 108 (middle row) and Ta = 5 × 108
+(bottom row). The curves shown correspond to the times denoted by the legend. Note that the axis ranges are held fixed as Ta is varied in order to facilitate
+quantitative comparisons between the three cases. (Insets are provided in two panels to illustrate the details of the low-amplitude variations.)
+(2008) and Silvers et al. (2009b), who conducted simulations of
+a fully compressible magnetohydrodynamic fluid under the influ-
+ence of a tachocline-like vertical shear flow. An imposed vertical
+magnetic field is stretched by the shear to produce a predominantly
+horizontal field, which then becomes unstable to magnetic buoy-
+ancy. The novel aspect of the work presented here is the inclusion
+of rotation in this system, which breaks the reflection symmetry of
+the buoyancy-driven flows, potentially enabling this instability to
+supply the necessary mean electromotive force (EMF) required for
+a Parker-like dynamo to operate.
+To assess the influence of rotation, we carried out a detailed
+quantitative comparison of simulations with different Taylor num-
+bers. Whilst rapid rotation does delay the development of the mag-
+netic buoyancy instability, the imposed shear still leads to the forma-
+tion of a magnetic layer that eventually becomes buoyantly unstable
+as the simulation progresses. If the rotation rate is high enough, the
+instability-induced perturbations (to the flow and magnetic fields)
+give rise to a systematic mean EMF, with a significant component,
+E𝑥, in the direction of the mean magnetic field (see Fig. 10). In
+agreement with previous analytical work (e.g. Davies & Hughes
+2011) we find that the magnitude of E𝑥 increases with increasing
+rotation rate. Like previous studies, we also find a significant E𝑦
+component of the mean EMF in all cases (regardless of the extent
+to which the layer is rotating).
+Following the ideas originally set out by Parker (1955b), a
+systematic E𝑥 (in combination with the imposed velocity shear)
+is likely to be conducive to dynamo action of 𝛼𝜔-type. However,
+the efficacy of any possible dynamo process will certainly depend
+on the magnitude of E𝑥, which would typically be expressed in
+terms of an 𝛼 coefficient in a standard mean-field dynamo model.
+As noted above, E𝑥 is certainly small (in a normalised sense) in
+all of these simulations, although it should be emphasised that it is
+growing throughout the final stages of the simulation in the most
+rapidly-rotating case (see Fig. 12). It is therefore plausible that this
+regenerative term will eventually become large enough to influence
+significantly the evolution of the mean magnetic field (see Eq. 12).
+Of course, some caution is needed when extrapolating results from
+any numerical simulations to conditions in the solar interior. Fur-
+thermore, full dynamo calculations (without a uniform imposed
+magnetic field) are needed in order to confirm the viability of this
+dynamo mechanism. Nevertheless, our initial results support the
+hypothesis that the magnetic buoyancy in the solar tachocline could
+be playing a crucial regenerative role in the solar dynamo.
+An obvious limitation of our current simulations is that the
+mean flow eventually diverges from the tachocline-like shear pro-
+MNRAS 000, 1–15 (2022)
+
+non-rotating
+ ×10-5
+×10-5
+6
+X10-6
+0
+4
+2
+10
+2 
+0
+0.5
+-20
+0
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+2
+2
+×10-5
+×10-6
+8
+2
+0
+0
+4
+0
+3
+10
+二
+2
+2
+0
+0.5
+-20
+a
+0
+0
+0.2
+0.4
+0.6
+0.8
+0
+0.2
+0.4
+0.6
+0.8
+1
+1
+2
+×10-5
+6×10-5
+0
+4
+X
+2
+3
+5
+-20
+0
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+a
+2
+～~.56
+110
+246
+310
+～ 50014
+C. D. Duguid et al.
+Figure 11. Comparisons between the individual terms, denoted by colour
+(see legend), that make up the EMF components E𝑥 (left) and E𝑦 (right).
+The comparisons are made for an individual time snapshot taken at 𝑡 ≈ 500.
+We make these comparisons for Ta = 0 (top row), Ta = 108 (middle row)
+and Ta = 5 × 108 (bottom row). For reference we show the total E𝑥 and E𝑦
+in each case as a dotted line.
+file that we seek to impose; this is the key determining factor in
+limiting the run time of these simulations to 𝑡 ≈ 500. While this
+limited run-time does not detract from the results we have presented,
+it is worth noting that for a dynamo calculation it would be desirable
+to run these simulations for a much longer time, ideally a significant
+fraction of the Ohmic timescale (which for our chosen parameters
+is 𝜏ohmic ≈ 2 × 105; see Table 1). The deviation from the imposed
+mean flow is an inevitable consequence of the Lorentz forces that
+result from the continual stretching of the imposed vertical field.
+This magnetic feedback perturbs the initial force balance, which
+smooths and then flattens the initial shear profile. In the rotating
+cases this also drives mean flows in the 𝑦-direction. Both of these
+changes to the mean flow are undesirable from the point of view
+of any dynamo calculation that seeks to mimic conditions in the
+solar tachocline. One way to resolve this issue would be to employ
+a shearing box model, in order to remove the back-reaction of the
+Lorentz force onto the mean flow (e.g. Barker et al. 2012). Alter-
+natively an adaptive forcing could be incorporated, rather than the
+fixed forcing F𝑠 used here, although this would inevitably introduce
+new dynamical effects into an already complex system. In the ab-
+sence of a complete understanding of the solar differential rotation,
+Figure 12. Time series of |E𝑥 (𝑧𝑚(𝑡), 𝑡) |/⟨𝐵𝑥 ⟩(𝑧𝑚(𝑡), 𝑡) for the three
+rotation rates shown in the legend.
+any model of the tachocline’s shear will necessarily be somewhat
+artificial. Fortunately, in a full dynamo simulation, without an im-
+posed vertical field, the winding up of the horizontal field is likely
+to become a self-limiting process. In that case, it may be possible
+to maintain a tachocline-like shear flow with the same kind of fixed
+forcing this is employed here.
+Another potential issue with our existing model is that the
+buoyantly rising magnetic field eventually reaches the top of the
+domain. At this point, the dynamics of the system start to become
+strongly dependent upon the choice of boundary conditions, which
+will never accurately replicate the conditions around the base of the
+solar convection zone (a fact that is true for any local model of the
+tachocline). A related point to note is that our magnetic boundary
+conditions allow magnetic fields to escape at the upper boundary,
+which might limit the efficiency of the dynamo if the field is able to
+escape faster than it can be regenerated in the tachocline. Whilst we
+know that magnetic flux can escape from the tachocline, the rate at
+which this happens in any local model will certainly depend upon
+the precise boundary conditions adopted. The rise of the horizontal
+field towards the surface could be countered by magnetic pumping
+(Moffatt 1978; Moffatt & Dormy 2019), either via a parameterised
+pumping term in the induction equation (e.g. Barker et al. 2012),
+or by including a convective layer at the top of the domain (e.g.
+Tobias et al. 1998, 2001; Brummell et al. 2002b; Silvers et al. 2009a;
+Weston 2020). We have started to carry out some preliminary studies
+on the viability of the convective analogue of magnetic pumping on
+the mean field, following a similar (parameterised) approach to that
+described by Barker et al. (2012), and plan to report on the results
+from these simulations in a future paper. If magnetic pumping, or
+the more technically challenging convective layer, can successfully
+pin down the toroidal field then it is possible this could increase the
+efficiency of any dynamo that this system might be able to excite.
+There are many other possible avenues for future work. All
+of the discussion in this paper has focused upon the case in which
+the rotation vector is aligned with gravity, which corresponds to the
+polar regions of the tachocline. By considering an inclined rotation
+vector, it is possible to consider the latitudinal dependence of this
+system. Given that active regions at the solar surface are confined
+to mid- to low-latitudes, this is a natural next step, not only for the
+imposed vertical magnetic field calculations that were considered in
+this paper, but also for the subsequent dynamo calculations. Another
+possible avenue of research is to explore possible sound-proof ap-
+proximations to this system. In a parallel study, we are investigating
+the extent to which various sound-proof approximations adequately
+describe the linear onset of magnetic buoyancy in an imposed mag-
+netic layer (Moss et al. 2022). Sound waves do not play a significant
+MNRAS 000, 1–15 (2022)
+
+y
+×10-6
+×10-5
+2
+non-rotatin
+5
+1
+-5
+-2
+0
+0.5
+1
+0
+0.5
+1
+×10-6
+×10-5
+C
+8
+5
+0
+11
+a
+-5
+T
+-5
+0
+0.5
+1
+0
+0.5
+1
+2
+8
+X10-5
+×10-4
+0
+2
+1
+5
+1
+X
+2
+0
+5
+-5
+-1
+11
+-2 
+a
+0
+0.5
+1
+0
+0.5
+1
+T
+2
+2
+(u,Br)
+<u,B")
+(u, B")
+<uB")X10-6
+magnitude
+1.5
+normalised
+1
+0.5
+0
+0
+100
+200
+300
+400
+500
+t
+Ta = 0
+Ta = 108
+Ta = 5 × 108Shear-driven magnetic buoyancy
+15
+role in the dynamics of this system, so sound-proof approximations
+(which the relax the requirement of having to resolve the acoustic
+timescale) could allow significantly larger time-steps to be taken,
+thus increasing the possible simulation time. This could be highly
+beneficial for any future dynamo simulations.
+ACKNOWLEDGEMENTS
+This work was supported by a Research Project Grant from the
+Leverhulme Trust (RPG-2020-109). This research made use of the
+Rocket High Performance Computing service at Newcastle Univer-
+sity.
+DATA AVAILABILITY
+The data underlying this article will be shared on reasonable request
+to the corresponding author.
+REFERENCES
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+Notices of the Royal Astronomical Society, 424, 115
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+Brummell N., Cline K., Cattaneo F., 2002a, Monthly Notices of the Royal
+Astronomical Society, 329, L73
+Brummell N. H., Clune T. L., Toomre J., 2002b, The Astrophysical Journal,
+570, 825
+Bushby P. J., Käpylä P. J., Masada Y., Brandenburg A., Favier B., Guervilly
+C., Käpylä M. J., 2018, Astronomy & Astrophysics, 612, A97
+Cattaneo F., Hughes D. W., 1988, Journal of Fluid Mechanics, 196, 323
+Cattaneo F., Hughes D. W., 2006, Journal of Fluid Mechanics, 553, 401
+Charbonneau P., 2020, Living Reviews in Solar Physics, 17, 4
+Chatterjee P., Mitra D., Rheinhardt M., Brandenburg A., 2011, Astronomy
+& Astrophysics, 534, A46
+Childress S., Soward A. M., 1972, Physical Review Letters, 29, 837
+Cline K. S., Brummell N. H., Cattaneo F., 2003a, The Astrophysical Journal,
+588, 630
+Cline K. S., Brummell N. H., Cattaneo F., 2003b, The Astrophysical Journal,
+599, 1449
+Cope L., 2021, PhD thesis, University of Cambridge
+Davies C. R., Hughes D. W., 2011, The Astrophysical Journal, 727, 112
+Favier B., Bushby P. J., 2013, Journal of Fluid Mechanics, 723, 529
+Ferriz-Mas A., Schmitt D., Schuessler M., 1994, Astronomy and Astro-
+physics, 289, 949
+Garaud P., Gagnier D., Verhoeven J., 2017, The Astrophysical Journal, 837,
+133
+Gilman P. A., 1970, The Astrophysical Journal, 162, 1019
+Gilman P. A., 2018, The Astrophysical Journal, 853, 65
+Howard L. N., 1961, Journal of Fluid Mechanics, 10, 509
+Hughes D. W., 2007, The solar tachocline. Cambridge University Press
+Hughes D. W., 2018, Journal of Plasma Physics, 84, 735840407
+Jensen E., 1955, Annales d’Astrophysique, 18, 127
+Kersalé E., Hughes D. W., Tobias S. M., 2007, The Astrophysical Journal,
+663, L113
+Käpylä P. J., Korpi M. J., Brandenburg A., 2009, The Astrophysical Journal,
+697, 1153
+Masada Y., Sano T., 2016, The Astrophysical Journal, 822, L22
+Matthews P. C., Proctor M. R. E., Weiss N. O., 1995a, Journal of Fluid
+Mechanics, 305, 281
+Matthews P. C., Hughes D. W., Proctor M. R. E., 1995b, The Astrophysical
+Journal, 448, 938
+Miles J. W., 1961, Journal of Fluid Mechanics, 10, 496
+Mizerski K. A., Davies C. R., Hughes D. W., 2013, The Astrophysical
+Journal Supplement Series, 205, 16
+Moffatt H. K., 1978, Magnetic field generation in electrically conducting
+fluids. Cambridge monographs on mechanics and applied mathematics,
+Cambridge University Press, Cambridge [Eng.] ; New York
+Moffatt
+K.,
+Dormy
+E.,
+2019,
+Self-Exciting
+Fluid
+Dynamos,
+1
+edn.
+Cambridge
+University
+Press,
+doi:10.1017/9781107588691,
+https://www.cambridge.org/core/product/identifier/
+9781107588691/type/book
+Moss J. B., Wood T. S., Bushby P. J., 2022, Physical Review Fluids, 7,
+103701
+Parker E. N., 1955a, The Astrophysical Journal, 121, 491
+Parker E. N., 1955b, The Astrophysical Journal, 122, 293
+Parker E. N., 1993, The Astrophysical Journal, 408, 707
+Schmitt D., 1984, in Guyenne T. D., Hunt J. J., eds, ESA Special Publication
+Vol. 220, ESA Special Publication. pp 223–224
+Silvers L. J., Bushby P. J., Proctor M. R. E., 2009a, Monthly Notices of the
+Royal Astronomical Society, 400, 337
+Silvers L. J., Vasil G. M., Brummell N. H., Proctor M. R. E., 2009b, The
+Astrophysical Journal, 702, L14
+Steenbeck M., Krause F., Rädler K.-H., 1966, Zeitschrift für Naturforschung
+A, 21, 369
+Thelen J. C., 2000a, Monthly Notices of the Royal Astronomical Society,
+315, 165
+Thelen J.-C., 2000b, Monthly Notices of the Royal Astronomical Society,
+315, 155
+Thompson M. J., Christensen-Dalsgaard J., Miesch M. S., Toomre J., 2003,
+Annual Review of Astronomy and Astrophysics, 41, 599
+Tobias S. M., Hughes D. W., 2004, The Astrophysical Journal, 603, 785
+Tobias S. M., Brummell N. H., Clune T. L., Toomre J., 1998, The Astro-
+physical Journal, 502, L177
+Tobias S. M., Brummell N. H., Clune T. L., Toomre J., 2001, The Astro-
+physical Journal, 549, 1183
+Vasil G. M., Brummell N. H., 2008, The Astrophysical Journal, 686, 709
+Vasil G. M., Brummell N. H., 2009, The Astrophysical Journal, 690, 783
+Weston D., 2020, PhD thesis, University of Leeds
+Wissink J. G., Hughes D. W., Matthews P. C., Proctor M. R. E., 2000,
+Monthly Notices of the Royal Astronomical Society, 318, 501
+Zahn J.-P., 1974, in Ledoux P., Noels A., Rodgers A. W., eds, , Stellar
+Instability and Evolution. Springer Netherlands, Dordrecht, pp 185–
+195, doi:10.1007/978-94-010-9794-9_34, http://link.springer.
+com/10.1007/978-94-010-9794-9_34
+APPENDIX A: FULL LIST OF SIMULATIONS
+For completeness we show the full list of simulations used in the
+analysis of this work. Table A1 lists the simulation which were used
+for the results presented in the main body of the paper, which had
+resolutions of (𝑁𝑥, 𝑁𝑦, 𝑁𝑧) = (192, 96, 192). These more compu-
+tationally expensive production runs were guided by an extensive
+low resolution parameter sweep. In particular, a number of processes
+change the timescales of the relevant dynamics of the system, such
+as the reduction of shear amplitude and/or amplitude of 𝐹 acting
+to slow the rate of toroidal field production and hence increase
+computational cost. Similarly, it was unclear a priori how rapid the
+rotation needed to be in order to produce a significant mean EMF in
+the mean field direction. The full list of low resolution simulations
+with (𝑁𝑥, 𝑁𝑦, 𝑁𝑧) = (128, 64, 128) can be found in Table A2.
+This paper has been typeset from a TEX/LATEX file prepared by the author.
+MNRAS 000, 1–15 (2022)
+
+16
+C. D. Duguid et al.
+F
+(×10−6)
+Taylor
+(×108)
+A
+F
+(×10−6)
+Taylor
+(×108)
+A
+0
+1
+0.02
+2.5
+0
+0.02
+0
+5
+0.02
+2.5
+1
+0.02
+0
+0
+0.02
+2.5
+5
+0.02
+0
+0
+0.05
+Table A1. A list of the field strength 𝐹, Taylor number, and shear amplitude
+𝐴 for the set of simulations explored in this work. These cases each have
+numerical resolutions of (𝑁𝑥, 𝑁𝑦, 𝑁𝑧) = (192, 96, 192) and all other
+parameters are fixed by the values defined in Table 1.
+F
+(×10−6)
+Taylor
+(×108)
+A
+F
+(×10−6)
+Taylor
+(×108)
+A
+0
+0
+0.02
+3.75
+0
+0.015
+0
+0
+0.015
+0.75
+0
+0.015
+0
+0
+0.05
+1.875
+0
+0.015
+0
+0
+0.01
+2.5
+0
+0.01
+0
+0.1
+0.02
+2.5
+0.1
+0.02
+0
+0.5
+0.02
+2.5
+0.5
+0.02
+0
+1
+0.02
+5
+1
+0.02
+0
+2
+0.02
+2.5
+1
+0.02
+0
+3
+0.02
+1
+1
+0.02
+0
+4
+0.02
+2.5
+2
+0.02
+0
+5
+0.02
+2.5
+3
+0.02
+0
+10
+0.02
+2.5
+4
+0.02
+12.5
+0
+0.05
+5
+5
+0.02
+5
+0
+0.02
+2.5
+5
+0.02
+1
+0
+0.02
+1
+5
+0.02
+2.5
+0
+0.02
+2.5
+10
+0.02
+Table A2. A list of the field strength 𝐹, Taylor number, and shear am-
+plitude 𝐴 for the set of low resolution simulations with (𝑁𝑥, 𝑁𝑦, 𝑁𝑧) =
+(128, 64, 128) that were used as an initial parameter sweep to guide this
+work. All other parameters are fixed by the values defined in Table 1.
+MNRAS 000, 1–15 (2022)
+

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@@ -0,0 +1,2253 @@
+arXiv:2301.11388v1  [math-ph]  26 Jan 2023
+PERTURBATION DETERMINANT AND LEVINSON’S FORMULA FOR
+SCHR¨ODINGER OPERATORS WITH GENERALIZED POINT INTERACTION
+MUHAMMAD USMAN AND MUHAMMAD DANISH ZIA
+Abstract. We consider the one dimensional Schr¨odinger operator with properly connecting gener-
+alized point interaction at the origin. We derive a trace formula for trace of difference of resolvents
+of perturbed and unperturbed Schr¨odinger operators in terms of a Wronskian which results into
+an explicit expression for perturbation determinant. Using the estimate for large time real argu-
+ment on the trace norm of the resolvent difference of the perturbed and unperturbed Schr¨odinger
+operators we express the spectral shift function in terms of perturbation determinant. Under cer-
+tain integrability condition on the potential function, we calculate low energy asymptotics for the
+perturbation determinant and prove an analog of Levinson’s formula.
+1. Introduction and main results
+Let HA
+V be the Schr¨odinger operator
+HA
+V = HA
+0 + V,
+HA
+0 = − d2
+dx2
+(1.1)
+on the real-line which is realized as a union of two positive semi-axis ej = [0, ∞), j = 1, 2 coupled at
+0. The potential function V is assumed to be real-valued and satisfies
+�
+ej
+|Vj(x)|dx < ∞,
+(1.2)
+where, Vj is the restriction of V on ej, j = 1, 2. The operator (1.1) is self-adjoint for all functions
+from the space H2([0, ∞)) ⊕ H2([0, ∞)) satisfying the conditions
+�ψ1(0)
+ψ′
+1(0)
+�
+= A
+�ψ2(0)
+ψ′
+2(0)
+�
+(1.3)
+where A = eiφ
+�a
+b
+c
+d
+�
+and ψ = (ψ1, ψ2)T ∈ L2([0, ∞)) ⊕ L2([0, ∞)) and the real parameters
+φ ∈ [−π/2, π/2], a, b, c, d ∈ R satisfying ad − bc = −1.
+The above conditions are commonly known as the generalized point interaction, first introduced
+by ˇSeba [18]. There are several other equivalent formulations for the generalized point interaction,
+see for example [9] and references therein. The free Hamiltonian HA
+0 with conditions (1.3) describes
+certain second order differential operators with generalized functions in coefficients. The Schr¨odinger
+operator with δ potential of strength α, for instance, corresponds to the unperturbed operator HA
+0
+with φ = 0, a = 1, b = 0, c = α and d = −1. Similarly, δ′ potential of strength β corresponds to
+φ = 0, a = −1, b = β, c = 0 and d = 1. Differential operators of this kind are closely related to
+exactly solvable models in quantum mechanics, atomic physics, and acoustics [1,7].
+2010 Mathematics Subject Classification. Primary: 34L25, 34L40; Secondary: 35P25, 81Q10.
+Key words and phrases. Schr¨odinger operators; Generalized point interaction; Trace formula; Perturbation deter-
+minant; Spectral shift function; Levinson’s formula.
+1
+
+2
+MUHAMMAD USMAN AND MUHAMMAD DANISH ZIA
+The present article is devoted to deriving explicit expression for perturbation determinant for
+the pair of operators HA
+V and HA
+0 and its relationship with spectral shift function and Levinson’s
+theorem. These mathematical objects play an important role in the study of direct and inverse scat-
+tering theory [10–13] as well as in solid state physics in connection with Friedel sum rule and excess
+charge [14]. Kr˘ein [15] (see also [17]) introduced the concept of perturbation determinant and it is
+an important tool in studying trace formulas of higher order. For a detailed study on these topics for
+Schr¨odinger operators on the half-line as well as on the whole real line we refer to the monograph [23]
+(see also reference [6]) and for a quantum star graph we refer to [5]. Our aim in this paper is to
+generalize these results for Schr¨odinger operators on the real-line, which can also be seen as a two
+edge star graph, satisfying the most general properly connecting self-adjoint matching conditions at
+zero.
+Using Kr˘ein’s resolvent formula (see [1] and [8]), we first prove a trace formula that expresses the
+trace of the difference between the perturbed and corresponding unperturbed resolvents in terms of a
+Wronskian. This trace formula allows us to derive an explicit expression for the perturbation deter-
+minant. The perturbation determinant is an interesting object in the spectral theory of Schr¨odinger
+operators, as the zeros of perturbation determinant coincide with the eigenvalues of the Schr¨odinger
+operator and the multiplicity of each eigenvalue is equal to the order of the corresponding zero of
+the perturbation determinant [3]. Estimates on the trace norm of the resolvent difference of the
+operators HA
+V and HA
+0 obtained in [21] allow one to express the spectral shift function in terms of
+perturbation determinant. For a real argument, perturbation determinant is used to define the phase
+shift function. The so called Levinson’s formula is derived for the Schr¨odinger operator HA
+V in terms
+of its phase shift function. This formula, also called a zero order trace formula gives a relationship
+between the number of negative eigenvalues and the scattering data via phase shift function.
+To state our main results we first consider the half-line Schr¨odinger operators HD
+ej = − d2
+dx2 + Vj,
+j = 1, 2, with Dirichlet boundary condition at 0 and let HD
+V = HD
+e1 ⊕ HD
+e2 denote the decoupled
+Schr¨odinger operator on the whole line. The resolvent (HD
+V − z)−1 is denoted by RD
+V (z). Under the
+condition (1.2) the differential equation
+−u′′
+j + Vju = ζ2u,
+j = 1, 2
+has two particular solutions, namely, the regular solution and the Jost solution. The regular solution
+φj is characterized by the conditions
+φj(0, ζ) = 0,
+φ′
+j(0, ζ) = 1
+and the Jost solution θj by the asymptotics θj(x, ζ) ∼ eiζx, as x → ∞. We denote the resolvent
+(HA
+V −z)−1 of the perturbed operator HA
+V by RA
+V (z) and the resolvent (HA
+0 −z)−1 of the unperturbed
+operator HA
+0 by RA
+0 (z). Moreover, the (modified) perturbation determinant is defined as
+D(z) := det(I +
+√
+V RA
+0 (z)
+�
+|V |),
+z ∈ ρ(HA
+0 ),
+where,
+√
+V = sgnV
+�
+|V |.
+Our first main result is the following trace formula for the difference of two resolvents in terms of
+Jost solutions θj and their derivatives
+Theorem 1.1. If the potential V satisfies (1.2) then the following trace formula holds
+Tr(RA
+V (z) − RA
+0 (z)) = − 1
+2ζ
+� d
+dζ ln
+�
+w1(ζ)w2(ζ)L(ζ)
+(a − d)ζ + (bζ2 + c)i
+��
+,
+ζ = z1/2,
+Im ζ > 0
+(1.4)
+
+PERTURBATION DETERMINANT— August 24, 2022
+3
+where
+L(ζ) = aθ′
+1(0, ζ)
+θ1(0, ζ) − dθ′
+2(0, ζ)
+θ2(0, ζ) + bθ′
+1(0, ζ)θ′
+2(0, ζ)
+θ1(0, ζ)θ2(0, ζ) − c
+and wj(ζ) = θj(0, ζ), j = 1, 2. Moreover, for z ∈ ρ(HV ) and Im z1/2 > 0, the perturbation determi-
+nant D(ζ) of the operator HA
+V with respect to HA
+0 is given by
+D(z) = L(√z)w1(√z)w2(√z)
+(a − d)√zi − (bz + c).
+(1.5)
+In Section 4 we study the behaviour of D(z) as |z| → 0 and prove the zero order trace formula,
+commonly known as the Levinson’s formula, for the operator HA
+V . We will need the following con-
+stants:
+α1 =
+�
+b
+�θ′
+1(0, 0)
+θ2(0, 0) + θ′
+2(0, 0)
+θ1(0, 0)
+�
+− aθ2(0, 0)
+θ1(0, 0) + dθ1(0, 0)
+θ2(0, 0)
+�
+i,
+(1.6)
+α2 = aθ′
+1(0, 0)θ2(0, 0) + bθ′
+1(0, 0)θ′
+2(0, 0),
+α3 = (cθ2(0, 0) + dθ′
+2(0, 0)) ˙θ1(0, 0),
+α4 = aθ′
+1(0, 0) ˙θ2(0, 0) − d ˙θ1(0, 0)θ′
+2(0, 0),
+where dot denotes the derivative with respect to ζ.
+Theorem 1.2. Assume that
+�
+ej
+(1 + x)|Vj(x)| dx < ∞,
+j = 1, 2
+(1.7)
+and let N be the number of negative eigenvalues of the operator HA
+V . Then, the following formula
+holds
+η(∞) − η(0) =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+π
+�
+N − P − Q
+2
+�
+,
+wj(0) ̸= 0, j = 1, 2,
+π
+�
+N − P − R
+2
+�
+,
+w1(0) = 0, w2(0) ̸= 0,
+π
+�
+N − P − S
+2
+�
+,
+w1(0) ̸= 0, w2(0) = 0,
+π
+�
+N − P − T
+2
+�
+,
+wj(0) = 0, j = 1, 2.
+Here, η is the so-called phase shift function, P, Q, R, S and T are functions which are defined as
+P =
+
+
+
+
+
+
+
+0
+c ̸= 0
+1
+c = 0, a − d ̸= 0
+2
+c = 0, a − d = 0, b ̸= 0,
+Q =
+
+
+
+
+
+
+
+0
+L(0) ̸= 0
+1
+L(0) = 0, α1 ̸= 0
+2
+L(0) = 0, α1 = 0, b ̸= 0,
+R =
+�
+0
+α2 ̸= 0
+1
+α2 = 0,
+S =
+�
+0
+α3 ̸= 0
+1
+α3 = 0,
+T =
+
+
+
+
+
+
+
+0
+b ̸= 0
+1
+b = 0, α4 ̸= 0
+2
+b = 0, α4 = 0, c ̸= 0.
+
+4
+MUHAMMAD USMAN AND MUHAMMAD DANISH ZIA
+2. Perturbation determinant
+Our main purpose in this section is to derive the trace formula (1.4) and the explicit expression
+(1.5) for the perturbation determinant of operator HA
+V with respect to H0 in terms of the Jost
+solutions θj and their derivatives θ′
+j. For this purpose we use Kr˘ein’s resolvent formula which allows
+us to describe the kernel of RA
+V (z) in terms of the kernel of RD
+V (z). More precisely, for Im ζ ≥ 0, the
+Krein’s formula states
+RA
+j,l(x, y; z) := RD
+j,l(x, y; z) + λj,lθj(x, ζ)θl(y, ζ),
+j, l = 1, 2,
+z = ζ2 ∈ ρ(HA
+V ) ∩ ρ(HD
+V ),
+(2.1)
+where RA
+j,l and RD
+j,l denote the kernel of 2 × 2 matrix integral operators RA
+V and RD
+V , respectively.
+The values of the coefficients λj,l are given by the following lemma
+Lemma 2.1. Let ψ(x) :=
+� ∞
+0
+RA(x, y; z)f(y) dy and f = (f1(x), f2(x))T ∈ L2([0, ∞)) ⊕ L2([0, ∞)).
+If ψ(x) satisfies HA
+V ψ = ζ2ψ + f and conditions (1.3) then λj,l in (2.1) are given by
+�λ1,1
+λ1,2
+λ2,1
+λ2,2
+�
+=
+
+
+−
+a + bθ′
+2(0, ζ)
+θ2(0, ζ)
+L(ζ)θ2
+1(0, ζ)
+(ad − bc)eiφ
+L(ζ)θ1(0, ζ)θ2(0, ζ)
+e−iφ
+L(ζ)θ1(0, ζ)θ2(0, ζ)
+−
+bθ′
+1(0, ζ)
+θ1(0, ζ) − d
+L(ζ)θ2
+2(0, ζ)
+
+
+where L(ζ) = a θ′
+1(0,ζ)
+θ1(0,ζ) − d θ′
+2(0,ζ)
+θ2(0,ζ) + b θ′
+1(0,ζ)θ′
+2(0,ζ)
+θ1(0,ζ)θ2(0,ζ) − c.
+Proof. It is a well-known fact (see for example [5]), that the resolvent of the operator HD
+ej is an
+integral operator with symmetric kernel
+RD
+ej(x, y; z) = φj(x, ζ)θj(y, ζ)
+wj(ζ)
+,
+x ≤ y,
+ζ = √z.
+(2.2)
+Therefore the resolvent RD
+V is a matrix integral operator with kernel
+RD
+j,l(x, y; z) = δj,lRD
+j (x, y; z)
+j, l = 1, 2.
+(2.3)
+For simplicity, we use the following notation
+RA
+j,l(x, y; z) = RA
+j,l,
+φj(x, ζ) = φj(x),
+θj(y, ζ) = θj(y)
+and
+wj(ζ) = wj.
+By substituting (2.2) and (2.3) into (2.1) and expressing it in matrix form we get
+�RA
+1,1
+RA
+1,2
+RA
+2,1
+RA
+2,2
+�
+=
+� φ1(x)θ1(y)+w1λ1,1θ1(x)θ1(y)
+w1
+λ1,2θ1(x)θ2(y)
+λ2,1θ2(x)θ1(y)
+φ2(x)θ2(y)+w2λ2,2θ2(x)θ2(y)
+w2
+�
+.
+Multiplying f from right and using the definition of function ψ the above expression becomes
+�ψ1(x)
+ψ2(x)
+�
+=
+�� ∞
+0 { φ1(x)θ1(y)
+w1
+f1(y) + �2
+i=1 λ1,iθ1(x)θi(y)fi(y)}dy
+� ∞
+0 { φ2(x)θ2(y)
+w2
+f2(y) + �2
+i=1 λ2,iθ2(x)θi(y)fi(y)}dy
+�
+which is equivalent to the following system of equations
+ψj(x) =
+∞
+�
+0
+�
+φj(x)θj(y)
+wj
+fj(y) +
+n
+�
+i=1
+λj,iθj(x)θi(y)fi(y)
+�
+dy,
+j = 1, 2.
+In order to determine λj,l uniquely, we first express conditions (1.3) as a system of two equations and
+first apply the sub-condition
+ψ1(0) = eiφ(aψ2(0) + bψ′
+2(0))
+
+PERTURBATION DETERMINANT— August 24, 2022
+5
+on ψ and Dirichlet condition on φ. This yields the following expression
+∞
+�
+0
+� 2
+�
+i=1
+λ1,iθ1(0)θi(y)fi(y)
+�
+dy =
+(2.4)
+eiφ
+
+a
+∞
+�
+0
+� 2
+�
+i=1
+λ2,iθ2(0)θi(y)fi(y)
+�
+dy + b
+∞
+�
+0
+�
+θ2(y)f2(y)
+θ2(0)
++
+2
+�
+i=1
+λ2,iθ′
+2(0)θi(y)fi(y)
+�
+dy
+
+ .
+Applying the second sub-condition
+ψ′
+1(0) = eiφ(cψ2(0) + dψ′
+2(0))
+on ψ and Dirichlet condition on φ, we obtain
+∞
+�
+0
+�
+θ1(y)f1(y)
+θ1(0)
++
+2
+�
+i=1
+λ1,iθ′
+1(0)θi(y)fi(y)
+�
+dy =
+(2.5)
+eiφ
+
+c
+∞
+�
+0
+� 2
+�
+i=1
+λ2,iθ2(0)θi(y)fi(y)
+�
+dy + d
+∞
+�
+0
+�
+θ2(y)f2(y)
+θ2(0)
++
+2
+�
+i=1
+λ2,iθ′
+2(0)θi(y)fi(y)
+�
+dy
+
+ .
+Comparing the coefficients of fi(y)θi(y) in (2.4) and (2.5), we get the following set of equations
+θ1(0)λ1,2 − eiφ
+�
+aθ2(0)λ2,2 − b
+�
+1
+θ2(0) + θ′
+2(0)λ2.2
+��
+= 0
+θ1(0)λ1,1 − eiφ (aθ2(0)λ2,1 + bθ′
+2(0)λ2,1) = 0
+θ′
+1(0)λ1,2 − eiφ
+�
+cθ2(0)λ2,2 − d
+�
+1
+θ2(0) + θ′
+2(0)λ2.2
+��
+= 0
+1
+θ1(0) + θ′
+1(0)λ1,1 − eiφ (cθ2(0)λ2,1 + dθ′
+2(0)λ2,1) = 0
+We obtain the values of λj,l j, l = 1, 2 by solving the above set of equations.
+□
+Using (2.3) and Lemma (2.1) we can express the resolvent kernel of HA
+V given by (2.1) as
+�RA
+1,1
+RA
+1,2
+RA
+2,1
+RA
+2,2
+�
+=
+
+
+RD
+e1 −
+a + bθ′
+2(0)
+θ2(0)
+L(ζ)θ2
+1(0) θ2
+1(x)
+(ad − bc)eiφ
+L(ζ)θ1(0)θ2(0)θ1(x)θ2(y)
+e−iφ
+L(ζ)θ1(0)θ2(0)θ2(x)θ1(y)
+RD
+e2 −
+bθ′
+1(0)
+θ1(0) − d
+L(ζ)θ2
+2(0) θ2
+2(x)
+
+
+.
+(2.6)
+Here, for simplicity, we used the notation RA
+i,j(x, y; ζ) = RA
+i,j, RD
+ei(x, y; ζ) = RD
+ei and θi(x, ζ) = θi(x).
+For unperturbed operator i.e., when Vj(x) ≡ 0, we may let
+θj(x, ζ) = eixjζ ⇒ L(ζ) = (a − d)iζ −
+�
+bζ2 + c
+�
+.
+Hence
+�
+RA
+0,(1,1)
+RA
+0,(1,2)
+RA
+0,(2,1)
+RA
+0,(2,2)
+�
+=
+�
+RD
+0,e1 −
+a+iζb
+(a−d)iζ−(bζ2+c)e2iζx
+(ad−bc)
+(a−d)iζ−(bζ2+c)ei(x+y+φ)ζ
+1
+(a−d)iζ−(bζ2+c)ei(x+y−φ)ζ
+RD
+0,e2 −
+iζb−d
+(a−d)iζ−(bζ2+c)e2iζx
+�
+.
+(2.7)
+we now prove Theorem (1.1).
+
+6
+MUHAMMAD USMAN AND MUHAMMAD DANISH ZIA
+Proof of Theorem 1.1. The operator difference RD
+V (z)−RD
+0 (z) is a trace class operator and the second
+term on the right hand side of (2.1) is a finite rank perturbation. Therefore, the difference RA
+V (z)−RA
+0
+is a trace class operator. The operator HA
+V can be seen as a particular case of a more general matrix
+valued Schr¨odinger operator considered in [21] on the half-line when potential is a diagonal matrix.
+The fact that RA
+V (z) − RA
+0 is trace class also follows from Lemma 9.1 of [21]. Representation (2.1)
+with values of λj,l given by lemma (2.1) allows us to compute the trace of the resolvent difference of
+operators HA
+V and HA
+0 .
+Using (2.6) and (2.7) the trace of the difference of RA
+V and RA
+0 can be written as
+Tr(RA
+V (z) − RA
+0 (z)) =
+2
+�
+j=1
+∞
+�
+0
+�
+RD
+ej(x, x, z) − RD
+0,ej(x, x, z)
+�
+dx
+−
+∞
+�
+0
+
+
+
+
+a + bθ′
+2(0)
+θ2(0)
+L(ζ)θ2
+1(0) θ2
+1(x) +
+bθ′
+1(0)
+θ1(0) − d
+L(ζ)θ2
+2(0) θ2
+2(x)
+
+
+
+ dx
++
+∞
+�
+0
+�
+(a − d + 2iζb)
+(a − d)iζ − (bζ2 + c)e2ixζ
+�
+dx.
+(2.8)
+The first term on the right hand side of (2.8) is given by (c.f. Remark 1.2 of [5])
+2
+�
+j=1
+∞
+�
+0
+�
+RD
+ej(x, x, z) − RD
+0,ej(x, x, z)
+�
+dx = − 1
+2ζ
+2
+�
+j=1
+˙wj(ζ)
+wj(ζ).
+(2.9)
+To compute the first term in the second integral on the right hand side of (2.8) we use the following
+equation which is true for any arbitrary solutions fj(x, ζ) and gj(x, ζ) of HD
+ejψj = ζ2ψj
+fj(x, ζ)gj(x, ζ) = (f ′
+j(x, ζ)˙gj(x, ζ) − fj(x, ζ)˙g′
+j(x, ζ))′
+2ζ
+.
+(2.10)
+If we let fj = gj = θj we obtain
+∞
+�
+0
+θ2
+j (x, ζ)dx = [θ′
+j(x, ζ) ˙θj(x, ζ) − θj(x, ζ) ˙θ′
+j(x, ζ)]∞
+0
+2ζ
+.
+If Vj is compactly supported potential then the Jost solution θj(x, ζ) = eiζx and hence
+θ′
+j(x, ζ) ˙θj(x, ζ) − θj(x, ζ) ˙θ′
+j(x, ζ) = ie2iζx.
+For Im ζ > 0, ie2iζx → 0 as x → ∞. Therefore, we are left with
+∞
+�
+0
+θ2
+j(x, ζ)dx = θj(0, ζ) ˙θ′
+j(0, ζ) − θ′
+j(0, ζ) ˙θj(0, ζ)
+2ζ
+= θ2
+j(0, ζ)
+2ζ
+d
+dζ
+θ′
+j(0, ζ)
+θj(0, ζ).
+
+PERTURBATION DETERMINANT— August 24, 2022
+7
+This implies
+∞
+�
+0
+
+
+
+
+a + bθ′
+2(0)
+θ2(0)
+L(ζ)θ2
+1(0) θ2
+1(x) +
+bθ′
+1(0)
+θ1(0) − d
+L(ζ)θ2
+2(0) θ2
+2(x)
+
+
+
+ dx
+=
+1
+2ζL(ζ)
+��
+a + bθ′
+2(0)
+θ2(0)
+� d
+dζ
+θ′
+1(0, ζ)
+θ1(0, ζ) +
+�
+bθ′
+1(0)
+θ1(0) − d
+� d
+dζ
+θ′
+2(0, ζ)
+θ2(0, ζ)
+�
+=
+˙L(ζ)
+2ζL(ζ).
+(2.11)
+By a density argument the above result can be extended to all potentials satisfying the condition
+� ∞
+0
+|Vj(x)| < ∞.
+Now it only remains to compute the last integral on the right hand side of (2.8), which is
+∞
+�
+0
+�
+(a − d + 2iζb)
+(a − d)iζ − bζ2 + ce2ixζ
+�
+dx =
+(a − d + 2iζb)
+2ζ((a − d)ζ + (bζ2 + c)i).
+(2.12)
+Substituting values from (2.9), (2.11) and (2.12) in (2.8), we get
+Tr(RA
+V (z) − RA
+0 (z)) = − 1
+2ζ
+
+
+2
+�
+j=1
+˙wj(ζ)
+wj(ζ) +
+˙L(ζ)
+L(ζ) −
+(a − d + 2iζb)
+(a − d)ζ + (bζ2 + c)i
+
+
+= − 1
+2ζ
+
+
+2
+�
+j=1
+d
+dζ ln(wj(ζ)) + d
+dζ ln(L(ζ)) − d
+dζ ln((a − d)ζ + (bζ2 + c)i)
+
+
+= − 1
+2ζ
+
+ d
+dζ ln
+
+
+L(ζ)
+(a − d)ζ + (bζ2 + c)i
+2
+�
+j=1
+wj(ζ)
+
+
+
+ .
+The second term on the right hand side of (2.1) with values of λj,l given by lemma (2.1) is a finite
+rank operator and if the potential function V satisfy condition (1.2) then the operator
+�
+|V |(HA
+0 )−1/2
+is Hilbert-Schmidt and therefore,
+√
+V RA
+0
+�
+|V | is trace class.
+This implies that the perturbation
+determinant D(z) is well defined. The perturbation determinant and trace of the resolvent difference
+RA
+V (z) − RA
+0 (z) are related by the following expression
+Tr(RA
+V (z) − RA
+0 (z)) =
+d
+dzD(z)
+D(z) ,
+z ∈ ρ(HA
+V ) ∩ ρ(HA
+0 ).
+In order to find an explicit expression for the perturbation determinant we choose ζ = √z such
+that Im (z) > 0. It follows from Theorem 1.1 that
+d
+dz (D(z))
+D(z)
+=
+d
+dz
+�
+L(√z)
+(a − d)√z + (bz + c)i
+2�
+j=1
+wj(√z)
+�
+L(√z)
+(a − d)√z + (bz + c)i
+2�
+j=1
+wj(√z)
+.
+From here we deduce that
+D(z) = A
+L(√z)
+(a − d)√z + (bz + c)i
+2
+�
+j=1
+wj(√z),
+A ∈ C.
+
+8
+MUHAMMAD USMAN AND MUHAMMAD DANISH ZIA
+To find the value of the complex coefficient A, we use the following asymptotic of perturbation
+determinant
+lim
+|Im(z)|→∞ D(z) = 1.
+This asymptotic holds if
+�
+|V |RA
+0 (z)−1 is Hilbert-Schmidt operator (see [23]). For the asymptotic
+behaviour of wj(√z) as |√z| → ∞ , we refer to [6], which provides
+wj(√z) = θj(0, √z) = 1 + O(|√z|−1).
+Further, it is easy to find that
+L(√z) = (a − d)√zi − (bz + c) + O(1),
+|√z| → ∞.
+(2.13)
+This implies, A = 1
+i .
+□
+2.1. Spectral shift function. In this section we briefly discuss the relationship between perturba-
+tion determinant D(z) and the spectral shift function ξ(λ; HA
+V , HA
+0 ). The spectral shift function for
+the pair of operators HA
+V and HA
+0 can be defined by the trace formula
+Tr(f(HA
+V − HA
+0 )) :=
+� ∞
+−∞
+ξ(λ; HA
+V , HA
+0 )f ′(λ) dλ
+(2.14)
+for all f ∈ C∞
+0 (R).
+Theorem 9.2 of [21] explicitly characterize the spectral shift function in the more general case of ma-
+trix valued Schr¨odinger operators with general self-adjoint conditions at the origin. It was proved that
+the trace formula (2.14) holds for a bigger class of functions satisfying f ′(λ) = O(λ−1/2−ε), f ′′(λ) =
+O(λ−1−ε), ε > 0, λ → ∞ and
+� ∞
+−∞
+|ξ(λ; HA
+V , HA
+0 )|(1 + |λ|)−1/2−ε dλ < ∞,
+ε > 0.
+One of the main ingredients in the proof of Theorem 9.2 of [21] is the following estimate1 (cf. inequality
+(9.2) of [21], see also (5.11) of [23] )
+||RA
+V (−t) − RA
+0 (−t)||1 ≤ Cεt− 3
+2 +ε,
+ε > 0,
+t → ∞.
+(2.15)
+The above estimate implies
+� ∞
+1
+t−m||RA
+V (−t) − RA
+0 (−t)||1 < ∞
+which holds for all m > −1/2. Proposition (3.3) of [5] then implies
+ξ(λ; HA
+V , HA
+0 ) = π−1 lim
+ε→0 arg D(λ + iε),
+where arg D(z) = Im ln D(z) is defined by the condition ln D(z) → 0 as dist{z, σ(HA
+0 )} → ∞.
+Moreover,
+ln D(z) =
+� ∞
+−∞
+ξ(λ; HA
+V , HA
+0 )(λ − z)−1 dλ,
+z ∈ ρ(HA
+0 ) ∩ ρ(HA
+V ).
+1The interested reader can consult Appendix (A) where explicit derivation of this estimate is provided.
+
+PERTURBATION DETERMINANT— August 24, 2022
+9
+3. Levinson’s formula
+In this section we derive a zero order trace formula, commonly known as Levinson’s formula, which
+describes the number of negative eigenvalues of an operator in terms of phase shift function which
+is denoted by η and it is defined, for real k, as η(k) := arg D(k). The Levinson’s formula derived in
+this section is closely related to the Levinson’s formulas obtained in Theorem 9.3 of [3] and Theorem
+9.3 of [21]. Levinson’s formula obtained in [3] relates the number of negative eigenvalues with the
+determinant of the scattering matrix and certain parameters that depend on the boundary condi-
+tions. Formula obtained in [21] relates the number of negative eigenvalues with the spectral shift
+function. The vertex conditions of this paper can be expressed as a particular case of more general
+conditions considered in references [3] and [21] and therefore the Levinson’s formulas obtained there
+also hold for the operator HA
+V . For higher order trace formulas of integer and half-integer order we
+refer to [19,22,23].
+We will need the following low-energy asymptotics of the perturbation determinant
+Lemma 3.1 (Low energy asymptotics). Let the potential V satisfies the condition
+�
+ej
+(1 + x)|Vj(x)|dx < ∞,
+1 ≤ j ≤ n.
+(3.1)
+Then as ζ → 0 the perturbation determinant satisfies
+D(ζ) =
+
+
+
+
+
+
+
+
+
+
+
+ζ(Q−P )(1 + o(1))
+wj(0) ̸= 0, j = 1, 2,
+ζ(R−P )(1 + o(1))
+w1(0) = 0, w2(0) ̸= 0,
+ζ(S−P )(1 + o(1))
+w1(0) ̸= 0, w2(0) = 0,
+ζ(T −P )(1 + o(1))
+wj(0) = 0, j = 1, 2.
+Where the functions P, Q, R, S and T are defined in Theorem (1.2).
+Proof. Expression for perturbation determinant is
+D(ζ) =
+L(ζ)
+2�
+j=1
+wj(ζ)
+(a − d)iζ − (bζ2 + c).
+We consider the following five different cases
+(1) wj(0) ̸= 0, j = 1, 2 and L(0) ̸= 0.
+(2) wj(0) ̸= 0, j = 1, 2 and L(0) = 0.
+(3) w1(0) = 0 and w2(0) ̸= 0.
+(4) w1(0) ̸= 0 and w2(0) = 0.
+(5) wj(0) = 0, j = 1, 2.
+
+10
+MUHAMMAD USMAN AND MUHAMMAD DANISH ZIA
+In the first case D(ζ) has no zeros and we can let
+L(0)
+2
+�
+j=1
+wj(0) = ˜c ̸= 0
+⇒ lim
+ζ→0 D(ζ) = lim
+ζ→0
+˜c
+(a − d)iζ − (bζ2 + c)
+⇒ lim
+ζ→0 D(ζ)((a − d)iζ − (bζ2 + c)) = ˜c
+⇒ lim
+ζ→0
+D(ζ)((a − d)iζ − (bζ2 + c)) − ˜c
+˜c
+= 0
+⇒ D(ζ) =
+˜c
+(a − d)iζ − (bζ2 + c)(1 + o(1)).
+In the second case, we make use of the following small energy asymptotics (c.f. Theorem 2.3 (iii)of [2])
+θ′
+j(0, ζ)
+θj(0, ζ) = θ′
+j(0, 0)
+θj(0, 0) −
+iζ
+θ2
+j(0, 0) + o(ζ)
+which implies
+L(ζ) =a
+�θ′
+1(0, 0)
+θ1(0, 0) −
+iζ
+θ2
+1(0, 0) + o(ζ)
+�
+− d
+�θ′
+2(0, 0)
+θ2(0, 0) −
+iζ
+θ2
+2(0, 0) + o(ζ)
+�
++ b
+�θ′
+1(0, 0)
+θ1(0, 0) −
+iζ
+θ2
+1(0, 0) + o(ζ)
+� �θ′
+2(0, 0)
+θ2(0, 0) −
+iζ
+θ2
+2(0, 0) + o(ζ)
+�
++ c
+=aθ′
+1(0, 0)
+θ1(0, 0) − dθ′
+2(0, 0)
+θ2(0, 0) + bθ′
+1(0, 0)
+θ1(0, 0)
+θ′
+2(0, 0)
+θ2(0, 0) − c + a
+�
+−iζ
+θ2
+1(0, 0) + o(ζ)
+�
+− d
+�
+−iζ
+θ2
+2(0, 0) + o(ζ)
+�
++ b
+��
+θ′
+1(0, 0)
+θ1(0, 0)θ2
+2(0, 0) +
+θ′
+2(0, 0)
+θ2
+1(0, 0)θ2(0, 0)
+�
+iζ
+�
++ b
+�
+o(ζ) + o(ζ2) −
+ζ2
+θ1(0, 0)θ2(0, 0)
+�
+This yields
+L(ζ) =L(0) + a
+�
+−iζ
+θ2
+1(0, 0) + o(ζ)
+�
+− d
+�
+−iζ
+θ2
+2(0, 0) + o(ζ)
+�
++ b
+��
+θ′
+1(0, 0)
+θ1(0, 0)θ2
+2(0, 0) +
+θ′
+2(0, 0)
+θ2
+1(0, 0)θ2(0, 0)
+�
+iζ + o(ζ) + o(ζ2) −
+ζ2
+θ1(0, 0)θ2(0, 0)
+�
+.
+Therefore,
+D(ζ) = α1ζ + o(ζ) + b(ζ2 + (o(ζ2)))
+(a − d)iζ − (bζ2 + c)
+,
+where, α1 =
+�
+b
+�θ′
+1(0, 0)
+θ2(0, 0) + θ′
+2(0, 0)
+θ1(0, 0)
+�
+− aθ2(0, 0)
+θ1(0, 0) + dθ1(0, 0)
+θ2(0, 0)
+�
+i.
+The above calculations show that the order of the numerator of D(ζ) depends on the value of the
+parameter b. If b = 0, then it will be of order ζ otherwise it will be of order ζ2.
+The study of first two cases can be summarized as follows
+• If wj(0) ̸= 0, j = 1, 2 and L(0) ̸= 0 then the numerator of D(ζ) is a non zero constant and it
+will have no zeros.
+• If wj(0) ̸= 0, j = 1, 2 and L(0) = 0 then the numerator of D(ζ) for small ζ will have one zero
+in case α1 ̸= 0 and it will have two zeros in case α1 = 0, b ̸= 0.
+This gives us the following low energy asymptotics in the first two cases
+D(ζ) = ζ(Q−P )(1 + o(1)),
+
+PERTURBATION DETERMINANT— August 24, 2022
+11
+where
+Q = Q(b, α1, L(0)) =
+
+
+
+
+
+
+
+0
+L(0) ̸= 0,
+1
+L(0) = 0, α1 ̸= 0
+2
+L(0) = 0, α1 = 0, b ̸= 0
+and
+P = P(a, b, c, d) =
+
+
+
+
+
+
+
+0
+c ̸= 0,
+1
+c = 0, a − d ̸= 0
+2
+c = 0, a − d = 0, b ̸= 0.
+The function P gives the number of poles of D(ζ) in the given cases.
+Let us now consider the third case, i.e., when w1(0) = 0 and w2(0) ̸= 0. We rewrite D(ζ) as
+D(ζ) =
+�
+aθ′
+1(0, ζ)
+θ1(0, ζ) − dθ′
+2(0, ζ)
+θ2(0, ζ) + b
+2�
+k=1
+θ′
+k(0, ζ)
+θk(0, ζ) − c
+�
+2�
+j=1
+θj(0, ζ)
+(a − d)iζ − (bζ2 + c)
+=
+�
+aθ′
+1(0, ζ)θ2(0, ζ) + bθ′
+1(0, ζ)θ′
+2(0, ζ) − dθ1(0, ζ)θ′
+2(0, ζ) − c
+2�
+j=1
+θj(0, ζ)
+�
+(a − d)iζ − (bζ2 + c)
+(3.2)
+and use the following asymptotic for θ1(x, ζ) (c.f. [4])
+θ1(0, ζ) = ζ ˙θ1(0, 0) + o(ζ).
+This implies
+D(ζ) =
+(α2 − c3ζ + o(ζ))
+(a − d)iζ − (bζ2 + c)
+Where α2 = aθ′
+1(0, 0)θ2(0, 0)+bθ′
+1(0, 0)θ′
+2(0, 0) and c3 = (cθ2(0, 0)+dθ′
+2(0, 0)) ˙θ1(0, 0). The numerator
+of D(ζ) is equal to α2 − c3ζ + o(ζ) as ζ → 0. If α2 ̸= 0 then the numerator of D(ζ) has no zeros for
+small ζ and if α2 = 0, then the numerator of D(ζ) has one zero for small ζ. Hence,
+D(ζ) = ζ(R−P )(1 + o(1)),
+ζ → 0,
+where,
+R = R(α2) =
+�
+0
+α2 ̸= 0,
+1
+α2 = 0.
+For the fourth case we substitute
+θ2(0, ζ) = ζ ˙θ2(0, 0) + o(ζ), ζ → 0
+into (3.2) and obtain
+D(ζ) =
+(α3 + c5ζ + o(ζ))
+(a − d)iζ − (bζ2 + c).
+Here, α3 = bθ′
+1(0, 0)θ′
+2(0, 0) − dθ1(0, 0)θ′
+2(0, 0) and c5 = (aθ′
+1(0, 0) − cθ1(0, 0)) ˙θ2(0, 0). The numerator
+of D(ζ) for small ζ has no zeros if α3 ̸= 0 and it has one zero if α3 = 0. Therefore,
+D(ζ) = ζ(S−P )(1 + o(1)),
+ζ → 0,
+where,
+S = S(α3) =
+�
+0
+α3 ̸= 0,
+1
+α3 = 0.
+
+12
+MUHAMMAD USMAN AND MUHAMMAD DANISH ZIA
+Finally, for the fifth case we substitute the asymptotics
+θ1(0, ζ) = ζ ˙θ1(0, 0) + o(ζ) and θ2(0, ζ) = ζ ˙θ2(0, 0) + o(ζ),
+ζ → 0
+into (3.2) and obtain
+D(ζ) = c
+�
+c6ζ2 + o(ζ2)
+�
++ α4ζ + bc8 + o(ζ)
+(a − d)iζ − (bζ2 + c)
+,
+ζ → 0.
+Here, c6 = ˙θ1(0, 0) ˙θ2(0, 0) ̸= 0, c8 = θ′
+1(0, 0)θ′
+2(0, 0) ̸= 0 (because θ(0, 0) = 0 ⇒ ˙θ(0, 0)θ′(0, 0) = −i,
+the proof is given in [5, Lemma 4.6.(1)], hence ˙θ(0, 0) ̸= 0 and θ′(0, 0) ̸= 0) α4 = aθ′
+1(0, 0) ˙θ2(0, 0) −
+d ˙θ1(0, 0)θ′
+2(0, 0). This implies the numerator of D(ζ) for small ζ has no zeros if b ̸= 0, it has one zero
+if b = 0, α4 ̸= 0 and it has two zeros if b = 0, α4 = 0, c ̸= 0. This yields
+D(ζ) = ζ(T −P )(1 + o(1)),
+ζ → 0,
+where
+T = T (b, c, α4) =
+
+
+
+
+
+
+
+0
+b ̸= 0,
+1
+b = 0, α4 ̸= 0
+2
+b = 0, α4 = 0, c ̸= 0.
+□
+Using the low energy asymptotics of D(ζ) we now prove the analogue of Levinson’s formula for
+the negative eigenvalues of HA
+V , stated in Theorem (1.2).
+Proof of Theorem 1.2. The function D(ζ) has a zero in ζ of order r if and only if ζ2 is an eigenvalue
+of multiplicity r of the operator HA
+V [3]. As HA
+V is a self-adjoint operator, so the zeros of D(ζ) lie on
+the positive imaginary axis and it may have only real eigenvalues.
+Let ΓR,ε denotes the contour (counterclockwise) consisting of semicircles C+
+R = {|ζ| = R,
+Im ζ ≥
+0} and C+
+ε = {|ζ| = ε,
+Im ζ ≥ 0} and the intervals (ε, R) and (−R, −ε). R and ε are chosen such
+that all N negative eigenvalues lie inside the contour ΓR,ε.
+The function wj(ζ) is analytic in the upper halfp-plane, therefore, D(ζ) is analytic inside and on
+the contour ΓR,ε. Hence,
+�
+ΓR,ε
+d
+dζ D(ζ)
+D(ζ) dζ = 2πiN.
+(3.3)
+Note that,
+lim
+Im ζ→∞
+D(ζ) =
+lim
+Im ζ→∞
+L(ζ)
+2�
+j=1
+wj(ζ)
+(a − d)iζ − (bζ2 + c) = 1 + O(|ζ|−1).
+(3.4)
+The branch of the function ln D(ζ) can be fixed by the condition ln D(ζ) → 0 as |ζ| → ∞. For k ∈ R,
+we set D(k) = a(k)eiη(k), where a(k) = |D(k)| and η(k) = arg D(k). Now
+varΓR,ε arg D(ζ) = η(R) − η(ε) + varC+
+R arg D(ζ) + {η(−ε) − η(−R)} + varC+
+ε arg D(ζ).
+It follows from the representation of D(k) that η(−k) = −η(k). Hence,
+varΓR,ε arg D(ζ) = 2(η(R) − η(ε)) + varC+
+R arg D(ζ) + varC+
+ε arg D(ζ)
+
+PERTURBATION DETERMINANT— August 24, 2022
+13
+Equation (3.3) implies that
+varΓR,ε arg D(ζ) = 2πN.
+This implies
+2πN = 2(η(R) − η(ε)) + varC+
+R arg D(ζ) + varC+
+ε arg D(ζ).
+Using (3.4), one can see that lim
+R→∞ varC+
+R arg D(ζ) = 0. We are left with
+η(∞) − η(ε) = πN − 1
+2varC+
+ε arg D(ζ).
+Now letting ε → 0 and using Lemma (3.1), we arrive at
+lim
+ε→0 varC+
+ε arg D(ζ) =
+
+
+
+
+
+
+
+
+
+
+
+−(Q − P)π
+wj(0) ̸= 0, j = 1, 2,
+−(R − P)π
+w1(0) = 0, w2(0) ̸= 0,
+−(S − P)π
+w1(0) ̸= 0, w2(0) = 0,
+−(T − P)π
+wj(0) = 0, j = 1, 2.
+Therefore,
+η(∞) − η(0) =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+π(N + Q − P
+2
+)
+wj(0) ̸= 0, j = 1, 2,
+π(N + R − P
+2
+)
+w1(0) = 0, w2(0) ̸= 0,
+π(N + S − P
+2
+)
+w1(0) ̸= 0, w2(0) = 0,
+π(N + T − P
+2
+)
+wj(0) = 0, j = 1, 2.
+□
+4. examples
+In this section we illustrate the trace formula and perturbation determinant obtained in theorem
+(1.1) with some examples.
+Example 4.1. For the δ interaction of strength α ∈ R, we can choose φ = 0, a = 1, b = 0, c = α
+and d = −1. Then according to theorem (1.1) the trace of the resolvent difference is given by
+Tr(RA
+V (ζ) − RA
+0 (ζ)) = − 1
+2ζ ln
+�θ′
+1(0, ζ)θ2(0, ζ) + θ1(0, ζ)θ′
+2(0, ζ) − αθ1(0, ζ)θ2(0, ζ)
+2ζ + iα
+�
+and the expression for perturbation determinant is
+D(ζ) =
+1
+2iζ − α (θ′
+1(0, ζ)θ2(0, ζ) + θ1(0, ζ)θ′
+2(0, ζ) − αθ1(0, ζ)θ2(0, ζ)) .
+For α = 0 the δ conditions are commonly known as Kirchhoff conditions and in this case the above
+expressions coincide with the results obtained in Theorem (1.1) and Corollary (3.1) of [5] with n = 2
+(see also [20]).
+Example 4.2. The δ′ conditions are obtained by choosing φ = 0, a = −1, b = β, c = 0 and d = 1 in
+(1.3). In this case theorem (1.1) implies
+Tr(RA
+V (ζ) − RA
+0 (ζ)) = − 1
+2ζ ln
+�θ′
+1(0, ζ)θ2(0, ζ) + θ1(0, ζ)θ′
+2(0, ζ) − βθ′
+1(0, ζ)θ′
+2(0, ζ)
+2ζ − iβζ2
+�
+and perturbation determinant is given by
+D(ζ) =
+1
+2iζ + βζ2 (θ′
+1(0, ζ)θ2(0, ζ) + θ1(0, ζ)θ′
+2(0, ζ) − βθ′
+1(0, ζ)θ′
+2(0, ζ)) .
+
+14
+MUHAMMAD USMAN AND MUHAMMAD DANISH ZIA
+In the next two examples we use the notation Dxφ = φ(1) for the generalized derivative in the
+sense of [16].
+Example 4.3. The Schr¨odinger operator with the singular density −Dx (1 + σδ) Dx is equivalent
+to the operator HA
+V with φ = 0, a = 1, b = −σ ∈ R, c = 0 and d = −1 and the perturbation
+determinant is given by
+D(ζ) =
+1
+2iζ + σζ2 (θ′
+1(0, ζ)θ2(0, ζ) + θ1(0, ζ)θ′
+2(0, ζ) − σθ′
+1(0, ζ)θ′
+2(0, ζ)) .
+Example 4.4. For real σ1 and σ2 the two parameters family of Schr¨odinger operators with singular
+potential −D2
+x + σ1δ + σ2δ(1) + V is equivalent to the operator HA
+V with φ = 0, a = 2+σ2
+2−σ2 , b = 0, c =
+4σ1
+4−σ2
+2 , d = −
+�
+2−σ2
+2+σ2
+�
+(c.f. Section 3.2.4 of [16]). Theorem (1.1) implies the following expression for
+the perturbation determinant
+D(ζ) =
+��
+2+σ2
+2−σ2
+�
+θ′
+1(0, ζ)θ2(0, ζ) +
+�
+2−σ2
+2+σ2
+�
+θ1(0, ζ)θ′
+2(0, ζ) −
+�
+4σ1
+4−σ2
+2
+�
+θ1(0, ζ)θ2(0, ζ)
+�
+.
+�
+8+2σ2
+2
+4−σ2
+2
+�
+iζ −
+�
+4σ1
+4−σ2
+2
+�
+5. conclusions
+The Schr¨odinger operators on the real-line with generalized point interaction at zero can be imag-
+ined as a 2 × 2 matrix-valued Schr¨odinger operator on semi-axis with general self-adjoint conditions
+at zero. Scattering theory of matrix-valued Schr¨odinger operators on semi-axis with general self-
+adjoint conditions at zero has been studied in much detail in the last few years. On the other hand,
+explicit derivation of perturbation determinants and its connections with spectral shift function and
+Levinson’s formulas has received relatively less attention. The present article is devoted to explicit
+derivation of perturbation determinant via trace formula for Schr¨odinger operators and expressing
+Levinson’s formula and spectral shift function in terms of perturbation determinant. Our results com-
+plement the previous studies in this regard. We used operator theoretic approach in our derivations
+similar to Demirel’s work which deals with star graphs with Kirchhoff conditions at the vertex. The
+expressions for perturbation determinants are further used to derive LevinsonˆA’s formulas connecting
+the number of negative eigenvalues with phase shift. Phase shift is defined in terms of perturbation
+determinant. Looking forward, it would be interesting to extend and generalize the results on pertur-
+bation determinant to matrix-valued Schr¨odinger operators with most general self-adjoint conditions
+at zero.
+Appendix A. Estimate on trace norm of resolvent difference
+Here we’ll provide and explicit derivation of the estimate
+||RA
+V (−t) − RA
+0 (−t)||1 ≤ Cεt− 3
+2 +ε
+for all ε > 0 and large t. We will need the following lemma.
+Lemma A.1 ( [5], Lemma 3.4 ). Let H be a Hilbert space and f, g ∈ H.
+Assume that R =
+(·, f)f − (·, g)g is an operator of rank two on H. Then, the trace norm of R is given by
+||R||1 =
+�
+(||f||2 + ||g||2)2 − 4|(f, g)|2.
+(A.1)
+Moreover, if we let g = f + h then,
+(||f||2 + ||g||2)2 − 4|(f, g)|2 ≤ 6||h||2||f||2 + 3||h||4.
+(A.2)
+Lemma A.2. Assume that the potential V satisfies condition (1.2). Then the resolvents RA
+V of HA
+V
+and RA
+0 of HA
+0 satisfy, for all ε > 0 and for large t,
+||RA
+V (−t) − RA
+0 (−t)||1 ≤ Cεt− 3
+2 +ε.
+
+PERTURBATION DETERMINANT— August 24, 2022
+15
+Proof. Let R1 = RA
+V (−t) − RD
+V (−t) + RD
+0 (−t) − RA
+0 (−t) and R2 = RD
+V (−t) − RD
+0 (−t) then,
+RA
+V (−t) − RA
+0 (−t) = R1 + R2.
+Hence
+||RA
+V (−t) − RA
+0 (−t)||1 ≤ ||R1||1 + ||R2||1.
+(A.3)
+Trace norm on R2 can be estimated as ||R2||1 ≤ ct− 3
+2 +ε (c.f. Lemma 4.5.6. [23]). We only need to
+find estimate on the trace norm of rank two operator R1. By adding and subtracting the quantity
+θ2
+j(xj, i
+√
+t)
+L(i
+√
+t)θ2
+j (0, i
+√
+t) +
+e−2xj
+√
+t
+(d − a)
+√
+t + bt − c
+j = 1, 2
+in the diagonal of R1 we can re-write it as
+R1 = S1 + S2,
+where
+S1 =
+
+
+(a−
+√
+tb+1)e−2x1
+√t
+(d−a)
+√
+t+bt−c
+−
+�
+a+b
+θ′
+2(0,i
+√
+t)
+θ2(0,i
+√
+t) +1
+�
+θ2
+1(x1,i
+√
+t)
+L(i
+√
+t)θ2
+1(0,i
+√
+t)
+0
+0
+(1−
+√
+tb−d)e−2x2
+√
+t
+(d−a)
+√
+t+bt−c
+−
+�
+b
+θ′
+1(0,i
+√
+t)
+θ1(0,i
+√
+t) −d+1
+�
+θ2
+2(x2,i
+√
+t)
+L(i
+√
+t)θ2
+2(0,i
+√
+t)
+
+
+and
+(S2)j,k =
+�
+θj(xj,i
+√
+t)θk(xk,i
+√
+t)
+L(i
+√
+t)θj(0,i
+√
+t)θk(0,i
+√
+t) −
+e−(xj +xk)
+√
+t
+(d−a)
+√
+t+bt−c
+�
+j, k = 1, 2.
+Now, as S1 is a diagonal matrix and in order to apply definition of trace norm, we can let S1 = IS1I,
+where I is second order identity matrix. This implies,
+||S1||1 = |σ1| + |σ2|
+where
+|σ1| =
+� ∞
+0
+������
+(a −
+√
+tb + 1)e−2x1
+√
+t
+(d − a)
+√
+t + bt − c
+−
+�
+a + b θ′
+2(0,i
+√
+t)
+θ2(0,i
+√
+t) + 1
+�
+θ2
+1(x1, i
+√
+t)
+L(i
+√
+t)θ2
+1(0, i
+√
+t)
+������
+dx1
+and
+|σ2| =
+� ∞
+0
+������
+(1 −
+√
+tb − d)e−2x2
+√
+t
+(d − a)
+√
+t + bt − c
+−
+�
+b θ′
+1(0,i
+√
+t)
+θ1(0,i
+√
+t) − d + 1
+�
+θ2
+2(x2, i
+√
+t)
+L(i
+√
+t)θ2
+2(0, i
+√
+t)
+������
+dx2.
+For t large enough, these expressions can be simplified to
+|σ1| ≈
+����
+(a −
+√
+tb + 1)
+(d − a)
+√
+t + bt − c
+����
+� ∞
+0
+���e−2x1
+√
+t − θ2
+1(x1, i
+√
+t)
+��� dx1
+=
+����
+(a −
+√
+tb + 1)
+(d − a)
+√
+t + bt − c
+����
+� ∞
+0
+���θ1(x1, i
+√
+t) − e−x1
+√
+t���
+���θ1(x1, i
+√
+t) + e−x1
+√
+t��� dx1
+and
+|σ2| ≈
+����
+(1 −
+√
+tb − d)
+(d − a)
+√
+t + bt − c
+����
+� ∞
+0
+���θ2(x2, i
+√
+t) − e−x2
+√
+t���
+���θ2(x2, i
+√
+t) + e−x2
+√
+t��� dx2.
+To find estimates for |σ1| and |σ2| we use the following estimates for large t
+���θk(xk, i
+√
+t) − e−
+√
+txk
+��� ≤ m1
+√
+t e−
+√
+txk
+(A.4)
+and
+���θk(xk, i
+√
+t) + e−
+√
+txk)
+��� ≤
+�m2
+√
+t + 2
+�
+e−
+√
+txk
+(A.5)
+
+16
+MUHAMMAD USMAN AND MUHAMMAD DANISH ZIA
+This yields the following bounds
+|σ1| ≤
+����
+(a −
+√
+tb + 1)
+(d − a)
+√
+t + bt − c
+����
+m1(m2 + 2
+√
+t)
+2t3/2
+= | − m1t−3/2 + O(t−2)|,
+|σ2| ≤
+����
+(1 −
+√
+tb − d)
+(d − a)
+√
+t + bt − c
+����
+m1(m2 + 2
+√
+t)
+2t3/2
+= | − m1t−3/2 + O(t−2)|.
+Therefore
+|σ1| = |σ2| = O(t−3/2)
+and hence
+||S1||1 = O(t−3/2).
+To find estimate on norm of S2, we use Lemma (A.1). Clearly, S2 = (·, f)f − (·, g)g is an operator
+of rank two, where
+fk(xk) =
+−e−xk
+√
+t
+�
+(d − a)
+√
+t + bt − c
+and
+gk(xk) =
+−θk(xk, i
+√
+t)
+�
+L(i
+√
+t)θk(0, i
+√
+t)
+.
+Let h = g − f, then
+hk(xk) =
+e−xk
+√
+t
+�
+(d − a)
+√
+t + bt − c
+−
+θk(xk, i
+√
+t)
+�
+L(i
+√
+t)θk(0, i
+√
+t)
+= −
+
+θk(xk,
+√
+ti) − e−
+√
+txk
+�
+L(i
+√
+t)θk(0, i
+√
+t)
++
+
+
+1
+�
+L(i
+√
+t)θk(0, i
+√
+t)
+−
+1
+�
+(d − a)
+√
+t + bt − c
+
+ e−
+√
+txk
+
+ .
+To compute estimate on ||h||2 =
+2�
+k=1
+∞
+�
+0
+|hk(xk)|2 dxk, we use (2.13) and (A.4) and after some simpli-
+fications we obtain
+||h||2 ≤
+m2
+1
+�
+(d − a)
+√
+t + bt − c
+�
+t3/2 .
+Similarly,
+||f||2 =
+2
+�
+k=1
+∞
+�
+0
+������
+−e−
+√
+txk
+�
+(d − a)
+√
+t + bt − c
+������
+2
+dxk =
+1
+�
+(d − a)
+√
+t + bt − c
+� √
+t.
+The estimate on the right hand side of inequality (A.2) is now given by
+6||h||2||f||2 + 3||h||4 ≤
+6m2
+1
+�
+(d − a)
+√
+t + bt − c
+�2 t2 +
+3m4
+1
+�
+(d − a)
+√
+t + bt − c
+�2 t3
+=
+3m4
+1 + 6m2
+1t
+�
+(d − a)
+√
+t + bt − c
+�2 t3
+= 6m2
+1
+b2t4 + 12m2
+1(a − d)
+b3t
+9
+2
++ O(t−5)
+= O(t−4).
+This implies
+||S2||1 = O(t−2)
+
+PERTURBATION DETERMINANT— August 24, 2022
+17
+and hence
+||R1||1 = O(t− 3
+2 ).
+□
+Declarations
+Ethics approval and consent to participate. Not applicable.
+Consent for publication. Not applicable.
+Availability of data and material. Not applicable
+Competing interests. The authors declare no competing interests.
+Funding. The research leading to these results received no funding.
+Authors’ contributions. M.U. conceptualized the study and drafted the manuscript.
+MD. Z.
+carried out the computations. All authors read and approved the final manuscript.
+References
+[1]
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+Chelsea Publishing, Providence, RI, second edition, 2005. With an appendix by Pavel Exner.
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+operator on the half line, J. Math. Phys. 54, 012108 (2013).
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+S. Demirel, The spectral shift function and Levinson’s theorem for quantum star graphs, J. Math. Phys. 53,
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+397–427 (2011).
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+Yu. N. Demkov, V. N. Ostrovskii, Zero-range potentials and their applications in atomic Physics, Leningrad
+Univ. Press, Leningrad, (1975) (in Russian) ; Plenum Press, New York/London, (1988) (English translation)
+[8]
+P. Exner, Weakly coupled states on branching graphs, Lett. Math. Phys., 38(3):313 320, (1996).
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+(1988).
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+[12]
+M. S. Harmer,
+The matrix Schr¨odinger operator and Schr¨odinger operator on graphs, In Ph. D. thesis.
+University of Auckland, New Zealand, (2004).
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+M.S. Harmer, Inverse scattering on matrices with boundary conditions, J. Phys. A 38, 4875ˆA–4885 (2005).
+[14]
+M. Kohmoto, T. Koma and S. Nakamura, The spectral shift function and the Friedel sum rule Ann. H.
+Poincare. 14, 1413ˆA–1424 (2013).
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+M. G. Krein, On the Trace Formula in Perturbation Theory, Mat. Sb. 33 (75), 597ˆA–626 (1953).
+[16]
+P. Kurasov, Distribution theory for discontinuous test functions and differential operators with generalized
+coefficients, Journal of Mathematical Analysis and Applications, Volume 201, Issue 1, 297-323, (1996).
+[17]
+S. T. Kuroda, On a generalization of the Weinstein Aronszajn formula and the infinite determinant, Sci.
+Papers Coll. Gen. Ed. Univ. Tokyo 11, 1ˆA–12 (1961).
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+P. ˇSeba, The generalized point interaction in one dimension, Czech. J. Phys., B36, 667ˆA–673 (1986).
+[19]
+M. Usman, A. A. Zaidi, Trace formulas for Schr¨dinger operators on star graphs with general matching
+conditions, J. Phys. A: Math. Theor. 51(36), 365301, (2018).
+[20]
+M. Usman, M.D. Zia, A trace formula, perturbation determinant and Levinson’s theorem for a class of
+quantum star graphs, Eur. Phys. J. Plus 137, 654 (2022).
+[21]
+R. Weder, Scattering theory for the matrix Schr¨odinger operator on the half line with general boundary
+conditions, J. Math. Phys. 56(9), 092103, (2015).
+
+18
+MUHAMMAD USMAN AND MUHAMMAD DANISH ZIA
+[22]
+R. Weder, Trace formulas for the matrix Schr¨odinger operator on the half-line with general boundary con-
+ditions, J. Math. Phys. 57(11), 112101, (2016).
+[23]
+D. R. Yafaev, Mathematical Scattering Theory, Analytic Theory, Mathematical Surveys and Monographs,
+vol. 158. American Mathematical Society, Providence (2010).
+Muhammad Usman, Department of Mathematics, Lahore University of Management Sciences (LUMS),
+Lahore, Pakistan
+Email address: usman@lums.edu.pk
+Muhammad Danish Zia, Department of Basic Sciences, School of Civil Engineering, National University
+of Sciences and Technology (NUST), Islamabad, Pakistan
+Email address: dazia@mce.nust.edu.pk
+

4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf

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+Date of current version January 1, 2023.
+Point Cloud-based
+Proactive Link Quality Prediction
+for Millimeter-wave Communications
+Shoki Ohta1, Student Member, IEEE, Takayuki Nishio1, Senior Member, IEEE,
+Riichi Kudo2, Member, IEEE, Kahoko Takahashi2, Hisashi Nagata2,
+1School of Engineering, Tokyo Institute of Technology, Tokyo 152-8550, Japan
+2NTT Network Innovation Laboratories, NTT Corporation, Yokosuka 239-0847, Japan
+Corresponding author: Takayuki Nishio (email: nishio@ict.e.titech.ac.jp).
+This work was supported in part by JSPS KAKENHI Grant Number JP22H03575.
+This article was presented in part at the 95th IEEE Vehicular Technology Conference (VTC2022-Spring).
+ABSTRACT This study demonstrates the feasibility of point cloud-based proactive link quality prediction
+for millimeter-wave (mmWave) communications. Image-based methods to quantitatively and determinis-
+tically predict future received signal strength using machine learning from time series of depth images
+to mitigate the human body line-of-sight (LOS) path blockage in mmWave communications have been
+proposed. However, image-based methods have been limited in applicable environments because camera
+images may contain private information. Thus, this study demonstrates the feasibility of using point clouds
+obtained from light detection and ranging (LiDAR) for the mmWave link quality prediction. Point clouds
+represent three-dimensional (3D) spaces as a set of points and are sparser and less likely to contain sensitive
+information than camera images. Additionally, point clouds provide 3D position and motion information,
+which is necessary for understanding the radio propagation environment involving pedestrians. This study
+designs the mmWave link quality prediction method and conducts two experimental evaluations using
+different types of point clouds obtained from LiDAR and depth cameras, as well as different numerical
+indicators of link quality, received signal strength and throughput. Based on these experiments, our proposed
+method can predict future large attenuation of mmWave link quality due to LOS blockage by human bodies,
+therefore our point cloud-based method can be an alternative to image-based methods.
+INDEX TERMS LiDAR, link quality prediction, machine learning, millimeter-wave communication, point
+cloud
+I. INTRODUCTION
+W
+ITH the rapid expansion of wireless communica-
+tion applications, the microwave frequency band is
+strained and the utilization of higher frequency bands, such
+as millimeter-wave (mmWave) is underway. mmWave com-
+munication is crucial for extremely high transmission rate
+in the fifth-generation (5G) mobile communication system
+and wireless local area network (WLAN) standard IEEE
+802.11ad/ay because mmWave communication can provide
+wide bandwidth [1]–[4]. This high transmission rate enables
+applications that require significant amounts of traffic, such
+as virtual and augmented realities (VR/AR), environment
+sensing, and ultra-high-definition (UHD) video streaming.
+Thus, mmWave communications greatly increase the possi-
+bilities of wireless communications and are expected to have
+a variety of applications.
+Despite its wide bandwidth, mmWave communication
+has technical challenges, such as sensitivity to line-of-sight
+(LOS) path blockage, radio directivity, significant path loss,
+and narrow beamwidth, owing to its short wavelengths. The
+link quality significantly deteriorates when the mmWave
+LOS path is blocked by a human body or vehicle [5].
+mmWave communications are expected to be used for indoor
+and dense urban environments, such as VR/AR applications
+in private residences, environment sensing and equipment
+control in factories, and UHD video streaming at event
+venues. Such indoor or dense urban environments are com-
+mon in residential or industrial spaces, and the mmWave
+LOS path blocked by human bodies or robots occurs fre-
+VOLUME ,
+1
+arXiv:2301.00752v1  [cs.NI]  2 Jan 2023
+
+Ohta et al.: Point Cloud-based Proactive Link Quality Prediction for Millimeter-wave Communications
+quently. When mmWave is used under these conditions,
+communication disconnections occur frequently and the
+average throughput significantly decreases compared with
+LOS communication. Therefore, it is effective to predict the
+future wireless communication environment and adaptively
+control communications in order to fully utilize the high
+transmission rate of mmWave communications.
+Computer vision-aided (CV-aided) mmWave communi-
+cations are gaining a lot of attention as a solution to
+the mmWave LOS blockage problem [6]–[8]. CV-aided
+mmWave communications employ camera and CV tech-
+niques, including machine learning (ML), to accurately pre-
+dict mmWave link quality (e.g. received signal strength and
+throughput). Camera images may include the geometry and
+dynamics of the mmWave communication radio propagation
+environment, such as locations and movements of obstacles.
+ML algorithms learn the mapping of camera images and
+link quality. Based on the accurate and deterministic link
+prediction, proactive communication controls such as trans-
+mission power control, base station handover, beamforming,
+frequency switching, and intelligent reflecting surface (IRS)
+control [9] are performed. Above proactive communication
+control can mitigate mmWave LOS blockage effects. Our
+previous work demonstrated that a deep learning-based
+method can predict mmWave received signal strength 500 ms
+ahead from depth camera images [7].
+However, images may contain confidential information,
+particularly in private residences, offices, and factories. This
+property limits the application scenarios of the existing
+CV-aided mmWave communication systems that leverage
+cameras. Therefore, alternative sources of information on the
+mmWave communications environment are required.
+This study proposes a link quality prediction method using
+point clouds as an alternative to images. A point cloud
+represents three-dimensional (3D) space as a set of points
+and can be obtained by light detection and ranging (LiDAR),
+or depth cameras. LiDAR estimates the distance to objects
+by measuring the time difference between the emission of
+light and the arrival of the reflected light [10]. Compared
+with images, point clouds are sparse and less likely to
+contain private information [11], [12]. Owing to privacy
+concerns, LiDAR is increasingly being installed in place of
+cameras for sensing. Point clouds have many applications
+such as robot operations [13], autonomous driving [14],
+[15], and digital twin [16], [17]. Wireless communications
+are expected to increase in value by integrating with these
+fields. Further, point clouds acquired from LiDAR are supe-
+rior to images regarding 3D position accuracy and lighting
+robustness. Cameras may not be able to observe objects
+at distant locations or accurately measure distances. Point
+clouds obtained from LiDAR have more detailed coordinate
+information of 3D space than images because the surface of
+an object can be accurately obtained as 3D information [18].
+Cameras are susceptible to sun glare, such as direct light
+and backlight [19], and using them in the dark is difficult.
+Our LiDAR point cloud-based system can operate the link
+quality prediction system without the influence of sunlight or
+lighting. Therefore, point clouds can be used as an alternative
+feature to images in predicting link quality.
+This study focuses on the feasibility of mmWave link
+quality prediction from point clouds. Previous studies [8],
+[20] showed that the link state, i.e., LOS or non-LOS
+(NLOS), can be predicted from the point cloud. However, the
+quantitative prediction of the future received signal strength
+or throughput (e.g. 500 ms or 1000 ms ahead), which enables
+fine-grade link control but is a more challenging task than
+classifying LOS or NLOS, was out of scope. Furthermore,
+the deep learning algorithms employed in the conventional
+image-based prediction [7] cannot be applied to point cloud-
+based prediction owing to the large data domain gap between
+point clouds and images. Therefore, we construct a prepro-
+cessing method of point clouds suitable for the link quality
+prediction, which transforms point clouds into a different
+representation of 3D space, voxel grids. We then selected
+regression ML algorithms for link quality prediction that can
+be applied to voxel grids.
+This study demonstrates the feasibility of the point cloud-
+based link quality prediction by conducting experiments in
+an environment closer to practical environments compared
+with existing study [7]. In terms of experimental equipment,
+commercially available IEEE 802.11ad-compliant devices
+were used for the access point (AP) and the station (STA). In
+terms of link quality, we used two numerical indicators with
+slightly different characteristics: received signal strength
+and throughput. In terms of point clouds, we used two
+types of point clouds with different properties acquired with
+different devices: LiDAR and depth cameras. In addition
+to regression-based loss function, root-mean-squared error
+(RMSE), the empirical distribution function of absolute
+errors was used to quantitatively evaluate the prediction error.
+The contributions of this study are summarized as follows:
+• We demonstrated the feasibility of proactive mmWave
+link quality prediction using point clouds, which rep-
+resent 3D spaces, are sparse and contain less private
+information than camera images through experiments.
+• We constructed a preprocessing method to extract fea-
+tures of point clouds for link quality prediction. We
+selected regression ML algorithms suitable to quanti-
+tatively and deterministically predict link quality and
+compared these methods.
+• We evaluated our point cloud-based method by con-
+ducting two kinds of experiments in indoor environ-
+ments, each with different point cloud acquisition de-
+vices and link quality indicators. One of the experi-
+ments was conducted in a more practical environment
+than that of the previous study [7].
+This study is an extension in terms of the prediction
+method and evaluation of our previous study [21]. In the
+evaluation using the depth camera point cloud and received
+2
+VOLUME ,
+
+signal strength dataset, link quality at more future time
+steps is predicted and prediction errors were evaluated in
+detail using the distribution of prediction errors. Moreover,
+a new mmWave communication experiment using LiDAR
+point cloud and throughput was conducted to evaluate the
+prediction method. These evaluations under two different
+experimental conditions show that our proposed point cloud-
+based link quality prediction method has high prediction
+accuracy and versatility.
+The remainder of this paper is organized as follows.
+Section II describes related works on mmWave link qual-
+ity prediction and point clouds application for wireless
+communication. Section III describes the system model,
+formulation, preprocessing method, and prediction methods
+of our point cloud-based mmWave link quality prediction.
+In Sections IV and V, our proposed method is evaluated
+through two experiments using depth camera point clouds
+and received signal strength datasets, and LiDAR point
+clouds and throughput datasets, respectively. Section VI
+discusses possible approaches other than our method for
+mmWave link quality prediction. Section VII concludes this
+paper.
+II. RELATED WORKS
+This section summarizes existing research on mmWave link
+quality prediction and applications of point clouds for wire-
+less communication. As mentioned in Section I, the link
+quality prediction task is critical in the proactive control of
+mmWave communications. Therefore, various methods have
+been proposed, including vision features such as cameras,
+radar, and LiDAR, as well as various assumed environ-
+ments such as indoor and outdoor. Related works on the
+application of point clouds for wireless communications are
+listed in Table 1. Our proposed point cloud-based mmWave
+link quality prediction method and evaluation approach are
+orthogonal to these studies and increase the potential for
+mmWave communication.
+Numerous studies on link quality prediction has been con-
+ducted [7], [8], [20], [22]–[27]. First, for related works [7],
+[8], [20], [22]–[24] with numerous similarities to this study,
+we demonstrated a comparison in Table 2 and discussed it
+in detail.
+Our previous study [7] used camera images as the key
+enabler of proactive mmWave link quality prediction. Cam-
+era images capture vision information about the environment
+and thus contain information necessary to predict mmWave
+communication LOS path blockage. The mmWave com-
+munication environment was captured by a depth camera,
+and ML was used to predict future received signal strength
+indicator (RSSI) using time series data of depth images.
+Three ML algorithms were used: two neural networks, in-
+cluding convolution and convolutional long short-term mem-
+ory (ConvLSTM) [30], and random forest [31]. Experiments
+are conducted indoors in a scenario in which a 60 GHz
+band mmWave communication LOS path is blocked by
+pedestrians. Experimental evaluation results show that large
+attenuations of the RSSI 500 ms ahead can be predicted. This
+result suggests that ML models can learn the relationship
+between the movement information of obstacles to the LOS
+path in the time series of camera images and future link
+quality. However, as mentioned in Section I, camera images
+may contain private information, such as human faces, texts
+of documents, or screens of computers. Therefore, the image-
+based RSSI prediction system [7] is difficult to apply in
+places with strict privacy-related constraints. In environments
+such as private homes, company offices, and hospitals, non-
+image-based link quality prediction methods can solve this
+problem. We solved this problem by using point clouds,
+which are sparser than images and can rarely identify per-
+sonal and sensitive information.
+Wu et al. [8] proposed a mmWave blockage prediction
+method using ML that converts point clouds obtained from
+a LiDAR observing a mmWave communication environment
+into a heat map image. Point clouds are converted to
+heatmaps by the distance to the reflection point for each
+horizontal angle and arranging them in the time direction.
+Link quality is predicted as a binary value of LOS or NLOS.
+An experiment was conducted in a scenario in which the
+mmWave communication LOS path is blocked by vehicles
+in an outdoor environment. The neural network (NN) model
+including convolution learns the relationship between the
+heatmap image and the binary label. Based on the eval-
+uation results, the system predicts blockages that occur
+within 100 ms with 95% accuracy and blockages that occur
+within 1000 ms with more than 80% accuracy. However,
+this experiment was conducted in an outdoor environment
+using vehicles, and this study does not focus on clarifying
+the possibility of predicting future blockage in an indoor
+environment where the human body blocks the LOS path of
+mmWave communications. Moreover, this method predicts
+whether the communication LOS path is blocked or not
+because this method uses the binary values of LOS or NLOS
+as the link quality, and predicting the degree of link quality
+degradation is beyond the scope.
+Marasinghe et al. [20] proposed a mmWave communi-
+cation LOS path blockage prediction method by detecting
+human position and motion from point clouds and predicting
+future positions. An NN model, including long short-term
+memory (LSTM) [32] was used to predict the future human
+bounding box, and a ray tracing algorithm was used to
+predict blockage. Link quality was predicted as a binary
+value of LOS or NLOS. This method was evaluated through
+computer simulations in a scenario in which the mmWave
+or terahertz (THz) communication LOS path was blocked by
+humans in an indoor environment. The system could predict
+future blockages with an accuracy of 87% while maintaining
+78% precision and 79% recall 300 ms ahead. However, this
+experiment was conducted as a computer simulation, and
+this study does not focus on whether link quality values can
+be predicted with equivalent accuracy in a real-world radio
+VOLUME ,
+3
+
+Ohta et al.: Point Cloud-based Proactive Link Quality Prediction for Millimeter-wave Communications
+TABLE 1. Summary of related works on the application of point clouds for wireless communications
+Existing works
+[7]
+[8]
+[20]
+[22]
+[23]
+[24]
+[25]
+[26]
+[27]
+[28]
+[29]
+Ours
+Link quality prediction
+√
+√
+√
+√
+√
+√
+√
+√
+√
+√
+mmWave
+√
+√
+√
+√
+√
+√
+√
+√
+√
+√
+Point cloud
+√
+√
+√
+√
+√
+√
+√
+√
+√
+Camera image
+√
+√
+Experimental evaluation
+√
+√
+√
+√
+√
+√
+√
+√
+√
+LOS blockage by pedestrians
+√
+√
+√
+√
+√
+Look ahead prediction
+√
+√
+√
+√
+TABLE 2. Comparison of link quality prediction methods for reliable mmWave communications
+Existing works
+[7]
+[8]
+[20]
+[22]
+[23]
+[24]
+Ours
+Vision sensor
+Depth camera
+LiDAR
+LiDAR
+LiDAR
+RGB camera
+mmWave radar
+Depth camera
+& LiDAR
+Raw data
+Depth image
+Point cloud
+Point cloud
+Point cloud
+RGB image
+Point cloud
+Point cloud
+Feature for prediction
+Depth image
+Heatmap image
+Bounding box
+3D histogram Bounding box Link quality map
+Voxel grid
+Frequency band
+60 GHz
+60 GHz
+(mmWave/THz)
+60 GHz
+28 GHz
+60 GHz
+60 GHz
+Evaluation method
+Simulation &
+Experiment
+Experiment
+Simulation
+Simulation
+Experiment
+Simulation &
+Experiment
+Experiment
+Evaluation environment
+Indoor
+Outdoor
+Indoor
+Outdoor
+Indoor
+Outdoor
+Indoor
+LOS path blocker
+Pedestrian
+Vehicle
+Pedestrian
+Vehicle
+Pedestrian
+Vehicle
+Pedestrian
+Prediction formulation
+Regression
+Classification
+Classification
+Classification
+Classification
+Regression
+Regression
+Link quality indicator
+RSSI
+LOS or
+NLOS
+LOS or
+NLOS
+LOS or
+NLOS
+LOS or
+NLOS
+RSSI
+RSSI &
+Throughput
+How far ahead to predict
+500 ms
+1000 ms
+300 ms
+—
+—
+—
+1000 ms
+propagation space. In addition, similar to [8], this system
+used binary values of LOS or NLOS as the link quality.
+Therefore, this method concentrates on demonstrating the
+feasibility of predicting whether the communication LOS
+path is blocked, not the actual attenuation value of the link
+quality in the real-world radio propagation space.
+Klautau et al. [22] proposed a point cloud-based LOS
+blockage prediction method for mmWave beam selection.
+In this method, LiDAR point clouds observed in a wire-
+less communication environment were converted into 3D
+histograms, and LOS probability was inferred using a con-
+volutional NN. This method has been evaluated to dis-
+criminate LOS with a 90% accuracy through simulations
+in a vehicle-to-infrastructure (V2I) scenario. However, this
+method predicts binary labels, LOS or NLOS, therefore this
+method does not focus on the quantitative and deterministic
+prediction of future link quality from LiDAR point clouds.
+Zhang et al. [23] demonstrated a platform for beam
+tracking and blockage prediction, using stereo cameras
+and LiDAR for mmWave communications and frequency
+switching from mmWave to sub-6 just before a blockage
+occurs. In particular, objects blocking the mmWave LOS
+path were detected based on RGB images, and the blockage
+was predicted using recurrent neural network (RNN) from
+the time series of bounding boxes. The transmitter (Tx)
+and receiver (Rx) were simultaneously detected from the
+LiDAR point cloud and used for beam tracking. However,
+this method does not use LiDAR point clouds for blockage
+prediction, and the feasibility of link quality prediction from
+point clouds is beyond the scope of the study. Moreover,
+similar to the aforementioned studies, the prediction of the
+degree of attenuation of link quality values quantitatively and
+deterministically was not discussed because this system uses
+binary values, LOS or NLOS, as the link quality.
+Asano et al. [24] proposed an RSSI prediction method
+of mmWave communications in the 60 GHz band to control
+the timing of communications from the viewpoint of power
+efficiency. In particular, a scenario in which a mmWave
+communication path was blocked by a large vehicle in an
+outdoor environment was assumed, and the vehicle path was
+first estimated theoretically. Subsequently, a dynamic link
+quality map was generated based on the static link quality
+map created in advance through simulations, overriding the
+link quality of the blocked area calculated from the pre-
+dicted position of the vehicle obtained from path prediction.
+Communication timing control was performed based on the
+predicted link quality. The average throughput value was 2.2
+times higher than that of the conventional method. However,
+the method does not focus on whether future link quality
+values can be quantitatively predicted for pedestrians in an
+indoor environment because experiments were conducted in
+an outdoor environment.
+Next, we summarize link quality LOS blockage prediction
+methods enumerated in Table 1 but not in Table 2. Yamazaki
+4
+VOLUME ,
+
+et al. [25] proposed the prediction method of the received
+path power for 25 GHz radio waves communication, instead
+of using images or point clouds. The transfer function is
+divided into a narrower bandwidth, and the first path obtained
+from the narrow-band channel is used to infer the received
+path power. Egi et al. [26] proposed an ML-based estimation
+method to estimate the path loss of 1.8 GHz radio waves from
+satellite images and point clouds in outdoor environments
+where static obstacles, such as trees and buildings block
+signals. J¨arvel¨ainen et al. [27] proposed and experimentally
+evaluated a LOS probability prediction method in outdoor
+environments using accurate environmental point clouds.
+In addition to link quality prediction, applications of
+point clouds for mmWave communications have been pro-
+posed [28], [29]. J¨arvel¨ainen et al. [28] proposed a point
+cloud-based ray-tracing simulation method for indoor prop-
+agation of mmWave channels. St´ephan et al. [29] evaluated
+the influence of the complexity of LiDAR-acquired geo-
+graphic data in point clouds format and propagation models
+in 60 GHz mmWave communications on the prediction of
+60 GHz mesh backhaul links in urban environments. In par-
+ticular, this method was proposed to predict LOS probabili-
+ties for outdoor environments using accurate environmental
+data from point clouds.
+Therefore, many existing studies have indicated the signif-
+icance of point clouds in wireless communications. However,
+these studies do not focus on whether the future link quality
+of mmWave communications, which dynamically changes
+due to human blockage, can be quantitatively and deter-
+ministically predicted from point clouds rather than camera
+images.
+III. POINT CLOUD-BASED LINK QUALITY PREDICTION
+A. SYSTEM MODEL
+The system model of point cloud-based link quality predic-
+tion is shown in Fig. 1. This system consists of APs and
+STAs for mmWave communication, LiDAR or depth camera
+to observe the environment and obtain point clouds, a pre-
+processing unit, a prediction unit, and a network controller.
+On the STA, applications requiring large amounts of data are
+in use, and the AP and STA generate large amounts of traffic
+through mmWave communications. LiDAR acquires and
+transmits point clouds to the preprocessing unit. The Prepro-
+cessing unit reduces data volume and noise of point clouds,
+and converts data format. The details of the preprocessing
+unit are described in Section III-D. The prediction unit infers
+future and current link quality values, such as RSSI or
+throughput, using ML. The details of the prediction unit are
+described in Section III-E. The AP measures and reports the
+link quality value to the network controller. The network
+controller instructs APs to take appropriate communication
+control actions, such as handover, beamforming, frequency
+switching, and IRS control based on the predicted link
+quality and the current link quality before the link quality
+deteriorates significantly due to LOS blockage. Above proac-
+mmWave AP
+LiDAR
+mmWave AP
+Preprocessing Unit
+Prediction Unit
+Communicating 
+with STA 
+mmWave STA
+Not Communicating 
+with STA 
+Network Controller
+Measured current
+link quality
+Predicted future
+link quality
+Predicted current 
+link quality
+Point cloud
+Time-series voxel grid
+Control instructions
+Control instructions
+Pedestrian
+LOS path
+FIGURE 1. The system model of proactive communication control for
+reliable mmWave communication based on link quality prediction using
+point clouds.
+tive communication control enables reliable mmWave com-
+munications. Note that network controls are not discussed in
+this paper because our focus is demonstrating the feasibility
+of link quality prediction. The system model in this study
+is consistent with the system that replaced the camera with
+LiDAR in our previous study [7].
+We assume an indoor scenario such as residences, offices,
+and public facilities. Pedestrians in the mmWave radio
+propagation space move aperiodically, and the mobility is
+observed as point clouds obtained by LiDAR. In mmWave
+communications, link quality (i.e., RSSI and throughput) is
+significantly degraded when the LOS path is blocked by
+pedestrians. This study assumes a simple case in which the
+AP and STA do not move, and the LOS between the AP and
+STA is within the field of view of the LiDAR. Therefore,
+the point cloud is expected to contain the essential visual
+information of mmWave radio propagation in mmWave
+communications.
+B. FORMULATION OF POINT CLOUD-BASED LINK
+QUALITY PREDICTION
+This study aims to quantitatively and deterministically pre-
+dict future link quality (i.e., RSSI or throughput) from time
+series data of point clouds representing radio propagation
+spaces of mmWave communications. This problem can be
+formulated as a regression that maps from the time series
+data of a point cloud to future link quality values. Let n be
+the number of points and pi = (xi, yi, zi) be the coordinates
+of the i-th point, the point cloud P is as follows:
+P =
+n−1
+�
+i=0
+{(xi, yi, zi)} .
+(1)
+Note that the points in the point cloud are in no particular
+order. Let Pt be the point cloud at timestep t, the time series
+data Dt,s of the point cloud for the previous s timesteps at
+some timestep t is as follows:
+Dt,s = (Pt−s+1, Pt−s+2, · · · , Pt) .
+(2)
+Let qt ∈ R be the link quality value at a particular timestep t,
+qt+k represents k timesteps ahead link quality value from a
+VOLUME ,
+5
+
+Ohta et al.: Point Cloud-based Proactive Link Quality Prediction for Millimeter-wave Communications
+particular timestep t. The regression task can be formulated
+as determining the following regression mapping f from the
+time series of point clouds Dt,s for k timesteps ahead of the
+link quality,
+qt+k = f(Dt,s).
+(3)
+C. PREDICTION METHOD DESIGN POLICY
+We use a supervised learning framework to solve the afore-
+mentioned regression task. Generally, supervised learning
+requires a large number of labeled datasets to obtain an
+accurate mapping. In the proposed system, labeling can be
+automatic, using the observed link quality, and constructing a
+large labeled dataset is easy, unlike tasks that require manual
+labelings, such as object detection and segmentation.
+The system using the supervised learning framework re-
+quires two phases: training phase and prediction
+phase. In the training phase, the ML model learns
+the correspondence between data and labels using a labeled
+dataset. In the prediction phase, labels are predicted
+from unlabeled data. In the point cloud-based link quality
+prediction task, data and labels are point clouds and link
+quality values, respectively. The method of generating la-
+beled datasets is described in Section III-D.
+Varieties of supervised learning algorithms exist, and we
+selected the appropriate algorithm for link quality predic-
+tion. A simple approach is to apply deep learning models
+specialized in point clouds, such as PointNet [33], [34] and
+VoteNet [35], which can directly input point clouds. These
+point cloud-based models can directly map from point clouds
+to the target values. However, based on our preliminary
+experiments, these models could not predict link quality.
+Therefore, this study adopted a method that is a 3D extension
+of the method used in the previous study [7]. Specifically,
+point clouds are converted into a data format that can be
+handled by a convolutional neural network (CNN).
+The proposed method consists of preprocessing that con-
+verts point clouds to a voxel format and an ML model to
+learn a mapping from the time series voxel grid to the future
+link quality value such as RSSI and throughput. Section III-
+D and Section III-E describe the preprocessing method and
+ML model, respectively.
+D. PREPROCESSING METHODS FOR POINT CLOUDS
+In the preprocessing unit, downsampling and denoising are
+applied to raw point clouds because raw point clouds ob-
+tained by LiDAR or depth cameras tend to be large in size
+and contain noise. In addition, preprocessing unit converts
+the point cloud data into voxel format so that the CNN-
+based algorithm can be applied. Thus, the proposed prepro-
+cessing method consists of the following six phases; cuboid
+cropping, random downsampling, statistical outlier removal,
+voxelization, time series concatenation, and labeling.
+The first three processes are used to reduce the data vol-
+ume and noise. Cropping removes redundant regions of the
+point cloud. Point clouds obtained from LiDAR sometimes
+cause inaccurate point observations due to the effects of
+reflective objects such as windows and mirrors. Limiting
+the region based on prior geographic knowledge can remove
+such obvious noise and points in regions that are irrelevant
+to sensing. We employed the cuboid cropping method to cut
+out a rectangular region from a 3D point cloud. Specifically,
+for each i-th point pi = (xi, yi, zi) in a point cloud, the point
+is removed when the (xi, yi, zi) coordinates are outside the
+rectangle region [xmin, ymin, zmin] × [xmax, ymax, zmax].
+Downsampling reduces the number of points in point
+clouds and the computational complexity of the following
+process. We employed random downsampling, which ran-
+domly removes points and is computationally less expensive.
+Random downsampling reduces the number of points by
+arranging the points in random order and only leaving the
+points corresponding to the indices up to the product of
+the original point cloud points and the reduction rate rd.
+A tradeoff exists; a low reduction rate reduces the number
+of points, thereby reducing the computational complexity of
+the following process, however, suppose too many points are
+removed, spatial features are also removed. The reduction
+rate rd should be determined by considering the number
+of points in the original point cloud because the number
+of points in the raw point cloud depends on measurement
+devices, such as LiDAR and depth cameras. In this study,
+we treated the reduction rate rd as a hyperparameter and
+experimentally determined it in Section IV and Section V.
+Other efficient determination methods of hyperparameters
+are beyond the scope of this study.
+Outlier removal removes noise points resulting from the
+measurements. Outlier removal enables an accurate under-
+standing of the 3D space. Statistical outlier removal is a
+method of removing points that are far from their neighbors
+by comparing the average distance between all points. Sta-
+tistical outlier removal is employed in this method because
+it can remove outliers independent of the scale of the
+region in which points exist. Two hyperparameters exist for
+statistical outlier removal: the number of nearest neighbor
+points no and the standard deviation ratio ro. First, the
+mean µ and standard deviation σ of the distances between
+all points are calculated. Subsequently, for each i-th point
+pi, the average distance di to no-nearest neighbor points is
+calculated. Finally, if µ+roσ < di, the point pi is removed.
+These two hyperparameters no and ro were experimentally
+determined in this study.
+Voxelization divides the space where point clouds exist
+by voxels, which are 3D extensions of pixels, and arranges
+them into a voxel grid, which is the regular grid in 3D
+space. A voxel grid can be represented as a 3D array on
+the computer, and the voxel grid can be input to the ML
+model proposed in Section III-E. The detailed voxelization
+method is described in Appendix A. The shape of the voxel
+grid is calculated using the voxel size sv and observation
+environment. To convert a point cloud into a voxel grid
+while preserving spatial characteristics, the voxel size sv
+6
+VOLUME ,
+
+must be appropriately determined while considering the
+observation space. In this study, the voxel size sv is treated
+as a hyperparameter and is experimentally determined owing
+to the capability of localizing pedestrians in the experimental
+environment. Open3D [36] and Point Cloud Library [37] are
+used for cuboid cropping, random downsampling, statistical
+outlier removal, and voxelization.
+Subsequently, the voxel grids are concatenated in the time
+direction to generate time series data. As formulated in
+Section III-C, the previous s timestep data is concatenated.
+Specifically, a time series data Dt,s is generated by concate-
+nating 3D arrays representing voxel data generated in the
+previous preprocessing steps. After the concatenation, the
+time series data Dt,s becomes a 4-dimensional array with the
+shape of (s, h, w, d). The parameter s represents the number
+of past frames concatenated for the time series input, and h,
+w, and d represent the height, width, and depth of the voxel
+grid, respectively.
+Finally, we describe the generation of labeled datasets
+for the training phase introduced in Section III-C. We
+used temporal difference labeling proposed in [7] for data
+annotation. Specifically, for all timesteps t, we map the voxel
+grid time series data Dt,s to the k timesteps ahead link
+quality value qt+k; thus, a labeled sample is generated as
+pair like (Dt,s, qt+k). The use of labeled datasets enables
+the training of an ML model that predicts k timesteps ahead
+of future link quality value in Section III-E.
+Table 3 shows an example of preprocessing results for
+the actual experimental data. In this paper, two types of
+point clouds are used for the experimental evaluation: depth
+camera point cloud and LiDAR point cloud. Both point
+clouds are observations of two people blocking LOS paths of
+mmWave communication in an indoor environment. Depth
+camera point clouds tend to have a lot of noise in the
+raw data. LiDAR point clouds tend to have noise points
+at locations that are far outliers. These noises disappeared
+by applying cuboid cropping, random downsampling, and
+statistical outlier removal. After voxelization, both point
+clouds were converted into voxel grids while preserving the
+shape of the human bodies.
+E. MACHINE LEARNING METHODS
+The voxel grid generated by preprocessing is used to train a
+model that maps the voxel grid to the link quality values by
+using ML. The voxel grids contain the location of objects,
+and motion information can also be obtained from the time
+series voxel grid. The location and motion information of
+objects is necessary for link quality prediction. Many ML
+models have already been proposed for the computer vision
+task of extracting spatio-temporal features from voxel grids.
+Among them, we used two ML algorithms: neural network
+(NN) and gradient boosting decision tree (GBDT). Although
+ML algorithms that outperform the algorithms used in this
+study may exist, a comprehensive investigation of the ML
+algorithms and hyperparameter tunings is beyond the scope
+TABLE 3. Examples of preprocessed point clouds and their bounding
+boxes
+Point cloud type
+Depth camera point cloud
+LiDAR point cloud
+Raw point cloud
+After
+cuboid cropping,
+random
+downsampling,
+statistical
+outlier removal
+After voxelization
+of this study, because this study aims to design a mechanism
+to predict the mmWave link quality value and to demonstrate
+its feasibility.
+The structure of the NN is shown in Fig. 2. This NN
+model is a 3D extension of the CNN, which is an ML
+model of the image-based RSSI prediction method [7]. We
+designed an NN model layer architecture based on the
+dense voxel-based method proposed in the field of object
+recognition and robot control [38]. The NN consists of
+the following layers: 3D convolution, rectified linear unit
+(ReLU), max pooling, flattening, dropout [39], and fully
+connected layer. First, 3D convolution is applied to the voxel
+grid to extract spatio-temporal features. Specifically, spatio-
+temporal feature extraction is performed simultaneously,
+while reducing computational complexity, by inputting time
+series into the voxel channels. ReLU and max pooling
+are layers used as a set with the convolution layer. After
+three convolution iterations, the four-dimensional tensor is
+flattened to one dimension. After flattening, a dropout layer
+is inserted to prevent overfitting and improve robustness. The
+NN implementation used Keras [40] in TensorFlow [41]. As
+a supplement, an RNN-based model ConvLSTM [30] has
+also been used in the existing work [7], but it is not employed
+in this study due to its large computational cost.
+Next, we describe the GBDT model. GBDT is an ensem-
+ble learning algorithm that has achieved remarkable results
+in various ML tasks [42]. A similar ensemble learning
+algorithm, random forests [31], was employed in the existing
+work [7]. Time series voxel grids are flattened to one-
+dimensional arrays and then input to GBDT because GBDT
+does not support the input of multidimensional tensors.
+VOLUME ,
+7
+
+Ohta et al.: Point Cloud-based Proactive Link Quality Prediction for Millimeter-wave Communications
+3×3×3 3D Convolution, 64
+ReLU
+2×2×2 3D Max Pooling
+Flattening
+50% Dropout
+Fully Connected, 1
+×3
+Input shape : (𝒉, 𝒘, 𝒅, 𝒔)
+Output : Link quality value
+FIGURE 2. NN structure. Parameters h, w, and d represent the height,
+width, and depth of the voxel grid, respectively, and s represents the
+number of past frames concatenated for the time series input.
+IV. EVALUATION USING DEPTH CAMERA POINT CLOUD
+AND RECEIVED SIGNAL STRENGTH DATASET
+A. DATASETS
+We evaluated our point cloud-based link quality prediction
+method using a depth camera point cloud dataset labeled
+with RSSI values of IEEE 802.11ad mmWave communica-
+tions. The depth camera point cloud dataset was generated
+from the depth image dataset originally created in [7] by
+converting depth images to point clouds using the method
+detailed in Appendix B. Depth camera point clouds represent
+spatial information in a specific direction, similar point
+clouds also can be obtained from solid-state LiDAR [43]. We
+compared the prediction result of our proposed point cloud-
+based method with that of the depth image-based method [7].
+The original dataset (i.e., depth image dataset before
+converting to point cloud) consists of a time series of pairs
+of mmWave RSSI and depth images acquired by a depth
+camera, with the time series measured at 30 frames per sec-
+ond (fps). The experimental environment for obtaining this
+dataset is shown on Fig. 3. The experimental environment
+included an AP, a response STA (R-STA) for communicat-
+ing with the AP, a measurement STA (M-STA) for RSSI
+measurement, and a depth camera at position A or B. The
+AP uses commercially available IEEE 802.11ad-compliant
+products. The M-STA measures the RSSI of IEEE 802.11ad
+frames sent from the AP to the R-STA without being
+affected by the beamforming operation, which varies among
+IEEE 802.11ad products [44]. Details of the experimental
+equipment are summarized in Table 4. The experiment was
+conducted in a room where two pedestrians aperiodically
+blocked the mmWave communication LOS path between the
+R-STA and the M-STA. The pedestrians block the LOS path
+aperiodically and the average interval between blockages per
+person was approximately 6 s. The average LOS blockage
+occurs approximately once every four seconds because two
+people may block the LOS simultaneously. When LOS
+blockage occurs, the RSSI is attenuated by approximately
+15 dBm, which is on the same level as the average value
+of the IEEE 802.11ad channel model, 13.4 dBm [45]. The
+dataset acquired in the situation where the camera position is
+A or B, respectively, is referred to as dataset A or B, respec-
+tively. Sample depth images of the two viewpoints in this
+dataset are shown on the left side of Fig. 3. Measurements
+!	(m)
+&	(m)
+AP
+STA
+MD
+Camera (A)
+Camera (B)
+Pedestrian
+5.34
+4.87
+Moving path
+1.15
+4.40
+3.25
+2.70
+0.60
+1.70
+2.65
+0.45
+1.80
+AP
+R-STA
+Camera (A)
+M-STA
+FIGURE 3. Experimental environment for obtaining original dataset (i.e.,
+depth image dataset before converting to point cloud). The AP and the
+response STA (R-STA) communicate using mmWave, and RSSI is
+measured by the measurement STA (M-STA). The two left images are
+samples of depth images acquired by cameras A and B, respectively.
+TABLE 4. Experimental equipment on the depth camera point cloud and
+received signal strength dataset
+mmWave AP
+Dell Wireless Dock D5000
+mmWave R-STA
+Dell Latitude E5540
+Wireless
+M-STA antenna
+Horn antenna, 24 dBi
+WLAN standard
+IEEE 802.11ad
+Channel
+60.48 GHz
+Bandwidth
+2.16 GHz
+Sensor
+Depth camera
+Microsoft Kinect Model 1656
+Frame rate
+30 fps
+were taken for approximately 10 min for camera positions A
+and B, and approximately 18,000 samples were measured.
+Depth camera point clouds were generated by applying the
+process detailed in Appendix B. Specifically, from depth im-
+ages with the shape of (512, 424) where each pixel represents
+depth values from 0 to 255, normalized point clouds existing
+in the region [0, 256)3 were generated. In this paper, depth
+camera point clouds are the aforementioned normalized
+point clouds. Generated depth camera point clouds contain
+information only in a specific direction because the depth
+camera observes a specific direction.
+B. PREPROCESSING AND MACHINE LEARNING SETUPS
+The preprocessing described in Section III-D is applied
+to the depth camera point cloud to generate the dataset
+corresponding to the time series voxel grids and RSSI values.
+We experimentally determined the values of hyperparameters
+for preprocessing; the values are shown in Table 5. In
+particular, cuboid cropping removes the space in the large
+z-coordinate range and paddings the blank space so that the
+space size returns to [0, 256)3 to remove the noise points in
+the foreground. The number of points in the depth camera
+point cloud is fixed at 217,088, and subsequent preprocessing
+would be time-consuming if the number of points is not
+reduced. Even if the link quality could be predicted 1000 ms
+8
+VOLUME ,
+
+TABLE 5.
+Preprocessing hyperparameters for the depth camera point
+cloud and received signal strength dataset
+Preprocessing phase
+Hyperparameter
+Value
+Cuboid cropping
+(xmin, ymin, zmin)
+(0, 0, 0)
+(xmax, ymax, zmax)
+(256, 256, 244)
+Random downsampling
+rd
+0.0921
+Statistical
+outlier removal
+no
+20
+ro
+2
+Voxelization
+sv
+8
+(i.e., the voxel grid shape is (32, 32, 32))
+Time series
+concatenation
+s
+16
+(i.e., using latest 500 ms time series data)
+k
+0, 15, 30
+(i.e., predicting 0, 500, 1000 ms ahead)
+ahead, if the inference results are not available until earlier
+than 1000 ms, the future prediction will be meaningless.
+The reduction rate for the random downsampling rd was
+experimentally determined to be 0.0921, which leaves ap-
+proximately 20,000 points. The values of no and ro were
+set to 20 and 2, respectively, with reference to Open3D [36]
+default values. The value of sv was set to 8 to be close to
+(40, 40), the input shape for the image-based method [7], to
+obtain a voxel grid with a shape of (32, 32, 32). Examples
+of preprocessing of a depth image point cloud are shown on
+the left column of Table 3.
+These datasets were acquired at 30 fps, and the model was
+input with a tensor that concatenated the voxel grids for
+16 frames, corresponding to the last 500 ms as well as our
+previous study [7]. The model predicts current and future
+values according to the system model shown in Section III-A.
+ML models predict RSSI 0 ms and 500 ms ahead, as well as
+our previous study [7]. In addition, ML models also predict
+1000 ms ahead. Since delays on the order of 100 ms occur
+when streaming UHD videos or VR content from a cloud
+server via the internet, communication control instructions
+can be given with time to spare by predicting 500 ms or
+1000 ms ahead, also taking into account the time of link
+quality inference and communication control. As described
+in Section III-D, temporal difference labeling [7] was used
+to create the time sequential dataset.
+Two point cloud-based ML models, NN and GBDT as
+proposed in Section III-E, were used to predict RSSI. The
+ML hyperparameters are as shown in Table 6. The loss
+function and objective criterion were both RMSE, which can
+be used for regression and has been used in our previous
+study [7]. Default values in Keras [40] and LightGBM [42]
+were used for the learning rate. In addition to these two point
+cloud-based models, two non-point cloud-based methods
+were prepared for comparative evaluation: an RSSI time
+series-based method and a depth image-based method. The
+RSSI time series-based method uses only the time series of
+TABLE 6. Hyperparameters for ML model training
+ML algorithm
+Hyperparameter
+Value
+NN
+Loss function
+RMSE
+Optimizer
+Adam [46]
+Learning rate
+0.001
+Early stopping epochs
+5
+GBDT
+Objective function
+RMSE
+Number of leaves
+31
+Learning rate
+0.1
+Early stopping boosting rounds
+20
+the previous RSSI as features and predicts RSSI values using
+the GBDT algorithm. The depth image-based method is the
+same as [7], and depth images before conversion to point
+clouds are used as the input feature. These two methods are
+the same as point cloud-based methods in that they use the
+latest 500 ms data to predict RSSI.
+We evaluated our method using two datasets, A and B.
+These two datasets both consist of time series data of approx-
+imately 10 min. We used the first 60% for training the ML
+model, the next 20% for validation during model training,
+and the last 20% for holdout validation used for evaluation.
+Cross-validation was not used to prevent data leakage of time
+series data and unbalanced or small training data volume. For
+NN training, we used ReduceLROnPlateau [40], which is a
+learning rate scheduler to quarter the learning rate if the loss
+function value of the validation data did not decrease by two
+epochs.
+C. EXPERIMENTAL RESULTS
+First, we qualitatively evaluated the link quality prediction
+method from a plot of measured and predicted RSSI values.
+Plots of the measured and predicted RSSI values are shown
+in Fig. 4. The RSSI values were predicted from time series
+voxel grids using NN and GBDT, presented in Section III-
+E. The left and right columns are the results for datasets
+A and B, that is, camera positions A and B, respectively.
+The first row is 0 ms ahead, i.e., the current prediction,
+while the second and third rows are future predictions
+500 ms and 1000 ms ahead, respectively. The measured RSSI
+values are significantly attenuated when pedestrians block
+the LOS of mmWave communications. Correspondingly, the
+predicted RSSI values also attenuate significantly, suggesting
+that the model can predict blockage. For camera positions
+A and B, the measured and predicted values appear to
+match in both cases. Comparing the two models, NN and
+GBDT, NN provided a better match better, albeit slightly.
+Based on comparisons of predictions 0, 500, and 1000 ms
+ahead, the discrepancy between the prediction and the actual
+measurement is greater when the prediction is further ahead
+in time. In particular, this tendency can be observed for large
+attenuations, such as LOS blockage, and may occur owing
+to the difficulty in predicting the time further ahead.
+VOLUME ,
+9
+
+Ohta et al.: Point Cloud-based Proactive Link Quality Prediction for Millimeter-wave Communications
+10
+15
+20
+25
+30
+−40
+−35
+−30
+−25
+−20
+Prediction 0 ms ahead for dataset A
+Prediction by NN
+Prediction by GBDT
+Measured value
+10
+15
+20
+25
+30
+−40
+−35
+−30
+−25
+−20
+Prediction 0 ms ahead for dataset B
+Prediction by NN
+Prediction by GBDT
+Measured value
+10
+15
+20
+25
+30
+−40
+−35
+−30
+−25
+−20
+Prediction 500 ms ahead for dataset A
+Prediction by NN
+Prediction by GBDT
+Measured value
+10
+15
+20
+25
+30
+−40
+−35
+−30
+−25
+−20
+Prediction 500 ms ahead for dataset B
+Prediction by NN
+Prediction by GBDT
+Measured value
+10
+15
+20
+25
+30
+−40
+−35
+−30
+−25
+−20
+Prediction 1000 ms ahead for dataset A
+Prediction by NN
+Prediction by GBDT
+Measured value
+10
+15
+20
+25
+30
+−40
+−35
+−30
+−25
+−20
+Prediction 1000 ms ahead for dataset B
+Prediction by NN
+Prediction by GBDT
+Measured value
+Time (s)
+RSSI (dBm)
+FIGURE 4. Predicted and measured RSSI values
+Next, we quantitatively evaluated link quality prediction
+errors. The RMSE was used to evaluate the prediction accu-
+racy, considering the accuracy of the prediction during LOS
+blockage, because the RMSE reflects the effects of outliers.
+The RMSE values in RSSI prediction using four methods
+are listed in Table 7. The top two methods in Table 7 are for
+comparison with point cloud-based methods as described in
+Section IV-B. The bottom two methods in Table 7 are our
+proposed point cloud-based methods proposed in Section III-
+E. Each model inputs 16 frames, corresponding to the last
+500 ms, as input and predicts the link quality values. For
+methods using NN, the average of four training results is
+shown in Table 7 to account for initial value dependence,
+whereas for methods using GBDT, one training result is
+shown because there is no need to consider randomness.
+We evaluated the values in the Table 7, in terms of
+input features and how far ahead to predict. The RSSI time
+series-based method has large error values compared with
+other methods and the LOS path blockage is considered
+unpredictable. This implies that LOS blockage prediction is
+a challenging task that cannot be accomplished from only
+previous link quality values. The point cloud-based method
+could predict 0 ms and 500 ms ahead with approximately
+2.3 dB and 3.2 dB, respectively, with almost the same errors,
+compared with those of existing depth image-based methods.
+Furthermore, for the prediction of 1000 ms ahead, the NN
+model of the point cloud-based method had an error of
+approximately 10% smaller than the depth image-based
+method and the GBDT model of the point cloud-based
+TABLE 7. RMSE values of RSSI prediction
+Ahead
+Feature
+Method
+RMSE (dB)
+Dataset A
+Dataset B
+0 ms
+RSSI time series
+GBDT
+—
+—
+Depth image
+NN
+2.319
+2.612
+Point cloud
+NN
+2.338
+2.374
+GBDT
+2.338
+2.324
+500 ms
+RSSI time series
+GBDT
+5.024
+4.973
+Depth image
+NN
+3.284
+3.594
+Point cloud
+NN
+3.236
+3.333
+GBDT
+3.428
+3.496
+1000 ms
+RSSI time series
+GBDT
+5.068
+5.053
+Depth image
+NN
+4.363
+4.262
+Point cloud
+NN
+3.992
+3.754
+GBDT
+4.311
+4.294
+method. This might be because the convolution layer in the
+NN of the point cloud-based method enables accurate spatio-
+temporal understanding. Accordingly, we conclude that the
+point cloud-based method can predict LOS path blockage as
+well as or better than the depth image-based method.
+Finally, we evaluated the characteristics of the RSSI pre-
+diction error distribution. The empirical distribution function
+of absolute errors and 80th and 95th percentile absolute error
+values were used to evaluate the distribution characteristics
+of the error. The plot of the empirical distribution function
+10
+VOLUME ,
+
+0
+5
+10
+15
+20
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0 (a) Prediction for dataset A by NN 
+Prediction 0 ms ahead
+Prediction 500 ms ahead
+Prediction 1000 ms ahead
+0
+5
+10
+15
+20
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0(b) Prediction for dataset A by GBDT
+Prediction 0 ms ahead
+Prediction 500 ms ahead
+Prediction 1000 ms ahead
+0
+5
+10
+15
+20
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0 (c) Prediction for dataset B by NN
+Prediction 0 ms ahead
+Prediction 500 ms ahead
+Prediction 1000 ms ahead
+0
+5
+10
+15
+20
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0(d) Prediction for dataset B by GBDT
+Prediction 0 ms ahead
+Prediction 500 ms ahead
+Prediction 1000 ms ahead
+Absolute error (dB)
+Empirical distribution function
+FIGURE 5. The empirical distribution function of absolute errors for RSSI
+prediction. Two horizontal dotted lines represent 0.8 and 0.95.
+about absolute errors is shown in Fig. 5. From Fig. 5, the
+80th percentile absolute error value is less than 5 dB for
+all the predictions 0, 500, and 1000 ms ahead, therefore the
+error is concentrated at 5 dB or less. In addition, the 95th
+percentile absolute error value is less than 12 dB for all
+predictions 0, 500, and 1000 ms ahead, therefore few errors
+above 12 dB occurred.
+We discussed the empirical distribution function of abso-
+lute errors from the percentage of LOS blockage time in the
+experiment. In this experiment, as described in Section IV-A,
+the LOS blockage by humans is on average once in 4 s; in
+addition, because the attenuation duration time is mostly 1 s,
+more than half of all timesteps can be considered as LOS
+communication. The RSSI value during LOS communication
+can be assumed as the median of all RSSI values in the
+test set because the RSSI value during LOS communication
+shows almost no fluctuation. From the RSSI value during
+LOS communication, the point in which the RSSI value is at-
+tenuated by more than 8 dBm, which is the 3 dBm attenuation
+presented in the mmWave channel model [45] plus a margin
+of 5 dBm, is considered as blockages. Timesteps with RSSI
+values below -33.05 dBm were assumed to be blockages
+because the median RSSI value in the test dataset was -
+25.05 dBm. In the actual test datasets, the percentages of all
+timesteps with RSSI values below -33.05 dBm were 13.0%
+and 11.9% for datasets A and B, respectively. Considering
+that 95% of the errors are less than 12 dB, and other errors
+not caused by blockage are also included, most of the
+blockage could be predicted, and the model is thought to
+rarely fail to predict complete blockage. This is also because
+the percentage of errors greater than 15 dB is less than 1%,
+which is almost non-existent. Therefore, we conclude that
+our point cloud-based link quality prediction method can
+predict RSSI attenuation due to LOS blockage.
+V. EVALUATION USING LIDAR POINT CLOUD AND
+THROUGHPUT DATASET
+A. EXPERIMENT FOR OBTAINING DATASET
+We evaluated our point cloud-based link quality prediction
+method using a LiDAR point cloud dataset labeled with
+throughput values of mmWave communication. We used
+mechanical rotation LiDAR, which scans spatial information
+in all directions, resulting in 360° point clouds. The transmis-
+sion control protocol (TCP) throughput of IEEE 802.11ad
+mmWave communications was used for link quality. TCP
+throughput values are influenced by many factors, such as
+beamforming, the transmission rate of the AP, and conges-
+tion control of the TCP. In this experiment, these controls
+were often set in motion by LOS blockage, which created
+more dynamics in the behavior of throughput values. We
+conducted such a new mmWave communication experiment
+to obtain the aforementioned dataset consisting of LiDAR
+point cloud for features and throughput for link quality.
+The indoor environment in which the experiment was
+conducted and the equipment and conditions used in the
+experiment are shown in Fig. 6 and Table 8, respectively.
+Both AP and STA used commercially available products.
+Furthermore, a smartphone was used for the STA to make
+the experimental environment more practical. mmWave com-
+munications between the AP and STA used 60 GHz IEEE
+802.11ad WLAN. We used iperf [47] to generate uplink
+TCP traffic, i.e., from the STA to the AP. Throughput was
+calculated using Wireshark [48] as the sum of the length
+of packets obtained by packet capture by tcpdump [49]
+in the last 100 ms. Two pedestrians aperiodically blocked
+the mmWave LOS path between the AP and the STA. As
+with Section IV-A, the two pedestrians block the LOS ap-
+proximately once in 4 s. The experimental environment was
+observed by LiDAR installed in the center of the room and
+obtained as point clouds in 3D Cartesian coordinates with
+the origin at the LiDAR installation position. An example
+of the obtained LiDAR point cloud data is shown in the
+figure on the right column of Table 3. In addition to LiDAR,
+the environment is also observed by a camera set up at the
+edge of the room for comparison with point clouds. The
+experiment was conducted at 10 fps for 30 min and 18,000
+samples were measured.
+B. PREPROCESSING AND MACHINE LEARNING SETUP
+Our proposed point cloud-based link quality prediction
+method was evaluated using LiDAR point clouds and
+throughput values obtained in the aforementioned experi-
+ment. As with Section IV-B, the preprocessing described
+in Section III-D was applied to the LiDAR point cloud
+to generate a voxel grid time series data and throughput
+corresponding dataset. We determined values of prepro-
+cessing hyperparameters, shown in Table 9. First, points
+outside the cuboid region with vertices (-5,-5,-5) and (5,5,5),
+which are obviously noise points outside the room, were
+removed because the size of the indoor environment used
+VOLUME ,
+11
+
+Ohta et al.: Point Cloud-based Proactive Link Quality Prediction for Millimeter-wave Communications
+3.5 m
+5.8 m
+1.5 m
+2.9 m
+1.75 m
+Pedestrian
+Access point
+Station
+LiDAR
+Depth camera
+mmWave
+Moving
+path
+FIGURE 6. Experimental environment for obtaining LiDAR point cloud and
+mmWave TCP throughput dataset
+TABLE 8. Experimental equipment and settings on the LiDAR point cloud
+and throughput dataset.
+Wireless
+mmWave AP
+NETGEAR Nighthawk X10
+mmWave STA
+ASUS ROG Phone
+WLAN standard
+IEEE 802.11ad
+Channel frequency
+60.48 GHz
+Channel bandwidth
+2.16 GHz
+iperf [47]
+Transport layer protocol
+TCP
+Traffic direction
+Uplink (i.e., from STA to AP)
+Sensor
+LiDAR
+Velodyne VLP-16
+Depth camera
+Intel RealSense L515
+Frame rate
+10 fps
+in this experiment was 5.8 m × 3.5 m. In the experimental
+environment, the average number of LiDAR point cloud
+points was 28,826, which allowed subsequent processing to
+be computed without the influence of far-ahead prediction.
+Therefore, random downsampling was not performed and rd
+was set to 1. The values of no and ro were set to 20 and 2,
+respectively, with reference to Open3D [36] default values.
+In the voxelization, sv was set to 0.2 m in order to divide as
+roughly as possible while preserving the human shape. The
+difference from Section IV-B is that cuboid cropping does
+not pad the blank space after cropping, but the space itself
+is narrowed. This is to prevent the large useless calculations
+of the following process caused by padding the blank space.
+The dataset consisted of 18,000 samples of 30 min at
+10 fps. The first 60% was used to train the ML model, the
+next 20% was used for validation during model training,
+and the last 20% was used for holdout validation to evaluate
+our link quality prediction method. The dataset was obtained
+at 10 fps, and features were input to the model for the six
+previous frames, corresponding to the last 500 ms as well as
+our previous study [7]. As with Section IV-B, ML models
+predict 0 ms ahead, 500 ms ahead, and 1000 ms ahead. We
+used temporal difference labeling [7] to create the dataset for
+training the ML model. In addition, we used the same two
+TABLE 9.
+Preprocessing hyperparameters for LiDAR point cloud and
+throughput dataset
+Preprocessing phase
+Hyperparameter
+Value
+Cuboid cropping
+(xmin, ymin, zmin)
+(-5 m, -5 m, -5 m)
+(xmax, ymax, zmax)
+(5 m, 5 m, 5 m)
+Random downsampling
+rd
+1
+Statistical
+outlier removal
+no
+20
+ro
+2
+Voxelization
+sv
+0.2 m
+(i.e., the voxel grid shape is (23, 33, 10))
+Time series
+concatenation
+s
+6
+(i.e., using latest 500 ms time series data)
+k
+0, 5, 10
+(i.e., predicting 0, 500, 1000 ms ahead)
+non-point cloud-based methods for comparison with point
+cloud-based methods described in Section IV-B.
+C. EXPERIMENTAL RESULTS
+First, we evaluated the plot of predicted and actual through-
+put values. Fig. 7 shows an example plot of predicted and
+measured throughput values from two point cloud-based
+ML models, NN and GBDT, which were introduced in
+Section III-E. The throughput value of mmWave during
+LOS communication was approximately 1.6 Gbit/s. When
+the LOS path was blocked by a pedestrian, the throughput
+value attenuated to 0 Gbit/s. When the measured values were
+significantly attenuated, the predicted values were signifi-
+cantly attenuated equally. Two consecutive throughput values
+degradations occurred around 108 s were also able to be
+predicted. Thus, the ML models can predict mmWave LOS
+blockage by the two pedestrians. However, the recovery of
+throughput values when the blockage ends is not predicted
+such as around 102 s. This is because, unlike RSSI, the
+throughput value is determined by a complex interplay of
+factors, such as beamforming, the transmission rate of the
+AP, and congestion control of TCP, and it is challenging to
+predict when the throughput value fully recovers from the
+spatial information of point clouds.
+Comparing the predictions at 0, 500, and 1000 ms ahead,
+the predicted values do not often deteriorate to 0 Gbit/s in
+the predictions at more future timesteps ahead. This may
+result from the difficulty in predicting future blockage, and
+because the prediction model is uncertain of the occurrence
+of blockage, thus outputting the expected value of RMSE
+to be as small as possible in anticipation of the case where
+blockage does not occur. Predictions using the NN model
+differ from that of the GBDT model in that the former
+predicts lower and more accurate values during blockage,
+whereas the latter often predicts smaller values compared
+with the actual values during LOS communication. The
+predictions capture spatial features more accurately owing
+12
+VOLUME ,
+
+95
+100
+105
+110
+115
+120
+125
+0.0
+0.5
+1.0
+1.5
+2.0
+Throughput (Gbit/s)
+Prediction 0 ms ahead
+Prediction by NN
+Prediction by GBDT
+Measured values
+95
+100
+105
+110
+115
+120
+125
+0.0
+0.5
+1.0
+1.5
+2.0
+Throughput (Gbit/s)
+Prediction 500 ms ahead
+Prediction by NN
+Prediction by GBDT
+Measured values
+95
+100
+105
+110
+115
+120
+125
+0.0
+0.5
+1.0
+1.5
+2.0
+Throughput (Gbit/s)
+Prediction 1000 ms ahead
+Prediction by NN
+Prediction by GBDT
+Measured values
+Time (s)
+FIGURE 7. Predicted and measured throughput values
+to using 3D convolution in the NN model. The discrepancy
+during LOS communication by GBDT is thought to result
+from the transmission rate control of AP, which is trained
+to reduce the expected value of the RMSE during LOS
+communication because transmission rate control sometimes
+results in 1.4 Gbit/s, such as in the interval between 120 s
+and 122 s.
+Next, we evaluated quantitatively from the numerical val-
+ues of prediction error. The RMSE values of the throughput
+prediction using four methods are shown in Table 7. As with
+the RSSI prediction comparison described in Section IV-B,
+the top two methods in Table 7 are for comparison with point
+cloud-based methods. Whereas, the bottom two methods in
+Table 7 are our proposed point cloud-based method proposed
+in Section III-E.
+We evaluated the values in the Table 10, in terms of input
+features and how far ahead to predict. Predictions of the
+throughput time series-based have larger error values than
+those of other methods, which may indicate that blockage
+could not be predicted. Similar to RSSI, predicting through-
+put values in a dynamically changing mmWave communica-
+tions environment solely based on previous time series link
+quality values is challenging. For point cloud-based methods,
+the LOS blockage prediction is as accurate as or more
+accurate than the depth image-based method. In particular,
+the LiDAR point cloud is better at predicting 1000 ms ahead.
+This might be because LiDAR point clouds have a wider
+field of view than the depth camera, thus the LiDAR point
+cloud has more information about the space held by the input
+features. The rate of increase in RMSE is lower than that of
+TABLE 10. RMSE values of throughput prediction
+Ahead
+Feature
+Method
+RMSE (Gbit/s)
+0 ms
+Throughput time series
+GBDT
+—
+Depth image
+NN
+0.2771
+Point cloud
+NN
+0.2747
+GBDT
+0.2767
+500 ms
+Throughput time series
+GBDT
+0.4435
+Depth image
+NN
+0.3178
+Point cloud
+NN
+0.2909
+GBDT
+0.2924
+1000 ms
+Throughput time series
+GBDT
+0.4497
+Depth image
+NN
+0.3756
+Point cloud
+NN
+0.3200
+GBDT
+0.3133
+the RSSI and depth camera point cloud datasets. This might
+be because the point cloud acquired from LiDAR can predict
+based on information from a wider field of view; thus, future
+conditions can be predicted more accurately. Therefore, the
+point cloud-based method can predict LOS path blockage
+similar to or better than the depth image-based method,
+especially when predicting blockages at a further time in
+the future.
+Finally, we evaluated the properties of the distribution
+of throughput prediction errors. The empirical distribution
+function of absolute prediction errors is shown in Fig. 8.
+In the experiments, the throughput value during LOS com-
+munication was approximately 1.6 Gbit/s, and when LOS
+blockage occurred, the throughput attenuated to 0 Gbit/s in
+many cases. Therefore, the attenuation due to blockage is
+1.6 Gbit/s. Suppose a blockage was not fully predicted, an
+absolute error of 1.6 Gbit/s occurred. In addition, an absolute
+error of approximately 0.3 Gbit/s occurred owing to the fluc-
+tuation of the transmission rate during LOS communication.
+From Fig. 8, the 80th and 95th percentile absolute error was
+less than 0.4 Gbit/s and 0.8 Gbit/s, respectively for all the 0,
+500, and 1000 ms ahead predictions.
+We discussed the empirical distribution function of abso-
+lute errors from the percentage of the LOS blockage time in
+the experiment as with Section IV-C. In this experimental
+environment, the throughput value was 1.6 Gbit/s during
+LOS communication, which is consistent with the maximum
+throughput value of 1.64 Gbit/s for all timesteps. Timesteps
+with a throughput value of 0.64 Gbit/s or less were con-
+sidered as LOS blockage because blockage causes attenua-
+tion of 1 Gbit/s or more. The percentages of all timesteps
+with throughput values below 0.64 Gbit/s was 11.2% in
+the actual test dataset. Because 95% of the errors are less
+than 0.8 Gbit/s and other errors not caused by blockage
+are included, we can assume that most of the blockage
+can be predicted. This is also because the percentage of
+errors above 1 Gbit/s is less than 1%, which is almost
+VOLUME ,
+13
+
+Ohta et al.: Point Cloud-based Proactive Link Quality Prediction for Millimeter-wave Communications
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(a) Prediction by NN
+Prediction 0 ms ahead
+Prediction 500 ms ahead
+Prediction 1000 ms ahead
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(b) Prediction by GBDT
+Prediction 0 ms ahead
+Prediction 500 ms ahead
+Prediction 1000 ms ahead
+Absolute error (Gbit/s)
+Empirical distribution function
+FIGURE 8. The empirical distribution function of absolute errors for
+throughput prediction. Two horizontal dotted lines represent 0.8 and 0.95.
+negligible. Therefore, we conclude our point cloud-based
+method can predict throughput values for both blockage
+and LOS communication because absolute errors are mostly
+concentrated in the 0.4 Gbit/s or less.
+VI. DISCUSSION
+We discuss other approaches for predicting mmWave link
+quality from point clouds. We employed a simple way where
+the point cloud is converted to the voxel data and well-
+established algorithms (i.e., 3D-CNN and GBDT) are applied
+because this study focuses on demonstrating the feasibility of
+the mmWave link quality prediction from point clouds. This
+method is advantageous in that it uses voxels as the data
+format, which are easy to extend and apply to image-based
+ML algorithms while retaining the 3D structure. However,
+other combinations of data formats and ML algorithms exist.
+We grouped other possible approaches into three categories:
+point clouds, 3D data representations other than point clouds,
+and hand-made features.
+As mentioned in Section III-C, the most straightforward
+approach is to use NN models designed for the point cloud,
+such as PointNet [33], [34] and VoteNet [35], which can
+directly input point clouds and extract features from the
+points. Our preliminary experiments leveraged the existing
+models (i.e., PointNet and VoteNet) for learning the direct
+mapping from point clouds to RSSI. However, they failed
+to predict the large attenuation of link quality induced by
+LOS blockage. This is because the models’ target task and
+required characteristics differ from ours. Generally, these
+existing models are, used for 3D object detection or segmen-
+tation, in which the translation invariant convolution in Point-
+Net properly functions to detect objects regardless of their
+positions. However, in link quality prediction, the positions
+of the objects are essential since mmWave communications
+are significantly affected by the mobility of the obstacles and
+the positions of reflectors. Thus, designing a NN architecture
+suitable for the prediction from point cloud to link quality
+can be a novel challenge in the vision-wireless ML task and
+has the potential to improve prediction accuracy.
+In addition to point clouds, 3D spaces can be represented
+using various data formats, such as meshes, octrees [37],
+and implicit function representations using NN models [50],
+[51]. These data formats can be converted from one format
+to another, although shape information is lost. For example,
+projecting the point cloud onto a fully 2D image [52], [53],
+transforming the point cloud into a pseudo-image [54], [55],
+converting the point cloud into sparse voxels [56] are pos-
+sible. Most methods that utilize these transformations have
+been proposed for robot control and autonomous driving,
+and are identical to the link quality prediction task in terms
+of recognizing and tracking objects in 3D space. Therefore,
+higher accuracy may be achieved or predictions can be made
+with a smaller data volume by taking these data format
+conversions and using ML methods proposed for each data
+format to predict link quality.
+Another approach is to improve the feature engineering
+method in the preprocessing unit. The previously mentioned
+new approaches assumed an end-to-end prediction system
+with trainable parameters, where the input is point clouds
+and the output is a link quality prediction. Alternatively, we
+can extract information from the point clouds through rule-
+based feature engineering and use it to predict link quality.
+For example, existing studies [8], [20], [23] converted to
+bounding boxes and heatmaps. The advantage of improving
+feature engineering is increasing the explainability of the pre-
+diction and reducing the data volume of features. However,
+these feature engineering methods are inferior to end-to-end
+prediction systems, such as the difficulty of inputting data
+into the prediction model when the number of people in the
+environment changes. Therefore, the proposition of a method
+that more effectively balances the trade-offs of mmWave link
+quality prediction with the objectives of the system can be
+a novel challenge.
+VII. CONCLUSION
+In this paper, we demonstrated that point clouds could
+be an alternative to camera images in terms of proac-
+tive link quality prediction for mmWave communications.
+Specifically, a preprocessing method consisting of cropping,
+downsampling, outlier removal, voxelization, time series
+concatenation, and labeling was constructed for the point
+clouds. Subsequently, we selected two ML methods, 3D-
+CNN and GBDT, which can extract spatio-temporal fea-
+tures from time series voxel grid. Our point cloud-based
+link quality prediction method was experimentally evaluated
+using two different numerical indicators of link quality,
+RSSI and throughput, as well as two different point clouds,
+depth camera point cloud and LiDAR point cloud, in a
+scenario where the mmWave LOS path was aperiodically
+blocked by pedestrians. These experimental results revealed
+that our point cloud-based method could quantitatively and
+deterministically predict large attenuation of link quality
+values up to 1000 ms ahead.
+APPENDIX A. VOXELIZATION ALGORITHM
+The detail of the voxelization algorithm is shown in Al-
+gorithm 1. First, the min bounds vector (xmin, ymin, zmin)
+14
+VOLUME ,
+
+and max bounds vector (xmax, ymax, zmax) of all points in
+the point cloud are calculated. Second, the voxel grid shape
+(h, w, d) is calculated and the 3D array V representing the
+voxel grid is initialized with 0. Finally, for each n-th point
+pn, the index (ix, iy, iz) of the corresponding voxel in the
+voxel grid is calculated and the voxel value is updated to 1.
+Algorithm 1 Voxelization
+Input: Number of points in point cloud N
+Input: Point cloud P =
+N−1
+�
+n=0
+{pn}
+Input: Voxel size sv > 0
+Output: 3D array representing the voxel grid V
+1: (x0, y0, z0) ← p0
+2: (xmin, ymin, zmin) ← (x0, y0, z0)
+3: (xmax, ymax, zmax) ← (x0, y0, z0)
+4: for n in 1 to N − 1 do
+5:
+(xn, yn, zn) ← pn
+6:
+xmin ← min(xmin, xn)
+7:
+ymin ← min(ymin, yn)
+8:
+zmin ← min(zmin, zn)
+9:
+xmax ← max(xmax, xn)
+10:
+ymax ← max(ymax, yn)
+11:
+zmax ← max(zmax, zn)
+12: end for
+13: h ←
+�xmax − xmin
+sv
+�
+14: w ←
+�ymax − ymin
+sv
+�
+15: d ←
+�zmax − zmin
+sv
+�
+16: V ← bool[h][w][d]
+17: for ix in 0 to h − 1 do
+18:
+for iy in 0 to w − 1 do
+19:
+for iz in 0 to d − 1 do
+20:
+V [ix][iy][iz] ← 0
+21:
+end for
+22:
+end for
+23: end for
+24: for n in 0 to N − 1 do
+25:
+(xn, yn, zn) ← pn
+26:
+ix ←
+�xn − xmin
+sv
+�
+27:
+iy ←
+�yn − ymin
+sv
+�
+28:
+iz ←
+�zn − zmin
+sv
+�
+29:
+V [ix][iy][iz] ← 1
+30: end for
+31: return V
+▷ ⌊x⌋ represents the greatest integer less than or equal to x.
+▷ ⌈x⌉ represents the least integer greater than or equal to x.
+APPENDIX B. CONVERSION FROM DEPTH IMAGE TO
+NORMALIZED POINT CLOUD
+The conversion procedure from a depth image to a normal-
+ized point cloud is shown in Algorithm 2. In this paper,
+this normalized point cloud is also referred to as a depth
+camera point cloud. Let M be a two-dimensional array
+representing a depth image. Let U and V be the width and
+height of the depth image, respectively, and let [0, D) be
+the range of depth value d. In the depth image used in this
+study, (U, V, D) = (512, 424, 256). A point p is assigned
+in the 3D Cartesian coordinate to each pixel in the depth
+image. This process fits all UV points into the [0, D)3 cubic
+region. All depth camera point clouds have 217,088 points in
+this study. The aforementioned method is applied to all the
+depth images to generate depth camera point clouds. This
+conversion method was constructed based on Open3D [36]
+and Pseudo-LiDAR [57].
+Algorithm 2 Conversion from depth image to normalized
+point cloud
+Input: Depth image M
+Input: Width of the depth image U > 0
+Input: Height of the depth image V > 0
+Input: Maximum value of depth D > 0
+Output: Point cloud P
+1: P ← ∅
+2: cu ← U − 1
+2
+3: cv ← V − 1
+2
+4: for u in 0 to U − 1 do
+5:
+for v in 0 to V − 1 do
+6:
+d ← M[u][v]
+7:
+x ← u − cu
+U − 1 d + D
+2
+8:
+y ← v − cv
+V − 1 d + D
+2
+9:
+z ← d
+10:
+p ← (x, y, z)
+11:
+P ← P ∪ {p}
+12:
+end for
+13: end for
+14: return P
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+Shoki Ohta received the B.E. degree in informa-
+tion and communications engineering from Tokyo
+Institute of Technology in 2022. He is currently
+studying toward the M.E. degree at the School
+of Engineering, Tokyo Institute of Technology. He
+received the IEEE Vehicular Technology Society
+(VTS) Japan Young Researcher’s Encouragement
+Award in 2022. He is a student member of the
+IEEE.
+Takayuki Nishio (S’11-M’14-SM’20) received
+the B.E. degree in electrical and electronic en-
+gineering and the master’s and Ph.D. degrees in
+informatics from Kyoto University in 2010, 2012,
+and 2013, respectively. He had been an assistant
+professor in the Graduate School of Informatics,
+Kyoto University from 2013 to 2020. From 2016
+to 2017, he was a visiting researcher in Wireless
+Information Network Laboratory (WINLAB), Rut-
+gers University, United States. He has been an
+associate professor in the School of Engineering,
+Tokyo Institute of Technology, Japan, since 2020. His current research
+interests include machine learning-based network control, machine learning
+in wireless networks, and heterogeneous resource management.
+Riichi Kudo received the B.S. and M.S. degrees
+in geophysics from Tohoku University, Japan, in
+2001 and 2003, respectively. He received the Ph.D.
+degree in informatics from Kyoto University in
+2010. In 2003, he joined NTT Network Innovation
+Laboratories, Japan. He was a visiting fellow at
+the Center for Communications Research (CCR),
+Bristol University, UK, from 2012 to 2013, and
+worked for NTT DOCOMO from 2015 to 2018.
+He is now working for NTT Network Innovation
+Laboratories. He received the Young Engineer
+Award from IEICE and IEEE AP-S Japan Chapter Young Engineer Award
+in 2006 and 2010, respectively. He is a member of IEICE and IEEE.
+Kahoko Takahashi received the B.S. degree in
+seismology and M.S. degree in Sensory Infor-
+mation Science from Yokohama City University,
+Japan, in 2017 and 2019, respectively. In 2019,
+she joined NTT Network Innovation Laboratories,
+Yokosuka, Japan. Her current research interests
+include machine learning. She is a member of
+IEICE.
+Hisashi Nagata received the B.S. and M.S. de-
+grees in physics from Tokyo University of Science,
+Japan, in 2012 and Osaka University, Japan, in
+2014, respectively. In 2014, he joined NTT Net-
+work Innovation Laboratories, Japan. He worked
+for NTT WEST from 2017 to 2021. He is now
+working for NTT Network Innovation Laborato-
+ries. He is a member of IEICE.
+VOLUME ,
+17
+

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@@ -0,0 +1,1195 @@
+A Simulation Study for the Expected
+Performance of Sharjah-Sat-1 payload
+improved X-Ray Detector (iXRD) in the
+Orbital Background Radiation
+Ali M. Altıng¨un1*, Emrah Kalemci1* and Efe ¨Oztaban1
+1*Faculty of Engineering and Natural Sciences, Sabanciı
+University, Orta Mah. Tuzla, Istanbul, 34956, Turkey.
+*Corresponding author(s). E-mail(s): aaltingun@sabanciuniv.edu;
+ekalemci@sabanciuniv.edu;
+Contributing authors: efeoztaban@sabanciuniv.edu;
+Abstract
+Sharjah-Sat-1 is a 3U cubesat with a CdZnTe based hard X-ray detec-
+tor, called iXRD (improved X-ray Detector) as a scientific payload
+with the primary objective of monitoring bright X-ray sources in the
+galaxy. We investigated the effects of the in-orbit background radia-
+tion on the iXRD based on Geant4 simulations. Several background
+components were included in the simulations such as the cosmic dif-
+fuse gamma-rays, galactic cosmic rays (protons and alpha particles),
+trapped protons and electrons, and albedo radiation arising from the
+upper layer of the atmosphere. The most dominant component is the
+albedo photon radiation which contributes at low and high energies
+alike in the instrument energy range of 20 keV - 200 keV. On the
+other hand, the cosmic diffuse gamma-ray contribution is the strongest
+between 20 keV and 60 keV in which most of the astrophysics source flux
+is expected. The third effective component is the galactic cosmic pro-
+tons. The radiation due to the trapped particles, the albedo neutrons,
+and the cosmic alpha particles are negligible when the polar regions
+and the South Atlantic Anomaly region are excluded in the analysis.
+The total background count rates are ∼0.36 and ∼0.85 counts/s for
+the energy bands of 20 - 60 keV and 20 - 200 keV, respectively. We
+performed charge transportation simulations to determine the spectral
+response of the iXRD and used it in sensitivity calculations as well.
+1
+arXiv:2301.02880v1  [physics.ins-det]  7 Jan 2023
+
+2
+Expected Performance of iXRD in the Orbital Background Radiation
+The simulation framework was validated with experimental studies. The
+estimated sensitivity of 180 mCrab between the energy band of 20 keV
+- 100 keV indicates that the iXRD could achieve its scientific goals.
+Keywords: iXRD, Geant4 simulations, THEBES, Background radiation in
+space, Sensitivity
+1 Introduction
+Sharjah-Sat-1 is a 3U cubesat X-ray satellite being developed in collaboration
+with the Sharjah Academy for Astronomy, Space Sciences, and Technology,
+the University of Sharjah, Istanbul Technical University, and Sabanci Univer-
+sity. Its scientific payload, the iXRD (improved X-Ray Detector), is a CdZnTe
+based hard X-ray detector with the main objective of monitoring bright X-ray
+sources [1]. The iXRD is an improved version of XRD (X-Ray Detector) that
+has been employed as a scientific payload for the 2U BeEagleSAT cubesat[2].
+While the XRD served as a demonstrator, the iXRD is a system with point
+source observing capability and better spectral and noise performance thanks
+to the improvements made in the readout electronics and mechanical design
+and the addition of a collimator [3]. The iXRD has 2.54×2.54×0.5 cm3 pixel-
+lated CdZnTe crystal produced by eV Products (Kromek). It consists of 256
+pixels (16 × 16) with 1.6 mm pitch size and a planar cathode. The energy
+range is from 20 keV to 200 keV, limited by the readout electronics. For read-
+out, RENA 3b ASIC (application-specific integrated circuit) with 36 channels
+is used[4]. Therefore, some of the pixels are connected into groups forming sin-
+gle (one pixel), small (6 pixels), medium (8-9 pixels), and large (10-12 pixels)
+channels. There is a square hole Tungsten collimator over the top of the crystal.
+Its structure also encircles the crystal for additional background protection.
+The collimator provides a 4.26◦ field of view (FoV) which enables the iXRD to
+observe point sources and reduces the cosmic X-ray background substantially.
+In addition, an aluminum plate of 0.3 mm is positioned on top of the colli-
+mator as an optical light blocker. Finally, a 2 mm thick tungsten plate (called
+back-shield) is placed under the crystal serving as a passive shielding against
+all radiation, especially the components arising at Earth’s albedo. A simple
+CAD drawing of the iXRD components is illustrated in Fig. 1. The details of
+the design and on-ground performance of the detector are given in [3].
+The cubesat is planned to be launched into a near-polar sun-synchronous
+orbit (SSO) with an altitude of 500 - 600 km. The harsh radiation envi-
+ronment in the low Earth orbit (LEO) has crucial effects on the operation
+of spacecrafts. High-energy particles and photons decrease the observation
+performance. Moreover, the charged particles cause radiation damage to
+the electronic systems and may degrade CdZnTe crystal properties in time.
+Therefore, understanding the in-orbit radiation effects plays an important
+role in the performance of the iXRD. The background radiation consists of
+
+Expected Performance of iXRD in the Orbital Background Radiation
+3
+Fig. 1 A plain CAD drawing of the iXRD system. Daughterboard carries the RENA and
+associated power supplies and motherboard carries the digital control electronics.
+two main components, prompt and delayed radiation. The prompt radiation
+is due to the trapped charged particles by the Earth’s magnetic field, cosmic
+diffuse gamma-rays, galactic cosmic rays, albedo radiation originating from
+the interactions of cosmic particles with the atmosphere. Long-term irradi-
+ation of the cosmic and trapped protons can produce radioactive isotopes
+within the material of the satellites. The emission from the induced isotopes
+is called delayed background radiation. Considering the size of Sharjah-Sat-1
+and the mission lifetime of 1-2 years, background due to the activation has
+not been considered in this work.
+We performed several Monte Carlo simulations to estimate the background
+effects by using GEANT4 software package [5], which allows us to calculate
+energy depositions in the CdZnTe crystal by the incoming radiation. In this
+way, we determined the contributions of each component to the total back-
+ground level. Also, an optimization study for the back-shield design to mitigate
+the effects of the albedo radiation has been carried out. Finally, by using the
+simulated background rates and charge transportation simulation results, we
+calculated the sensitivity of the iXRD.
+
+> Optical Light Blocker
+Tungsten
+Collimator
+Daughter Board
+CdZnTe Crystal <
+Tunsgten
+Mother
+Mid-Shield
+Board4
+Expected Performance of iXRD in the Orbital Background Radiation
+2 Components of the background radiation
+environment in the LEO
+In our work, we considered several background components such as the cos-
+mic diffuse gamma-rays, galactic cosmic rays, trapped particles due to the
+Earth’s magnetic field, and albedo particles and photons originating from the
+interactions of cosmic particles with the atmosphere. In the following sections,
+we describe the spectrum of each component that is used as an input for the
+Geant4 simulations.
+2.1 Cosmic diffuse gamma-rays
+The first evidence for the cosmic diffuse gamma-rays (CDGR) was reported
+by a rocket-borne large area Geiger counters in 1962[6]. At soft X-ray energies,
+primary contributors are active galactic nuclei (AGNs) [7] and X-ray Binaries
+(XRB) in the distant galaxies [8], while at hard X-ray and gamma-ray energies,
+the contributions are mainly due to the AGNs components, supernovas and
+blazars[9]. The differential flux of the CDGR in the LEO is given by Eq. 1 for
+the energies between 10 keV to 100 MeV[9]. The given flux is in the unit of
+particles/cm2/s/sr/keV.
+f
+�
+E
+�
+dE
+=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+7.877× E−1.29 exp
+� −E
+41.13
+�
+for 3 keV ≤ E ≤ 60 keV
+4.32×10−4 � E
+60
+�−6.5
++ 8.4×10−3 � E
+60
+�−2.58
+for E≥ 60 keV
++ 4.8×10−4 � E
+60
+�−2.05
+(1)
+2.2 Trapped particles
+Unlike the other terrestrial planets in the Solar System, there is an immense
+magnetic field around the Earth which serves as a strong shield for the charged
+particles originating from the Sun and outer space. The particles streaming
+toward the Earth are trapped by the magnetic field and move along the field
+lines for long periods. Although the first significant studies on the existence
+of the trapped charged particles were carried out towards the end of the 19th
+century, it was first confirmed by the Explorer 1 spacecraft equipped with
+a Geiger Muller tube in 1958 [10]. The trapped charged particle zones are
+called Van Allen radiation belts. The Van Allen belts consist of the inner belt,
+which has a population of MeV protons and tens to hundreds of keV electrons,
+and the outer belt which is mostly dominated by MeV electrons [11]. The
+population of the trapped particles is not stable and varies in the orbit due
+to effects such as solar activity and the local changes in the Earth’s magnetic
+field. The South Atlantic Anomaly (SAA) is the area where the magnetic
+
+Expected Performance of iXRD in the Orbital Background Radiation
+5
+field lines dip down to low altitudes, which is surmised to be caused by the
+complicated motion of the molten metals in the Earth’s outer core and the tilt
+of the Earth’s magnetic field axis with respect to the rotation axis. The trapped
+proton population is quite high inside the SAA region. Rigidity of a charged
+particle is a measure of the particle stiffness against the magnetic field to be
+bent. Particles with the same rigidity, charge, and initial conditions will follow
+the same path under a static magnetic field. For each coordinate on the Earth,
+there is a rigidity value, called geomagnetic cut-off rigidity, that describes the
+access of a charged particle to the Earth. Particles with rigidity less than the
+cut-off rigidity can not penetrate the magnetosphere and are deflected. Due to
+the low cut-off rigidities in the polar regions, the trapped electron fluxes can
+increase substantially at high latitudes. Since regular science observations are
+not possible due to high particle fluxes, the iXRD will be inoperative while
+passing through the SAA and the polar regions.
+2.2.1 Trapped electrons
+Trapped electron flux averaged over one year mission was obtained from SPEN-
+VIS by using the AE8 MAX model in the energy range of 40 keV to 7 MeV.
+Average trapped electron exposure on the cubesat along the 1-day orbital
+trajectory is shown in Fig. 2. The squares in the figure indicate the orbital
+positions of the cubesat and each position corresponds to approximately 60
+seconds of flight time. The average electron flux is extremely high in the SAA
+region and at high altitudes. The orbit-averaged energy spectra for the trapped
+electrons excluding and including the high count rate areas are given in Fig. 3.
+It can be easily deduced from the spectra that the effect of trapped elec-
+trons is considerably reduced when the regions with high background rates are
+excluded in the planning of observations.
+2.2.2 Trapped protons
+Average proton flux over one year mission was calculated by using the model
+for the trapped proton population at minimum solar activity, AP8 MIN [12].
+The energy range of the trapped protons is from 100 keV to 400 MeV in the
+LEO. In Fig. 4, average trapped proton exposure on the cubesat along the 1-
+day orbital trajectory is illustrated and one can see that the trapped protons
+are highly populated inside the SAA region. Fig. 5shows the orbit-averaged
+trapped proton flux and the background flux significantly drops when the SAA
+area is excluded.
+2.3 Galactic cosmic rays
+Galactic cosmic rays (GCRs) originate from the outside of the Solar System.
+GCRs consist of approximately 98% high energetic, completely ionized nuclei
+ranging from hydrogen to uranium and around 2% electrons and positrons.
+The nuclei population consists of ∼87% hydrogen ions (protons), ∼12% helium
+nuclei (alpha particles), and ∼1% of much heavier nuclei [13]. In this work,
+
+6
+Expected Performance of iXRD in the Orbital Background Radiation
+)
+-1
+ s
+-2
+AE-8 MAX Integral Flux > 0.04 MeV (cm
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+10
+Longitude
+150
+−
+100
+−
+50
+−
+0
+50
+100
+150
+Latitude
+80
+−
+60
+−
+40
+−
+20
+−
+0
+20
+40
+60
+80
+Fig. 2 The average trapped electron flux to which Sharjah-Sat-1 is exposed during one-day
+orbital trajectory.
+we only considered the hydrogen ions and helium nuclei. The fluxes for the
+GCRs were retrieved from SPENVIS and the ISO-15390 standard model was
+implemented. The GCRs are considerably affected by solar winds. We used
+the following parameters for the ISO-15390 model: mission epoch for solar
+activity, magnetic shielding on, all directions, stormy magnetosphere, Størmer
+with eccentric dipole method, and magnetic field moment unchanged. Fig. 6
+shows the differential fluxes for cosmic protons and alpha particles averaged
+over the spacecraft orbit obtained by the ISO-15390 model1 for the SSO orbit
+at 550 km altitude.
+2.4 Albedo Radiation
+The atmosphere of the Earth obstructs energetic particles from the Sun and
+outer space. From the outer layers, some fraction of the incident particles is
+reflected as well as create secondary (albedo) particles contributing to the
+radiation environment in the Earth’s orbits. In this paper, we considered two
+secondary components, albedo photons, and neutrons. The given fluxes are in
+the unit of particles/cm2/s/sr/keV.
+1https://www.spenvis.oma.be/help/background/gcr/gcr.html#ISO
+
+Expected Performance of iXRD in the Orbital Background Radiation
+7
+Fig. 3 The integral spectra for the orbit-averaged trapped electrons with AE8 MAX model
+including/excluding count rates inside the SAA and polar regions.
+2.4.1 Albedo photons
+Albedo photons are produced as a result of cosmic ray interactions with the
+Earth’s atmosphere, as well as the reflection of the CDGR from the atmo-
+sphere. The photons with energies above 50 MeV are produced by the decay
+of mesons due to hadronic interactions, while the photons with lower energies
+(< 50 MeV) are the production of bremsstrahlung of cosmic electrons and
+positrons with the atmospheric atoms. The details of the model implemented
+in this work are given in [14]. The energy range for the albedo gamma-rays is
+considered between 10 keV and 100 MeV, as with the CDGR.
+f
+�
+E
+�
+dE
+=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1.87×10−2
+�
+E
+33.7
+�−5.0
++
+�
+E
+33.7
+�1.72
+for E ≤ 200 keV
+1.01×10−4 �
+E
+MeV
+�−1.34
+for 200 keV ≤ E ≤ 20 MeV
+7.29×10−4 �
+E
+MeV
+�−2.0
+for E ≥ 20 keV
+2.4.2 Albedo neutrons
+Another component originating from the atmosphere is the albedo neutrons.
+Bombardment of the cosmic rays on the atmospheric nuclei produces neutron
+emission which contributes the background radiation environment in the LEO.
+The energy range for the albedo neutron is 10 keV - 1 GeV in the simulations.
+The spectrum for the albedo neutrons is presented as [14]:
+
+Trapped Electron
+Integral Spectra - SPENVIS
+107
+-Including SAA and Polar Regions
+s-1)
+106
+Excluding SAA and Polar Regions
+105
+104
+103
+Flux
+102
+10
+1
+10-1
+10-2
+10-3
+10-1
+1
+10
+Energy
+(MeV)8
+Expected Performance of iXRD in the Orbital Background Radiation
+)
+-1
+ s
+-2
+AP-8 MAX Integral Flux > 0.10 MeV (cm
+5
+10
+4
+10
+3
+10
+2
+10
+1
+10
+Longitude
+150
+−
+100
+−
+50
+−
+0
+50
+100
+150
+Latitude
+80
+−
+60
+−
+40
+−
+20
+−
+0
+20
+40
+60
+80
+Fig. 4 The averaged trapped proton flux to which Sharjah-Sat-1 is exposed during one-day
+orbital trajectory.
+f
+�
+E
+�
+dE
+=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+9.98×10−8 �
+E
+GeV
+�−0.5
+for 10 keV ≤ E ≤ 1 MeV
+3.16×10−9 �
+E
+GeV
+�−1.0
+for 1 MeV ≤ E ≤ 100 MeV
+3.16×10−10 �
+E
+GeV
+�−2.0
+for 100 MeV ≤ E ≤ 100 GeV
+3 Simulations
+Monte Carlo-based simulation toolkit, Geant4, was employed for the back-
+ground radiation simulations. The Geant4 mass model of Sharjah-Sat-1 used
+in simulations is shown in Fig. 7. For the geometric mass model, all crucial
+components are included, such as the CdZnTe crystal, tungsten collimator
+and tungsten back-shield underneath the crystal, optical light blocker, PCBs,
+aluminum shielding boxes for the electronic components, batteries, and all
+structural supports. Geant4 offers a wide range of physics processes and mod-
+els. Choosing a physics list (physics models for particle interactions with matter
+in Geant4) that is appropriate for the required application is very important.
+In our work, we implemented the Shielding Physics List2 in the simulations.
+2https://www.slac.stanford.edu/comp/physics/geant4/slac physics lists/shielding/physlistdoc.
+html
+
+Expected Performance of iXRD in the Orbital Background Radiation
+9
+Fig. 5 The integral spectra for the orbit-averaged trapped protons with AP8 MIN model
+including/excluding count rates inside the SAA and polar regions.
+Fig. 6 The differential spectra for the orbit-averaged cosmic hydrogen ions (protons) and
+helium ions (alpha particles) calculated by using ISO-15390 model in SPENVIS. The dis-
+continuities in the spectra could be related with the lag of the flux variations of the GCRs
+relative to the solar activity variations.
+This list consists of all necessary electromagnetic and hadronic physics pro-
+cesses required for the simulations of the background radiation environment
+in outer space. For each background component, position and energy informa-
+tion of the particle interactions in the crystal, named as events, were recorded
+for further analysis.
+
+Trapped Proton
+Integral Spectra - SPENVIS
+105
+Including SAA and Polar Regions
+s-1)
+104
+Excluding SAA and Polar Regions
+2
+103
+(cm
+102
+Flux
+10
+Integral
+1
+10-1
+10-2
+10-3
+10-1
+1
+10
+102
+103
+Energy
+(MeV)Differential Spectra of Galactic Cosmic Rays - SPENvIs
+(MeV/n)-1)
+10-3
+- CR Protons
+10-4
+- CR Alpha Particles
+10-5
+s-1
+T-T:
+10-6
+s
+10-7
+(cm
+icle
+10-9
+Part:
+P
+10-10
+1
+10
+102
+103
+104
+105
+Energy
+(MeV)10
+Expected Performance of iXRD in the Orbital Background Radiation
+Fig. 7 The drawing of Geant4 mass model of Sharjah-Sat-1. The side panel that includes
+an aluminum plate, solar panels, and corresponding PCBs is not shown to reveal inner
+structure.
+3.1 Primary particle generation
+Together with the mass model and the physics lists, we need to determine
+the spectral energy distributions, positions, directions, and orientations of the
+primary particles in order to perform simulations in Geant4. We used General
+Particle Source (GPS) class3 for the primary particle specifications. For each
+background component, the spectral models described in Section 2 were given
+3https://geant4-userdoc.web.cern.ch/UsersGuides/ForApplicationDeveloper/html/
+GettingStarted/generalParticleSource.html
+
+Expected Performance of iXRD in the Orbital Background Radiation
+11
+as inputs in Geant4. Since the astrophysics sources can be considered as
+distant point sources, particles emerging from these sources were handled as
+a parallel beam that envelopes the satellite geometry. On the other side, pri-
+mary particles created for the background components were emitted inward
+from a virtual sphere with radius RS with a cosine-law angular distribution.
+The radius of the sphere was set to 90 cm in the simulations in order to
+maintain the cosine distribution of the primary particles. The positions and
+the directions of the primary particles on the sphere were calculated randomly
+by Geant4. The mass model of the cubesat was located at the center of the
+sphere and the top of the collimator was considered to be pointing the zenith.
+The CDGR photons and the GCRs are considered to be coming from outer
+space, so they were radiated from the upper part of the sphere that covers a
+solid angle of 8.64 sr. Regarding the altitude between 500-600 km, approx-
+imately 3.93 sr is occulted by the Earth’s atmosphere. For this reason, the
+albedo particles and photons were emitted from the bottom part of the sphere
+which encases a solid angle of 3.93 sr. In the case of the trapped protons and
+electrons, the particles were radiated the satellite from the solid angle of 4π sr.
+3.2 Normalization
+In order to calculate the detected count rates, we normalized the simulation
+results as follows [14], [15]. The energy spectrum for each background compo-
+nent was divided into N number of bins equal in logarithmic scale. For each
+energy bin width, Ej ( j is from 1 to N), we created Np primary particles,
+from a surface area of A, over a solid angle Ω subtended by the satellite. For
+each component, Np is considered individually in order to obtain good enough
+statistics. Since the differential particle flux unit is defined as particles per
+cm2 per second per steradian per keV, particle rate (P - particles\s) for each
+energy bin Ei can be calculated by integrating the differential flux as:
+Pj =
+�
+dA
+�
+dΩ
+� f
+�
+E
+�
+dE dEj
+(particles
+s−1)
+where dA = R2
+S
+� 2π
+0
+dφ
+� θ
+0 sinθ′dθ′. The angle θ depends on the area of the
+spherical surface that the particles are radiated from. Ω is the solid angle
+of a cone that its apex is the vertex of the incoming particle. dΩ =
+� 2π
+0 dΦ
+� δ
+0 cosδ′sinδ′dδ′ and δ angle ranges from 0 to π/2. In order to decrease the
+CPU time and to increase the simulation statistics, one can use a smaller δ
+angle less than π/2.
+As a result, the observation time in second, T, is obtained by dividing the Np
+by the calculated particle rate per energy bin.
+Tj = Np
+Pj
+
+12
+Expected Performance of iXRD in the Orbital Background Radiation
+The total detected count rate, C, is calculated by the summation of the ratio
+of the number of the deposited particles per energy bin, Mj, to the observation
+time Tj.
+C =
+N
+�
+j=1
+Mi
+Tj
+3.3 Sensitivity Calculations
+Sensitivity is one of the most important parameters of a detector system
+defined as the limiting intensity of the detector for a weak source observation
+at a particular significance level. It depends on the background radiation and
+detector properties such as its effective area, energy resolution, observation
+time, and dead time. The continuum sensitivity can be estimated with the
+following formula [16].
+Fmin = SNR ·
+�
+2.33 · B(E)
+flive · Aeff(E) ·
+√
+T ·
+�
+FWHM(E)
+where Fmin is the sensitivity of the iXRD at a given energy E, SNR is
+the signal-to-noise ratio, B is the simulated background radiation counts cm−2
+s−1 keV−1 at a given energy E, flive is the livetime fraction, Aeff
+is the
+effective area of the iXRD at a given energy E, T is the total observing time
+and FWHM is the full width at maximum for the energy resolution of the
+iXRD at a given energy E.
+For a realistic sensitivity estimation, one has to consider the iXRD spectral
+response which depends on the crystal’s internal properties, the transporta-
+tion of the charge carriers created by the source particles, the electric field
+configuration inside the crystal, and the readout electronic noise. For this rea-
+son, we conducted simulations of the charge transportation inside the CdZnTe
+by using a C++ based simulator THEBES (Transporter of Holes and Elec-
+trons By Electric field in Semiconductor Detectors) that was developed by our
+group. The details of the THEBES simulator can be found in [17]. The total
+background spectrum, B(E), used in the sensitivity calculations was obtained
+by employing the Geant4 simulations and the THEBES simulator together.
+4 Results
+In the following, the optimization study for the back-shield design, the back-
+ground simulation results, count rates for each background component, and
+sensitivity estimation results are presented.
+4.1 Back-Shield Design
+The Tungsten collimator enclosing the CdZnTe crystal helps to minimize the
+effects of the background particles coming from outer space. In addition to
+
+Expected Performance of iXRD in the Orbital Background Radiation
+13
+that, the simulations indicate that another shield is required, especially to
+reduce the effect of albedo radiation from the atmosphere. As it is illustrated
+in Fig. 1, the readout electronics design comprises two PCBs, the motherboard
+and the daughterboard where the daughterboard carries the noise-sensitive
+analog readout circuitry, and the motherboard carries most of the digital and
+communication circuitry [3]. A metal plate acting as a passive shielding (called
+the back-shield) has been placed between those two PCBs. Due to the prox-
+imity of the PCBs, we were limited to using a back-shield with a maximum
+thickness of 2 mm. To achieve optimal performance, we conducted simulations
+to quantify background rates due to albedo photons by using several materi-
+als (W, Pb, Sn, Cd, Cu). The most promising ones out of many configurations
+are a lead (Pb) shield, a Tungsten (W) shield, and three graded-z shieldings.
+The results are presented in Table 1.
+Making use of the back-shield considerably reduces albedo photon count
+rates. For the low energy range (20 - 60 keV), all material configurations yield
+almost similar count rates. However, the 2 mm thick Tungsten provides the
+lowest counting rate for the total energy range of the iXRD. Tungsten was
+preferred as the back-shield, because it not only shows good background reduc-
+tion performance but also was relatively easy to procure compared with the
+graded-Z configurations.
+Table 1 The fraction of count rates for the albedo γ-rays for different back-shield designs.
+For each shielding design, the fractions are given as the count rates obtained using the
+corresponding shielding divided by the count rates with no shield. The thickness of the
+back-shield is 2 mm. For the graded-Z shielding designs, the thicknesses are 0.5 mm, 1.0
+mm, and 0.5 mm respectively.
+Energy Range
+W
+SnWCd
+Pb
+SnWCu
+PbSnCu
+20 - 60 keV
+0.16
+0.17
+0.16
+0.18
+0.21
+20 - 200 keV
+0.14
+0.16
+0.16
+0.17
+0.19
+4.2 Total background spectrum and count rates
+In order to obtain the background in orbit, seven background components were
+considered (see Section 2). Fig. 8 shows the simulated background spectra
+obtained using the deposited energies of the background particles in Geant4
+simulations. It is worth noting that the detector’s spectral response is not
+included in the background spectra shown. The Crab spectrum is also shown.
+A power law distribution with parameters Γ = 2.08 and N = 8.97 (photons
+cm−2 s−1 keV−1) was used to model the Crab spectrum with the energy range
+between 10 keV and 1 MeV. [18]. The CDGR, albedo photons, and the CR
+protons are the most dominant components. The main contributor to the total
+background in the energy range of interest (20 - 200 keV) is the albedo photon
+radiation, which corresponds to roughly ∼40% of the total flux. The energy
+
+14
+Expected Performance of iXRD in the Orbital Background Radiation
+Fig. 8 Spectra of the background radiation components obtained with the help of the
+deposited energies of the background particles in the crystal calculated by Geant4 simula-
+tions. The total background is shown and labeled in black color. The crab spectrum is also
+included in the plot and labeled in black. Tungsten is used as the back-shield material in
+the simulations.
+distribution of the albedo photons is almost flat in the respective energy range.
+Also, the CDGR and the GC protons significantly contribute to the total
+background by around ∼30% and ∼25%, respectively. Between ∼20 and ∼60
+keV, for which a larger number of source photons are expected due to typi-
+cal power-law spectra of hard X-ray sources, the contribution of the CDGR
+photons to the total background is the largest. However, beyond ∼70 keV, the
+CDGR becomes less important than the other most dominant components,
+the albedo photons and the CR protons. Finally, one can see that the contri-
+butions of the albedo neutrons, galactic cosmic alpha particles, and trapped
+particles are insignificant. In the case of the trapped protons and electrons, the
+particle fluxes inside the SAA and the geometric polar regions are excluded
+(see Section 2.2). Further, the characteristic X-ray emissions from the tung-
+sten (K-line emissions at 57.9 keV, 59.3 keV, and 67.2 keV) are very significant
+in the background spectra as well. There are also three distinct lines due to
+
+Background Radiation
+10
+CXB
+AlbedoPhotons
+Crab
+AlbedoNeutrons
+Trapped Protons
+Trapped Electrons
+10-2
+CRProtons
+CRAlphaParticles
+Total Spectrum
+10
+10
+10-
+30
+40
+50
+60
+708090100
+200
+Energy (keV)Expected Performance of iXRD in the Orbital Background Radiation
+15
+the gamma-ray emissions of the excited tungsten nuclei as a result of inelas-
+tic neutron and proton scattering in the spectrum, between 100 keV and 130
+keV[19], [20], but their contribution is insignificant in the overall spectrum.
+The background rates are given in Table 2 for each component. The total
+count rate (for the particles deposited energies greater than 20 keV) is ∼ 7
+counts/s. For the energy range of the iXRD payload (20 - 200 keV), the count
+rate becomes ∼0.85 counts/s, while the rate for the particles with energies
+between 20 keV and 60 keV is approximately 0.36 counts/s. The count rates
+for the trapped protons and electrons are also calculated (Table 3) for the
+SAA and the polar regions both included and excluded. One can see that
+the particle count rates are extremely large inside those areas for operating
+the iXRD. Excluding the high particle-populated regions reduces the trapped
+particle rates significantly. However, the fact that the iXRD will be turned
+off during the passage through the SAA and the polar regions will result in a
+break in scientific observations. The period for one orbit is around 90 minutes
+(∼15 orbits per day). The time period that the cubesat spends inside the SAA
+and the polar regions varies between 30 and 40 minutes. This provides a time
+frame of approximately one hour for the iXRD to operate per orbit. Since the
+power, telemetry, and control constraints allow around 10-minute operation of
+iXRD in each orbit, this break can be managed.
+4.3 Sensitivity
+In order to estimate the sensitivity curve, we performed THEBES simulations
+using the Geant4 background simulation outputs as input. The positions and
+the energy deposition of the electron-hole pairs from Geant4 were imported
+into the THEBES simulator to calculate the induced signals per background
+event. To validate the dedicated Geant4 and THEBES simulations, a compar-
+ative study between an experiment and the corresponding simulation set was
+conducted. We carried out a set of experiments with 57Co (122.1 keV, 136.5
+keV) source for on-ground calibration of the energy response of the iXRD.
+The iXRD system was housed in a metal box. The radioactive source was then
+placed on the top of the box, facing the top of the collimator. The details of the
+calibration setup are reported in [3]. More than 5 ×105 events were registered
+in all channels.
+In the meantime, we modeled the experimental setup in Geant4 as well.
+The Shielding Physics List was chosen and approximately 4.5 ×105 events
+were obtained using a 57Co source. The positions and the deposited energies
+of the electron-hole pairs created by the incident particles were registered. The
+induced signals were calculated with the THEBES simulator for four different
+channel groups as a final step. Fig. 9 illustrates an example from the compar-
+ison results for a single channel. The electronic noise level for the single pixel
+was assigned as 4% and for the planar cathode, the noise level was 6%. The
+simulations are in good agreement with the measurements.
+The THEBES simulation parameters that provided the best simulated 57Co
+spectra compared to the measured ones are given in Table 4.
+
+16
+Expected Performance of iXRD in the Orbital Background Radiation
+Table 2 The background count rates for the corresponding background components in the orbit. The count rates are calculated by using the
+deposited energies of the background particles in Geant4 simulations.
+Energy Range
+CDGR
+Albedo γ
+GCRs p+
+GCRs α
+(0.01-100 MeV)
+(0.01-100 MeV)
+(0.001-100 GeV)
+(0.001-100 GeV)
+(cnt/s)
+(cnt/s)
+(cnt/s)
+(cnt/s)
+≥20 keV
+0.39
+1.58
+2.70
+0.30
+20 - 60 keV
+0.17
+0.12
+0.04
+0.005
+20 - 200 keV
+0.27
+0.37
+0.13
+0.01
+Energy Range
+Trapped p+
+Trapped e−
+Albedo n0
+Crab
+(0.1-400 MeV)
+(0.04-7 MeV)
+(10 keV - 1 GeV)
+(10 keV - 1 MeV)
+(cnt/s)
+(cnt/s)
+(cnt/s)
+(cnt/s)
+≥20 keV
+2.03
+0.01
+0.02
+1.10
+20 - 60 keV
+0.02
+0.002
+0.005
+0.81
+20 - 200 keV
+0.05
+0.005
+0.01
+1.08
+
+Expected Performance of iXRD in the Orbital Background Radiation
+17
+Table 3 The background count rates for the
+trapped particles in the orbit inside and
+outside of the SAA and the polar regions. The
+count rates are calculated for the particles
+that deposit energy greater than 20 keV in the
+detector volume.
+Region
+Trapped p+
+Trapped e−
+(0.1-400 MeV)
+(0.04-7 MeV)
+(cnt/s)
+(cnt/s)
+Including
+SAA and Polar Regions
+9047.7
+1240.0
+Excluding
+SAA and Polar Regions
+2.03
+0.02
+Table 4 The simulation parameters to obtain
+the iXRD background spectrum.
+THEBES Parameters
+Values
+Electron mobility (mm2/V s)
+1.0 ×105
+Hole mobility (mm2/V s)
+1.05 ×104
+Electron trapping time (s)
+3.0 ×10−6
+Hole trapping time (s)
+1.0 ×10−6
+Electronic noise level for anode channels
+4-7%
+Electronic noise level for planar cathode
+6%
+Minimum detectable energy for anodes
+20 keV
+Minimum detectable energy for cathode
+40 keV
+In Fig. 10, the calculated sensitivity curve of the iXRD is presented with
+the Crab spectrum. We considered a background simulation of 9000 seconds
+of integration time (indicating one day) due to the 600 seconds operation time
+of the iXRD in each orbit. The induced background radiation signals for all
+channel groups were then calculated in the THEBES simulator. According to
+the experimental data, the noise levels were set to be 4%, 5%, 6%, and 7% for
+the single, small, medium, and large channels, respectively. The effective area
+of the iXRD was simulated using a circular source radiating a mono-energetic
+beam of photons ranging from 10 keV to 300 keV. Finally, a SNR of 3, a
+live-time fraction of 0.9, and simulated energy resolution values for the energy
+range of 20 keV to 240 keV were considered in the sensitivity calculations.
+The simulations showed that the continuum sensitivity of the iXRD is
+about 180 mCrab (∼0.07 photons cm−2 s−1) between 20 to 100 keV energy
+band. This sensitivity is enough to monitor very bright X-ray sources in the
+Galaxy (such as Cyg X-1 and GRS 1915+105).
+
+18
+Expected Performance of iXRD in the Orbital Background Radiation
+Fig. 9 The comparison of the energy spectra for the simulations and experiments for a
+single channel. For comparative purposes, the simulation result is normalized by utilizing
+the 122 keV peak count from the measurements.
+5 Discussion and Conclusion
+In this paper, we presented a simulation study on the estimation of the effects
+of the background radiation on the performance of the iXRD, a CdZnTe-based
+scientific payload of Sharjah-Sat-1 cubesat. The albedo photon radiation is the
+most dominant and flat in the iXRD energy range, while the CDGR radiation
+contribution is the highest in the energy range of most of the astrophysical
+sources, from ∼20 keV to ∼60 keV. Another dominant component is the galac-
+tic cosmic protons, while the radiation effects of the trapped particles, the
+albedo neutrons, and the cosmic alpha particles are negligible. In addition, we
+did not consider the secondary protons, electrons, and positrons created by the
+interaction of the cosmic rays with the atmosphere since their contributions
+are insignificant.
+The radiation environment in LEO has a dynamical nature and the inten-
+sity of the components can vary considerably. The albedo photon background
+contribution depends on the orientation of the cubesat. For the background
+simulations, the cubesat was oriented to be pointing the zenith, in which the
+albedo radiation would be quite effective. Orienting the cubesat at an angle of
+45◦ relative to the zenith decreases the total count rate for the iXRD energy
+range because less number of photons are registered due to the tilt of the
+detector plane with respect to the zenith, although the fluorescence emission
+increases due to the exposure of the collimator to more albedo photons from
+the side of the crystal. However, a zenith angle of 45◦ reduces the albedo pho-
+ton count rate by only around 5% in the energy band of 20-200 keV. This
+
+140
+Experiment
+120
+Simulation
+100
+Counts
+80
+60
+40
+20
+0
+20
+40
+60
+80
+100
+120
+140
+Energy (keV)Expected Performance of iXRD in the Orbital Background Radiation
+19
+Fig. 10 Simulated sensitivity curve of iXRD with an observation time of 9000 seconds, a
+live-time fraction of 90% and a SNR of 3.
+shows that albedo photons are the most dominant and almost immutable back-
+ground component. The effect of the cosmic rays depends on the geomagnetic
+cutoff rigidity. Due to the low cutoff rigidities in the high geomagnetic lati-
+tudes, the effect of the charged particles becomes more significant [21]. Since
+the operation of the iXRD will be halted during the passage through regions
+with high latitudes, the cosmic ray count rates will also decrease. Therefore,
+the worst-case scenario is considered for the effects of the background radi-
+ation in this work. Also, the total count rate is around 7 counts/s, which is
+critical to determine the live-time fraction of iXRD[3].
+This study does not consider the delayed background radiation due to the
+activation of radioactive isotopes within the satellite material. However, when
+considering large count rates for the trapped particles (see Table 3) during
+the passages through the SAA and the polar regions for the anticipated orbit
+and high-density materials used in the iXRD system, the spectral performance
+of the iXRD can suffer from the emissions from short-lived induced isotopes
+[22]. The trapped proton simulations indicate proton-induced radioisotopes
+with half-lives in the order of minutes such as 109m,107m,105mAg, 104,111mCd
+formed in the CdZnTe crystal and 176W, 177m,179m,183mTa, 170m,172m,173Lu in
+the collimator and the back-shield. Since in each orbit the operation period
+of the iXRD is expected to be around 10 minutes and the time period that
+the iXRD will be outside of the SAA and the polar regions is approximately
+1 hour per orbit (see Section 4.2), the effects of the delayed emission can be
+eliminated by activating the iXRD within a reasonable time after leaving those
+areas. This will be part of the future planning of observations.
+
+iXRD continuum sensitivity (3-sigma in 9000 sec)
+iXRD sensitivity
+1 Crab
+1O
+Intensity (photon/cm?/s/keV)
+180 mCrab
+10-4
+20
+60
+100
+200
+Energy(keV)20
+Expected Performance of iXRD in the Orbital Background Radiation
+The expected detection sensitivity was obtained with the help of the back-
+ground simulations and THEBES charge transportation simulations. It is
+around 180 mCrab between 20-100 keV energy band in one day. The sensitiv-
+ity result will assist us in creating a strategic plan for observations in order to
+attain the scientific objectives of the iXRD.
+Finally, the iXRD was designed and developed from the ground up by our
+group and the simulation studies have provided us a considerable amount of
+information about a detector system that can work in the space environment.
+This great deal of technological know-how will have significant contributions
+for prospective future projects with extensive science goals.
+Acknowledgments.
+The development of iXRD has been supported by the
+University of Sharjah, Sabancı University and T¨ubitak Project 116F151.
+Author Contributions.
+AMA and EK prepared, analyzed, and interpreted
+simulation and experimental data. E¨O prepared Fig. 11 and interpreted data
+for sensitivity calculations. The first draft of the manuscript was written by
+AMA and all authors commented on previous versions of the manuscript. All
+authors read and approved the final manuscript.
+Funding.
+This work is supported by the University of Sharjah, Sabancı
+University and T¨ubitak Project 116F151.
+Data Availability.
+The measured and analyzed data of this work are
+available from the corresponding author upon reasonable request.
+Code Availability.
+The code used for this work is custom-made and
+available from the corresponding author upon reasonable request.
+Declarations
+Conflicts of interest.
+The authors have no relevant financial or non-
+financial interests to disclose.
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+

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+PAPER PREPRINT – CURRENTLY UNDER REVIEW
+Towards Learned Emulation of Interannual Water Isotopo-
+logue Variations in General Circulation Models
+Jonathan Wider1,2*
+,
+Jakob Kruse1,2,
+Nils Weitzel1,3
+,
+Janica C. Bühler3
+,
+Ullrich Köthe2
+and Kira Rehfeld1,3,4
+1Institut für Umweltphysik, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany
+2Interdisziplinäres Zentrum für Wissenschaftliches Rechnen, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany
+3Department of Geosciences, University of Tübingen, Tübingen, Germany
+4Department of Physics, University of Tübingen, Tübingen, Germany
+*Corresponding author. Email: jonathan.wider@ufz.de
+Keywords: Climate Models; Convolutional Neural Networks; Spherical Networks; Paleoclimate
+Abstract
+Simulating abundances of stable water isotopologues, i.e. molecules differing in their isotopic composition, within
+climate models allows for comparisons with proxy data and, thus, for testing hypotheses about past climate and
+validating climate models under varying climatic conditions. However, many models are run without explicitly
+simulating water isotopologues. We investigate the possibility to replace the explicit physics-based simulation of
+oxygen isotopic composition in precipitation using machine learning methods. These methods estimate isotopic
+composition at each time step for given fields of surface temperature and precipitation amount. We implement
+convolutional neural networks (CNNs) based on the successful UNet architecture and test whether a spherical
+network architecture outperforms the naive approach of treating Earth’s latitude-longitude grid as a flat image.
+Conducting a case study on a last millennium run with the iHadCM3 climate model, we find that roughly 40%
+of the temporal variance in the isotopic composition is explained by the emulations on interannual and monthly
+timescale, with spatially varying emulation quality. A modified version of the standard UNet architecture for flat
+images yields results that are equally good as the predictions by the spherical CNN. We test generalization to
+last millennium runs of other climate models and find that while the tested deep learning methods yield the best
+results on iHadCM3 data, the performance drops when predicting on other models and is comparable to simple
+pixel-wise linear regression. An extended choice of predictor variables and improving the robustness of learned
+climate–oxygen isotope relationships should be explored in future work.
+Impact Statement
+Information on the hydrological cycle is imprinted onto the isotopic composition of precipitation, which sub-
+sequently is oftentimes preserved in natural climate archives like speleothems or ice deposits. Some climate
+models, so-called isotope-enabled general circulation models (iGCMs), simulate isotopes explicitly and, thus,
+allow comparing climate model output under paleoclimate scenarios to samples taken from natural climate
+archives. However, isotopes are not included in most climate simulations due to computational constraints or
+the complexity of their implementation. We test the possibility of using machine learning methods to infer
+the isotopic composition from surface temperature and precipitation amounts, which are standard outputs for
+a wide range of climate models.
+© The Author(s) 2023. Currently under review and subject to potential changes. This is an Open Access article, distributed under the terms
+of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and
+reproduction in any medium, provided the original work is properly cited.
+arXiv:2301.13462v1  [physics.ao-ph]  31 Jan 2023
+
+2
+Wider et al.
+1. Introduction
+Reliable analysis of current climate change, as well as robust prediction of future Earth system behavior,
+have become a crucial foundation for all endeavors to protect humanity’s prosperity, mitigate ecological
+disasters, or formulate plans for adaptation (Langsdorf et al., 2022). This analysis hinges on an accu-
+rate understanding and modeling of complex mechanisms in the climate system, which in turn relies
+on knowledge of the system’s past behavior. To analyze past climatic conditions outside the compar-
+atively short period of instrumental measurements, we depend on environmental processes recording
+and preserving information on the climate system in natural “climate archives”. One way to recover
+past climate information from such archives is to measure the relative abundance of isotopes, par-
+ticularly of the isotopes of the constituents of water molecules (Mook, 2000). Due to differences in
+mass, molecules with varying isotopic compositions, so-called isotopologues, differ in their behavior
+in chemical reactions and phase transitions. For the special case of water, molecules containing heavy
+18O atoms, further denoted heavy isotopes, evaporate slower but condensate faster than ones containing
+the lighter 16O. These effects are imprinted on the global hydrological cycle. The resulting patterns of
+the isotopic composition of precipitation depend on many variables such as precipitation amount, tem-
+perature, relative humidity, and the circulation of the atmosphere (Dansgaard, 1964). This makes heavy
+isotopes in water an important tracer of the hydrological cycle and consequently a valuable proxy for
+past climatic changes.
+Isotopic abundances are canonically expressed in the delta notation. For stable oxygen isotopes 18O
+and 16O, this is given by
+𝛿18O =
+� [18Osample]
+[16Osample]
+�
+[18Oreference]
+[16Oreference]
+�
+− 1[�].
+(1)
+Here the ratio of concentrations of the isotopic species in a given sample is compared to a defined
+reference standard. For 𝛿18O of precipitation, this standard is an artificially created sample with an
+isotopic composition that is typical for ocean surface water (Baertschi, 1976).
+One important task in paleoclimatology is to test whether hypotheses about the past climate are
+compatible with proxy data like 𝛿18O measured in natural climate archives (e.g. Bühler et al., 2022).
+To compare simulations of hypothetical climate states to those measurements, a special sub-type of
+climate models, so-called isotope-enabled General Circulation Models (iGCMs), was developed. They
+explicitly simulate isotopic compositions by following the isotopic water species through the hydrolog-
+ical cycle (Brady et al., 2019; Tindall et al., 2009; Yoshimura et al., 2008; Colose et al., 2016; Werner
+et al., 2016). However, many climate models and climate model simulations exist that do not include
+information on water isotopologues. Simulating 𝛿18O is costly because it typically requires duplicat-
+ing large parts of the water cycle for each simulated water species (Tindall et al., 2009). In light of
+recent advances in data science, the question arises whether this isotopic output can instead be emulated
+using machine learning (ML) models that infer the 𝛿18O at each location from other climate variables
+after a model run is finished. We thus call this approach “offline-emulation”. Conducting the emulation
+“offline”, i.e. not coupled to the climate simulation, is possible because isotopes are passive tracers of
+the hydrological cycle that have no feedback onto the climate system.
+Within this study, we narrow the broad task of “offline-emulation” by making a number of choices
+for the learned isotope emulation. The first choice is to only emulate the isotopic composition of pre-
+cipitation. This is the quantity that is also output by iGCMs; it neglects all processes that might disturb
+the signal until it is stored in a climate archive (see e.g. Casado et al., 2018). Because observations of
+isotopes in precipitation are sparse and only exist starting from the 1960s (IAEA/WMO, 2020), we con-
+duct our experiments entirely on simulated data. We limit ourselves to using surface temperature and
+precipitation amount as the two fundamental predictor variables since these variables possess strong
+correlations to 𝛿18O that are well known experimentally (Dansgaard, 1964) and from simulations (see
+Figure 2, C) and are frequently simulated in climate models.
+
+3
+We decide to emulate yearly 𝛿18O data from last millennium (850 CE to 1849 CE) climate simula-
+tions. This is motivated by the combination of the high data availability of simulation runs of sufficient
+length, and the archiving resolution of paleoclimate records during this time period which is typically
+between monthly and sub-decadal. We also contrast the yearly emulation results with experiments using
+monthly resolution.
+As a measure of emulator performance, we will use the 𝑅2 score, which measures the fraction of
+explained temporal variance, as detailed in Section 2.2.5. While we use ML methods that exploit spatial
+correlations in the data by design, we leave explicit incorporation of temporal correlations largely to
+future investigation.
+Working within these constraints, our paper presents the following contributions:
+• We train a deep neural network to estimate stable oxygen isotopes in precipitation (𝛿18O) given
+surface temperature and precipitation and compare to common regression baselines.
+• To respect the underlying geometry of the climate model data, we investigate the performance of
+a spherical network architecture.
+• We present cross-model results, where a regressor trained on simulated data from one climate
+model is used to emulate 𝛿18O in a run from a different model.
+2. Data and Methodology
+Our approach to emulating 𝛿18O is sketched in Figure 1. For each time step, we start with variables
+that we know are statistically related to 𝛿18O, namely surface temperature and precipitation amount.
+All variables are standardized pixel-wise, i.e. we subtract the mean and divide by the standard devi-
+ation, both computed from the training set. We then estimate the standardized spatial field of 𝛿18O
+from the predictor variables by training a machine learning (ML) regression model. Subsequently, the
+standardization for the inferred 𝛿18O is reversed, resulting in our estimate for the isotopic composition.
+2.1. Data
+We use data from the isotope-enabled version of the Hadley Center Climate Model version 3 climate
+model (hereafter iHadCM3, Tindall et al., 2009). iHadCM3 is a fully coupled atmosphere-ocean gen-
+eral circulation model (AOGCM). The horizontal resolution of iHadCM3 is 3.75° in the longitudinal
+direction, and 2.5° in the latitudinal direction. We exclude -90° and 90° from the latitudinal values
+because 𝛿18O is not simulated at these latitudes. We focus on the last millennium (850 CE to 1849 CE),
+which is characterized by a stable climate with variability on interannual-to-centennial timescales, but
+no major trends (Jungclaus et al., 2017). Additionally, the last millennium is well documented in cli-
+mate archives and observations (PAGES2k-Consortium, 2019; Konecky et al., 2020; Comas-Bru et al.,
+2020; Morice et al., 2012).
+Diagnostics of the iHadCM3 data set are visualized in Figure 2. As can be seen from Figure 2 B, the
+standard deviation of the simulated 𝛿18O is large over dry regions like the Sahara desert or the Arabian
+peninsula. This is partly related to the way 𝛿18O is computed in the climate models: in these regions
+the abundances of 18O and 16O are both small because of generally low precipitation amounts, leading
+to numerically unstable ratios and missing values on the monthly time scale. Overall, 0.3% of the 𝛿18O
+values are missing on the monthly timescale, with a strong clustering in the regions with numerical
+instabilities described above (compare Figure A.10). We take this into account by adapting the loss we
+use to train our ML methods to deal with missing values, as described in Section 2.2.3.
+To test the extrapolation and robustness of our emulator, we use last millennium simulations of three
+other climate models: AGCM Scripps Experimental Climate Prediction Center’s Global Spectral Model
+(hereafter isoGSM, Yoshimura et al., 2008), iCESM1 version 1.2 (hereafter iCESM, Brady et al., 2019),
+
+4
+Wider et al.
+Figure 1. Our approach to the emulation of 𝛿18O in precipitation: for each time step, we use surface
+temperature and precipitation amount as predictor variables. Subsequently, the data is standardized
+pixel-wise by subtracting the mean and dividing it by the standard deviation at each pixel (top right).
+Means and standard deviations are based on the training set of the investigated climate model simu-
+lation. We use a machine learning emulation model (ML Regressor) to obtain a standardized estimate
+for 𝛿18O. The emulator output (bottom right) is then de-standardized using the training set mean and
+standard deviation of 𝛿18O at every pixel, to arrive at the final emulation result (bottom left). When
+applying the ML model to data from climate models other than the one that was used for training (e.g.
+in the cross-comparison experiment in Section 3.4) we use the mean and standard deviation from the
+training set of the new model.
+and ECHAM5/MPI-OM (hereafter ECHAM5-wiso, Werner et al., 2016). While iCESM and ECHAM5-
+wiso are fully coupled AOGCMs, isoGSM is an atmospheric GCM forced by sea surface temperatures
+and sea ice distributions of a last millennium run with the CCSM4 climate model (Landrum et al.,
+2013). We re-grid the other climate model simulations to the iHadCM3 grid using bilinear interpolation
+from the CDO tool set (Schulzweida, 2020).
+All data sets are freely available at https://doi.org/10.5281/zenodo.7516327 and described in detail
+in Bühler et al. (2022).1
+2.1.1. Pre-processing
+We apply the following pre-processing steps to the climate simulation data:
+• We set valid ranges for all variables, thereby excluding implausibly large or small values, using
+the following choices: surface temperature range: [173, 373] K, 𝛿18O range: [−100, 100]�,
+precipitation amount: [−1, 10000] mm
+month. Wide ranges are chosen because we aim to exclude only
+implausible values that might deteriorate emulator performance without artificially removing
+model deficiencies. Thus, we also keep small negative precipitation values that climate models
+might produce due to numerical inaccuracies in rare occasions.
+• Time steps with missing values in the predictor variables are excluded from the data set. This
+leads to the exclusion of 31 of the 12000 monthly time steps of iHadCM3.
+1Bühler et al. (2022) also investigate a fifth climate model, GISS ModelE2-R (Colose et al., 2016), which we excluded from
+our study because of physically implausible trends in polar regions in the corresponding model run.
+
+5
+Figure 2. Statistical properties of the iHadCM3 𝛿18O data: (A) mean state of isotopic composition
+(𝛿18O) in the precipitation in precipitation and (B) standard deviation of 𝛿18O on an annual timescale.
+(C) absolute correlations of 𝛿18O with temperature (green) and precipitation amount (brown) on
+interannual timescale; for each grid cell only the stronger of the two is shown.
+• We form yearly averages from monthly data. Missing 𝛿18O data points are omitted in the yearly
+averaging. We argue that this does not impact our results negatively, because the invalid 0.3% of
+𝛿18O values cluster in regions, where due to numerical instabilities in the “ground truth”
+iHadCM3 simulation, learning a physically consistent emulation would not have been possible
+anyway (compare Figure A.10).
+• We re-grid the yearly data sets to the irregular grid on which the investigated spherical network
+operates (see Section 2.2) using a first-order conservative remapping scheme (Schulzweida,
+2020).
+• We split the data into test and training sets. We use 850-1750 CE for training and 1751-1849 CE
+for testing. The data are split chronologically instead of randomly to make the test and training
+set as independent as possible, and prevent the network from exploiting auto-correlations from
+previous or subsequent time steps. If a validation set (used for making choices of ML
+hyperparameters) is needed, we split off 10% of the training set randomly unless specified
+otherwise.
+• Before the ML methods are applied, the data are standardized pixel-wise by subtracting the
+training set mean and dividing by the standard deviation of the corresponding climate model, as
+visualized in Figure 1.
+2.2. Methodology
+To obtain a spatially consistent emulation, and to utilize the fact that the local statistical relations
+between 𝛿18O and the predictor variables are similar in many grid boxes on Earth’s surface, we choose
+
+Mean state, 5180 [%o]
+Standard deviation, §18o [%o]
+(A)
+(B)
+-30
+-24
+-18
+-12
+9-
+0
+6
+0
+1
+2
+3
+4
+5
+6
+7
+Absolute correlation
+(C)
+0.00
+0.25
+0.50
+0.75
+1.00
+Temperature
+0.00
+0.25
+0.50
+0.75
+1.00
+Precipitation amount6
+Wider et al.
+two approaches based on Convolutional Neural Networks (CNNs). Both utilize the successful UNet
+architecture (Ronneberger et al., 2015), whose multi-scale analysis can simultaneously capture fine
+structure variations and utilize large-scale contextual information. UNet architectures have been suc-
+cessfully applied in a climate science context before (e.g. Kadow et al., 2020). The first of our two
+approaches treats data on the latitude-longitude grid as a flat image. The second explicitly incorporates
+the spherical geometry of the data.
+2.2.1. Flat network
+Because our data naturally lie on the surface of a sphere, distortions arise when treating the equally
+spaced longitude-latitude grid as a flat image using e.g. a plate carrée projection (lat/lon projection).
+We test if we can still obtain reasonable results with this naive setup. Furthermore, we try to par-
+tially remedy the effects of the distortions within the “flat” approach, by modifying the standard UNet
+architecture in three ways :
+• We use area-weighted loss functions.
+• We use periodic padding in the longitudinal direction, i.e. we append the rightmost column to the
+very left of the plate carrée map (and vice versa) before computing convolutions. Thereby we
+assure continuity along the 0°-360° coordinate discontinuity.
+• We incorporate CoordConv (Liu et al., 2018), a tweak to convolutional layers that appends the
+coordinates to the features input into each convolution, thus, allowing networks to learn to break
+translational symmetry if necessary.
+2.2.2. Spherical network
+As a more sophisticated technique, a multitude of approaches to directly incorporate the spherical
+nature of data into a neural network architecture has been proposed (Cohen et al., 2019, 2018; Coors
+et al., 2018; Defferrard et al., 2020; Esteves et al., 2018; Lam et al., 2022). We reproduce the approach
+of Cohen et al. (2019), where the network operates on an icosahedral grid, with grid boxes centered on
+the vertices. Using the icosahedron offers a straightforward way to increase or decrease resolution for a
+UNet-like design, as we can recursively subdivide each of its triangles into four smaller triangles, pro-
+jecting all newly created vertices onto the sphere again. We denote the number of recursive refinements
+of the grid as 𝑟, with 𝑟 = 0 identifying the grid containing only the twelve vertices of the regular icosa-
+hedron. As the refined icosahedral grid is locally very similar to a flat hexagonal grid, we can use an
+appropriately adapted implementation of the usual efficient way to compute convolutions. Additionally,
+the architecture of Cohen et al. (2019) is equivariant to a group of symmetry transformations, meaning
+that if the input to the CNN is transformed by an element of the symmetry group, the output transforms
+accordingly. This fits well with the approximate symmetries present in the Earth system, like symmetry
+to reflections on the equatorial plane or rotations around the polar axis. We validate our implementation
+of the method on a toy problem described by Cohen et al. (2019): the classification of handwritten dig-
+its projected onto a spherical surface. We obtain results that are comparable to those reported by Cohen
+et al. (2019), see Appendix A.1.1 for more details.
+2.2.3. Loss function
+To train our UNet architectures for isotope emulation, we use a weighted mean squared error loss
+between the standardized 𝛿18O ground truth 𝑌 and the predicted values ˆ𝑌:
+𝐿(𝑌, ˆ𝑌) = 1
+𝑏
+𝑏
+∑︁
+𝑖=1
+1
+|G𝑖|
+|G𝑖 |
+∑︁
+𝑗 ∈G𝑖
+𝑤 𝑗
+�
+𝑌𝑖, 𝑗 − ˆ𝑌𝑖, 𝑗
+�2
+,
+(2)
+where the loss is averaged over a batch of size 𝑏 and the set of valid grid boxes G𝑖 at time step 𝑖. A grid
+box is valid if the simulated ground truth data has no missing value at this time step in this grid box. |G𝑖|
+denotes the cardinality of G𝑖 and 𝑤 𝑗 are weighting coefficients. For the convolutional UNet working on
+
+7
+the plate carrée projection, we choose 𝑤 𝑗 to be proportional to the cosine of the latitude of the center
+of grid cell 𝑗, which is an approximation of the physical size of the grid cell. We rescale the weights,
+such that they sum to the total number of grid boxes. For the icosahedral UNet, all grid boxes are of
+approximately equal size. Therefore, no weighting is applied and 𝑤 𝑗 is a constant independent of 𝑗.
+2.2.4. Baselines
+In addition to the UNet models, we implement three simple baseline models to assess the relative benefit
+of complex and deep models in our emulation problem. These baselines are:
+• Grid-box-wise linear regression, the simplest conceivable model: regress 𝛿18O on temperature
+and precipitation amount in a separate model for each grid box.
+• Grid-box-wise random forest regression model: in contrast to the linear regression baseline, we
+train a single random forest (Breiman, 2001) to make predictions on all grid boxes. To allow the
+model to learn spatially varying relationships, we include the coordinates as predictor variables.2
+• Grid-to-grid approach (PCA regression): relations between 𝛿18O and other climatic variables
+tend to behave similarly over large areas (see Figure 2 C), justifying a dimension reduction of the
+input and output spaces before applying a multivariate linear regression. This is implemented by
+computing the principal components of the input and output spaces. Schematically, the
+computation goes as follows: 𝑋
+PCA𝑋
+↦→ 𝐶𝑋
+lin. reg.
+↦→
+ˆ𝐶𝑌
+PCA−1
+𝑌
+↦→
+ˆ𝑌. Approximately optimal numbers of
+principal components are obtained as follows: we iterate over a 50 × 50 logarithmically spaced
+grid of candidate values for the number of input and output principal components. For each
+configuration, the emulation model is trained and its performance is measured on a held-out
+validation set. We then select the combination of numbers of input and output principal
+components which yields the best results on the validation set. As a last step, the selected model
+is retrained, now including the validation set data. Principal component analysis can be
+performed on arbitrary grids, which makes it equally applicable to the projected 2D data and the
+icosahedral representation.
+2.2.5. Metrics
+The metric we use for evaluating emulation approaches is the 𝑅2 score, also called the “coefficient of
+determination”, which quantifies what fraction of the temporal variance in the test set is explained by
+the ML estimate in each grid box. The 𝑅2 score compares the 𝛿18O ground truth 𝑌𝑗 and an estimate ˆ𝑌𝑗
+in a given grid box 𝑗 as
+𝑅2(𝑌𝑗, ˆ𝑌𝑗) = 1 − MSE(𝑌𝑗, ˆ𝑌𝑗)
+𝜎2
+𝑗
+,
+(3)
+where MSE(𝑌𝑗, ˆ𝑌𝑗) is the mean squared error and 𝜎2
+𝑗 the variance of the test set ground truth, both
+taken over the time axis at grid box 𝑗. A value of 𝑅2 = 1 indicates perfect emulation, while a model
+that simply outputs the temporal mean at every time step has 𝑅2 = 0. The score can become arbitrarily
+negative.
+Additionally, we compute the Pearson correlation coefficient between the true and emulated time
+series at some grid boxes. To select time steps in which a method’s performance is particularly strong
+or weak, we calculate the Anomaly Correlation Coefficient (ACC) between emulation and ground truth.
+ACC is defined as the Pearson correlation coefficient between the true and emulated anomaly patterns
+for a given time step. Anomalies are computed with respect to the training set mean.
+If error intervals on performance metrics are given, unless stated otherwise, they are 1𝜎 intervals
+computed over a set of ten runs. Thus, the uncertainties only account for the uncertainty of the stochastic
+aspects of the ML model parameter optimization, discarding any uncertainty that is related to the data.
+2We encode each longitude 𝜙 as two values sin(𝜙) and cos(𝜙) to avoid the discontinuity at 0°/360°.
+
+8
+Wider et al.
+Implementation details for training and configuration of the ML methods are provided in
+Appendix A.1 and code to reproduce our experiment is freely available at https://github.com/
+jonathanwider/isoEm
+3. Results
+We structure the Results section as follows. First, we give a detailed spatiotemporal overview to illus-
+trate the characteristics of the ML-based emulation results. To this purpose, we use the best performing
+emulation method as an example. Subsequently, we compare emulation methods amongst each other,
+contrasting deep architectures and baselines as well as “flat” and “spherical” approaches. We follow up
+with a range of sensitivity experiments, and conclude by conducting a cross-model experiment, i.e. we
+train a ML model on data from one climate model and then use the trained model to emulate 𝛿18O in
+other climate model simulations.
+3.1. Spatiotemporal Overview of Emulation Results
+In Section 3.3, we will discover that the best performing ML emulation method, a deeper version of
+the flat UNet architecture, reaches an average 𝑅2 score of 0.389 ± 0.006 on the plate carrée grid. This
+means that in the global average almost 40% of the temporal variance in the test set is explained by our
+emulation on the interannual timescale. We use this best ML method to introduce spatial and temporal
+characteristics of the emulation.
+The prediction quality varies spatially, as shown in Figure 3A. 𝑅2 scores of 0.6 or larger are reached
+in 18.5% of the grid cells, and 𝑅2 ≤ 0 for only 5.4% of grid cells. The best results are achieved
+over tropical oceans, which are regions with strong correlations of 𝛿18O and precipitation amounts.
+Performance is good over large parts of the Arctic and over western Antarctica as well, which is impor-
+tant because these regions are especially relevant for the comparison with 𝛿18O measurements from
+ice cores. We illustrate the performance in these regions by comparing emulated and ground truth
+time series in the grid boxes closest to two ice core drilling sites in panels B and C of Figure 3: the
+North Greenland Ice Core Project (“NGRIP”, 75.1° N, 42.3° W, Berggren et al., 2009) and the West
+Antarctic Ice Sheet Divide ice core project (“WAIS Divide”, 79.5° S, 112.1° W, Buizert et al., 2015).
+For these drilling sites, the correlation between our emulation and the exact output time series of an
+isotope-enabled climate model exceeds 70%.
+In general, spatial variations in performance follow the correlation structure between 𝛿18O and the
+predictor variables (Figure 2C): in regions with strong absolute correlations between 𝛿18O and surface
+temperature or precipitation amount, the 𝑅2 scores are higher than in regions where none of the predic-
+tor variables is strongly correlated with 𝛿18O. Thus, performance is worse over landmasses, especially
+in the low and mid-latitudes.
+Next, we visualize emulation and climate model output for individual time steps. For a year with typ-
+ical emulator performance3, we plot emulated (panel A) and simulated (panel B) anomalies in Figure 4.
+We can see that the large-scale patterns match well between emulation and simulation: there are strong
+positive anomalies over the Arctic, related to positive temperature anomalies in this time step, and the
+large-scale structure over the Pacific is captured as well. Strong negative anomalies over parts of South
+America and northern India and Pakistan are reproduced. Emulation and ground truth simulation dif-
+fer in their fine-scale structure: the ground truth is generally less smooth than the emulation and seems
+particularly noisy over some dry regions like the Sahara and the Arabic desert. In these regions, there
+is a potential for numerical inaccuracies in the isotopic component of climate models, due to small
+abundances of each isotopic species, and it is hard to untangle which parts of the “noisy” signal have
+a climatic origin and which parts are simulation artifacts. A part of the overall smoother nature of the
+UNet regression results can be attributed to the MSE Loss giving a large (quadratic) penalty for strong
+3We chose the median in terms of anomaly correlation coefficient (ACC).
+
+9
+Figure 3. Test set emulation performance of the best ML emulation method. The bluer the colors, the
+better the emulation. Blue colors indicate regions in which the performance is better than a trivial
+baseline model that returns the correct test set mean at every time step. This plot displays the average
+of the 𝑅2 scores over ten runs. Additionally, we show the time series of the ML emulation (green, mean
+over ten runs) and the true simulation data (black) for grid boxes next to two ice core drilling sites.
+Panel (B) "NGRIP" (Greenland). Panel (C) "WAIS Divide" (West Antarctica).
+deviations from the true values, thus, priming the network against predicting values in the tails of the
+distribution. Figure A.4 compares emulation and simulation for three additional time steps: time steps
+in which the emulation works particularly well or poorly, and a climatically interesting year: 1816 CE,
+the “year without a summer” (Luterbacher and Pfister, 2015), which is caused by a volcanic eruption
+included in the volcanic forcing of the iHadCM3 simulation. For 1816 CE, we observe that the emu-
+lator reproduces a strong negative 𝛿18O anomaly in regions where 𝛿18O is primarily influenced by
+temperature, namely in the Arctic, Northern North America and Siberia.
+3.2. Comparing Machine Learning Methods
+The ML emulation models (UNet architectures and simpler baselines) differ in the quality of their
+emulation. In the following, we compare the methods amongst each other. For details on the training
+procedures, network architectures, and method implementations, see Appendix A.1. We also address
+the question of whether using an inherently spherical approach is beneficial over treating the latitude-
+longitude grid as “flat”. However, the comparison is not trivial: the approaches are developed for data
+
+iHadCM3, R2score
+(A)
+0.8
+0.4
+score
+0.0
+2.
+R
+-0.4
+-0.8
+Time series at WAls Divide
+Time series at NGRIP
+(B)
+28
+32
+30
+34
+[%] Ogt9
+-32
+-36
+-38
+-34
+Corr: 0.79 +/- 0.01
+Corr: 0.70 +/- 0.01
+0
+20
+40
+60
+80
+100
+0
+20
+40
+60
+80
+100
+timestep in test set
+timestep in test set10
+Wider et al.
+Figure 4. Typical emulation results on iHadCM3 data set: We show anomalies as they are output by
+the ML-emulator (“Emulation”) and the “true” result in the simulation data set (“Ground truth”). The
+anomalies are computed with respect to the training set mean. For the selected time step, the anomaly
+correlation coefficient (ACC) reaches its median value.
+Table 1. Globally averaged 𝑅2 scores for the different ML-emulation methods. Results are calculated for
+the icosahedral grid that the method of Cohen et al. (2019) operates on and the plate carrée grid. When a
+method works with data on the other grid, the emulated data is interpolated. "Flat UNet, unmodified" and
+"Flat UNet, modified" refer to the flat network architecture described in Section 2.2.1, either not applying
+or applying the modifications to remedy projection artifacts described in that chapter.
+Emulation Method
+𝑅2score, plate carrée grid
+𝑅2score, icosahedral grid
+Flat UNet, unmodified
+0.352 ± 0.015
+0.374 ± 0.017
+Flat UNet, modified
+0.377 ± 0.005
+0.402 ± 0.006
+Flat random forest baseline
+0.212
+0.256
+Flat linear regression baseline
+0.251
+0.274
+Flat PCA regression baseline
+0.303
+0.332
+Icosahedral UNet
+0.126 ± 0.011
+0.396 ± 0.009
+Icosahedral PCA regression baseline
+0.076
+0.339
+on different grids (plate carrée and icosahedral) and the necessary interpolations may deteriorate per-
+formance. Thus, we compute performances on both grids, interpolating the predictions from one grid to
+another. Results for the globally averaged 𝑅2 scores are shown in Table 1. The best model in the com-
+parison is the “modified” version of the flat UNet that includes the three modifications (area-weighted
+loss, adapted padding, CoordConv) described in Section 2.2. The effect of the individual modifications
+is detailed in Table A.2 and Figure A.6.
+All UNet architectures outperform all baseline architectures that operate on the same grid. The best
+UNet method explains 7% more of the test set variance than the best baseline model, PCA regression.
+The other baseline models perform worse. In particular, it seems that the random forest baseline, which
+regresses on a pixel-to-pixel level is not able to capture the spatially varying relationships between
+𝛿18O and the predictor variables surface temperature and precipitation amount well enough, even when
+including coordinates as additional inputs. The spatial performance differences between the UNet meth-
+ods and the best baselines are visualized in Figure A.5. The improvements by the UNets are largest over
+oceans.
+On the icosahedral grid, the icosahedral UNet and the modified flat UNet achieve 𝑅2 scores that
+are not significantly different. On the plate carrée grid, however, the results of the icosahedral UNet
+are much worse. This drop can largely be attributed to the interpolation method (see Figure A.7): on
+the plate carrée grid, neither training data nor results of the flat UNet are interpolated, while for the
+icosahedral UNet interpolations are necessary in both cases.
+
+Emulation
+Ground truth
+3
+(A)
+(B)
+5180 anomaly [%o]
+2
+2
+311
+3.3. Sensitivity experiments
+We conduct a range of sensitivity experiments, to test a) the influence of each predictor variable on
+the results, b) whether we can further improve the performance of our ML method and c) whether
+emulation quality varies with timescale.
+First, we use the modified flat UNet architecture as employed in Section 3.2 and test, how the results
+differ if we exclude one of the predictor variables. The globally averaged 𝑅2 score on the plate carrée
+grid drops from 0.377 ± 0.005 if both precipitation and temperature are used to 0.327 ± 0.006 when
+using only precipitation and to 0.251 ± 0.004 when only using temperature. The spatial differences in
+emulation quality follow the large-scale behavior of the correlation structure in panel C of Figure 2.
+When precipitation is excluded, the performance decreases most over low latitudes, while the 𝑅2 score
+drops over polar regions without temperature. This is visualized in Figure A.8.
+To potentially improve the emulation results even further, we create variations of the modified flat
+UNet architecture: a “wider” version in which the number of computed features per network layer is
+doubled (𝑅2 = 0.386 ± 0.008, plate carrée grid), and a “deeper” version with six additional network
+layers4, which obtains 𝑅2 = 0.389±0.006 (plate carrée grid), both improving over the default choice by
+roughly 0.01 . Additionally, we test, whether results could be improved by tuning the learning rate of the
+employed optimizer by testing a grid of 20 logarithmically spaced values between 10−4 and 10−1. The
+performance is best for learning rates between 10−3 and 10−2. However, no substantial improvements
+over the default parameter choice were reached in the limited range of tested values.
+The monthly timescale differs from the interannual scale by a pronounced seasonal cycle of 𝛿18O in
+many regions. Thus, even a simple climatology can explain a part of the variability in 𝛿18O. To exclude
+this trivially explainable part from the computation of the 𝑅2 score, we compute the score separately
+for each month. Results are similar to the results on the interannual scale with roughly 40% of variance
+explained. The higher time resolution suggests exploring whether the emulation can profit from taking
+the temporal context into account. We test this by including not only the temperature and precipitation
+of the current time step but also of the previous month as inputs to the emulation of 𝛿18O. Results do not
+improve strongly, however, possibly because the investigated timescale is still larger than the average
+atmospheric moisture residence time (Trenberth, 1998).
+3.4. Cross-Model Comparison
+For practical applicability, it is essential that an emulator’s performance is robust under varying cli-
+matic conditions and under potential biases of the climate model that produces the training data for the
+emulator. We address these questions by testing how well our emulation generalizes to data generated
+with different climate models (iCESM, ECHAM5-wiso, isoGSM). To do so, we train the best model
+architecture so far, the deeper modified flat UNet, on data from iHadCM3. Subsequently, the trained
+network is used to emulate 𝛿18O for the test sets of the other climate model data sets. Results of the
+emulation are visualized in Figure 5. For all data sets the mean 𝑅2 score is positive, meaning that in the
+global average, the emulation is preferable to predicting the mean state of the corresponding training
+set. The 𝑅2 score is highest for the ECHAM5-wiso simulation and lowest for isoGSM, where 80% less
+variance is explained than on iHadCM3.
+In all three cross-prediction cases, the performance drops strongly in the Pacific Ocean west of South
+America, a region that is important for the El Niño–Southern Oscillation (ENSO). This might hint at
+inter-model differences in the spatial pattern of ENSO variability. For isoGSM, the emulation quality
+over Antarctica is considerably worse than for all other models. The Antarctic in isoGSM is much less
+depleted in 𝛿18O (less negative 𝛿18O) than in the other models while showing similar equator to pole
+temperature gradients (Bühler et al., 2022). This can potentially impact the relationship between the
+temporal variations of temperature and 𝛿18O.
+4I.e. one additional “depth step” in Figure A.11
+
+12
+Wider et al.
+Figure 5. Results for the cross-prediction task: A UNet is trained on the iHadCM3 training data set. The
+performance is then evaluated on the test set of various climate models, shown 𝑅2 scores are averages
+over ten runs.
+For isoGSM and iCESM, 𝑅2 is negative over large areas of the mid-latitude oceans. As synoptic-
+scale variability of moisture transport pathways might be an important factor for 𝛿18O in the
+mid-latitudes, adding predictor variables that encode information on the atmospheric circulation in the
+respective models could improve the results. The independence of the isoGSM and iCESM runs in these
+regions must be assessed carefully: isoGSM is forced by sea-surface temperatures and sea-ice distri-
+butions of a last millennium run with CCSM4, which is a predecessor model of iCESM. Therefore,
+characteristics of iCESM might also be present in the isoGSM results.
+We also test, how well the baseline ML models generalize when employed to estimate 𝛿18O for other
+climate models. The very simplistic pixel-wise linear regression yields better results than the PCA-
+regression baseline. Figure A.9 shows the cross-model performance of the linear regression baseline.
+While the 𝑅2 score for iHadCM3 itself is significantly smaller than for the UNet model, the loss of
+performance when doing cross-prediction is much smaller, for iCESM the results are even better than
+those obtained with the UNet model. Especially over mid-latitude oceans, the 𝑅2 scores of the linear
+regression are better the ones obtained with the UNet.
+4. Discussion
+In a first step towards data-driven emulation of water isotopes in precipitation, we show that in a
+simulated data set 40% of the interannual 𝛿18O variance can be explained by ML models. The emula-
+tion quality follows patterns of the correlation between 𝛿18O and the predictor variables precipitation
+amount and surface temperature. This hints at the possibility of further improving the emulation
+by including other variables that are statistically connected to 𝛿18O as predictors. 𝛿18O composition
+depends on atmospheric moisture transport, which in turn, depends on atmospheric circulation. Thus,
+variables encoding information on atmospheric circulation such as sea-level pressure are promising
+
+iHadCM3, R2 = 0.386
+ECHAM5-wis0, R2 = 0.267
+(A)
+(B)
+0.8
+0.4
+score
+0.0
+is0GSM. R2 = 0.084
+iCESM, R2 = 0.162
+R
+(C)
+(D)
+-0.4
+-0.813
+candidates which should be explored in future research. This could be particularly relevant in the mid-
+latitudes, where the comparably poor performance of the emulators might be due to synoptic-scale
+moisture transport variability which is not well captured by annual or monthly means of precipitation
+and temperature. In addition, relative humidity seems a promising candidate as it is important for the
+evolution of 𝛿18O during the evaporation process.
+It should be noted that correlation structures between predictor variables and 𝛿18O are likely
+timescale dependent. Our results suggest that temperature, precipitation and atmospheric circulation
+variations due to internal variability in the climate system and short-scale external forcing such as
+volcanic eruptions and solar variability are the most important factors controlling interannual 𝛿18O vari-
+ability. On the other hand, changes in long-term external forcings such as greenhouse gas concentrations
+and Earth’s orbital configuration, and variations in oceanic circulation have been found to explain 𝛿18O
+changes on millennial and orbital (10,000 years and longer) timescales (He et al., 2021). This varying
+importance of factors controlling climate variations can also result in timescale-dependent relationships
+between the predictor variables surface temperature and precipitation amount (Rehfeld and Laepple,
+2016), which limits the generalization of emulators between timescales. Meanwhile, on timescales
+from hours to weeks, the memory in the atmosphere is higher. Thus, taking into account previous time
+steps and explicitly tracking moisture pathways for example in tropical or extratropical cyclones could
+improve the emulation performance. On these timescales, ML methods to model sequences of data, like
+long short-term memory (LSTM), recurrent neural networks (RNNs), or transformer models could be
+good alternatives.
+A tested spherical CNN architecture shows no clear benefit over a modified version of the stan-
+dard flat UNet for our task of emulating 𝛿18O in precipitation globally. We suppose that this is partly
+due to the strong latitudinal dependence of the statistical relationships between 𝛿18O and the predictor
+variables (as indicated by correlations in Figure 2C). Thus, the strength of the spherical network archi-
+tecture, namely its equivariance to rotations, possibly does not offer a strong benefit. Additionally, the
+interpolation between the plate carrée grid and the icosahedral grid deteriorates the results. This might
+be remedied by “differentiating through” the interpolation or directly learning the interpolation, as is
+done in Lam et al. (2022). Using ML architectures that are equivariant to approximate symmetries in
+the Earth system might still be beneficial in many applications, since adapting the ML approach to sym-
+metries of the problem reduces overfitting and the demands for training data. One might for example
+use Cohen and Welling (2016) as a starting point and test a network that is equivariant under rotations
+around the polar axis and reflections on the equatorial plane.
+The cross-model emulations can be seen as a supplement to test for the generalization to the (unavail-
+able) real-world 𝛿18O data. Assuming that each model possesses deficiencies in its 𝛿18O simulation,
+robustness under varying models would hint at robustness in the generalization to real-world data. Addi-
+tionally, reliable 𝛿18O emulations for climate models that do not possess an implementation of water
+isotopologues would ideally be done with an emulator that does not overfit to a certain climate model it
+was trained on. Two reasons that might make an ML emulator perform poorly under cross-emulations
+are a) weak statistical connections between 𝛿18O and the predictor variables in the training set and
+b) differences in the statistical connections of 𝛿18O and the predictor variables between climate mod-
+els. We investigate whether drops in cross-prediction performance can be attributed to these causes in
+Appendix A.2 and Figure A.3. Indeed, most regions, in which there is a drop in emulation performance,
+coincide with regions of differing correlation structures or weak correlations between 𝛿18O and the pre-
+dictor variables in the iHadCM3 data set (Figure A.3, B4 to D4 and B2 to D2). In regions with weak
+correlations between 𝛿18O and the predictor variables in the iHadCM3 data set such as the Southern
+Hemisphere mid-latitudes, the UNet has to predict 𝛿18O based on spatial similarity structures (tele-
+connections). The poor performance in the Southern Hemisphere mid-latitudes in IsoGSM and iCESM
+suggest that the spatial similarity structures differ between those two GCMs and iHadCM3. Here, pre-
+dictors that encode atmospheric circulation more directly such as sea-level pressure could be beneficial
+in future studies.
+
+14
+Wider et al.
+This interpretation is supported by a much sharper drop in performance of the UNet architectures
+than simple linear regression when methods were trained on the iHadCM3 climate model and then
+used to emulate other climate model data. As a result, the 𝑅2 scores on the other climate models were
+comparable between UNet and linear regression. This suggests that the UNet might overfit to the spatial
+anomaly patterns in iHadCM3 given the limited information provided by the predictor variables. This
+overfitting will partly reproduce deficiencies of the respective data set used for the training of the
+emulator. It was shown previously that the models used in our study differ in their mean climate state.
+For example, iHadCM3 and ECHAM5-wiso show a similar global temperature state but iHadCM3
+𝛿18O is much more negative in the global mean (Bühler et al., 2022). Similar differences in the spatial
+anomaly patterns between models need to be explored further to understand their contribution to poor
+cross-model emulation performance. To obtain a more robust emulator that is applicable across models,
+one might utilize data from multiple climate models and climate states (e.g. Last Glacial Maximum,
+mid-Holocene, Pliocene) in the training set.
+Spatially, ML estimates are smoother than the true simulated data. The ground truth data show very
+noisy behavior over dry regions, part of which is likely due to numerical instabilities in the computation
+of 𝛿18O for very low precipitation amounts. Missing data points also occur more frequently in these
+regions, thus, potentially biasing the emulator and its measured performance. Because of these incon-
+sistencies in the input data, it might be beneficial to focus on particular regions when developing an
+emulator with the aim of comparing to a certain natural climate archive. Examples are the polar regions
+for comparisons to ice core data or the mid-latitudes for speleothem records. Restricting the spatial
+extent would also alleviate artifacts of the map projection and render spherical approaches unneces-
+sary. Alternatively, one might think about the application of ML to do in-painting of missing values of
+𝛿18O for the training of emulators, similar to Kadow et al. (2020). In this case, the incomplete 𝛿18O
+would serve as an input to the ML method in addition to precipitation and temperature.
+Training an isotope emulator on real-world data would avoid uncertainties originating from climate
+models and the implementation of isotopes within them. Databases of observed 𝛿18O in precipitation
+(IAEA/WMO, 2020) or 𝛿18O from natural climate archives (Konecky et al., 2020) are publicly avail-
+able. However, challenges arise from the spatial scarcity and unequal distribution of data, and the short
+temporal coverage of observations. Here, using graph networks like the one developed by Defferrard
+et al. (2020) might be an option, and likely strong prior constraints would need to be used to compen-
+sate for small data set sizes. For the future goal of comparing emulations to 𝛿18O measured in natural
+climate archives, archive-specific processes need to be taken into account. This is because 𝛿18O in pre-
+cipitation is not archived directly, but always as the response of a sensor of the archiving medium.
+For example, precipitation 𝛿18O is archived in speleothem records as calcite carbonate in accumulating
+layers that form from cave drip water (Fairchild and Baker, 2012).
+We calculate yearly 𝛿18O as the unweighted average of monthly 𝛿18O. In most natural climate
+archives, yearly 𝛿18O is weighted by precipitation amount. We tested the influence of such a weighting
+and find that it does not impact the emulator performance negatively (not shown). However, climate
+archives can also show seasonal preference in their sensitivity to 𝛿18O (Wackerbarth et al., 2010;
+Fohlmeister et al., 2017; Baker et al., 2019) such that there is likely no optimal way for computing
+yearly values. Including archive-specific processes could either be a second step in a two-step approach,
+where an ML emulator is trained to predict 𝛿18O in precipitation and then a proxy system model (Evans
+et al., 2013) is used to forward-model archive-specific processes. Alternatively, one might include a
+differentiable proxy system model in the ML pipeline. This would make it possible to train the ML
+architecture directly with proxy data instead of 𝛿18O measured in precipitation.
+5. Conclusion
+In this study, we explored the ability of machine learning methods to emulate oxygen isotopes as sim-
+ulated by isotope-enabled general circulation models (GCMs). Focussing on interannual variability in
+
+15
+a last millennium simulation, we show that UNet neural networks improve the emulation performance
+compared to baseline methods such as pixel-wise linear regression and PCA-regression. Averaged over
+all grid-cells, our best-performing UNet architecture explains ∼40% of the temporal 𝛿18O variance.
+The emulation performs best in polar regions, where 𝛿18O is strongly controlled by surface temper-
+ature variations, and in low latitude ocean areas, where 𝛿18O is highly correlated with precipitation
+amounts. Lowest performances occur in arid regions, partly because of numerical instabilities in the
+simulation of 𝛿18O for very low precipitation amounts. Using a spherical network architecture does not
+improve the results compared to a modified flat architecture which accounts better for Earth’s spheri-
+cal geometry than a default UNet architecture. This might be because our spherical UNet architecture
+is not optimized to capture latitudinal dependences in the relationships between 𝛿18O and the predictor
+variables.
+We tested the generalization of the emulator trained on output from the iHadCM3 GCM to last mil-
+lennium simulations with other GCMs. While the performance is better than predicting the model’s
+climatology for all GCMs, the explained variance is substantially lower than for iHadCM3. Perfor-
+mances are especially poor in regions where the correlation structure between 𝛿18O and the predictor
+variables differs from the correlation structure in iHadCM3 and in regions with low correlations
+between 𝛿18O and the predictor variables in iHadCM3. In the latter case, the UNet architecture learns
+spatial dependence structures to improve the emulation of 𝛿18O. This improves the performance within
+iHadCM3 compared to pixel-wise regression. However, these spatial structures seem to differ too much
+between GCMs to facilitate skillful cross-model emulations, especially in the mid-latitudes where
+encoding synoptic-scale circulation variations could be important to capture 𝛿18O variations.
+To further improve emulation performance, adding more predictor variables could be a promis-
+ing next step. In particular, variables such as sea level pressure, which capture characteristics of
+the atmospheric circulation more directly than surface temperature and precipitation amount, could
+help in regions with currently poor performance. To compare emulated isotopes to 𝛿18O measured in
+natural climate archives such as ice cores and speleothems, a way of incorporating archive-specific
+processes needs be investigated. This could be done by incorporating differentiable proxy system mod-
+els into UNet architectures or by applying proxy system models to the emulator output in a two-step
+approach. For comparison with 𝛿18O measurements in natural climate archives, the timescales of varia-
+tions recorded by the archives are important. While we focused on interannual timescales in this study,
+shorter as well as longer timescales could be explored in future research to understand the importance
+of synoptic-scale processes, local predictor variables and external forcings for 𝛿18O emulation across
+timescales.
+Acknowledgments. We thank Nadine Theisen for help with the implementation and testing of baseline models. We are grateful
+for the contribution and standardization of model data from Jesper Sjolte, Kei Yoshimura, Madhavan Midhun, Martin Werner,
+and Josefine Axelsson.
+Funding Statement. This research was supported by grants from the Deutsche Forschungsgemeinschaft (DFG, German
+Research Foundation) through the STACY (project no. 395588486) and CLIMAIC (project no. 442926051) projects.
+Competing Interests. None.
+Data Availability Statement. The data used in this study can be freely downloaded here: https://doi.org/10.5281/zenodo.
+7516327. Code to reproduce our experiments is publicly available at https://github.com/jonathanwider/isoEm, it is subject to
+the license statements in the GitHub repository.
+Ethical Standards. The research meets all ethical guidelines, including adherence to the legal requirements of the study country.
+Author Contributions. Conceptualization: UK, KR, JW, NW; Data curation: JB, KR; Formal Analysis: JW; Funding Acquisi-
+tion: KR; Investigation: JW; Methodology: JK, UK, KR, JW, NW; Project administration: UK, KR; Resources: KR; Software:
+JW; Supervision: JB, JK, UK, KR, NW; Validation: JK, JW; Visualization: JW; Writing original draft: JK, JW; Writing – review
+& editing: all authors. All authors approved the final submitted draft.
+Supplementary Material. Supplementary material has only been provided as an Appendix to this document.
+
+16
+Wider et al.
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+temperature and rainfall in asia. Earth and Planetary Science Letters, 436:1–9.
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+N., Hornegger, J., Wells, W. M., and Frangi, A. F., editors, Medical Image Computing and Computer-Assisted Intervention –
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+Schulzweida, U. (2020). CDO User Guide.
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+Oscillation and the tropical amount effect. Journal of Geophysical Research, 114(D4):D04111.
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+Wackerbarth, A., Scholz, D., Fohlmeister, J., and Mangini, A. (2010). Modelling the delta 18 O value of cave drip water and
+speleothem calcite. Earth and Planetary Science Letters, 299(3-4):387–397.
+Wanner, H., Brönnimann, S., Casty, C., Gyalistras, D., Luterbacher, J., Schmutz, C., Stephenson, D. B., and Xoplaki, E. (2001).
+North Atlantic Oscillation–concepts and studies. Surveys in geophysics, 22:321–381.
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+HDO and deuterium excess–results from the fully coupled ECHAM5/MPI-OM Earth system model. Geoscientific Model
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+Journal of Geophysical Research, 113(D19):D19108.
+
+18
+Wider et al.
+Figure A.1. Validation task for the icosahedral neural network: shown are randomly selected examples
+of handwritten digits from the MNIST data set, projected onto the icosahedral grid, in this case the
+refinement level of the icosahedral grid 𝑟 equals 4. For the validation tasks, three versions of this data
+set are generated: One where the digits are not rotated ("N"), one in which rotations of the symmetry
+group of the icosahedron are applied ("I") and one in which we apply rotations of the group of rotations
+of the sphere("R").
+A. Supplementary Material
+A.1. Implementation Details
+We use pytorch (Paszke et al., 2019) to implement the CNNs and scikit-learn (Pedregosa et al., 2011)
+for the baseline models. Our code is publically available on GitHub.5
+A.1.1. Validation Experiment
+The validation experiment is a variation of one of the seminal tasks in ML: the classification of the
+MNIST data set of handwritten digits (LeCun and Cortes, 2005). As a toy example, Cohen et al. (2019)
+project the digits onto the southern hemisphere of the icosahedral grid, some examples are visualized
+in Figure A.1. Then, three types of data sets are produced: the digits can either be left as they are
+(“non-rotated”) - or the icosahedral grid can be rotated before the digits are projected onto it, either
+by a symmetry transformation of the icosahedron or by an unconstrained rotation (“fully-rotated”).
+Because the network architecture of Cohen et al. (2019) is equivariant to rotations of the icosahedron,
+the network should be able to classify digits that have been rotated by elements of this symmetry group
+with the same accuracy as non-rotated digits, even if during training it was never shown any rotated
+digits.
+The data set creation process, our network architecture (Figure A.2), and the training procedure
+for this task follow Cohen et al. (2019) as closely as possible. We train for 60 epochs on the non-
+rotated and fully-rotated data sets and for one epoch on the icosahedrally rotated data set, which is
+60 times larger than the other sets. We use a cross-entropy loss function and the Adam optimizer
+(Kingma and Ba, 2014) with a learning rate of 0.001 and the other parameters at their default values
+𝛽1,2 = (0.9, 0.999), 𝜖 = 10−8.
+Our results in Table A.1 are comparable or better than those reported in Cohen et al. (2019). Notably,
+they demonstrate the equivariance of our implementation - the classification results for digits rotated by
+an icosahedral symmetry are almost identical to those of non-rotated digits.6 Small differences between
+the results of our implementation and the implementation of Cohen et al. (2019) can partly be attributed
+to the fact that none of our hyperparameters were tuned - and to implementation details not described
+in Cohen et al. (2019), like the batch size or the used optimizer.
+A.1.2. Isotope Emulation Experiment
+UNet Training.
+To optimize the UNet models, we use the Adam optimizer (Kingma and Ba, 2014),
+with lr = 0.001 and the other parameters at their default values: 𝛽1,2 = (0.9, 0.999), 𝜖 = 10−8. We use
+early stopping with a patience of 5, i.e. we abort training if no global minimum of the validation set loss
+5https://github.com/jonathanwider/isoEm.
+6We did not fix a random seed, therefore, the results of N/N and N/I are not exactly identical.
+
+19
+Figure A.2. Sketch of the architecture used to validate our implementation of the icosahedral network
+of Cohen et al. (2019). Spatial resolutions (green text) relate to the refinement level of the icosahedral
+grid. The architecture was chosen according to the details given in the supplementary material of
+the original paper. "S2R" and "R2R" are specific types of convolutional layers developed to maintain
+equivariance, and IcoBN is an adapted version of batch normalization. For details, see Cohen et al.
+(2019).
+Table A.1. Classification accuracies (fraction of correctly classified to the total number of tested digits)
+for the icosahedral MNIST validation experiment (Cohen et al., 2019). "N", "I" and "R" indicate data
+sets, where either no rotations, rotations of the symmetry group of the icosahedron, or rotations of the
+symmetry group of the sphere were applied. Shown results are averages over three runs.
+Training set rot. type/Test set rot. type
+N/N
+N/I
+N/R
+I/I
+I/R
+R/R
+Cohen et al.
+99.43
+99.43
+69.99
+99.38
+66.26
+99.31
+Ours
+99.23
+99.34
+68.57
+99.27
+69.31
+99.37
+is reached in 5 consecutive epochs. As a result, the training runs are stopped after roughly 20 epochs.
+We use a batch size of 8. Training a model takes on the order of minutes to tens of minutes for a single
+run on a basic graphics card (NVIDIA GeForce GTX1650).
+For both icosahedral and flat UNets, we use batch normalization (BN, Ioffe and Szegedy, 2015)
+and ReLU-activations after all but the last convolutional layer. We use 2x2 max-pooling to go to
+coarser spatial resolution and nearest neighbor interpolation (plate carrée projection) and a modified
+form of bilinear interpolation (icosahedral grid) to increase resolution. Data from skip connections and
+upsampling are simply concatenated.
+Icosahedral UNet Approach.
+We create the icosahedral data set using an icosahedral grid at refinement
+level 𝑟 = 5, meaning we apply 5 recursive steps in which each of the triangular faces of the icosahedron
+is divided into four smaller triangles, with the grid points subsequently projected back to the spherical
+surface. This results in a grid with 5×22𝑟+1 +2 = 10242 vertices, while the iHadCM3 latitude-longitude
+grid encompasses 6816 grid points. Because the areas covered by the iHadCM3 grid cells vary with
+latitude, the resolution of the icosahedral grid is higher than the resolution of the iHadCM3 grid close
+to the equators, while close to the poles iHadCM3 data is better resolved than the icosahedral grid.
+The architecture used for the icosahedral grid is visualized in Figure A.12. It is inspired by an
+architecture in Cohen et al. (2019). There is an implementation uncertainty remaining that relates to
+
+20
+Wider et al.
+Table A.2. Effect of modifications to flat UNet architecure, globally averaged 𝑅2 scores. Shown are
+standard deviations and mean over ten runs.
+Use CoordConv
+Use area-weighted loss
+Use cylindrical padding
+𝑅2score
+No
+No
+No
+0.352 ± 0.015
+No
+No
+Yes
+0.357 ± 0.015
+No
+Yes
+No
+0.365 ± 0.005
+Yes
+No
+No
+0.367 ± 0.010
+Yes
+Yes
+Yes
+0.377 ± 0.005
+the treatment of the vertices of the icosahedron at 𝑟 = 0, i.e. the 12 corners of the unrefined grid. In
+opposition to all the other pixels in the icosahedral grid, these possess only five neighboring pixels
+(instead of six), thus introducing irregularities in the convolution patterns. Unable to extract the exact
+treatment of these corner pixels in Cohen et al. (2019), we made the choice to set these corner pixel
+values to zero in every layer. This will likely introduce biases at very coarse resolutions (i.e. 𝑟 small).
+We use the MSE-Loss from Equation (2) without weighting, because all pixels in the icosahedral
+grid cover an approximately equal area. Padding is done similarly to Cohen et al. (2019) in a way that
+asserts continuity between the faces of the icosahedral grid.
+Flat UNet.
+The default architecture for the flat UNets is visualized in Figure A.11. By default,
+we use zero padding to keep the resolution before and after convolutions identical, in most experi-
+ments, however, only the latitudes get zero-padded, while we pad the longitudes cyclically to avoid
+discontinuities.
+A.2. Investigating reasons for cross-prediction performance drops
+We investigate the drops in 𝑅2 score that occur when predicting with a model trained on iHadCM3
+data on simulation data from other climate models. This experiment and its results were described in
+Section 3.4. As described in the main text, a possible explanation might be differences in the statistical
+connections between 𝛿18O and the predictor variables amongst the different climate models. To estimate
+these differences, we calculate the root of the squared differences in correlation coefficients between
+each model and iHadCM3 grid-box wise:
+��
+𝑟Model
+d18O,prec − 𝑟iHadCM3
+d18O,prec
+�2
++
+�
+𝑟Model
+d18O,tsurf − 𝑟iHadCM3
+d18O,tsurf
+�2� 1
+2
+(A.1)
+Here 𝑟 indicates the (temporal) Pearson correlation coefficients computed for each model and grid box.
+The results are visualized in panels (B2) to (D2) of Figure A.3.
+Additionally, the generalization to other models might be impaired in regions where there are weak
+or no statistical relationships between 𝛿18O and the predictor variables in the iHadCM3 data. This may
+occur due to iHadCM3 misrepresenting the 𝛿18O relationships in the Earth system - or there might just
+not be strong connections between 𝛿18O and the predictor variables in these regions. In a simplistic
+approach, we assess this by plotting regions in which neither the correlation of temperature to 𝛿18O
+nor the correlation of precipitation amount and 𝛿18O exceeds an absolute value of 0.25 as hatches in
+Figure A.3 (B4) to (D4).
+In panels (B4) to (D4), we plot the difference in cross-prediction performance between each model
+and iHadCM3, whose training set was used to train the emulator. We observe that especially for the
+ENSO region west of South America and over the North Atlantic Ocean (a region relevant for the
+North Atlantic Oscillation, see Wanner et al., 2001) a decline in performance coincides with changes
+in the correlation structure between the models. Over the southern oceans, there are both changes in
+correlation structure between the models and weak correlations for iHadCM3.
+
+21
+Figure A.3. Investigating reasons for emulation quality decrease in cross-prediction experiments: (A1)
+to (D1) show the absolute correlations between 𝛿18O and the predictor variables. For each grid cell,
+only the larger of the two absolute correlation coefficients is shown. (B2) to (D2) show the differences
+in the correlation structure between the models as computed in Equation (A.1). (A3) to (D3) show
+the cross-prediction 𝑅2 scores of the best deeper flat UNet model and are identical to the content of
+Figure 5. (B4) to (D4) show 𝑅2 scores differences between each model and iHadCM3. In hatched
+regions no correlation coefficient has an absolute value bigger than 0.25 in the iHadCM3 data set.
+
+iHadCM3, abs. correlations
+iHadCM3, R2 iHadCM3
+(A1)
+(A3)
+ECHAM5-wiso, abs. correlations
+ECHAM5-wiso, corr. diff. to iHadCM3
+ECHAM5-wiso, R2 iHadCM3
+ECHAM5-wiso, R2 diff. to iHadCM3
+(B1)
+(B2)
+(B3)
+(B4)
+isoGSM, abs. correlations
+isoGSM, corr. diff. to iHadCM3
+isoGSM, R2 iHadCM3
+isoGSM, R2 diff. to iHadCM3
+(C1)
+(C2)
+(C3)
+(C4)
+iCESM, abs. correlations
+iCESM, corr. diff. to iHadCM3
+iCESM, R2 iHadCM3
+iCESM, R2 diff. to iHadCM3
+(D1)
+(D2)
+(D3)
+(D4)
+0.00
+0.25
+0.50
+0.75
+1.00
+Temperature
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+-0.8-0.4
+0.0
+0.4
+0.8
+-1.0
+-0.5
+0.0
+0.5
+1.0
+R2 score
+R2 score difference
+0.00
+0.25
+0.50
+0.75
+1.00
+Precipitation amount22
+Wider et al.
+Figure A.4. Emulation results on iHadCM3 data set: We show anomalies produced by the ML-emulator
+(“emulation”) and the “true” result in the iHadCM3 data set (“ground truth”). The anomalies are com-
+puted as the difference to the training set mean. We select time steps in which the emulator performs
+especially strong (“90th percentile”) and weak (“10th percentile”), as measured by the anomaly cor-
+relation coefficient (ACC). Additionally, we show the time step, for which the ACC reaches its median
+value, and a year with a pronounced climatic anomaly: 1816 CE, the “year without a summer”.
+
+1oth percentile, emulation
+Median, emulation
+(A1)
+(B1)
+3
+2
+1oth percentile, ground truth
+Median, ground truth
+(A2)
+(B2)
+180 anomaly [%o]
+0
+90th percentile, emulation
+1816 CE, emulation
+(C1
+(D1)
+9oth percentile, ground truth
+1816 CE, ground truth
+(C2)
+(D2)23
+Figure A.5. The difference in 𝑅2 score between the UNet model and the PCA-based regression baseline
+for each grid type,(A): plate carrée grid, (B): icosahedral grid. On the plate carrée grid, we use the
+“modified” version of the flat UNet, including the modifications remedy distortions due to the spherical
+nature of the data. Green colors indicate better performance of the UNet compared to the baseline
+model. Before computing the differences, 𝑅2 scores of ten runs are averaged for each configuration.
+Figure A.6. Effects of modifications to the “standard” UNet, which treats the longitude-latitude grid
+as a flat image: In each panel, we show the difference in 𝑅2 score with respect to the unmodified
+version. We investigate the following modifications: (A) A loss function, in which the contribution of
+each grid box is weighted by the area it covers on Earth’s surface. (B) CoordConv (Liu et al., 2018), a
+modification to CNNs, that allows the network to access the coordinates of locations in the image and
+break translational equivariance. (C) padding the longitudes cyclically instead of using zero padding
+and (D) the joint effect of the modifications. Shown in each panel are differences between averages over
+the 𝑅2 scores of ten runs.
+
+Flat UNet - Flat PCA baseline
+co UNet - Ico PCA baseline
+0.4
+(A)
+(B)
+: difference
+0.2
+0.0
+score
+-0.2
+R
+-0.4Area-weighted loss
+CoordConv
+(A)
+(B)
+0.10
+0.05
+ difference
+0.00
+Cyclically padding longitudes
+All modifications
+score
+(C
+(D)
+R
+-0.05
+-0.1024
+Wider et al.
+Figure A.7. Interpolation between grids degrades results. (A) Performance of the icosahedral UNet
+architecture, here the results were interpolated to the flat grid. (B) results of the flat UNet architec-
+ture, after interpolating the predictions to the icosahedral grid and back to the plate carrée grid. The
+prediction quality is almost indistinguishable from the results of the icosahedral UNet. Additionally,
+the results are bad in regions, where we expect interpolations to have a negative effect: Over coastal
+and in the polar regions, where the icosahedral grid cells are larger then the iHadCM3 grid cells
+and therefore data partly gets “averaged” when interpolating from the finer iHadCM3 grid and to the
+coarser icosahedral grid and back. Thus, we can attribute a large part of the 𝑅2 score differences to
+the interpolations between the grids.
+Figure A.8. Difference in emulation quality (𝑅2 score) between using only one of the predictor vari-
+ables (surface temperature and precipitation amount) and using both simultaneously. Regions shaded
+in purple indicate a drop in performance when leaving out the corresponding predictor variable. Before
+computing the differences, averages over the 𝑅2 scores of ten runs were formed for each configuration..
+
+Ico UNet, interpolated
+Flat UNet, interpolated twice
+(A)
+(B)
+0.8
+0.4
+score
+0.0
+2.
+-0.4
+R
+-0.8Difference: Excluding precipitation amount
+Difference: Excluding temperature
+0.4
+(A)
+(B)
+difference
+0.2
+0.0
+score
+-0.2
+R
+-0.425
+Figure A.9. Results for the cross-prediction task with a pixel-wise linear regression model. The model
+was trained on the iHadCM3 data set, where the performance is worse than with the UNet model. For
+cross-predictions on other data sets, however, the emulation quality of the linear model is comparable
+to or even better than the performance of the UNet, as can be seen by comparing the globally averaged
+𝑅2scores in panels (B) to (D) to the corresponding panels of Figure 5.
+Figure A.10. Investigating missing 𝛿18O values: Spatial distribution of missing 𝛿18O data for iHadCM3
+on monthly timescale. While for over 80% of the gridboxes, not a single value is missing, missing values
+are clustering mainly in hot and dry regions. For the worst grid box, 𝛿18O is missing in 25% of the time
+steps.
+
+iHadCM3, R2 = 0.251
+ECHAM5-wis0, R2 = 0.223
+(A)
+(B)
+0.8
+0.4
+score
+0.0
+is0GSM. R2 = 0.061
+iCESM, R2 = 0.249
+R
+(C)
+(D)
+-0.4
+-0.8Fraction of missing §i8o values in the iHadCM3 data set
+0
+1
+5
+10
+15
+20
+25
+30
+Fraction of missiong 5180 values [%]26
+Wider et al.
+Figure A.11. Sketch of our default flat UNet architecture. The parameters were chosen to roughly match
+in size with the icosahedral UNet architecture. Before the data is input into the network, it is resized
+in order to assure divisibility during down-and up-scaling, thus the input resolution (green) to the first
+layer is 72 × 96 instead of 71 × 96 as in the iHadCM3 grid. In the end, the output of the architecture
+is scaled back to the original grid. The number of computed features per layer is written below the
+corresponding data blocks (black). The number of input features (𝑛𝑖𝑛) depends on the number of chosen
+variables (it equals 2 if using both temperature and precipitation amount), and the number of output
+features 𝑛𝑜𝑢𝑡 equals 1 for all our applications. This graphic doesn’t include the coordinate features that
+get appended to the data-tensors before each convolution, if CoordConv (Liu et al., 2018) is used. BN is
+short for batch normalization, and 3x3 conv indicates a convolutional layer with a 3 × 3 convolutional
+kernel.
+
+ReLU(BN(3x3 conv))
+skip-connection
+> 2x2 max-pool
+upsampling
+1x1 conv27
+Figure A.12. Sketch of the default icosahedral UNet architecture that takes into account the spher-
+ical nature of our data. The parameters are chosen similar to details given in the paper of Cohen
+et al. (2019). The icosahedral refinement level 𝑟 is given in green, the number of features per layer in
+black below the corresponding blocks. “S2R” (scalar-to-regular) and “R2R” (regular-to-regular) are
+convolutional layers defined by Cohen et al. (2019) to achieve equivariance to a group of symmetry
+transformations.
+
+→ ReLU(IcoBN(R2R 3x3 conv))
+→ Skip-connection
+> 2x2 icosahedral max-pool
+→ Icosahedral upsampling
+> Max-Pool-orientations(R2R 3x3 conv)
+> S2R 3x3 conv

EtE1T4oBgHgl3EQfWwQ3/content/tmp_files/2301.03117v1.pdf.txt

@@ -0,0 +1,2381 @@
+Dihadron production in DIS at small x at next to leading order: transverse photons
+Filip Bergabo1, 2, ∗ and Jamal Jalilian-Marian1, 2, †
+1Department of Natural Sciences, Baruch College, CUNY,
+17 Lexington Avenue, New York, NY 10010, USA
+2City University of New York Graduate Center, 365 Fifth Avenue, New York, NY 10016, USA
+We calculate the next to leading order corrections to dihadron production in Deep Inelastic Scat-
+tering (DIS) at small x using the Color Glass Condensate formalism for the case when the virtual
+photon is transverse polarized. Similar to the case of longitudinal photon exchange all UV and
+soft singularities cancel while the collinear divergences are absorbed into quark and antiquark-
+hadron fragmentation functions. Rapidity divergences lead to JIMWLK evolution of dipoles and
+quadrupoles which describe multiple-scatterings of the quark antiquark dipole on the target pro-
+ton/nucleus and contain all the QCD dynamics of the target leading to a finite final result for the
+dihadron production cross section.
+I.
+INTRODUCTION
+A hadron or nucleus wave function at high energy (equivalently, small x) contains a large number of predominantly
+gluons leading to the phenomenon of gluon saturation [1–5]. Inclusive and diffractive two-particle production and
+angular correlations in high energy hadronic/nuclear collisions is a sensitive probe of gluon saturation in a proton
+or nucleus at small x [6–40]. The disappearance of the away side peak in proton (deuteron)-nucleus collisions in the
+forward rapidity region at RHIC [41, 42] as predicted by gluon saturation models [8] provides the strongest hint for the
+presence of gluon saturation in the wave function of the target nucleus at small x. Nevertheless due to complications
+arising from further interactions and radiation from both initial and final states an unambiguous interpretation of
+the RHIC results remains illusive. DIS offers the cleanest environment in which the dynamics of gluon saturation
+can be investigated theoretically as the virtual photon probing the inner structure of the target does not interact
+strongly. The proposed Electron-Ion Collider (EIC) will allow precision studies of the observables [43, 44] in which
+gluon saturation is expected to play a dominant role and as such establish the presence of saturation and clarify the
+kinematics in which it is the main QCD effect. Due to this fact it is imperative that the existing predictions for
+saturation effects are made more precise by calculating higher orders in αs corrections.
+Next to leading order calculations for many processes using the Color Glass Condensate effective theory of QCD
+at small x have recently become available [45–56]. In a recent paper [52] we calculated the one-loop corrections to
+inclusive dihadron production in DIS at small x for the case when the exchanged virtual photon is longitudinal. In this
+work we extend our studies of this process and calculate dihadron production in DIS with transverse photon echange.
+As the calculational methods are identical to our earlier work we will skip a lot of the details of the calculation and refer
+the reader to [52]. As before there are several divergences that appear at the next to leading order. All divergences
+either cancel or can be absorbed into evolution of physical quantities. Our final results are then completely finite and
+can be used to calculate inclusive dihadron production and angular correlations in DIS at small x.
+In the small x limit of DIS the virtual photon (transverse or longitudinal) splits into a quark antiquark pair (a
+dipole), which then multiply scatters from the target hadron or nucleus. To leading order (LO) accuracy the double
+inclusive production cross section can be written as
+dσγ∗A→q¯qX
+d2p d2q dy1 dy2
+= e2Q2(z1z2)2Nc
+(2π)7
+δ(1 − z1 − z2)
+�
+d8x [S122′1′ − S12 − S1′2′ + 1]
+eip·(x′
+1−x1)eiq·(x′
+2−x2)
+�
+4z1z2K0(|x12|Q1)K0(|x1′2′|Q1) +
+(z2
+1 + z2
+2) x12 · x1′2′
+|x12||x1′2′| K1(|x12|Q1)K1(|x1′2′|Q1)
+�
+.
+(1)
+where the first and second terms in the curly bracket above correspond to the contribution of the longitudinal and
+transverse polarizations of the virtual photon.
+The production cross section is a convolution of the probability
+∗ fbergabo@gradcenter.cuny.edu
+† jamal.jalilian-marian@baruch.cuny.edu
+arXiv:2301.03117v1  [hep-ph]  8 Jan 2023
+
+2
+for a photon to split into a quark at transverse position x1 and an anti-quark at position x2 represented by the
+Bessel functions, with the probability for this quark antiquark pair to scatter from the target encoded in the dipoles
+Sij and quadrupoles Sijkl. The virtual photon has momentum lµ with l2 = −Q2 and we have set the transverse
+momentum of the photon to zero without any loss of generality.
+Furthermore pµ (qµ) is the momentum of the
+outgoing quark (antiquark) and z1 (z2) is its longitudinal momentum fraction relative to the photon. x1 (x2) is the
+transverse coordinate of the quark (antiquark), and primed coordinates are used in the conjugate amplitude. Quark
+and antiquark rapidities y1 and y2 are related to their momentum fractions z1 and z2 via dyi = dzi/zi. For convenience
+we also define and use the following shorthand notations:
+Qi = Q
+�
+zi(1 − zi),
+xij = xi − xj,
+d8x = d2x1 d2x2 d2x1′ d2x2′.
+(2)
+Dipoles Sij and quadrupoles Sijkl are normalized correlation functions of two and four Wilson lines defined as
+Sij = 1
+Nc
+tr
+�
+ViV †
+j
+�
+,
+Sijkl = 1
+Nc
+tr
+�
+ViV †
+j VkV †
+l
+�
+,
+(3)
+which contain the full dynamics of gluon saturation. Here index i refers to the transverse coordinate xi and we use
+the following notation for Wilson lines
+Vi = ˆP exp
+�
+ig
+�
+dx+A−(x+, xi)
+�
+,
+(4)
+which resum the multiple scatterings of the quark and antiquark from the target hadron or nucleus.
+S
+II.
+ONE-LOOP CORRECTIONS
+In [47, 48, 51, 52, 55] spinor helicity formalism was used to calculate the contribution of real diagrams to next to
+leading order corrections to the leading order results. The real corrections are shown in Fig. (1) and involve radiation
+of a gluon either by the quark or antiquark before they scatter from the target in which case the gluon also scatters
+from the target [57–59], or after they scatter from the target in which case the radiated gluon does not scatter from
+the target,
+l
+p
+q
+k1
+l − k1
+a
+k
+iAa
+1
+l
+p
+k
+k1
+l − k1
+q
+a
+iAa
+2
+l
+p
+q
+k1
+l − k1
+a
+k
+k2
+k1 − k2
+iAa
+3
+l
+l − k1
+k1
+k2
+k1 − k2
+k
+a
+p
+q
+iAa
+4
+FIG. 1: The real corrections iAa
+1, ..., iAa
+4. The arrows on Fermion lines indicate Fermion number flow, all momenta
+flow to the right. The thick solid line indicates interaction with the target.
+
+3
+The virtual corrections are shown in figure 2 and involve radiation of a gluon by either quark or antiquark which
+is then absorbed by the quark or antiquark line still in the amplitude [52, 55],
+l
+p
+q
+l − k1
+k1
+k2
+k3
+k3 − p
+k2 − k1
+iA5
+l
+p
+l − k1
+iA6
+k1
+k2
+k3
+k3 − q
+k2 − k1
+q
+l
+p
+q
+iA7
+l − k1
+k1
+k3
+k3 − p
+k2
+k2 − k1
+l
+p
+q
+k1
+l − k1
+iA8
+k2
+k3 − q
+k2 − k1
+k3
+l
+p
+q
+l − k1
+k1
+iA9
+k2
+k2 − p
+l
+p
+q
+k2
+k2 − q
+iA10
+k1
+l − k1
+l
+p
+q
+l − k1
+k1
+iA11
+k2
+k2 − k1
+l
+p
+l − k1
+q
+k1
+k2 − k1
+k2
+iA12
+l
+p
+q
+k1
+l − k1
+iA13
+k2 + p
+k2
+q − k2
+l
+p
+q
+k1
+l − k1
+iA14
+k2
+k1 − k2
+l − k2
+FIG. 2: The ten virtual NLO diagrams iA5, ..., iA14. The arrows on fermion lines indicate fermion number flow, all
+momenta flow to the right, except for gluon momenta. The thick solid line indicates interaction with the target.
+We refer the reader to [52] for the explicit expressions for the amplitudes for real and virtual corrections given by
+eqs. (5) and (8 − 14) respectively, and eqs. (6) and (14 − 19) for the Dirac numerators. In this study, we focus on the
+contribution from transversely polarized photons and we compute the numerators using the spinor helicity formalism.
+The needed real numerators are in table I and the virtual numerators are in Eq. 5 - 15.
+
+4
+A.
+Dirac numerators for real diagrams
+Numerator λγ; λq, λg N
+λγ;λq,λg
+i
+N1
++; +, +
+−(z1)3/2√z2(1 − z2) [(z1k−z3p)·ϵ]
+(z1k−z3p)2 (k1 · ϵ)
++; +, −
+−√z1z2(1 − z2)2 [(z1k−z3p)·ϵ∗]
+(z1k−z3p)2
+(k1 · ϵ)
++; −, +
+(z2)3/2√z1(1 − z2) [(z1k−z3p)·ϵ]
+(z1k−z3p)2 (k1 · ϵ)
++, −, −
+(z1z2)3/2 [(z1k−z3p)·ϵ∗]
+(z1k−z3p)2
+(k1 · ϵ)
+N2
++; +, +
+−(z1)3/2√z2(1 − z1) [(z2k−z3q)·ϵ]
+(z2k−z3q)2 (k1 · ϵ)
++; +, −
+−(z1z2)3/2 [(z2k−z3q)·ϵ∗]
+(z2k−z3q)2
+(k1 · ϵ)
++; −, +
+(z2)3/2√z1(1 − z1) [(z2k−z3q)·ϵ]
+(z2k−z3q)2 (k1 · ϵ)
++, −, −
+√z1z2(1 − z1)2 [(z2k−z3q)·ϵ∗]
+(z2k−z3q)2
+(k1 · ϵ)
+N3
++; +, +
+(z1)3/2√z2(1 − z2)
+�
+k2·ϵ
+z3 − k1·ϵ
+1−z2
+�
+k1 · ϵ
++; +, −
+√z1z2(1 − z2)2 �
+k2·ϵ∗
+z3
+− k1·ϵ∗
+1−z2
+�
+k1 · ϵ
++; −, +
+−(z2)3/2√z1(1 − z2)
+�
+k2·ϵ
+z3 − k1·ϵ
+1−z2
+�
+k1 · ϵ
++, −, −
+−(z1z2)3/2 ��
+k2·ϵ∗
+z3
+−
+k1·ϵ∗
+(1−z2)
+�
+k1 · ϵ +
+k2
+1+z2(1−z2)Q2
+2z2(1−z2)
+�
+N4
++; +, +
+(z1)3/2√z2(1 − z1)
+�
+k2·ϵ
+z3 − k1·ϵ
+1−z1
+�
+k1 · ϵ
++; +, −
+(z1z2)3/2 ��
+k2·ϵ∗
+z3
+−
+k1·ϵ∗
+(1−z1)
+�
+k1 · ϵ +
+k2
+1+z1(1−z1)Q2
+2z1(1−z1)
+�
++; −, +
+−(z2)3/2√z1(1 − z1)
+�
+k2·ϵ
+z3 − k1·ϵ
+1−z1
+�
+k1 · ϵ
++, −, −
+−√z1z2(1 − z1)2 �
+k2·ϵ∗
+z3
+− k1·ϵ∗
+1−z1
+�
+k1 · ϵ
+TABLE I: The minimal set of transverse photon numerators N1 to N4 in momentum fraction notation. Complex
+conjugation results in the numerator with all helicities flipped (while leaving longitudinal helicities unchanged), and
+any numerator where λq = λ¯q is zero.
+N +;+
+5
+=25(l+)2z3/2
+1
+√z2
+(z1 − z3)2
+�
+z2
+1
+��
+k3 − z3
+z1
+p
+�
+· ϵ
+� ��
+k2 − z3
+z1
+k1
+�
+· ϵ∗
+�
++ z2
+3
+��
+k3 − z3
+z1
+p
+�
+· ϵ���
+� ��
+k2 − z3
+z1
+k1
+�
+· ϵ
+� �
+k1 · ϵ,
+N +;−
+5
+=−25(l+)2z3/2
+2
+√z1
+(z1 − z3)2
+�
+z2
+1
+��
+k3 − z3
+z1
+p
+�
+· ϵ∗
+� ��
+k2 − z3
+z1
+k1
+�
+· ϵ
+�
++ z2
+3
+��
+k3 − z3
+z1
+p
+�
+· ϵ
+� ��
+k2 − z3
+z1
+k1
+�
+· ϵ∗
+� �
+k1 · ϵ
++ 24(l+)2√z2z2
+3
+√z1(z1 − z3) [k2
+1 + z1z2Q2]
+�
+k3 − z3
+z1
+p
+�
+· ϵ,
+(5)
+N +;+
+6
+=25(l+)2z3/2
+1
+√z2
+(z2 − z3)2
+�
+z2
+3
+��
+k3 − z3
+z2
+q
+�
+· ϵ
+� ��
+k2 − z3
+z2
+k1
+�
+· ϵ∗
+�
++ z2
+2
+��
+k3 − z3
+z2
+q
+�
+· ϵ∗
+� ��
+k2 − z3
+z2
+k1
+�
+· ϵ
+� �
+k1 · ϵ
+− 24(l+)2√z1z2
+3
+√z2(z2 − z3) [k2
+1 + z1z2Q2]
+�
+k3 − z3
+z2
+q
+�
+· ϵ,
+N +;−
+6
+=−25(l+)2z3/2
+2
+√z1
+(z2 − z3)2
+�
+z2
+3
+��
+k3 − z3
+z2
+q
+�
+· ϵ∗
+� ��
+k2 − z3
+z2
+k1
+�
+· ϵ
+�
++ z2
+2
+��
+k3 − z3
+z2
+q
+�
+· ϵ
+� ��
+k2 − z3
+z2
+k1
+�
+· ϵ∗
+� �
+k1 · ϵ
+(6)
+
+5
+N +;+
+7
+= −25(l+)2z3√z1z2
+(1 − z3)(z1 − z3)2
+�
+z1z2
+��
+k3 − z3
+z1
+p
+�
+· ϵ
+� ��
+z2k1 − (1 − z3)k2
+�
+· ϵ∗�
++ z3(1 − z3)
+��
+k3 − z3
+z1
+p
+�
+· ϵ∗
+� ��
+z2k1 − (1 − z3)k2
+�
+· ϵ
+��
+k1 · ϵ
+− 24(l+)2(z1z2)3/2
+(z1 − z3)(1 − z3)
+�
+k2
+1 + z3(1 − z3)Q2� ��
+k3 − z3
+z1
+p
+�
+· ϵ
+�
+,
+N +;−
+7
+=25(l+)2√z1z2
+(z1 − z3)2
+�
+z1z2
+��
+k3 − z3
+z1
+p
+�
+· ϵ∗
+� ��
+z2k1 − (1 − z3)k2
+�
+· ϵ
+�
++ z3(1 − z3)
+��
+k3 − z3
+z1
+p
+�
+· ϵ
+� ��
+z2k1 − (1 − z3)k2
+�
+· ϵ∗��
+k1 · ϵ,
+(7)
+N +;+
+8
+=−25(l+)2√z1z2
+(z2 − z3)2
+�
+z3(1 − z3)
+��
+k3 − z3
+z2
+q
+�
+· ϵ
+� ��
+z1k1 − (1 − z3)k2
+�
+· ϵ∗�
++ z1z2
+��
+k3 − z3
+z2
+q
+�
+· ϵ∗
+� ��
+z1k1 − (1 − z3)k2
+�
+· ϵ
+��
+k1 · ϵ,
+N +;−
+8
+= 25(l+)2z3√z1z2
+(1 − z3)(z2 − z3)2
+�
+z3(1 − z3)
+��
+k3 − z3
+z2
+q
+�
+· ϵ∗
+� ��
+z1k1 − (1 − z3)k2
+�
+· ϵ
+�
++ z1z2
+��
+k3 − z3
+z2
+q
+�
+· ϵ
+� ��
+z1k1 − (1 − z3)k2
+�
+· ϵ∗��
+k1 · ϵ.
++ 24(l+)2(z1z2)3/2
+(z2 − z3)(1 − z3)
+�
+k2
+1 + z3(1 − z3)Q2� ��
+k3 − z3
+z2
+q
+�
+· ϵ
+�
+,
+(8)
+N +;+
+9
+=24(l+)2z3/2
+1
+√z2
+�
+k2
+2 + (2z1 − z)
+z
+(k2 − p)2 − [z2
+1 + (z1 − z)2]
+z1z
+p2
+�
+k1 · ϵ,
+N +;+
+9
+= − 24(l+)2z3/2
+2
+√z1
+�
+k2
+2 + (2z1 − z)
+z
+(k2 − p)2 − [z2
+1 + (z1 − z)2]
+z1z
+p2
+�
+k1 · ϵ,
+(9)
+N +;+
+10
+= − 24(l+)2z3/2
+1
+√z2
+�
+k2
+2 + (2z2 − z)
+z
+(k2 − q)2 − [z2
+2 + (z2 − z)2]
+z2z
+q2
+�
+k1 · ϵ,
+N +;−
+10
+=24(l+)2z3/2
+2
+√z1
+�
+k2
+2 + (2z2 − z)
+z
+(k2 − q)2 − [z2
+2 + (z2 − z)2]
+z2z
+q2
+�
+k1 · ϵ,
+(10)
+N +;+
+11
+=24(l+)2z3/2
+1
+√z2
+��
+(k+
+2 )2 + (p+)2�
+p+(k+
+2 − p+) k2
+1 + (p+ + k+
+2 )
+(p+ − k+
+2 )k2
+2 + (k1 − k2)2
+�
+k1 · ϵ,
+N +;−
+11
+= − 24(l+)2z3/2
+2
+√z1
+��
+(k+
+2 )2 + (p+)2�
+p+(k+
+2 − p+) k2
+1 + (p+ + k+
+2 )
+(p+ − k+
+2 )k2
+2 + (k1 − k2)2
+�
+k1 · ϵ
+− 24z3/2
+2
+z3(l+)2
+√z1(z1 − z3) k2
+1(z3k1 − z1k2) · ϵ,
+(11)
+
+6
+N +;+
+12
+=24(l+)2z3/2
+1
+√z2
+��
+(k+
+2 )2 + (q+)2�
+q+(k+
+2 − q+)
+k2
+1 + (q+ + k+
+2 )
+(q+ − k+
+2 )k2
+2 + (k1 − k2)2
+�
+k1 · ϵ
++ 24(l+)2z3/2
+1
+z3
+√z2(z2 − z3) k2
+1(z3k1 − z2k2) · ϵ,
+N +;+
+12
+= − 24(l+)2z3/2
+2
+√z1
+��
+(k+
+2 )2 + (q+)2�
+q+(k+
+2 − q+)
+k2
+1 + (q+ + k+
+2 )
+(q+ − k+
+2 )k2
+2 + (k1 − k2)2
+�
+k1 · ϵ,
+(12)
+N +;+
+13
+= 25(l+)2(z1 + z)√z1z2
+�
+z1z
+�p · ϵ
+z1
+− q · ϵ
+z2
+� �p · ϵ∗
+z1
+− k2 · ϵ∗
+z
+�
++ z2
+�k2 · ϵ
+z
+− q · ϵ
+z2
+� �p · ϵ∗
+z1
+− k2 · ϵ∗
+z
+�
+− z2z
+�k2 · ϵ
+z
+− q · ϵ
+z2
+� �p · ϵ∗
+z1
+− q · ϵ∗
+z2
+�
+− p · q − (z1 + z)
+2z
+(k2 − q)2 + (z2 − z)
+2z
+(k2 + p)2
+�
+k1 · ϵ,
+N +;−
+13
+= −25(l+)2(z2 − z)√z1z2
+�
+z1z
+�p · ϵ∗
+z1
+− q · ϵ∗
+z2
+� �p · ϵ
+z1
+− k2 · ϵ
+z
+�
++ z2
+�k2 · ϵ∗
+z
+− q · ϵ∗
+z2
+� �p · ϵ
+z1
+− k2 · ϵ
+z
+�
+− z2z
+�k2 · ϵ∗
+z
+− q · ϵ∗
+z2
+� �p · ϵ
+z1
+− q · ϵ
+z2
+�
+− p · q − (z1 + z)
+2z
+(k2 − q)2 + (z2 − z)
+2z
+(k2 + p)2
+�
+k1 · ϵ,
+(13)
+N +;+
+14(1) = −25(l+)2√z1z2
+(1 − z3)(z1 − z3)
+�
+z1z2
+2
+�
+k2
+1 + z3(1 − z3)Q2�
++ z3(1 − z3)
+2
+�
+k2
+2 + z1z2Q2�
++ (z1 − z3)
+��
+z2k1 + z3k2
+�
+· ϵ
+���
+z1k1 + (1 − z3)k2
+�
+· ϵ∗��
+k1 · ϵ
++
+24(l+)2(z1z2)3/2
+z3(z1 − z3)(1 − z3)[k2
+1 + z3(1 − z3)Q2](z1k1 − z3k2) · ϵ,
+N +;−
+14(1) =25(l+)2√z1z2
+z3(z1 − z3)
+�
+z1z2
+2
+�
+k2
+1 + z3(1 − z3)Q2�
++ z3(1 − z3)
+2
+�
+k2
+2 + z1z2Q2�
++ (z1 − z3)
+��
+z2k1 + z3k2
+�
+· ϵ∗���
+z1k1 + (1 − z3)k2
+�
+· ϵ
+��
+k1 · ϵ,
+N +;+
+14(2) = −25(l+)2√z1z2
+(1 − z3)(z3 − z1)
+�
+z1z2
+2
+�
+k2
+1 + z3(1 − z3)Q2�
++ z3(1 − z3)
+2
+�
+k2
+2 + z1z2Q2�
++ (z3 − z1)
+��
+z2k1 + z3k2
+�
+· ϵ
+���
+z1k1 + (1 − z3)k2
+�
+· ϵ∗��
+k1 · ϵ,
+(14)
+N +;−
+14(2) =25(l+)2√z1z2
+z3(z3 − z1)
+�
+z1z2
+2
+�
+k2
+1 + z3(1 − z3)Q2�
++ z3(1 − z3)
+2
+�
+k2
+2 + z1z2Q2�
++ (z3 − z1)
+��
+z2k1 + z3k2
+�
+· ϵ∗���
+z1k1 + (1 − z3)k2
+�
+· ϵ
+��
+k1 · ϵ
++
+24(l+)2(z1z2)3/2
+z3(z1 − z3)(1 − z3)[k2
+1 + z3(1 − z3)Q2](z2k1 − (1 − z3)k2) · ϵ.
+(15)
+In all these expressions, the momentum fractions z and z3 are all defined in the same was as in [52].
+
+7
+III.
+RESULTS
+To calculate the O(αs) corrections to the production cross section we need to multiply the helicity amplitudes with
+the corresponding conjugate amplitudes. We’ll write the real corrections as σi×j for i, j = 1, ..., 4 and the virtual
+corrections as σi for i = 5, ..., 14. The details are shown in [52] and here we just show the final results. The T label
+signifies that we are including contributions only from transversely polarized photons, and imply that we have summed
+over all outgoing polarizations. Furthermore and for the sake of brevity here we omit a factor of δ(1 − z1 − z2 − z)
+in the real corrections and δ(1 − z1 − z2) in the virtual corrections and restore them at the end. In many cases, it is
+easiest to write the results in coordinate space with the radiation kernel ∆(3)
+ij defined as follows.
+∆(3)
+ij = x3i · x3j
+x2
+3ix2
+3j
+.
+(16)
+The next to leading order corrections are then,
+dσT
+1×1
+d2p d2q dy1 dy2 = e2g2Q2N 2
+c z2
+2(1 − z2)[z2
+1z2
+2 + (z2
+1 + z2
+2)(1 − z2)2 + (1 − z2)4]
+2(2π)10z1
+� dz
+z
+�
+d10x[S122′1′ − S12 − S1′2′ + 1]
+eip·x1′1eiq·x2′2K1(|x12|Q2)K1(|x1′2′|Q2) x12 · x1′2′
+|x12||x1′2′|e
+i z
+z1 p·x1′1∆(3)
+1′1.
+(17)
+dσT
+2×2
+d2p d2q dy1 dy2 = e2g2Q2N 2
+c z2
+1(1 − z1)[z2
+1z2
+2 + (z2
+1 + z2
+2)(1 − z1)2 + (1 − z1)4]
+2(2π)10z2
+� dz
+z
+�
+d10x[S122′1′ − S12 − S1′2′ + 1]
+eip·x1′1eiq·x2′2K1(|x12|Q1)K1(|x1′2′|Q1) x12 · x1′2′
+|x12||x1′2′|e
+i z
+z2 q·x2′1∆(3)
+2′2.
+(18)
+dσT
+1×2
+d2p d2q dy1 dy2 = e2g2Q2N 2
+c
+�
+z1z2(1 − z1)(1 − z2)
+2(2π)10
+� dz
+z
+�
+d10x[S12S1′2′ − S12 − S1′2′ + 1]eip·x1′1eiq·x2′2
+K1(|x12|Q2)K1(|x1′2′|Q1)4 Re
+�
+(x12 · ϵ)(x1′2′ · ϵ∗)
+|x12||x1′2′|
+�
+(z2
+1 + z2
+2)(1 − z1)(1 − z2)(x31 · ϵ)(x2′3 · ϵ∗)
+x2
+31x2
+2′3
++ z1z2((1 − z1)2 + (1 − z2)2)(x31 · ϵ∗)(x2′3 · ϵ)
+x2
+31x2
+2′3
+��
+e
+i z
+z1 p·x31e
+i z
+z2 q·x2′3.
+(19)
+dσT
+3×3
+d2p d2q dy1 dy2 = e2g2Q2N 2
+c z1z3
+2
+2(2π)10
+� dz
+z
+�
+d10x[S11′S22′ − S13S23 − S1′3S2′3 + 1]eip·x1′1eiq·x2′2
+K1(QX)K1(QX′)
+XX′
+4 Re
+�
+(z2
+1 + z2
+2)(x31 · ϵ)(x31′ · ϵ∗)
+x2
+31x2
+31′
+[(z1x12 + zx32) · ϵ][(z1x1′2′ + zx32′) · ϵ∗]
++
+�
+(1 − z2)2 + (z1z2)2
+(1 − z2)2
+� (x31 · ϵ∗)(x31′ · ϵ)
+x2
+31x2
+31′
+[(z1x12 + zx32) · ϵ][(z1x1′2′ + zx32′) · ϵ∗]
+−
+z2
+1z2z
+2(1 − z2)2
+�(x31 · ϵ∗)
+x2
+31
+[(z1x12 + zx32) · ϵ] + (x31′ · ϵ)
+x2
+31′
+[(z1x1′2′ + zx32′) · ϵ∗]
+�
++
+z2
+1z2
+4(1 − z2)2
+�
+.
+(20)
+dσT
+4×4
+d2p d2q dy1 dy2 = e2g2Q2N 2
+c z2z3
+1
+2(2π)10
+� dz
+z
+�
+d10x[S11′S22′ − S13S23 − S1′3S2′3 + 1]eip·x1′1eiq·x2′2
+K1(QX)K1(QX′)
+XX′
+4 Re
+�
+(z2
+1 + z2
+2)(x32 · ϵ)(x32′ · ϵ∗)
+x2
+32x2
+32′
+[(z2x21 + zx31) · ϵ][(z2x2′1′ + zx31′) · ϵ∗]
++
+�
+(1 − z1)2 + (z1z2)2
+(1 − z1)2
+� (x32 · ϵ∗)(x32′ · ϵ)
+x2
+32x2
+32′
+[(z2x21 + zx31) · ϵ][(z2x2′1′ + zx31′) · ϵ∗]
+−
+z2
+2z1z
+2(1 − z1)2
+�(x32 · ϵ∗)
+x2
+32
+[(z2x21 + zx31) · ϵ] + (x32′ · ϵ)
+x2
+32′
+[(z2x2′1′ + zx31′) · ϵ∗]
+�
++
+z2
+2z2
+4(1 − z1)2
+�
+.
+(21)
+dσT
+3×4
+d2p d2q dy1 dy2 = e2g2Q2N 2
+c (z1z2)2
+2(2π)10
+� dz
+z
+�
+d10x[S11′S22′ − S13S23 − S1′3S2′3 + 1]eip·x1′1eiq·x2′2
+K1(QX)K1(QX′)
+XX′
+4 Re
+�
+(z2
+1 + z2
+2)(x31 · ϵ)(x32′ · ϵ∗)
+x2
+31x2
+32′
+[(z1x12 + zx32) · ϵ][(z2x2′1′ + zx31′) · ϵ∗]
++
+z1z2
+(1 − z1)(1 − z2)[(1 − z1)2 + (1 − z2)2](x31 · ϵ∗)(x32′ · ϵ)
+x2
+31x2
+32′
+[(z1x12 + zx32) · ϵ][(z2x2′1′ + zx31′) · ϵ∗]
+
+8
+− z2z(1 − z2)
+2(1 − z1)
+(x31 · ϵ∗)
+x2
+31
+[(z1x12 + zx32) · ϵ] − z1z(1 − z1)
+2(1 − z2)
+(x32′ · ϵ)
+x2
+32′
+[(z2x2′1′ + zx31′) · ϵ∗]
+�
+.
+(22)
+dσT
+1×3
+d2p d2q dy1 dy2 = −e2g2Q2N 2
+c z5/2
+2
+√1 − z2
+2(2π)10
+� dz
+z
+�
+d10x[S122′3S1′3 − S1′3S2′3 − S12 + 1]eip·x1′1eiq·x2′2
+K1(|x12|Q2)K1(QX′)
+X′
+4 Re
+�
+(1 − z2)(z2
+1 + z2
+2)(x12 · ϵ)(x31′ · ϵ∗)
+|x12|x2
+31′
+(x31 · ϵ)[(z1x1′2′ + zx32′) · ϵ∗]
+x2
+31
++
+�
+(1 − z2)3 + (z1z2)2
+1 − z2
+� (x12 · ϵ)(x31′ · ϵ)
+|x12|x2
+31′
+(x31 · ϵ∗)[(z1x1′2′ + zx32′) · ϵ∗]
+x2
+31
+−
+z2
+1z2z
+2(1 − z2)
+(x12 · ϵ)
+|x12|
+(x31 · ϵ∗)
+x2
+31
+�
+e
+i z
+z1 p·x31.
+(23)
+dσT
+1×4
+d2p d2q dy1 dy2 = −e2g2Q2N 2
+c z1z3/2
+2
+√1 − z2
+2(2π)10
+� dz
+z
+�
+d10x[S122′3S1′3 − S1′3S2′3 − S12 + 1]eip·x1′1eiq·x2′2
+K1(|x12|Q2)K1(QX′)
+X′
+4 Re
+�
+(1 − z2)(z2
+1 + z2
+2)(x12 · ϵ)(x32′ · ϵ∗)
+|x12|x2
+32′
+(x31 · ϵ)[(z2x2′1′ + zx31′) · ϵ∗]
+x2
+31
++ z1z2
+1 − z1
+�
+(1 − z1)2 + (1 − z2)2� (x12 · ϵ)(x32′ · ϵ)
+|x12|x2
+32′
+(x31 · ϵ∗)[(z2x2′1′ + zx31′) · ϵ∗]
+x2
+31
+− z2z(1 − z2)2
+2(1 − z1)
+(x12 · ϵ)
+|x12|
+(x31 · ϵ∗)
+x2
+31
+�
+e
+i z
+z1 p·x31.
+(24)
+dσT
+2×3
+d2p d2q dy1 dy2 = e2g2Q2N 2
+c z2z3/2
+1
+√1 − z1
+2(2π)10
+� dz
+z
+�
+d10x[S1231′S2′3 − S1′3S2′3 − S12 + 1]eip·x1′1eiq·x2′2
+K1(|x12|Q1)K1(QX′)
+X′
+4 Re
+�
+(1 − z1)(z2
+1 + z2
+2)(x12 · ϵ)(x31′ · ϵ∗)
+|x12|x2
+31′
+(x32 · ϵ)[(z1x1′2′ + zx32′) · ϵ∗]
+x2
+32
++ z1z2
+1 − z2
+�
+(1 − z1)2 + (1 − z2)2� (x12 · ϵ)(x31′ · ϵ)
+|x12|x2
+31′
+(x32 · ϵ∗)[(z1x1′2′ + zx32′) · ϵ∗]
+x2
+32
+− z1z(1 − z1)2
+2(1 − z2)
+(x12 · ϵ)
+|x12|
+(x32 · ϵ∗)
+x2
+32
+�
+e
+i z
+z2 q·x32.
+(25)
+dσT
+2×4
+d2p d2q dy1 dy2 = e2g2Q2N 2
+c z5/2
+1
+√1 − z1
+2(2π)10
+� dz
+z
+�
+d10x[S1231′S2′3 − S1′3S2′3 − S12 + 1]eip·x1′1eiq·x2′2
+K1(|x12|Q1)K1(QX′)
+X′
+4 Re
+�
+(1 − z1)(z2
+1 + z2
+2)(x12 · ϵ)(x32′ · ϵ∗)
+|x12|x2
+32′
+(x32 · ϵ)[(z2x2′1′ + zx31′) · ϵ∗]
+x2
+32
++
+�
+(1 − z1)3 + (z1z2)2
+1 − z1
+� (x12 · ϵ)(x32′ · ϵ)
+|x12|x2
+32′
+(x32 · ϵ∗)[(z2x2′1′ + zx31′) · ϵ∗]
+x2
+32
+−
+z2
+2z1z
+2(1 − z1)
+(x12 · ϵ)
+|x12|
+(x32 · ϵ∗)
+x2
+32
+�
+e
+i z
+z2 q·x32.
+(26)
+dσT
+5
+d2p d2q dy1 dy2
+= e2g2Q2N 2
+c z5/2
+2
+√z1
+2(2π)10
+� z1
+0
+dz
+z d10x [S322′1′S13 − S13S23 − S1′2′ + 1]
+K1(QX5)K1(|x1′2′|Q1)
+X5x2
+31|x1′2′|
+eip·(x′
+1−x1)eiq·(x′
+2−x2)e−i z
+z1 p·(x3−x1)
+x1′2′ ·
+�
+(z2
+1 + z2
+2)(z2
+1 + (z1 − z)2)x32 + (z1 − z)
+�
+z1(z2
+1 + (z1 − z)2) + z2[z1z2 + (z1 − z)(z2 + z)]
+�
+x13
+�
+.
+(27)
+dσT
+6
+d2p d2q dy1 dy2
+= −e2g2Q2N 2
+c z5/2
+1
+√z2
+2(2π)10
+� z2
+0
+dz
+z d10x[S132′1′S23 − S13S23 − S1′2′ + 1]
+K1(QX6)K1(|x1′2′|)
+X6|x1′2′|x2
+32
+eip·(x′
+1−x1)eiq·(x′
+2−x2)e−i z
+z2 q·(x3−x2)
+x1′2′ ·
+�
+(z2
+1 + z2
+2)(z2
+2 + (z2 − z)2)x31 + (z2 − z)[z2(z2
+2 + (z2 − z)2) + z1(z1z2 + (z2 − z)(z1 + z))]x23
+�
+.
+(28)
+
+9
+dσT
+7
+d2p d2q dy1 dy2
+= e2g2Q2N 2
+c (z1z2)3/2
+2(2π)10
+� z1
+0
+dz (z1 − z)
+z
+d10x [S322′1′S13 − S13S23 − S1′2′ + 1]K1(QX5)K1(|x1′2′|Q1)
+X5x2
+31|x1′2′|
+�
+4 Re
+x2
+32
+�
+(x1′2′ · ϵ∗)
+��
+x31 +
+z2
+z2 + z x23
+�
+· ϵ
+� �
+z2(z1 − z)
+z1
+�
+z2
+1 + (z2 + z)2�
+(x31 · ϵ)(x32 · ϵ∗)
++ (z2 + z)(z2
+2 + (z1 − z)2)(x32 · ϵ)(x31 · ϵ∗)
+��
+− z1z2z
+z2 + z x31 · x1′2′
+�
+eip·(x′
+1−x1)eiq·(x′
+2−x2)e−i z
+z1 p·(x3−x1).
+(29)
+dσT
+8
+d2p d2q dy1 dy2
+= −e2g2Q2N 2
+c (z1z2)3/2
+2(2π)10
+� z2
+0
+dz (z2 − z)
+z
+d10x [S132′1′S23 − S13S23 − S1′2′ + 1]K1(QX6)K1(|x1′2′|Q1)
+X6|x1′2′|x2
+32
+�
+4 Re
+x2
+31
+�
+(x1′2′ · ϵ∗)
+��
+x32 +
+z1
+z1 + z x13
+�
+· ϵ
+� �
+(z1 + z)(z2
+1 + (z2 − z)2)(x31 · ϵ)(x32 · ϵ∗)
++ z1(z2 − z)
+z2
+�
+z2
+2 + (z1 + z)2�
+(x32 · ϵ)(x31 · ϵ∗)
+��
+− z1z2z
+z1 + z x32 · x1′2′
+�
+eip·(x′
+1−x1)eiq·(x′
+2−x2)e−i z
+z2 q·(x3−x2).
+(30)
+dσT
+9
+d2p d2q dy1 dy2
+=−e2g2Q2N 2
+c (z1z2)2(z2
+1 + z2
+2)
+4(2π)8
+�
+d8x
+�
+S122′1′ − S12 − S1′2′ + 1
+� x12 · x1′2′
+|x12||x1′2′|K1(|x12|Q1)K1(|x1′2′|Q1)
+× eip·(x′
+1−x1)eiq·(x′
+2−x2)
+� z1
+0
+dz
+z
+�z2
+1 + (z1 − z)2
+z2
+1
+� �
+d2k
+(2π)2
+1
+�
+k − z
+z1 p
+�2 .
+(31)
+dσT
+10
+d2p d2q dy1 dy2
+=−e2g2Q2N 2
+c (z1z2)2(z2
+1 + z2
+2)
+4(2π)8
+�
+d8x
+�
+S122′1′ − S12 − S1′2′ + 1
+� x12 · x1′2′
+|x12||x1′2′|K1(|x12|Q1)K1(|x1′2′|Q1)
+× eip·(x′
+1−x1)eiq·(x′
+2−x2)
+� z2
+0
+dz
+z
+�z2
+2 + (z2 − z)2
+z2
+2
+� �
+d2k
+(2π)2
+1
+�
+k − z
+z2 q
+�2 .
+(32)
+dσT
+11
+d2p d2q dy1 dy2
+= ie2g2QN 2
+c z3/2
+2
+√z1(z2
+1 + z2
+2)
+2(2π)7
+�
+d8x
+�
+S122′1′ − S12 − S1′2′ + 1
+�K1(|x1′2′|Q1)
+|x1′2′|
+eip·(x′
+1−x1)eiq·(x′
+2−x2)
+� z1
+0
+dz
+z2
+[(z1 − z)2 + z2
+1]
+(z1 − z)
+�
+d2k2
+(2π)2
+�
+d2k1
+(2π)2
+k1 · x1′2′ eik1·(x1−x2)
+�
+k2
+1 + Q2
+1
+� �
+Q2 +
+k2
+1
+z1z2 +
+z1
+z(z1−z)k2
+2
+�
+(33)
+dσT
+12
+d2p d2q dy1 dy2
+= −ie2g2QN 2
+c z3/2
+1
+√z2(z2
+1 + z2
+2)
+2(2π)7
+�
+d8x
+�
+S122′1′ − S12 − S1′2��� + 1
+�K1(|x1′2′|Q1)
+|x1′2′|
+eip·(x′
+1−x1)eiq·(x′
+2−x2)
+� z2
+0
+dz
+z2
+[(z2 − z)2 + z2
+2]
+(z2 − z)
+�
+d2k2
+(2π)2
+�
+d2k1
+(2π)2
+k1 · x1′2′ eik1·(x2−x1)
+�
+k2
+1 + Q2
+1
+� �
+Q2 +
+k2
+1
+z1z2 +
+z2
+z(z2−z)k2
+2
+�
+(34)
+
+10
+dσT
+13(1)
+d2p d2q dy1 dy2
+= e2g2Q2N 2
+c (z1z2)3/2
+2(2π)8
+� z2
+0
+dz
+�
+(z1 + z)(z2 − z)
+�
+d8x [S12S1′2′ − S12 − S1′2′ + 1]eip·x1′1eiq·x2′2
+K1
+�
+|x12|Q
+�
+(z1 + z)(z2 − z)
+�
+|x12|
+K1(|x1′2′|Q1)
+|x1′2′|
+�
+d2k
+(2π)2 eik·x214 Re
+�
+(x12 · ϵ)(x1′2′ · ϵ∗)
+� z2(z2−z)[z1(z1+z)+z2(z2−z)]
+2z
+(z2k − zq)2
++ z1z2z
+�
+�
+z1z
+�
+p·ϵ
+z1 − q·ϵ
+z2
+� �
+p·ϵ∗
+z1 − k·ϵ∗
+z
+�
++ z2 �
+k·ϵ
+z − q·ϵ
+z2
+� �
+p·ϵ∗
+z1 − k·ϵ∗
+z
+�
+− z2z
+�
+k·ϵ
+z − q·ϵ
+z2
+� �
+p·ϵ∗
+z1 − q·ϵ∗
+z2
+�
+− p · q
+(z2k − zq)2
+�
+(z1k−zp)2
+z1(z1+z) − (z2k−zq)2
+z2(z2−z)
+�
+�
+�
++ z2
+2z(z2 − z)
+�
+�
+z1z
+�
+p·ϵ∗
+z1 − q·ϵ∗
+z2
+� �
+p·ϵ
+z1 − k·ϵ
+z
+�
++ z2 �
+k·ϵ∗
+z
+− q·ϵ∗
+z2
+� �
+p·ϵ
+z1 − k·ϵ
+z
+�
+− z2z
+�
+k·ϵ∗
+z
+− q·ϵ∗
+z2
+� �
+p·ϵ
+z1 − q·ϵ
+z2
+�
+− p · q
+(z1 + z)(z2k − zq)2
+�
+(z1k−zp)2
+z1(z1+z) − (z2k−zq)2
+z2(z2−z)
+�
+�
+�
+��
+.
+(35)
+dσT
+13(2)
+d2p d2q dy1 dy2
+= e2g2Q2N 2
+c (z1z2)3/2
+2(2π)8
+� z1
+0
+dz
+�
+(z1 − z)(z2 + z)
+�
+d8x [S12S1′2′ − S12 − S1′2′ + 1]eip·x1′1eiq·x2′2
+K1
+�
+|x12|Q
+�
+(z1 − z)(z2 + z)
+�
+|x12|
+K1(|x1′2′|Q1)
+|x1′2′|
+�
+d2k
+(2π)2 eik·x124 Re
+�
+(x12 · ϵ)(x1′2′ · ϵ∗)
+� z1(z1−z)[z1(z1−z)+z2(z2+z)]
+2z
+(z1k − zp)2
++ z1z2z
+�
+�
+z1z
+�
+p·ϵ∗
+z1 − q·ϵ∗
+z2
+� �
+p·ϵ
+z1 − k·ϵ
+z
+�
+− z2 �
+k·ϵ∗
+z
+− q·ϵ∗
+z2
+� �
+p·ϵ
+z1 − k·ϵ
+z
+�
+− z2z
+�
+k·ϵ∗
+z
+− q·ϵ∗
+z2
+� �
+p·ϵ
+z1 − q·ϵ
+z2
+�
++ p · q
+(z1k − zp)2
+�
+(z1k−zp)2
+z1(z1−z) − (z2k−zq)2
+z2(z2+z)
+�
+�
+�
++ z2
+1z(z1 − z)
+�
+�
+z1z
+�
+p·ϵ
+z1 − q·ϵ
+z2
+� �
+p·ϵ∗
+z1 − k·ϵ∗
+z
+�
+− z2 �
+k·ϵ
+z − q·ϵ
+z2
+� �
+p·ϵ∗
+z1 − k·ϵ∗
+z
+�
+− z2z
+�
+k·ϵ
+z − q·ϵ
+z2
+� �
+p·ϵ∗
+z1 − q·ϵ∗
+z2
+�
++ p · q
+(z2 + z)(z1k − zp)2
+�
+(z1k−zp)2
+z1(z1−z) − (z2k−zq)2
+z2(z2+z)
+�
+�
+�
+��
+.
+(36)
+dσT
+14(1)
+d2p d2q dy1 dy2
+= −ie2g2QN 2
+c (z1z2)3/2
+2(2π)7
+� z1
+0
+dz
+z d8xK1(|x1′2′|Q1)
+|x1′2′|
+[S122′1′ − S1′2′ − S12 + 1]ei(p·x1′1+q·x2′2)
+�
+d2k1
+(2π)2
+d2k2
+(2π)2 eik2·x12
+�
+[z1(z1 − z) + z2(z2 + z) − z(1 − z)] (k2 · x1′2′)
+�
+k2
+2 + Q2
+1
+� ��
+k1 − z1−z
+z1 k2
+�2
++ z(z1−z)
+z2z2
+1
+k2
+2 + z
+z1 (z1 − z)Q2
+�
++
+(z1−z)
+z1
+(1 + z2 − 2z2(z1 − z))(k1 · x1′2′)
+�
+k2
+1 + (z1 − z)(z2 + z)Q2
+� ��
+k1 − z1−z
+z1 k2
+�2
++ z(z1−z)
+z2z2
+1
+k2
+2 + z
+z1 (z1 − z)Q2
+�
+−
+Q2 (z1−z)
+z1
+�
+2z1z2z(k1 · x1′2′) + z(z + z2 − z1)2(k2 · x1′2′)
+�
+�
+k2
+1 + (z1 − z)(z2 + z)Q2
+��
+k2
+2 + Q2
+1
+� ��
+k1 − z1−z
+z1 k2
+�2
++ z(z1−z)
+z2z2
+1
+k2
+2 + z
+z1 (z1 − z)Q2
+�
+�
+(37)
+dσT
+14(2)
+d2p d2q dy1 dy2
+= −ie2g2QN 2
+c (z1z2)3/2
+2(2π)7
+� z2
+0
+dz
+z d8xK1(|x1′2′|Q1)
+|x1′2′|
+[S122′1′ − S1′2′ − S12 + 1]ei(p·x1′1+q·x2′2)
+�
+d2k1
+(2π)2
+d2k2
+(2π)2 eik2·x12
+�
+[z2(z2 − z) + z1(z1 + z) − z(1 − z)] (k2 · x1′2′)
+�
+k2
+2 + Q2
+1
+� ��
+k1 − z2−z
+z2 k2
+�2
++ z(z2−z)
+z1z2
+2
+k2
+2 + z
+z2 (z2 − z)Q2
+�
++
+(z2−z)
+z2
+(1 + z2 − 2z1(z2 − z))(k1 · x1′2′)
+�
+k2
+1 + (z2 − z)(z1 + z)Q2
+� ��
+k1 − z2−z
+z2 k2
+�2
++ z(z2−z)
+z1z2
+2
+k2
+2 + z
+z2 (z2 − z)Q2
+�
+
+11
+−
+Q2 (z2−z)
+z2
+�
+2z1z2z(k1 · x1′2′) + z(z + z1 − z2)2(k2 · x1′2′)
+�
+�
+k2
+1 + (z2 − z)(z1 + z)Q2
+��
+k2
+2 + Q2
+1
+� ��
+k1 − z2−z
+z2 k2
+�2
++ z(z2−z)
+z1z2
+2
+k2
+2 + z
+z2 (z2 − z)Q2
+�
+�
+(38)
+These expressions constitute the full result for the one-loop corrections to inclusive quark anti-quark production cross
+section with transverse photon exchange. We have written these results all in terms of the dipole and quadrupole
+functions defined in Eq. (3) in the large Nc limit and ignored all subleading Nc terms.
+We have also used the following notation for the coordinate dependence of some of the Bessel functions:
+X =
+�
+z1z2x2
+12 + z1zx2
+13 + z2zx2
+23,
+X5 =
+�
+z2(z1 − z)x2
+12 + z(z1 − z)x2
+13 + z2z x2
+23,
+X6 =
+�
+z1(z2 − z)x2
+12 + z1z x2
+13 + z(z2 − z)x2
+23.
+(39)
+Note that when z → 0 these all become |x12|√z1z2. The primed version X′ that appears in some real corrections is
+the same as X above but with x1, x2 → x′
+1, x′
+2.
+IV.
+DIVERGENCES
+The above expressions are formal in the sense that they contain divergences that render them ill-defined unless
+regulated. As in the case of longitudinal exchange there are 4 types of divergences:
+• Ultraviolet (UV) divergences when loop momentum k → ∞ or equivalently in coordinate space, when the transverse
+coordinate of the radiated gluon approaches the transverse coordinate xi of either quark or antiquark when integrated,
+i.e. x3 → xi such that |x3−xi| → 0. The UV structure of the production cross section with transverse photon exchange
+is identical to that of longitudinal photon exchange so that cancellations are identical, i.e.
+[dσ5 + dσ11]UV = 0,
+[dσ6 + dσ12]UV = 0,
+�
+dσ9 + dσ10 + dσ14(1) + dσ14(2)
+�
+UV = 0.
+(40)
+with the rest of the contributions being UV finite.
+• Soft divergences when kµ → 0, which in this context corresponds to both transverse momentum in the loop k and
+the radiated gluon momentum fraction z go to zero simultaneously, k, z → 0. Both the real and virtual corrections
+contain soft divergences, however all soft divergences cancel between real and virtual corrections as shown below,
+[dσ1×1 + 2 dσ9]soft = 0,
+[dσ2×2 + 2 dσ10]soft = 0,
+�
+dσ1×2 + dσ13(1) + dσ13(2)
+�
+soft = 0,
+[dσ3×3 + dσ4×4 + 2 dσ3×4]soft = 0,
+[dσ1×3 + dσ1×4]soft = 0,
+[dσ2×3 + dσ2×4]soft = 0,
+[dσ5 + dσ7]soft = 0,
+[dσ6 + dσ8]soft = 0,
+�
+dσ11 + dσ14(1)
+�
+soft = 0,
+�
+dσ12 + dσ14(2)
+�
+soft = 0.
+(41)
+• Collinear divergences when the radiated gluon momentum becomes parallel to either quark or anti-quark momentum
+at finite k and z. They are present in diagrams iA1, iA2 (real corrections) and in iA9, iA10 (virtual corrections). These
+collinear divergences are absorbed into quark-hadron and antiquark-hadron fragmentation functions which makes the
+fragmentation functions scale dependent, for example
+Dh1/q(zh1, µ2) =
+� 1
+zh1
+dξ
+ξ D0
+h1/q
+�zh1
+ξ
+� �
+δ(1 − ξ) + αs
+2π Pqq(ξ) log
+� µ2
+Λ2
+� �
+,
+(42)
+
+12
+defined using a cutoff scheme or
+Dh1/q(zh1, µ2) =
+� 1
+zh1
+dξ
+ξ D0
+h1/q
+�zh1
+ξ
+� �
+δ(1 − ξ) + αs
+π Pqq(ξ)
+�1
+ϵ − log (πeγEµ|x′
+1 − x1|)
+� �
+,
+(43)
+when using dimensional regularization scheme. We refer the reader to [52] for full details.
+• Rapidity divergences when the momentum fraction z of the gluon goes to zero while the transverse momentum k
+of the gluon remains finite. These are handled by introducing a longitudinal momentum fraction factorization scale
+zf and dividing the z integration into two regions: z > zf and z < zf,
+� 1
+0
+dz
+z f(z) =
+�� zf
+0
+dz
+z +
+� 1
+zf
+dz
+z
+�
+f(z).
+(44)
+The rapidity divergences are present only in the first term above and lead to evolution (renormalization) the dipoles
+and quadrupoles according to the BK and JIMWLK evolution equations [60–68]. The second term contains no rapidity
+divergences, it is completely finite and is part of the next to leading order corrections.
+Our final result for the regulated dihadron production cross section can then be symbolically written as sum of
+several terms (Eq. 45) as shown below
+dσγ∗A→h1h2X = dσLO ⊗ JIMWLK + dσLO ⊗ Dh1/q(zh1, µ2) ⊗ Dh2/¯q(zh2, µ2) + dσfinite
+NLO
+(45)
+The first term contains the z integration region below zf where the leading order cross section is evolved with the
+BK/JIMWLK evolution equations. The second term includes the integration region z > zf where the leading order
+cross section is convoluted with the DGLAP evolved fragmentation functions for both quark and antiquark. Finally
+the last term constitutes all the remaining contributions to the NLO cross section which is finite. Presence of the bare
+fragmentation functions in the first and last terms is implied.
+In summary, we have calculated the one-loop corrections to inclusive quark antiquark production in DIS at small x
+for transverse photons. We have shown the production cross section factorizes: all divergences that appear at the
+one-loop level are either canceled or absorbed into JIMWLK evolution of dipoles and quadrupoles, and into DGLAP
+evolution of parton-hadron fragmentation functions. These results are well suited for further phenomenological studies
+of angular correlations of the dihadrons produced in DIS at small x [51].
+a.
+Acknowledgements:
+We gratefully acknowledge support from the DOE Office of Nuclear Physics through Grant
+No. DE-SC0002307 and by PSC-CUNY through grant No. 63158-0051. We would like to thank T. Altinoluk, G. Beuf,
+R. Boussarie, P. Caucal, L. Dixon, Y. Kovchegov, C. Marquet, Y. Mulian, F. Salazar, M. Tevio, R. Venugopalan, W.
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+

FtE5T4oBgHgl3EQfVg81/content/tmp_files/2301.05551v1.pdf.txt

@@ -0,0 +1,633 @@
+DYNAMIC DATA ASSIMILATION OF MPAS-O AND THE GLOBAL
+DRIFTER DATASET
+A PREPRINT
+Derek DeSantis
+CCS-2, W-13
+Los Alamos National Laboratory
+ddesantis@lanl.gov
+Ayan Biswas
+CCS-3
+Los Alamos National Laboratory
+ayan@lanl.gov
+Earl Lawrence
+CCS-6
+Los Alamos National Laboratory
+earl@lanl.gov
+Phillip J. Wolfram
+W-13
+Los Alamos National Laboratory
+pwolfram@lanl.gov
+January 16, 2023
+Keywords Data assimilation ⋅ Dynamic Mode Decomposition ⋅ Ocean Dynamics ⋅ E3SM ⋅ GDP
+Abstract
+In this study, we propose a new method for combining in situ buoy measurements with Earth system models (ESMs) to
+improve the accuracy of temperature predictions in the ocean. The technique utilizes the dynamics and modes identified
+in ESMs to improve the accuracy of buoy measurements while still preserving features such as seasonality. Using this
+technique, errors in localized temperature predictions made by the MPAS-O model can be corrected. We demonstrate
+that our approach improves accuracy compared to other interpolation and data assimilation methods. We apply our
+method to assimilate the Model for Prediction Across Scales Ocean component (MPAS-O) with the Global Drifter
+Program’s in-situ ocean buoy dataset.
+1
+Introduction
+In the field of Earth System Sciences, it is important to constantly improve the accuracy of measurements and models in
+order to better understand the Earth’s climate. One way to analyze climate is through the use of powerful Earth system
+models (ESMs), such as the DOE’s Energy Exascale Earth System Model (E3SM) [6]. ESMs provide estimates of
+state variables across a large number of grid cells, such as the Model for Prediction Across Scales Ocean component
+(MPAS-O), which is used in the E3SM and provides temperature estimates at different sized ocean cells [6, 9, 11].
+Another method for studying climate is through direct in situ observations, such as those collected by satellites, towers,
+or ocean buoys. These observations are more accurate at specific locations, but are limited in coverage. An example
+of a highly resolved ocean buoy dataset is the Global Drifter dataset program (GDP) overseen by NOAA [5, 4]. In
+many cases, scientists use interpolation techniques to estimate state variables, such as temperature or salinity, between
+observations. Each of these approaches has its own advantages and disadvantages, and integrating both types of data
+can provide the greatest flexibility and predictive power [10].
+arXiv:2301.05551v1  [physics.ao-ph]  11 Jan 2023
+
+A PREPRINT - JANUARY 16, 2023
+Basis function models, which use linear combinations of continuous functions to estimate a state, are a flexible and
+computationally efficient method for solving non-stationary systems [2]. In these models, a set of basis functions are
+chosen or discovered, and the associated coefficient weights for an unknown function in the space are learned. In this
+work, we propose a new basis function model for data integration and compare its effectiveness to other data integration
+methods. We will explore three different methods for obtaining basis functions: 1) basis functions fit purely from
+in situ observations, 2) basis functions discovered through ESMs and fit to in situ data, and 3) our newly developed
+dynamic model, which builds on the second approach by weighting the basis functions according to extracted dynamical
+information from the ESM.
+In the following sections, we provide background information and detail the techniques used in this study. In Section
+Three, we present the results of these methods for assimilating GDP data with MPAS-O and show that the dynamic
+method provides the best overall results in terms of mean squared error. We also discuss how the dynamic method is
+able to correct for biases within the MPAS model while simultaneously adjusting for temporal variations, which the
+other basis methods fail to capture.
+2
+Methods
+2.1
+Notation, Data set dimensions and variables
+Throughout this study, let M denote the physical domain of interest in the ocean, specifically the Atlantic Ocean
+between latitudes 15○ − 55○ N and longitudes 260○ − 20○ E. The discrete times for buoy and model observations are
+denoted by tj for j = 1,2,...,T, with a time increment of five days between each slice (tj+1 − tj). The number
+of time slices will vary depending on the training problem, but examples we will consider are T=6, 18, and 74,
+corresponding to one month, one season, and one year, respectively. At each time tj, there are Nj buoy measurements,
+and the total number of buoy measurements over the entire time period from t1 to tT is denoted by N = ∑T
+j=1 Nj. We
+use {xi}N
+i=1 ⊂ M and {yi}N
+i=1 ⊂ R to represent the complete set of buoy locations and temperature measurements,
+respectively. For emphasis, the buoy measurements indexed by {(xi,yi)}Nj+1−1
+i=Nj
+correspond to all the data at time tj,
+and we use (xi(tj),yi(tj)) to refer to a single datum at time tj. For a fixed set of sequential temporal observations, we
+let [t1 ∶ tT ] = {t1,t2,...,tT }.
+For the ESM data, there are L fixed grid center locations {wl}L
+l=1 ⊂ M that provide temperature measurements at each
+time t1,...tT . These grid cells cover the entire domain M , and each buoy position {xi}N
+i=1 ⊂ M belongs to one of the
+grid cells from the ESM. We let ˆxi denote the index of the cell that buoy xi belongs to. Similar to the notation used for
+the buoy data, we use {zl(tj)}L
+l=1 to represent the ESM temperature measurements at time tj.
+In this study, we will use the The Model for Prediction Across Scales Ocean component (MPAS-O) as our ESM and the
+Global Drifter Program (GDP) for ocean buoy data. MPAS-O is an ocean model that simulates ocean systems on time
+scales ranging from months to millenia and spatial scales as small as sub 1 kilometer [6]. Specifically, we will be using
+the V1 historical run of MPAS-O [3]. The GDP is a large array of more than 1000 satellite-tracked ocean buoys that
+measure ocean variables such as drift and sea surface temperature, and are commonly used in weather prediction [5, 4].
+These two datasets have different temporal resolutions, so some time averaging within the GDP is required to match the
+MPAS-O. See the appendix for more details.
+2.2
+Basis Function Interpolation for Data Integration
+Basis function interpolation is a useful method for data integration. In general, interpolation involves finding a function
+F ∶ M → R such that F(xi) ≅ yi for i = 1,...,N, where M is the domain of interest, {xi} are known points in M,
+and {yi} are corresponding values. One approach to constructing F is to represent it as a linear combination of simple
+basis functions φj ∶ M → R, belonging to the same class as F, as follows:
+F(x) =
+M
+∑
+j=1
+ajφj(x).
+This reduces the interpolation problem to a regression problem, where we seek to determine the coefficients {aj} by
+fitting the function F to the data points {(xi,yi)}.
+To perform a basis function interpolation, we first select or discover the basis functions {φj}M
+j=1, and then solve the
+regression problem:
+yi ≅ F(xi) =
+M
+∑
+j=1
+ajφj(xi).
+2
+
+A PREPRINT - JANUARY 16, 2023
+This is a regression problem that can be solved by selecting a set of basis functions {φj}M
+j=1 and finding the coefficients
+{aj} that best fit the data points {(xi,yi)}. This can be written in matrix form as:
+⃗y ≅ Φ⃗a,
+(1)
+where Φi,j = φj(xi), ⃗y = (y1,...,yN)T , and ⃗a = (a1,...,aM)T . The least-square solution to Equation 1 is obtained
+by taking the Moore-Penrose inverse of Φ on both sides:
+⃗a ≅ Φ†⃗y.
+The Moore-Penrose inverse can be computed using the singular value decomposition (SVD) of Φ. If the SVD of Φ
+is given by Φ = UΣV T , then the Moore-Penrose inverse is Φ† = V Σ†U T , where Σ† is found by taking the transpose
+and inverting each entry on the diagonal of Σ. In many cases, the matrix Φ may be ill-conditioned, in which case
+regularization is needed. Optimization is performed over a range of regularization parameters to find the best fit on the
+training data. Once the coefficients ⃗a have been obtained, the model can be tested for accuracy on the test data.
+Data integration is achieved through basis function interpolation by selecting basis functions derived from the ESM
+and interpolating them over the buoy observations {(xi,yi)}. In this work, we will consider different types of basis
+functions and different temporal ranges for the interpolation. The basis functions do not need to be orthogonal. In the
+next section, we will describe our choice of models in more detail.
+2.3
+Radial Basis Functions - Purely Buoy Data Driven
+One approach to deriving F is to use only the observations {(xi,yi)} and ignore the ESM data completely. In this case,
+a natural choice is to use radial basis functions (RBFs) for φj. Specifically, we can define φj(x) = ψ(∣x − µj∣), where
+ψ ∶ M → R is a fixed function (such as a Gaussian) and µj ∈ M are the mean or center points. Often, the RBF chosen
+has a tuning parameter ϵ, such as the standard deviation in a Gaussian. In addition to the RBFs, it is also common to
+include a polynomial term P(x) of degree D to capture mean behavior. Therefore, F ∶ M → R can be expressed as:
+F(x) = P(x) +
+M
+∑
+j=1
+ajφj(x)
+(2)
+In this approach, the user must choose the number of basis functions M, the centers µj, the basis functions φj, and the
+degree of the interpolating polynomial D. A common choice for M and µj is to use the size of the training set N and
+the locations of the training set, and is the practice adopted here. For more information on RBFs, see [1, 12].
+2.4
+Static ESM Mode Decompositions for Basis
+In this subsection, we describe how to obtain basis functions {φj}j = 1M from the ESM data. We consider two methods
+for extracting spatially coherent functions, or modes of variability, that are faithful to the spatial grid {wl}L
+l=1 provided
+by the ESM. First, we describe how to obtain these modes, and then discuss how to use them to produce basis functions.
+For each time t ∈ [t1 ∶ tT ], the ESM data zl(t)L
+l=1 provides an estimate of the temperature distribution over the spatial
+domain M at time t. This data can be arranged into an L × T matrix Z.
+The Singular Value Decomposition (SVD) can be used to directly obtain modes [7]. Let Z ≅ UΣV T be the rank-M
+SVD of Z. The M left singular vectors in the columns of U = [U1,U2,...,UM] ∈ RL,M form a set of (orthogonal)
+spatial modes for Z.
+Another method for extracting modes and their associated oscillation frequencies is the Dynamic Mode Decomposition
+(DMD) [8, 13]. Unlike SVD, the derived modes are not orthogonal, which allows DMD to capture more physically
+relevant modes within the decomposition (at the potential cost of parsimony). The process of computing DMD modes
+{Uj}M
+j=1 is more involved than SVD and is briefly summarized in the appendix.
+Since each point x ∈ M belongs to one of the grid cells {wl}L
+l=1 of the ESM, any mode decomposition of Z can be
+used for interpolation by evaluation. Specifically, let ˆx denote the cell index l = 1,...,L that the point x belongs to, and
+{Uj}M
+j=1 ⊂ RL be modes of Z. We define the static (SVD or DMD) mode basis as
+φj(x) ∶= Uˆx,j
+for each j = 1,...,M. In other words, the static mode basis is simply the spatial modes obtained through the modal
+decomposition (SVD or DMD).
+3
+
+A PREPRINT - JANUARY 16, 2023
+2.5
+Dynamic ESM Mode Decompositions for Basis
+Both of the mode-based decompositions discussed in the previous subsection also include additional dynamical
+information that has not been utilized. In this subsection, we present a method for incorporating this information into
+basis function interpolation.
+In SVD, the left singular vectors U = [U1,...,UM] ∈ RL,M represent spatial patterns, while the right singular vectors
+V = [V1,...,VT ] ∈ RT,M represent the intensity of each of the M patterns across the T time slices. Similarly, DMD
+produces its M DMD modes U = [U1,...,UM] ∈ RL,M and their associated dynamics (intensity across time), as
+described in the appendix. For a mode decomposition, let αj(t) describe the dynamics of the DMD mode Uj.
+Each buoy measurement (xi,yi) is taken at a specific time t ∈ [t1 ∶ tT ]. More generally, we can associate each point
+x ∈ M with a time t ∈ [t1 ∶ tT ] at which we want to provide interpolation. We define the dynamic (SVD or DMD)
+mode basis as
+φj(x) ∶= αj(t)Uˆx,j
+for each j = 1,...,M 1. In other words, the dynamic mode basis is obtained from the static mode basis by weighting
+the static mode at an observation by its intensity at the time of the observation.
+2.6
+MPAS-O - Purely ESM Data Driven
+As a baseline, we will compare the performance of our basis models to the ESM, which provides an approximation of
+the surface temperature over M. Ideally, the ESM would accurately represent the in-situ buoy measurements. To do
+this, we can compute the ESM baseline by finding the ESM cell that each buoy belongs to and comparing the model’s
+output to the buoy measurement:
+z ˆ
+wi(tj) − yi(tj).
+3
+Results
+Figure 1 displays the test errors for the different models for three different now-cast time lengths T. Each model from
+Section 2 is fit for lengths T = 6,18 or 74 to represent one month, three months, or one year. The model is fit on 80% of
+the data and tested on the remaining 20% to produce an error measurement. A distribution of errors for each model, and
+each time T is created by choosing different starting times t1. A total of 100 start times t1 are chosen uniformly spaced
+one month apart from one another. The left side of Figure 1 displays the box and whisker plot for each of the error
+distributions of each model at each time length T. The right side of Figure 1 displays the top three best performing
+models.
+Figure 1 shows the test errors for the different models for three different now-cast time lengths T. Each model from
+Section 2 is fit for lengths T = 6,18 or 74 to represent one month, three months, or one year. The model is fit on 80% of
+the data and tested on the remaining 20% to produce an error measurement. A distribution of errors for each model and
+each time T is created by choosing different starting times t1. A total of 100 start times t1 are chosen uniformly spaced
+one month apart from one another. The left side of Figure 1 displays the box and whisker plot for each of the error
+distributions of each model at each time length T. The right side of Figure 1 displays the top three best-performing
+models.
+One key takeaway from Figure 1 is that in each scenario, the dynamic basis interpolation using DMD modes outperforms
+MPAS across all time scales. While the RBF model performs better than the fully dynamic DMD model on the short
+timescale of one month, it performs worse on longer timescales. Another important observation is that the SVD methods
+perform better on longer timescales. On the short timescale of one month, both SVD methods have a poor fit, but on the
+one-year fit, the dynamic SVD begins to outperform the RBF model. Given the dominance of the fully dynamic DMD
+model over the other basis models, we will focus on its benefits in future analysis.
+Next, we will perform a deeper analysis of the model performance over a single year. For demonstration purposes, we
+have selected the date range of Jan. 1st, 2008 to Jan 1st, 2009 to display here 2. Figure 2(a) shows how the test error
+1Note that since the dynamic information αj(t) has already been captured, the added complexity compared to the static method
+is O(1) per observation
+2While we focus on Jan. 1st, 2008 to Jan 1st, 2009, the results presented here are representative of the general year-long
+phenomenon. The following discussion and analysis can be applied to any time slice, since the fully dynamic models have temporal
+information
+4
+
+A PREPRINT - JANUARY 16, 2023
+is improved by adding the dynamic component to the DMD mode decomposition. The DMD basis decomposition
+has a somewhat bimodal error, indicating that the model has over- or undercompensated for SST in some regions.
+Figure 2(b) illustrates that this is due to seasonality - the basic DMD model has discovered a mean state to represent the
+temperature, missing the extremes of both summer and winter. Weighting the modes by their dynamics removes these
+biases, producing a normally distributed error with a much lower standard deviation. Figure 2(c) shows that MPAS has
+a hot temperature bias, pushing the distribution’s mean into the negatives. This is further seen in Figure 2(d), where
+MPAS appears to have a negative temperature bias across the whole year.
+Finally, we will analyze the spatial distribution of the discovered temperature field to ensure that it adheres to known
+physics. Figure 3(a) shows the average yearly temperature field interpolation provided by the fully dynamic DMD
+model, while figure (b) displays the difference between the fully dynamic DMD and MPAS. Figure 3(c) shows the
+temperature distribution for both the fully dynamic DMD basis model and MPAS. Figure 4(a) plots the location and
+temperature of the test buoys across the year. Figure 4(b) and (c) show the test errors of the fully dynamic DMD and
+MPAS models, respectively. The buoy errors in Figure 4 show that MPAS has a clear latitudinal temperature bias,
+whereas the fully dynamic DMD has relatively uniform errors. Figures 3 and 4 show that the interpolation method
+discovered by the fully dynamic DMD is not dissimilar from the physical model MPAS-O. The primary difference is a
+lower temperature in the fully dynamic DMD model in the higher latitudes (the location where MPAS-O has a large
+temperature bias). This suggests that the dynamic model is learning something that is physically consistent with MPAS,
+while simultaneously improving upon its ground truth accuracy as measured by the GDD.
+4
+Discussion
+RBF methods can provide an arbitrary, unrealistic goodness of fit by adding more nodes. Since the RBF centers are
+chosen from the buoy locations, this model is less sensitive to time scales. As noted above, on longer time lengths RBF
+begins to perform more poorly compared to other models. This is likely due to the fact that RBF works to find the best
+fits at the time the buoy measurements are made. As a result, if longer timescales are included, the RBF model will fit a
+dataset with nearby buoy measurements with seasonal variability, such as Winter and Summer measurements in close
+proximity. This lack of physical knowledge makes the fit rather unrealistic, as nearby nodes in the model compete for
+what the "real" value should be, resulting in high spatial variability.
+The SVD works to reduce noise, and in noise-dominated cases can be effectively swamped in such a way that it does
+not provide clear basis modes. As discussed in the previous section, the SVD models perform extremely poorly on
+short timescales of one to three months. On such timescales, the high-frequency data can dominate the signal, causing
+poor predictions. This might be due in part to the fact that SVD assumes orthogonal, non-interacting modes, which are
+poor at capturing shorter timescales.
+DMD provides better estimates of spatio-temporal evolution than SVD. This could be due in part to the fact that
+DMD modes have dynamics described by the growth and decay of eigenvalues. As such, the spatial modes are
+unambiguously associated with particular months, seasons, cycles, etc. Consequently, DMD can intrinsically adapt to
+seasonal changes better. While SVD does provide dynamic information, DMD is explicitly built to extract different
+modes of variability. Climate change identification and predictions can be made with DMD, whereas the SVD’s lack of
+flexibility and susceptibility to noise make it less accurate because it is not temporally adaptable. That said, the static
+mode decomposition using DMD modes appears to get worse as you increase the time window. This is due to the fact
+that the weight of the mean state still dominates, as shown in Figure 2 and discussed above.
+The key discovery from the previous section is that adding available dynamics, specifically to the DMD method of
+interpolation, improves predictive skill while correcting for mean biasing (Figure 1). We want to use the dynamic
+models when the time includes seasonal features (longer time scales). This is because the fully dynamic DMD captures
+different time scales and reweights them into the model, whereas other basis models either do an arbitrary fit (RBF) or
+capture means (Figure 2). The fully dynamic DMD method also improves the biases of MPAS as seen by combining
+the buoy and spatial pictures of Figures 3 and 4. As noted above, the fully dynamic DMD model corrects for the
+temperature bias within MPAS-O at higher latitudes. The discovered DMD modes capture the dynamics of MPAS-O,
+while having the flexibility to be reweighted to match the real observational data.
+5
+Conclusion
+In this paper, we compared various methods for interpolating ocean buoy data. In particular, we are interested in the data
+assimilation of the ESM ocean data MPAS-O with in situ GDP buoy data. Our analysis showed that the performance
+of an interpolation method depends on the time scale being considered. Some methods may be more effective for
+short time scales, while others may perform better on longer time scales. We found that adding dynamic information,
+5
+
+A PREPRINT - JANUARY 16, 2023
+specifically to the DMD method, improves predictive accuracy and corrects for biases on longer time scales with
+seasonal dynamics. The DMD with dynamics was consistently the best performer among all methods, improving the
+ESM MPAS-O. We demonstrated that the DMD with dynamics is not overfitting and is making reasonable spatial
+predictions while also correcting for biases in MPAS. Therefore, we conclude that the DMD with dynamics is the
+superior method among those examined.
+This type of dynamic decomposition for interpolating ocean data that can adapt to both high and low frequency
+information show great potential. The dynamic data assimilation technique discussed here allow for more data-rich
+analysis that could be useful for understanding evolving dynamics. Further applications of these approaches may be
+useful and application to in-situ analysis or data assimilation appear most promising.
+6
+Acknowledgements
+Research presented in this article was supported by the Laboratory Directed Research and Development program of Los
+Alamos National Laboratory under project number 20200065DR. DD was supported by the DOE Office of Science
+Biological and Environmental Research (BER), as a contribution to the HiLAT-RASM project. MPAS-O V1 were
+obtained from the Energy Exascale Earth System Model project, sponsored by the U.S.Department of Energy, Office of
+Science, Office of Biological and Environmental Research. Satellite-tracked drifting buoy data are available from the
+Global Drifter Program (GDP), with support from their website (ftp://ftp.aoml.noaa.gov/pub/phod/buoydata/).
+7
+Appendix
+7.1
+Time Filtering GDD Data to MPAS-O
+The MPAS-O and GDP datasets are on different temporal resolutions (Five day averages versus six hour). We therefore
+coarse-grained the GDP data in time to fit the temporal resolution of MPAS. Given two sequential MPAS-O time
+snapshots t and t + 1, all the GDP buoy data is collected within each of those five days. For each buoy, the average
+spatial location and temperature is then recorded and used as {xi(t)} and {yi(t)}.
+7.2
+DMD Short Summary
+In the Dynamic Mode Decomposition (DMD), the system is assumed to be evolved in an approximately linear fashion:
+⃗zt+1 ≅ A⃗zt.
+(3)
+This is effectively the solution of a forward-in-time spatially discrete solver, where A can contain physical operators
+such as advection, diffusion, etc. The value in this form is that growth, decay, and oscillatory behavior is all immediately
+represented, albeit pseudo-linearly from one time step to the next. Given this interpretation, one can think of A as
+effectively a learned, empirical physically-meaningful discretization.
+Let Z1 and Z2 be the matrices given by Z1 = [⃗z1⃗z2 ... ⃗zT −1] and Z2 = [⃗z2⃗z3 ... ⃗zT ]. The goal is to find a good
+approximation ˜A of the matrix A that represents this system. This can be cast as the following matrix problem:
+Z2 ≅ AZ1
+(4)
+The least square solution to Equation 4 is found by taking the Moore-Penrose inverse:
+A ≅ Z2Z†
+1.
+Since A is a square matrix, one can consider the eigen-decomposition of A:
+AU = UΛ
+(5)
+The eigenvectors and eigenvalues (uj,λj) of the A are the DMD modes and eigenvalues respectively.
+For systems with a large number of data points L, this direct method isn’t tractable since A is a L × L square matrix.
+Therefore different methods are designed to 1) solve Equation 4 while 2) providing a reduced order model at the same
+time. Among the most simple and popular methods for doing this is the SVD-based method:
+1. Compute the rank M SVD of Z1 = WΣV T .
+2. Consider ˜A ∶= W T AW. Then ˜A is M × M, and since it is unitarily equivalent to A, has the same eigenvalues
+with eigenvectors ξ of ˜A related to A via Wξ.
+6
+
+A PREPRINT - JANUARY 16, 2023
+3. Note that since Z2 ≅ AZ1 = AWΣV T , after multiplying by W T and rearranging we have
+˜A = W T AW = W T Z2V Σ−1.
+In other words, ˜A can be computed directly from the data and SVD of Z1.
+4. Compute eigenvectors and eigenvalues {˜uj,λj} of the much smaller M × M matrix ˜A.
+5. The j′th DMD mode is computed as uj ∶= W ˜uj.
+Other methods for approximating the DMD modes based off SVD exist, such as exact DMD. See [13].
+The DMD modes and eigenvalues can be used to derive dynamical information analogous with the right singular vectors
+of SVD. By iteratively applying Equation 3, we find that
+⃗zt+1 ≅ At⃗z1
+(6)
+Write ⃗z1 in the basis provided by the modes Ψ:
+⃗z1 = Ψ⃗b.
+(7)
+Then combining Equations 5, 6 and 7 we see that
+⃗zt+1 ≅ At⃗z1 = AtΨ⃗b = ΨΛt⃗b.
+Hence, we have represented the state of the system in terms of a DMD mode expansion with temporal evolution on the
+DMD eigenvalues. The time series {(λt
+jbj)T
+t=1}M
+j=1 are referred to as the DMD dynamics for the j’th mode ψj.
+References
+[1] Martin D Buhmann. Radial basis functions. Acta numerica, 9:1–38, 2000.
+[2] Noel Cressie, Matthew Sainsbury-Dale, and Andrew Zammit-Mangion. Basis-function models in spatial statistics.
+arXiv preprint arXiv:2202.03660, 2022.
+[3] DOE E3SM Project. Energy exascale earth system model v1.0. [Computer Software] https://doi.org/10.
+11578/E3SM/dc.20180418.36, apr 2018.
+[4] S Elipot, A Sykulski, R Lumpkin, L Centurioni, and M Pazos. Hourly location, current velocity, and temperature
+collected from global drifter program drifters world-wide. Accession, 248584:v1, 2022.
+[5] Shane Elipot, Adam Sykulski, Rick Lumpkin, Luca Centurioni, and Mayra Pazos. A dataset of hourly sea surface
+temperature from drifting buoys. arXiv preprint arXiv:2201.08289, 2022.
+[6] Jean-Christophe Golaz, Peter M Caldwell, Luke P Van Roekel, Mark R Petersen, Qi Tang, Jonathan D Wolfe,
+Guta Abeshu, Valentine Anantharaj, Xylar S Asay-Davis, David C Bader, et al. The doe e3sm coupled model
+version 1: Overview and evaluation at standard resolution. Journal of Advances in Modeling Earth Systems,
+11(7):2089–2129, 2019.
+[7] Nicholas J Higham. Accuracy and stability of numerical algorithms. SIAM, 2002.
+[8] J Nathan Kutz, Steven L Brunton, Bingni W Brunton, and Joshua L Proctor. Dynamic mode decomposition:
+data-driven modeling of complex systems. SIAM, 2016.
+[9] Mark R Petersen, Xylar S Asay-Davis, Anne S Berres, Qingshan Chen, Nils Feige, Matthew J Hoffman, Douglas W
+Jacobsen, Philip W Jones, Mathew E Maltrud, Stephen F Price, et al. An evaluation of the ocean and sea ice
+climate of e3sm using mpas and interannual core-ii forcing. Journal of Advances in Modeling Earth Systems,
+11(5):1438–1458, 2019.
+[10] Rolf H Reichle. Data assimilation methods in the earth sciences. Advances in water resources, 31(11):1411–1418,
+2008.
+[11] Todd Ringler, Mark Petersen, Robert L Higdon, Doug Jacobsen, Philip W Jones, and Mathew Maltrud. A
+multi-resolution approach to global ocean modeling. Ocean Modelling, 69:211–232, 2013.
+[12] Claude Sammut and Geoffrey I Webb. Encyclopedia of machine learning. Springer Science & Business Media,
+2011.
+[13] Jonathan H Tu. Dynamic mode decomposition: Theory and applications. PhD thesis, Princeton University, 2013.
+7
+
+A PREPRINT - JANUARY 16, 2023
+(a) Monthly errors - all models
+(b) Monthly errors - top models
+(c) Three month errors - all models
+(d) Three month errors - top models
+(e) Yearly errors - all models
+(f) Yearly errors - top models
+Figure 1: Figures (a), (c), (e) display the distribution of errors for all the models fit over one month, three month, and one year time
+windows. Figures (b), (d), (f) show the top three models from distributions.
+8
+
+300
+250 -
+200-
+Temp.
+150
+100 -
+50 -
+0
+-
+8
+MPAS
+DMD_dyn
+DMD
+SVD_dyn
+SVD
+RBF
+Model10
+8
+0
+C
+Temp.
+4 
+2
+MPAS
+DMD_dyn
+RBF
+Model175
+150
+125
+100
+Temp.
+75
+50 
+25 -
+0
+MPAS
+DMD_dyn
+DMD
+SVD_dyn
+SVD
+RBF
+Model8
+7 -
+6
+Temp.
+0
+4
+m
+2 
+1 
+0
+MPAS
+DMD_dyn
+RBF
+Model20.0
+0
+17.5
+15.0
+12.5
+Temp.
+10.0
+7.5
+5.0
+2.5 
+0.0
+MPAS
+DMD_dyn
+DMD
+SVD_dyn
+SVD
+RBF
+Model5
+0
+C
+Temp.
+m
+2 
+1 -
+MPAS
+DMD_dyn
+SVD_dyn
+ModelA PREPRINT - JANUARY 16, 2023
+(a) Errors of DMD basis versus adding dynamics
+(b) Error over time - DMD basis versus adding dynamics
+(c) Error distribution of top models
+(d) Error over time - top models
+Figure 2: Errors of models for time period Jan. 1st 2008-Jan 1st 2009. Plots (a) and (c) display error distributions for different models.
+Plots (b) and (d) compute the average error on each of the five day time slices to explore inability to capture seasonality/baises.
+9
+
+1.0
+0.5
+0.0
+0.5
+1.0
+-1.5
+2.0
+Errors: yi - F(xi)
+MPAS
+2.5
+DMD dyn
+RBF
+0
+10
+20
+30
+40
+50
+60
+70
+Time StepErrors: yi - F(xi)
+250
+DMD_dyn
+DMD
+200-
+100 -
+50 -
+0
+7.5
+5.0
+2.5
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+Temperature C3
+0
+Errors: yi - F(xi)
+DMD
+DMD_dyn
+-3
+0
+10
+20
+30
+40
+50
+60
+70
+Time Step250
+Errors: yi - F(xi)
+MPAS
+DMD_dyn
+200-
+RBF
+150 -
+Count
+100-
+50 -
+0
+7.5
+5.0
+2.5
+0.0
+2.5
+5.0
+7.5
+10.0
+Temperature CA PREPRINT - JANUARY 16, 2023
+(a) Fully dynamic DMD basis interpolation
+(b) Difference between fully dynamic DMD basis and MPAS: FMPAS(xi) − FDMD_dyn(xi)
+(c) SST distribution of MPAS versus fully dynamic DMD
+basis
+Figure 3: Average yearly temperature comparison of fully dynamic DMD versus MPAS-O for time period Jan. 1st 2008-Jan 1st
+2009
+10
+
+0
+5
+10
+15
+20
+25
+30
+Temperature C"-4
+0
+2
+Temperature CSST Distribution: 2008-01-01
+800-
+MPAS
+DMD_dyn
+700
+600
+Count
+500
+400
+300-
+200-
+100
+0
+5
+10
+15
+20
+25
+Temperature C*A PREPRINT - JANUARY 16, 2023
+(a) Buoy observations
+(b) Fully dynamic DMD errors on buoys: yi − F(xi)
+(c) MPAS errors on buoys: yi − F(xi)
+Figure 4: (a) Buoy observations, with (b) Fully dynamic DMD errors and (c) MPAS errors on test buoys for year Jan. 1st 2008-Jan
+1st 2009.
+11
+
+Buoy Test
+5
+10
+15
+20
+25
+30
+Temperature CTest error
+-6
+-4
+2
+0
+2
+4
+TemperatureCTest error
+-8
+-6
+-4
+-2
+0
+2
+4
+6
+8
+TemperatureC

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+page_content='gov  Phillip J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Wolfram  W-13  Los Alamos National Laboratory  pwolfram@lanl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='gov  January 16, 2023  Keywords Data assimilation ⋅ Dynamic Mode Decomposition ⋅ Ocean Dynamics ⋅ E3SM ⋅ GDP  Abstract  In this study, we propose a new method for combining in situ buoy measurements with Earth system models (ESMs) to  improve the accuracy of temperature predictions in the ocean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' The technique utilizes the dynamics and modes identified  in ESMs to improve the accuracy of buoy measurements while still preserving features such as seasonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Using this  technique, errors in localized temperature predictions made by the MPAS-O model can be corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' We demonstrate  that our approach improves accuracy compared to other interpolation and data assimilation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' We apply our  method to assimilate the Model for Prediction Across Scales Ocean component (MPAS-O) with the Global Drifter  Program’s in-situ ocean buoy dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  1  Introduction  In the field of Earth System Sciences, it is important to constantly improve the accuracy of measurements and models in  order to better understand the Earth’s climate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' One way to analyze climate is through the use of powerful Earth system  models (ESMs), such as the DOE’s Energy Exascale Earth System Model (E3SM) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' ESMs provide estimates of  state variables across a large number of grid cells, such as the Model for Prediction Across Scales Ocean component  (MPAS-O), which is used in the E3SM and provides temperature estimates at different sized ocean cells [6, 9, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  Another method for studying climate is through direct in situ observations, such as those collected by satellites, towers,  or ocean buoys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' These observations are more accurate at specific locations, but are limited in coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' An example  of a highly resolved ocean buoy dataset is the Global Drifter dataset program (GDP) overseen by NOAA [5, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' In  many cases, scientists use interpolation techniques to estimate state variables, such as temperature or salinity, between  observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Each of these approaches has its own advantages and disadvantages, and integrating both types of data  can provide the greatest flexibility and predictive power [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='05551v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='ao-ph]  11 Jan 2023  A PREPRINT - JANUARY 16, 2023  Basis function models, which use linear combinations of continuous functions to estimate a state, are a flexible and  computationally efficient method for solving non-stationary systems [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' In these models, a set of basis functions are  chosen or discovered, and the associated coefficient weights for an unknown function in the space are learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' In this  work, we propose a new basis function model for data integration and compare its effectiveness to other data integration  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' We will explore three different methods for obtaining basis functions: 1) basis functions fit purely from  in situ observations, 2) basis functions discovered through ESMs and fit to in situ data, and 3) our newly developed  dynamic model, which builds on the second approach by weighting the basis functions according to extracted dynamical  information from the ESM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  In the following sections, we provide background information and detail the techniques used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' In Section  Three, we present the results of these methods for assimilating GDP data with MPAS-O and show that the dynamic  method provides the best overall results in terms of mean squared error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' We also discuss how the dynamic method is  able to correct for biases within the MPAS model while simultaneously adjusting for temporal variations, which the  other basis methods fail to capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  2  Methods  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='1  Notation, Data set dimensions and variables  Throughout this study, let M denote the physical domain of interest in the ocean, specifically the Atlantic Ocean  between latitudes 15○ − 55○ N and longitudes 260○ − 20○ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' The discrete times for buoy and model observations are  denoted by tj for j = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=',T, with a time increment of five days between each slice (tj+1 − tj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' The number  of time slices will vary depending on the training problem, but examples we will consider are T=6, 18, and 74,  corresponding to one month, one season, and one year, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' At each time tj, there are Nj buoy measurements,  and the total number of buoy measurements over the entire time period from t1 to tT is denoted by N = ∑T  j=1 Nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' We  use {xi}N  i=1 ⊂ M and {yi}N  i=1 ⊂ R to represent the complete set of buoy locations and temperature measurements,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' For emphasis, the buoy measurements indexed by {(xi,yi)}Nj+1−1  i=Nj  correspond to all the data at time tj,  and we use (xi(tj),yi(tj)) to refer to a single datum at time tj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' For a fixed set of sequential temporal observations, we  let [t1 ∶ tT ] = {t1,t2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=',tT }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  For the ESM data, there are L fixed grid center locations {wl}L  l=1 ⊂ M that provide temperature measurements at each  time t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='tT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' These grid cells cover the entire domain M , and each buoy position {xi}N  i=1 ⊂ M belongs to one of the  grid cells from the ESM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' We let ˆxi denote the index of the cell that buoy xi belongs to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Similar to the notation used for  the buoy data, we use {zl(tj)}L  l=1 to represent the ESM temperature measurements at time tj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  In this study, we will use the The Model for Prediction Across Scales Ocean component (MPAS-O) as our ESM and the  Global Drifter Program (GDP) for ocean buoy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' MPAS-O is an ocean model that simulates ocean systems on time  scales ranging from months to millenia and spatial scales as small as sub 1 kilometer [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Specifically, we will be using  the V1 historical run of MPAS-O [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' The GDP is a large array of more than 1000 satellite-tracked ocean buoys that  measure ocean variables such as drift and sea surface temperature, and are commonly used in weather prediction [5, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  These two datasets have different temporal resolutions, so some time averaging within the GDP is required to match the  MPAS-O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' See the appendix for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='2  Basis Function Interpolation for Data Integration  Basis function interpolation is a useful method for data integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' In general, interpolation involves finding a function  F ∶ M → R such that F(xi) ≅ yi for i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=',N, where M is the domain of interest, {xi} are known points in M,  and {yi} are corresponding values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' One approach to constructing F is to represent it as a linear combination of simple  basis functions φj ∶ M → R, belonging to the same class as F, as follows:  F(x) =  M  ∑  j=1  ajφj(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  This reduces the interpolation problem to a regression problem, where we seek to determine the coefficients {aj} by  fitting the function F to the data points {(xi,yi)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  To perform a basis function interpolation, we first select or discover the basis functions {φj}M  j=1, and then solve the  regression problem:  yi ≅ F(xi) =  M  ∑  j=1  ajφj(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  2  A PREPRINT - JANUARY 16, 2023  This is a regression problem that can be solved by selecting a set of basis functions {φj}M  j=1 and finding the coefficients  {aj} that best fit the data points {(xi,yi)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' This can be written in matrix form as:  ⃗y ≅ Φ⃗a,  (1)  where Φi,j = φj(xi), ⃗y = (y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=',yN)T , and ⃗a = (a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=',aM)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' The least-square solution to Equation 1 is obtained  by taking the Moore-Penrose inverse of Φ on both sides:  ⃗a ≅ Φ†⃗y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  The Moore-Penrose inverse can be computed using the singular value decomposition (SVD) of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' If the SVD of Φ  is given by Φ = UΣV T , then the Moore-Penrose inverse is Φ† = V Σ†U T , where Σ† is found by taking the transpose  and inverting each entry on the diagonal of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' In many cases, the matrix Φ may be ill-conditioned, in which case  regularization is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Optimization is performed over a range of regularization parameters to find the best fit on the  training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Once the coefficients ⃗a have been obtained, the model can be tested for accuracy on the test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  Data integration is achieved through basis function interpolation by selecting basis functions derived from the ESM  and interpolating them over the buoy observations {(xi,yi)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' In this work, we will consider different types of basis  functions and different temporal ranges for the interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' The basis functions do not need to be orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' In the  next section, we will describe our choice of models in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='3  Radial Basis Functions - Purely Buoy Data Driven  One approach to deriving F is to use only the observations {(xi,yi)} and ignore the ESM data completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' In this case,  a natural choice is to use radial basis functions (RBFs) for φj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Specifically, we can define φj(x) = ψ(∣x − µj∣), where  ψ ∶ M → R is a fixed function (such as a Gaussian) and µj ∈ M are the mean or center points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Often, the RBF chosen  has a tuning parameter ϵ, such as the standard deviation in a Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' In addition to the RBFs, it is also common to  include a polynomial term P(x) of degree D to capture mean behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Therefore, F ∶ M → R can be expressed as:  F(x) = P(x) +  M  ∑  j=1  ajφj(x)  (2)  In this approach, the user must choose the number of basis functions M, the centers µj, the basis functions φj, and the  degree of the interpolating polynomial D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' A common choice for M and µj is to use the size of the training set N and  the locations of the training set, and is the practice adopted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' For more information on RBFs, see [1, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='4  Static ESM Mode Decompositions for Basis  In this subsection, we describe how to obtain basis functions {φj}j = 1M from the ESM data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' We consider two methods  for extracting spatially coherent functions, or modes of variability, that are faithful to the spatial grid {wl}L  l=1 provided  by the ESM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' First, we describe how to obtain these modes, and then discuss how to use them to produce basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  For each time t ∈ [t1 ∶ tT ], the ESM data zl(t)L  l=1 provides an estimate of the temperature distribution over the spatial  domain M at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' This data can be arranged into an L × T matrix Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  The Singular Value Decomposition (SVD) can be used to directly obtain modes [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Let Z ≅ UΣV T be the rank-M  SVD of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' The M left singular vectors in the columns of U = [U1,U2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=',UM] ∈ RL,M form a set of (orthogonal)  spatial modes for Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  Another method for extracting modes and their associated oscillation frequencies is the Dynamic Mode Decomposition  (DMD) [8, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Unlike SVD, the derived modes are not orthogonal, which allows DMD to capture more physically  relevant modes within the decomposition (at the potential cost of parsimony).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' The process of computing DMD modes  {Uj}M  j=1 is more involved than SVD and is briefly summarized in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  Since each point x ∈ M belongs to one of the grid cells {wl}L  l=1 of the ESM, any mode decomposition of Z can be  used for interpolation by evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Specifically, let ˆx denote the cell index l = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=',L that the point x belongs to, and  {Uj}M  j=1 ⊂ RL be modes of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' We define the static (SVD or DMD) mode basis as  φj(x) ∶= Uˆx,j  for each j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=',M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' In other words, the static mode basis is simply the spatial modes obtained through the modal  decomposition (SVD or DMD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  3  A PREPRINT - JANUARY 16, 2023  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='5  Dynamic ESM Mode Decompositions for Basis  Both of the mode-based decompositions discussed in the previous subsection also include additional dynamical  information that has not been utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' In this subsection, we present a method for incorporating this information into  basis function interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  In SVD, the left singular vectors U = [U1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=',UM] ∈ RL,M represent spatial patterns, while the right singular vectors  V = [V1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=',VT ] ∈ RT,M represent the intensity of each of the M patterns across the T time slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Similarly, DMD  produces its M DMD modes U = [U1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=',UM] ∈ RL,M and their associated dynamics (intensity across time), as  described in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' For a mode decomposition, let αj(t) describe the dynamics of the DMD mode Uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  Each buoy measurement (xi,yi) is taken at a specific time t ∈ [t1 ∶ tT ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' More generally, we can associate each point  x ∈ M with a time t ∈ [t1 ∶ tT ] at which we want to provide interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' We define the dynamic (SVD or DMD)  mode basis as  φj(x) ∶= αj(t)Uˆx,j  for each j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=',M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' In other words, the dynamic mode basis is obtained from the static mode basis by weighting  the static mode at an observation by its intensity at the time of the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='6  MPAS-O - Purely ESM Data Driven  As a baseline, we will compare the performance of our basis models to the ESM, which provides an approximation of  the surface temperature over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Ideally, the ESM would accurately represent the in-situ buoy measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' To do  this, we can compute the ESM baseline by finding the ESM cell that each buoy belongs to and comparing the model’s  output to the buoy measurement:  z ˆ  wi(tj) − yi(tj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  3  Results  Figure 1 displays the test errors for the different models for three different now-cast time lengths T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Each model from  Section 2 is fit for lengths T = 6,18 or 74 to represent one month, three months, or one year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' The model is fit on 80% of  the data and tested on the remaining 20% to produce an error measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' A distribution of errors for each model, and  each time T is created by choosing different starting times t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' A total of 100 start times t1 are chosen uniformly spaced  one month apart from one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' The left side of Figure 1 displays the box and whisker plot for each of the error  distributions of each model at each time length T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' The right side of Figure 1 displays the top three best performing  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  Figure 1 shows the test errors for the different models for three different now-cast time lengths T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Each model from  Section 2 is fit for lengths T = 6,18 or 74 to represent one month, three months, or one year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' The model is fit on 80% of  the data and tested on the remaining 20% to produce an error measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' A distribution of errors for each model and  each time T is created by choosing different starting times t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' A total of 100 start times t1 are chosen uniformly spaced  one month apart from one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' The left side of Figure 1 displays the box and whisker plot for each of the error  distributions of each model at each time length T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' The right side of Figure 1 displays the top three best-performing  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  One key takeaway from Figure 1 is that in each scenario, the dynamic basis interpolation using DMD modes outperforms  MPAS across all time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' While the RBF model performs better than the fully dynamic DMD model on the short  timescale of one month, it performs worse on longer timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Another important observation is that the SVD methods  perform better on longer timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' On the short timescale of one month, both SVD methods have a poor fit, but on the  one-year fit, the dynamic SVD begins to outperform the RBF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Given the dominance of the fully dynamic DMD  model over the other basis models, we will focus on its benefits in future analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  Next, we will perform a deeper analysis of the model performance over a single year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' For demonstration purposes, we  have selected the date range of Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' 1st, 2008 to Jan 1st, 2009 to display here 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Figure 2(a) shows how the test error  1Note that since the dynamic information αj(t) has already been captured, the added complexity compared to the static method  is O(1) per observation  2While we focus on Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' 1st, 2008 to Jan 1st, 2009, the results presented here are representative of the general year-long  phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' The following discussion and analysis can be applied to any time slice, since the fully dynamic models have temporal  information  4  A PREPRINT - JANUARY 16, 2023  is improved by adding the dynamic component to the DMD mode decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' The DMD basis decomposition  has a somewhat bimodal error, indicating that the model has over- or undercompensated for SST in some regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  Figure 2(b) illustrates that this is due to seasonality - the basic DMD model has discovered a mean state to represent the  temperature, missing the extremes of both summer and winter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Weighting the modes by their dynamics removes these  biases, producing a normally distributed error with a much lower standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Figure 2(c) shows that MPAS has  a hot temperature bias, pushing the distribution’s mean into the negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' This is further seen in Figure 2(d), where  MPAS appears to have a negative temperature bias across the whole year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  Finally, we will analyze the spatial distribution of the discovered temperature field to ensure that it adheres to known  physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Figure 3(a) shows the average yearly temperature field interpolation provided by the fully dynamic DMD  model, while figure (b) displays the difference between the fully dynamic DMD and MPAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Figure 3(c) shows the  temperature distribution for both the fully dynamic DMD basis model and MPAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Figure 4(a) plots the location and  temperature of the test buoys across the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Figure 4(b) and (c) show the test errors of the fully dynamic DMD and  MPAS models, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' The buoy errors in Figure 4 show that MPAS has a clear latitudinal temperature bias,  whereas the fully dynamic DMD has relatively uniform errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Figures 3 and 4 show that the interpolation method  discovered by the fully dynamic DMD is not dissimilar from the physical model MPAS-O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' The primary difference is a  lower temperature in the fully dynamic DMD model in the higher latitudes (the location where MPAS-O has a large  temperature bias).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' This suggests that the dynamic model is learning something that is physically consistent with MPAS,  while simultaneously improving upon its ground truth accuracy as measured by the GDD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  4  Discussion  RBF methods can provide an arbitrary, unrealistic goodness of fit by adding more nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Since the RBF centers are  chosen from the buoy locations, this model is less sensitive to time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' As noted above, on longer time lengths RBF  begins to perform more poorly compared to other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' This is likely due to the fact that RBF works to find the best  fits at the time the buoy measurements are made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' As a result, if longer timescales are included, the RBF model will fit a  dataset with nearby buoy measurements with seasonal variability, such as Winter and Summer measurements in close  proximity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' This lack of physical knowledge makes the fit rather unrealistic, as nearby nodes in the model compete for  what the "real" value should be, resulting in high spatial variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  The SVD works to reduce noise, and in noise-dominated cases can be effectively swamped in such a way that it does  not provide clear basis modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' As discussed in the previous section, the SVD models perform extremely poorly on  short timescales of one to three months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' On such timescales, the high-frequency data can dominate the signal, causing  poor predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' This might be due in part to the fact that SVD assumes orthogonal, non-interacting modes, which are  poor at capturing shorter timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  DMD provides better estimates of spatio-temporal evolution than SVD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' This could be due in part to the fact that  DMD modes have dynamics described by the growth and decay of eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' As such, the spatial modes are  unambiguously associated with particular months, seasons, cycles, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Consequently, DMD can intrinsically adapt to  seasonal changes better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' While SVD does provide dynamic information, DMD is explicitly built to extract different  modes of variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Climate change identification and predictions can be made with DMD, whereas the SVD’s lack of  flexibility and susceptibility to noise make it less accurate because it is not temporally adaptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' That said, the static  mode decomposition using DMD modes appears to get worse as you increase the time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' This is due to the fact  that the weight of the mean state still dominates, as shown in Figure 2 and discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  The key discovery from the previous section is that adding available dynamics, specifically to the DMD method of  interpolation, improves predictive skill while correcting for mean biasing (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' We want to use the dynamic  models when the time includes seasonal features (longer time scales).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' This is because the fully dynamic DMD captures  different time scales and reweights them into the model, whereas other basis models either do an arbitrary fit (RBF) or  capture means (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' The fully dynamic DMD method also improves the biases of MPAS as seen by combining  the buoy and spatial pictures of Figures 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' As noted above, the fully dynamic DMD model corrects for the  temperature bias within MPAS-O at higher latitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' The discovered DMD modes capture the dynamics of MPAS-O,  while having the flexibility to be reweighted to match the real observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  5  Conclusion  In this paper, we compared various methods for interpolating ocean buoy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' In particular, we are interested in the data  assimilation of the ESM ocean data MPAS-O with in situ GDP buoy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Our analysis showed that the performance  of an interpolation method depends on the time scale being considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Some methods may be more effective for  short time scales, while others may perform better on longer time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' We found that adding dynamic information,  5  A PREPRINT - JANUARY 16, 2023  specifically to the DMD method, improves predictive accuracy and corrects for biases on longer time scales with  seasonal dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' The DMD with dynamics was consistently the best performer among all methods, improving the  ESM MPAS-O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' We demonstrated that the DMD with dynamics is not overfitting and is making reasonable spatial  predictions while also correcting for biases in MPAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Therefore, we conclude that the DMD with dynamics is the  superior method among those examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  This type of dynamic decomposition for interpolating ocean data that can adapt to both high and low frequency  information show great potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' The dynamic data assimilation technique discussed here allow for more data-rich  analysis that could be useful for understanding evolving dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Further applications of these approaches may be  useful and application to in-situ analysis or data assimilation appear most promising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  6  Acknowledgements  Research presented in this article was supported by the Laboratory Directed Research and Development program of Los  Alamos National Laboratory under project number 20200065DR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' DD was supported by the DOE Office of Science  Biological and Environmental Research (BER), as a contribution to the HiLAT-RASM project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' MPAS-O V1 were  obtained from the Energy Exascale Earth System Model project, sponsored by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='Department of Energy, Office of  Science, Office of Biological and Environmental Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Satellite-tracked drifting buoy data are available from the  Global Drifter Program (GDP), with support from their website (ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='aoml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='noaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='gov/pub/phod/buoydata/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  7  Appendix  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='1  Time Filtering GDD Data to MPAS-O  The MPAS-O and GDP datasets are on different temporal resolutions (Five day averages versus six hour).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' We therefore  coarse-grained the GDP data in time to fit the temporal resolution of MPAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Given two sequential MPAS-O time  snapshots t and t + 1, all the GDP buoy data is collected within each of those five days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' For each buoy, the average  spatial location and temperature is then recorded and used as {xi(t)} and {yi(t)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='2  DMD Short Summary  In the Dynamic Mode Decomposition (DMD), the system is assumed to be evolved in an approximately linear fashion:  ⃗zt+1 ≅ A⃗zt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  (3)  This is effectively the solution of a forward-in-time spatially discrete solver, where A can contain physical operators  such as advection, diffusion, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' The value in this form is that growth, decay, and oscillatory behavior is all immediately  represented, albeit pseudo-linearly from one time step to the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Given this interpretation, one can think of A as  effectively a learned, empirical physically-meaningful discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  Let Z1 and Z2 be the matrices given by Z1 = [⃗z1⃗z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' ⃗zT −1] and Z2 = [⃗z2⃗z3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' ⃗zT ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' The goal is to find a good  approximation ˜A of the matrix A that represents this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' This can be cast as the following matrix problem:  Z2 ≅ AZ1  (4)  The least square solution to Equation 4 is found by taking the Moore-Penrose inverse:  A ≅ Z2Z†  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  Since A is a square matrix, one can consider the eigen-decomposition of A:  AU = UΛ  (5)  The eigenvectors and eigenvalues (uj,λj) of the A are the DMD modes and eigenvalues respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  For systems with a large number of data points L, this direct method isn’t tractable since A is a L × L square matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  Therefore different methods are designed to 1) solve Equation 4 while 2) providing a reduced order model at the same  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Among the most simple and popular methods for doing this is the SVD-based method:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Compute the rank M SVD of Z1 = WΣV T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Consider ˜A ∶= W T AW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Then ˜A is M × M, and since it is unitarily equivalent to A, has the same eigenvalues  with eigenvectors ξ of ˜A related to A via Wξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  6  A PREPRINT - JANUARY 16, 2023  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Note that since Z2 ≅ AZ1 = AWΣV T , after multiplying by W T and rearranging we have  ˜A = W T AW = W T Z2V Σ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  In other words, ˜A can be computed directly from the data and SVD of Z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Compute eigenvectors and eigenvalues {˜uj,λj} of the much smaller M × M matrix ˜A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' The j′th DMD mode is computed as uj ∶= W ˜uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  Other methods for approximating the DMD modes based off SVD exist, such as exact DMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' See [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  The DMD modes and eigenvalues can be used to derive dynamical information analogous with the right singular vectors  of SVD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' By iteratively applying Equation 3, we find that  ⃗zt+1 ≅ At⃗z1  (6)  Write ⃗z1 in the basis provided by the modes Ψ:  ⃗z1 = Ψ⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  (7)  Then combining Equations 5, 6 and 7 we see that  ⃗zt+1 ≅ At⃗z1 = AtΨ⃗b = ΨΛt⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  Hence, we have represented the state of the system in terms of a DMD mode expansion with temporal evolution on the  DMD eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' The time series {(λt  jbj)T  t=1}M  j=1 are referred to as the DMD dynamics for the j’th mode ψj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
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+page_content='  [9] Mark R Petersen, Xylar S Asay-Davis, Anne S Berres, Qingshan Chen, Nils Feige, Matthew J Hoffman, Douglas W  Jacobsen, Philip W Jones, Mathew E Maltrud, Stephen F Price, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' An evaluation of the ocean and sea ice  climate of e3sm using mpas and interannual core-ii forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Journal of Advances in Modeling Earth Systems,  11(5):1438–1458, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  [10] Rolf H Reichle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Data assimilation methods in the earth sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Advances in water resources, 31(11):1411–1418,  2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  [11] Todd Ringler, Mark Petersen, Robert L Higdon, Doug Jacobsen, Philip W Jones, and Mathew Maltrud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' A  multi-resolution approach to global ocean modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Ocean Modelling, 69:211–232, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  [12] Claude Sammut and Geoffrey I Webb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Encyclopedia of machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Springer Science & Business Media,  2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  [13] Jonathan H Tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Dynamic mode decomposition: Theory and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' PhD thesis, Princeton University, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  7  A PREPRINT - JANUARY 16, 2023  (a) Monthly errors - all models  (b) Monthly errors - top models  (c) Three month errors - all models  (d) Three month errors - top models  (e) Yearly errors - all models  (f) Yearly errors - top models  Figure 1: Figures (a), (c), (e) display the distribution of errors for all the models fit over one month, three month, and one year time  windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Figures (b), (d), (f) show the top three models from distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  8  300  250 -  200-  Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  150  100 -  50 -  0 8  MPAS  DMD_dyn  DMD  SVD_dyn  SVD  RBF  Model10  8  0  C  Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  4  2  MPAS  DMD_dyn  RBF  Model175  150  125  100  Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  75  50  25 -  0  MPAS  DMD_dyn  DMD  SVD_dyn  SVD  RBF  Model8  7 -  6  Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  0  4  m  2  1  0  MPAS  DMD_dyn  RBF  Model20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='0  0  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='5  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='0  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='5  Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='0  MPAS  DMD_dyn  DMD  SVD_dyn  SVD  RBF  Model5  0  C  Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  m  2  1 -  MPAS  DMD_dyn  SVD_dyn  ModelA PREPRINT - JANUARY 16, 2023  (a) Errors of DMD basis versus adding dynamics  (b) Error over time - DMD basis versus adding dynamics  (c) Error distribution of top models  (d) Error over time - top models  Figure 2: Errors of models for time period Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' 1st 2008-Jan 1st 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' Plots (a) and (c) display error distributions for different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  Plots (b) and (d) compute the average error on each of the five day time slices to explore inability to capture seasonality/baises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='0  Errors: yi - F(xi)  MPAS  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='5  DMD dyn  RBF  0  10  20  30  40  50  60  70  Time StepErrors: yi - F(xi)  250  DMD_dyn  DMD  200-  100 -  50 -  0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='0  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='5  Temperature C3  0  Errors: yi - F(xi)  DMD  DMD_dyn  3  0  10  20  30  40  50  60  70  Time Step250  Errors: yi - F(xi)  MPAS  DMD_dyn  200-  RBF  150 -  Count  100-  50 -  0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='0  Temperature CA PREPRINT - JANUARY 16, 2023  (a) Fully dynamic DMD basis interpolation  (b) Difference between fully dynamic DMD basis and MPAS: FMPAS(xi) − FDMD_dyn(xi)  (c) SST distribution of MPAS versus fully dynamic DMD  basis  Figure 3: Average yearly temperature comparison of fully dynamic DMD versus MPAS-O for time period Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' 1st 2008-Jan 1st  2009  10  0  5  10  15  20  25  30  Temperature C"-4  0  2  Temperature CSST Distribution: 2008-01-01  800-  MPAS  DMD_dyn  700  600  Count  500  400  300-  200-  100  0  5  10  15  20  25  Temperature C*A PREPRINT - JANUARY 16, 2023  (a) Buoy observations  (b) Fully dynamic DMD errors on buoys: yi − F(xi)  (c) MPAS errors on buoys: yi − F(xi)  Figure 4: (a) Buoy observations, with (b) Fully dynamic DMD errors and (c) MPAS errors on test buoys for year Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content=' 1st 2008-Jan  1st 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}
+page_content='  11  Buoy Test  5  10  15  20  25  30  Temperature CTest error  6  4  2  0  2  4  TemperatureCTest error  8  6  4  2  0  2  4  6  8  TemperatureC' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE5T4oBgHgl3EQfVg81/content/2301.05551v1.pdf'}

G9AzT4oBgHgl3EQfHfuN/content/tmp_files/2301.01046v1.pdf.txt

@@ -0,0 +1,1120 @@
+Critical scaling law for the deposition efficiency of inertia-driven particle collisions with a cylinder
+in high Reynolds number air flow
+Matthew R. Turner1, ∗ and Richard P. Sear1, †
+1School of Mathematics and Physics, University of Surrey, Guildford, GU2 7XH, United Kingdom
+(Dated: January 4, 2023)
+The Earth’s atmosphere is an aerosol, it contains suspended particles. When air flows over an obstacle such
+as an aircraft wing or tree branch, these particles may not follow the same paths as the air flowing around the
+obstacle. Instead the particles in the air may deviate from the path of the air and so collide with the surface of
+the obstacle. It is known that particle inertia can drive this deposition, and that there is a critical value of this
+inertia, below which no point particles deposit. Particle inertia is measured by the Stokes number, St. We show
+that near the critical value of the Stokes number, Stc, the amount of deposition has the unusual scaling law of
+exp(-1/(St-Stc)1/2). The scaling is controlled by the stagnation point of the flow. This scaling is determined by
+the time for the particle to reach the surface of the cylinder varying as 1/(St-Stc)1/2, together with the distance
+away from the stagnation point (perpendicular to the flow direction) increasing exponentially with time. The
+scaling law applies to inviscid flow, a model for flow at high Reynolds numbers. The unusual scaling means
+that the amount of particles deposited increases only very slowly above the critical Stokes number. This has
+consequences for applications ranging from rime formation and fog harvesting to pollination.
+I.
+INTRODUCTION
+The Earth’s atmosphere is an aerosol, in that it contains sus-
+pended particles, with sizes up to tens of micrometres [1]. For
+example, clouds and fogs are aerosols of water droplets sus-
+pended in air. When air flows over an obstacle such as an
+aircraft wing or tree branch, the suspended particles may fol-
+low the air flow around the obstacle, or they may deposit on
+the surface of the obstacle. For particles tens of micrometres
+in diameter, deposition on the surface of the obstacle is largely
+due to the particle’s inertia. While the air flow curves to move
+around the obstacle, the particle’s inertia means the particle
+tends to move in straight lines, and so collides with the obsta-
+cle, see Fig. 1. Here we study these inertia-driven collisions.
+Deposition of particles from flowing air onto obstacles oc-
+curs in many contexts. These include filtration [2–4], harvest-
+ing water from fog [5–7], pollination [8, 9], and rime (ice)
+formation [10–12]. It has been studied for approximately a
+hundred years. A lot of the earliest work considered appli-
+cations where the air was flowing rapidly, at large Reynolds
+numbers, which is relevant to applications such as rime form-
+ing on the leading edges of aircraft wings. The large Reynolds
+number limit is also the limit we consider here. Rime forms
+by water droplets below the freezing temperature (T = 0 ◦C)
+depositing on a surface and then freezing. This can coat a
+wing with ice, risking a crash.
+In 1931 Albrecht [13] found a threshold in the inertia, be-
+low which no (point) particles deposited on the obstacle. A
+minimum amount of inertia is needed before any particles are
+deposited. Then in the 1940s, first G.I. Taylor [14] (also in
+the scientific papers of G.I. Taylor[15]), and then Langmuir
+and Blodgett [16] calculated this critical value of the inertia.
+Here we confirm this result, and build on it. We determine the
+critical scaling of the deposition efficiency, and show how this
+∗ m.turner@surrey.ac.uk; http://personal.maths.surrey.ac.uk/st/M.Turner/
+† r.sear@surrey.ac.uk; https://richardsear.me/
+relates to the flow field around the obstacle.
+There has been considerable work on this problem since
+the 1940s [17, 18], motivated by its many applications, but
+the critical scaling has not been studied before, for the high
+Reynolds number flow field considered by Taylor [14, 15], and
+by Langmuir and Blodgett [16]. How the threshold varies with
+Reynolds number has been studied by Phillips and Kaye[19],
+and Araújo et al.[20] determined the critical scaling for zero-
+Reynolds-number flow.
+Most of the work has been computational or theoretical, as
+experiments on collisions in high- Reynolds-number flows are
+challenging. However, Wong and coworkers did obtain some
+experimental data on deposition efficiency at Reynolds num-
+bers of hundreds [21]. This was for an aerosol with particles
+with a narrow distribution of sizes. They found no measur-
+able deposition below a threshold near that predicted by Tay-
+lor [14, 15], and by Langmuir and Blodgett [16]. So these
+experiments agree with theoretical predictions (for a simpli-
+fied model) that a threshold exists. Makkonen and coworkers
+[22–24] have measured ice deposition on cylinders. This is for
+the typical case in the environment, where the droplets have
+a broad range of sizes, which complicates comparison with
+theoretical predictions. Here we mostly consider aerosols of
+identical particles but we do look at the affect of a distribution
+of particle sizes, in order to compare with this work.
+A.
+Obstacles in high Reynolds number flow
+The air flow over an aircraft wing is fast, speeds U of order
+100ms−1 (≃ 360kmh−1). This speed, combined with a wing
+leading edge radius R of order 10cm, means that the Reynolds
+number of the flow over the wing is Re = UR/ν ∼ 106 ≫ 1
+with ν ≃ 10−5ms−2, the kinematic viscosity of air. So, fol-
+lowing Langmuir and Blodgett and many others, we approxi-
+mate the airflow over a wing by inviscid, incompressible flow
+over an infinite cylinder, where we have a simple analytic ex-
+pression for the flow field [25]. This is a simple model of
+arXiv:2301.01046v1  [physics.flu-dyn]  3 Jan 2023
+
+2
+FIG. 1. Plot of the cross-section of the cylinder (yellow), the flow
+field (streamlines in blue), and three trajectories. The green and black
+trajectories collide with the cylinder, while the red trajectory misses.
+Here St = 0.7 and the initial conditions are at Cartesian coordinate
+xC(t = 0) = −10, with initial yC values given in the key. The black
+trajectory is the one that defines the edge of the region where particles
+collide, and so its value of yC = 0.2793 = ηd. N.B. Numerical errors
+mean we cannot determine ηd to four significant figures, so the yC
+value for black curve should not be taken as accurate to four figures.
+high-Reynolds-number flow.
+The cylinder is of radius R, with its axis along the zC-axis,
+directed into the page in Fig. 1. The cylinder is taken to be at
+rest in the frame of reference. Far from the cylinder, flow is in
+the xC direction: Ui. Note that we denote the Cartesian coor-
+dinates as (xC,yC,zC), with corresponding unit vectors i, j, k,
+because below we will use x and y to indicate distances from
+the stagnation point. In cylindrical polar coordinates, the flow
+field u(r) is given by
+u
+U =
+�
+1− R2
+r2
+�
+cos(θ)�r−
+�
+1+ R2
+r2
+�
+sin(θ)�θ,
+(1)
+where (r,θ) are plane polar coordinates in the (xC,yC)-plane.
+Streamlines of the flow field are illustrated in Fig. 1.
+In the remainder of this paper, we will set the radius of the
+cylinder R = 1 and the flow field speed U = 1. So for exam-
+ple, both the Cartesian coordinate xC and the distance along
+the Cartesian coordinate from the stagnation point, x, are in
+units such that the cylinder radius R = 1. This choice also
+means that time is measured in units such that it takes unit
+time to move a distance of the cylinder’s radius, when moving
+at speed U.
+B.
+Particles in air flowing around an obstacle: The effect of
+inertia on deposition
+We approximate the particles by point particles, they are
+about ten thousand times smaller than the wing leading edge.
+Point particles that follow the streamlines of fluid flow per-
+fectly never collide with the obstacle. However, if the parti-
+cles have inertia then when the air flow changes direction to
+flow around the obstacle, the particle’s inertia may cause it to
+go straight on potentially leading it to crash into the obstacle.
+So particle inertia can cause collisions.
+The inertia of a particle is quantified by its Stokes number
+St, defined for a particle of mass mp by
+St = mpUB
+�
+R
+(2)
+with B the particle mobility and R the lengthscale of the obsta-
+cle. The Stokes expression for the mobility is B = 1/(6πηap)
+with η the dynamic viscosity of air, and ap the radius of the
+particle. The Stokes number is the dimensionless ratio be-
+tween the inertia of a particle – which tends to cause the par-
+ticle to move in straight lines – and the friction between the
+particle and the surrounding air – which tends to make the par-
+ticle follow streamlines. For a wing of width 0.1m, the Stokes
+number is of order one for droplets micrometres in diameter,
+so cloud droplets of size micrometres, and tens of microme-
+tres will deposit on the wing.
+In the St → ∞ limit, particles move in straight lines and so
+in that limit a cylinder of radius R sweeps out a strip of air
+of thickness 2R, collecting all the particles in this strip. This
+allows us to define the deposition or collection efficiency
+ηd =
+maximum displacement from cylinder axis
+perpendicular to the flow direction, for
+which a particle collides with cylinder surface
+�
+R
+which varies from zero when no particles collide, to one when
+St → ∞. The displacement is taken far upstream of the cylin-
+der. Calculation of ηd is done by starting particle trajectories
+far upstream of the cylinder at varying values of the displace-
+ment yC normal to the flow direction. Then ηd is defined by
+the largest initial displacement along yC, for which the particle
+collides. This is illustrated by the black trajectory in Fig. 1.
+Numerical results for ηd as a function of St, together with
+the fit of Langmuir and Blodgett [16], are shown in Fig. 2.
+Note that at small values of St, the collection efficiency in-
+creases very slowly with increasing inertia. This is what we
+will explain here. It follows directly from the high Reynolds
+number flow field. In the low-Reynolds-number limit, where
+the flow field is very different, ηd increases much more rapidly
+above the critical Stokes number [3, 20, 26]. Langmuir and
+Blodgett’s fit is in Appendix A, and the details of our numeri-
+cal calculations are in Appendix B. See Appendix B 1 for the
+details of the fit in Fig. 2.
+
+yc(t = 0) = 0.4
+yc(t = 0) = 0.2793
+Yc(t = 0) = 0.1
+stagnation point3
+FIG. 2. Plot of the deposition or collection efficiency ηd as a function
+of the Stokes number. Shown are numerical results (blue circles), the
+function of Langmuir and Blodgett [16] (green curve) and our fit to
+the region of small δ = St−Stc (dashed red curve).
+II.
+NEWTON’S EQUATION FOR A PARTICLE IN
+FLOWING AIR
+For a particle suspended in flowing air we assume that the
+only force on the particle is the friction with the surrounding
+air, which is taken to be proportional to the difference between
+particle’s velocity v and the local flow velocity u. This force
+is taken to act on the particle’s centre of mass. Then Newton’s
+equation for the particle motion is
+Stdv
+dt = −(v−u).
+(3)
+In cylindrical polar coordinates, this is
+St
+�
+¨r −r
+� ˙θ
+�2�
+= −(˙r −ur),
+(4a)
+St
+�
+¨θ + 2˙r ˙θ
+r
+�
+= −
+� ˙θ −uθ
+�
+,
+(4b)
+where u = urˆr+uθ ˆθ.
+III.
+THE CYLINDER’S FORWARD STAGNATION POINT
+We are interested in the behaviour near the critical Stokes
+number of Langmuir and Blodgett [16]. Here, particle tra-
+jectories pass close to the stagnation point at the front of the
+cylinder. This stagnation point is at r = 1 and θ = π, see
+Fig. 1. So we will study behaviour near this stagnation point.
+We start by changing variables to the distance to contact with
+the cylinder x = r − 1 ≪ 1, and the angle from the angle of
+the stagnation point y = π −θ ≪ 1. Note that x and y are not
+the conventional Cartesian coordinates, which we denote by
+xC and yC.
+In these new coordinates, Newton’s equations for the parti-
+cle, Eq. (4), become
+¨x−(1+x)˙y2 = − ˙x−ur
+St
+,
+(5a)
+¨y+
+2
+1+x ˙x˙y = − ˙y+uθ
+St
+.
+(5b)
+Note that ˙y = − ˙θ and ¨y = − ¨θ etc. Near the stagnation point,
+we can expand the flow field in Eq. (1) as a series in x and y
+u =
+�
+−2x+3x2 +O(cubic terms)
+� ˆr
++(−2y+2xy+O(cubic terms)) ˆθ.
+(6)
+It is worth noting that as we do not have stick boundary con-
+ditions here, uθ is not zero at the cylinder surface, except at
+the stagnation point; ur is zero at the surface is because there
+is no flow into the cylinder.
+We substitute the flow field of Eq. (6) into the equation for
+the particle trajectory (Eq. (5)). Then if we keep only linear
+and quadratic terms we obtain
+¨x− ˙y2 = − 1
+St
+�
+˙x+2x−3x2�
+,
+(7a)
+¨y+2˙x˙y = − 1
+St (˙y−2y+2xy),
+(7b)
+We now solve these equations to find out which are the trajec-
+tories of particles that collide with the cylinder.
+IV.
+PARTICLE TRAJECTORIES ON AXIS, AND THE
+CRITICAL STOKES NUMBER
+We start by considering particle trajectories precisely on
+axis, where θ = π and y = 0. This will enable us to deter-
+mine the critical value of the Stokes number, below which no
+particles collide with the cylinder. We follow Taylor [14, 15],
+and Langmuir and Blodgett [16] here. On axis the system re-
+duces to a one dimensional problem.
+Near the stagnation point x ≪ 1, and we can approximate
+the flow by keeping only the leading order terms in x. Then
+Eq. (7a) is just
+St¨x = −˙x−2x,
+(8)
+which as both Taylor, and Langmuir and Blodgett realised,
+is just the differential equation for damped simple harmonic
+motion (SHM). It has solutions
+x(t) = A0 exp(λ1t)+B0 exp(λ2t),
+(9)
+λ1,λ2 = −1±√1−8St
+2St
+,
+(10)
+where A0 and B0 are fixed by the initial conditions, for exam-
+ple x(t = 0) = x0 and ˙x(t = 0) = u0. See Appendix C.
+For 0 < St < 1/8 both λ are real and negative (overdamped
+SHM solutions), and so the particle approaches the cylinder
+surface at a speed which decays exponentially. There is only
+
+0.10
+L&B
+0.08
+numerics
+fit
+0.06
+0.04
+0.02
+0.0Q
+0.1
+0.3
+0.2
+0.4
+0.0
+St4
+FIG. 3. Plot of the trajectories in Fig. 1 but in the xy plane. The green
+and black trajectories collide with the cylinder, while the red trajec-
+tory misses. Here St = 0.7 and the initial conditions are at Cartesian
+coordinate xC(t = 0) = −10, with initial yC values given in the key.
+Note that the trajectory which just collides does so tangentially to the
+surface.
+a collision in the t → ∞ limit, i.e., no collision at finite time.
+But if St > 1/8 then we have complex λ (underdamped SHM
+solutions), and the collision will occur in finite time. The crit-
+ical value of the Stokes number is therefore Stc = 1/8. We
+can define the distance from this critical value as
+δ = St−Stc.
+(11)
+Note that there are initial conditions for Eq. (9), for which
+the particle collides in finite time[27]. However, they do not
+appear to be physically relevant, as Ingham et al.[27] discuss.
+A.
+Time to collision, on axis
+The collision occurs when x(tCOLL) = 0 and is set by the
+angular frequency — the imaginary part of Eq. (10) — ω =
+√1−8St/(2St) ≃ 8
+√
+2δ 1/2. As expected (see Appendix C
+for details) the time to collide tCOLL is half the period
+tCOLL ≃
+π
+8
+√
+2
+δ −1/2
+δ ≪ 1.
+(12)
+As the critical Stokes number is approached from above, the
+time to reach the cylinder surface and collide diverges as
+1/δ 1/2. Numerical calculations using the full flow field agree
+with this observation, see Appendix B 1.
+V.
+PARTICLE TRAJECTORIES OFF AXIS, AND THE
+CRITICAL SCALING OF THE DEPOSITION EFFICIENCY
+We now consider the full two-dimensional case, near the
+stagnation point. Off axis we also need an equation for y.
+Retaining only the leading order linear terms in Eq. (7b) for y
+leads to
+St¨y = −˙y+2y,
+(13)
+in which there is no coupling with the x direction. This equa-
+tion is almost the damped SHM equation again, but in SHM
+language, the ‘force’ term has the opposite sign, so it is not a
+restoring force but drives exponential growth of y. The solu-
+tion is
+y(t) = C0 exp(µ1t)+D0 exp(µ2t)
+(14)
+with
+µ1,µ2 = −1±√1+8St
+2St
+(15)
+Now, µ1 > 0 and µ2 < 0 so at long times the µ1 solution dom-
+inates; it increases exponentially while the other solution de-
+cays to zero. The constants C0 and D0 are related to the initial
+conditions. If y(t = 0) = y0 and ˙y(t = 0) = v0, then
+C0 = v0 − µ2y0
+µ1 − µ2
+and D0 = −v0 − µ1y0
+µ1 − µ2
+.
+(16)
+Having determined the leading order behaviour of y, we
+return to Eq. (7a) for x. The leading order y term in Eq. (7a)
+is the ˙y2 term. Retaining only this term, we have
+¨x+ ˙x+2x
+St
+= ˙y2.
+(17)
+The new term acts like an effective force that pushes the par-
+ticle away from a collision.
+The general solution to Eq. (17) is given in Eq. (D2) in Ap-
+pendix D. We take this equation, expand for δ ≪ 1, keep only
+leading-order terms, and then set x = 0 at the collision time of
+Eq. (12). This yields
+x(tCOLL) = −A0 exp
+�
+− π
+2
+√
+2
+δ −1/2
+�
++E0 exp
+�
+(
+√
+2−1)π
+√
+2
+δ −1/2
+�
+= 0,
+(18)
+with E0 = C2
+0(11−6
+√
+2)/49. This equation is a function of δ
+and the initial boundary conditions on x and y.
+Equation (18) has two terms. The first term is negative and
+is the same for the on-axis case. It is this term that results
+in a collision after a time ∼ 1/δ 1/2. The second term is new
+for the off-axis case, it increases exponentially with δ −1/2,
+and the prefactor scales with C2
+0 ∼ (v0 − µ2y0)2, i.e., with the
+initial conditions on y.
+We want to estimate ηd, which is set by the largest dis-
+placement y0 for which the particle collides with the cylinder,
+which is when Eq. (18) is satisfied. Near the critical Stokes
+number y0 will be small, and the exponential variation of the
+terms in Eq. (18) suggests it needs to be exponentially small,
+so we write C0 = ζ exp[α(δ)] where ζ is a constant to leading
+order in δ, and α is a function of δ.
+Then we insert C0 = ζ exp(α) into Eq. (18). For small δ
+solutions are only possible when the exponential exponents
+are equal. Equating the exponents of the two terms gives us
+−
+π
+2
+√
+2
+δ −1/2 = 2α + (
+√
+2−1)π
+√
+2
+δ −1/2,
+(19)
+
+0.45
+Yc(t = 0) = 0.4
+0.40
+Yc(t = 0) = 0.2793
+0.35
+yc(t = 0) = 0.1
+0.30
+0.25
+0.20
+0.15
+0.10
+0.05
+0.00
+0.4
+0.6
+0.8
+0.0
+0.2
+1.0
+y=-5
+and
+α = −(4−
+√
+2)
+8
+πδ −1/2 ≃ −1.015δ −1/2.
+(20)
+Collisions only occur at small δ when y0 scales as
+exp(−1.015δ −1/2). This sets the width of the strip of air over
+which particles collide with the cylinder, and so the deposition
+efficiency is
+ηd ∼
+�
+0
+δ ≤ 0
+exp
+�
+− (4−
+√
+2)
+8
+πδ −1/2�
+0 < δ ≪ 1
+(21)
+Above the critical Stokes number, the collection efficiency
+has the unusual scaling exp(−1/δ 1/2). This means that the
+deposition efficiency increases only slowly above the critical
+Stokes number, see Fig. 2. Numerical calculations with the
+full flow field confirm the result. They are in Appendix B 1.
+Limited experimental data [21] near Stc means that a quanti-
+tative comparison with our scaling result is not possible.
+The scaling is a direct consequence of the form of the flow
+field near the stagnation point, Eq. (6). From the fact that the
+flow field varies linearly with x and y. The flow field leads
+to the time to collision scaling as 1/δ 1/2, and to the y coor-
+dinate increasing exponentially with time, which in turn gives
+us Eq. (21).
+Numerical results for particle trajectories as functions of x
+and y are plotted in Fig. 3. The trajectory that just collides,
+and so defines ηd, is in black. Note the rapidly increasing y
+as the collision is approached, and that the collision occurs
+tangential to the cylinder surface.
+VI.
+COMPARISON WITH PARTICLE DEPOSITION IN
+STOKES FLOW
+Our results are for a flow field that neglects viscosity, in ef-
+fect the infinite-Reynolds-number limit. The opposite limit is
+where inertia is negligible and viscosity dominates. This is the
+zero-Reynolds-number or Stokes-flow limit, and it is relevant
+to the filtration of aerosol particles from air by fibrous filters,
+where the Reynolds number is small [2, 3, 28]. For particles
+in Stokes flow, there is also a critical value of the Stokes num-
+ber, but the scaling of the deposition efficiency is ηd ∼ δ 1/2,
+which is completely different scaling. Langmuir and Blodgett
+suspected that there were critical Stokes numbers for a num-
+ber of different flow fields including spheres in Stokes flow,
+but this scaling was first found for spheres in Stokes flow by
+Araújo et al.[20].
+The particle trajectories are very different for the inviscid
+and Stokes flow fields. For example, with the slip boundary
+conditions in the inviscid limit, the trajectories where the par-
+ticle just collides with the cylinder surface do so tangentially,
+see Fig. 3. This is because the particle’s velocity normal to
+the cylinder surface tends to zero at the collision, while the
+tangential component is non-zero. Particles in Stokes flow do
+not collide tangentially because here the stick boundary con-
+ditions mean that the tangential fluid-flow velocity is zero at
+contact [26].
+FIG. 4. Plot of the mean deposition or collection efficiency, ηd, as
+a function of the median Stokes number. Shown are numerical re-
+sults for monodisperse droplets (blue circles), and for polydisperse
+droplets whose Stokes numbers obey a log-normal distribution with
+width parameter σ = 1/2 (orange curve).
+At intermediate Reynolds numbers, there will be a bound-
+ary layer of thickness ∼ 1/Re1/2. Within this boundary layer
+the flow field is approximately viscous flow, outside it is closer
+to inviscid flow. So as the Reynolds nunber is increased the
+boundary layer thins and the behaviour changes continuously
+from the Stokes to the inviscid limit.
+Work by Robinson
+and coworkers suggests that the deposition behaviour changes
+smoothly between the limits [26].
+VII.
+DEPOSITION FROM AN AEROSOL OF PARTICLES
+WITH A RANGE OF DIAMETERS
+In natural aerosols, the particles are almost never all of the
+same diameter. A broad distribution of particle sizes is typi-
+cal [1, 22, 24]. Within our simplified model, each particle is
+characterised by a single parameter: the Stokes number. A
+distribution of particle sizes results in a distribution of Stokes
+numbers. The Stokes number of a particle is given by Eq. (2).
+When mass mp ∝ a3
+p and the Stokes mobility B ∝ 1/ap, the
+scaling of the Stokes number is St ∝ a2
+p, i.e., the Stokes num-
+ber scales with the square of the particle radius. Thus a distri-
+bution of particle radii gives a distribution of Stokes numbers.
+An aerosol of particles with a range of radii will have particles
+with a distribution of Stokes numbers: p(St).
+We need a model probability distribution function, p. We
+use the standard log-normal distribution
+p(St;StMED,σ) =
+1
+Stσ(2π)1/2 exp
+�
+−(lnSt− µ)2
+2σ2
+�
+(22)
+Here, the median Stokes number StMED = exp(µ) provides
+a measure of the typical Stokes number. The width of the
+distribution is set by σ: the ratio of the standard deviation to
+the median value is exp(σ2/2)(exp(σ2)−1)1/2.
+
+mono.
+g= 1/2
+0.4
+0.2
+0.5
+1.0
+1.5
+StMED6
+FIG. 5. A quantity M, which is proportional to the mass deposited,
+plotted as a function of the median Stokes number. Shown are nu-
+merical results for monodisperse droplets (dark blue circles) and for
+polydisperse droplets whose Stokes numbers obey a log-normal dis-
+tribution with width parameter σ = 1/2 (magenta curve).
+Within our model, the particles are independent. So the
+mean deposition efficiency, ηd, for the aerosol is simply
+ηd(StMED,σ) =
+�
+η(St)p(St;StMED,σ)dSt
+(23)
+In Fig. 4, we compare the mean deposition efficiency as a
+function of median Stokes number, of monodisperse particles,
+and particles with a log-normal distribution with width param-
+eter σ = 1/2. This distribution has a ratio of standard devia-
+tion to the median of 0.60. Near the critical value of the Stokes
+number, the convolution of deposition efficiency with the dis-
+tribution of Stokes numbers of the particles smooths out the
+transition at Stc. For a distribution of particle sizes and hence
+and Stokes numbers, the deposition efficiency is never zero,
+unless all particles have Stokes numbers less than Stc. Thus
+when the median Stokes number is below Stc polydisperse
+aerosols have larger deposition efficiencies than monodisperse
+ones.
+But regardless of polydispersity the deposition effi-
+ciency tends to one at larges values of the median Stokes num-
+ber. So at large median Stokes numbers a broad distribution
+of particle sizes has little effect on the mean deposition effi-
+ciency.
+In experiment, the total mass deposited on an obstacle can
+be measured [22, 24]. As the mass of a particle scales as a3
+p,
+the mass of a particle scales as St3/2. So the three-halves mo-
+ment of the efficiency times the distribution of particle Stokes
+numbers gives us a quantity proportional to the total mass de-
+posited. We call this quantity M:
+M(StMED,σ) =
+�
+η(St)p(St;StMED,σ)St3/2dSt
+(24)
+We have plotted this deposited mass as a function of median
+Stokes number in Fig. 5. As with the deposition efficiency in
+Fig. 4, when a range of particle sizes are present, the transition
+is smoothed over. Mass is deposited at all values of the median
+Stokes number. This is consistent with the work of Makkonen
+and coworkers [22, 24], who find that some mass is deposited
+under all their experimental conditions. However, the mass
+deposited is always larger than for monodisperse droplets with
+the same median Stokes number. The total mass of aerosol
+particles is larger in the polydisperse case. (at the same me-
+dian Stokes number). In addition, the large Stokes number tail
+of the distribution contributes large amounts to the mass de-
+posited as here the deposition efficiency is high and these large
+particles contribute large amounts to the total mass deposited.
+VIII.
+CONCLUSION
+We have built on Taylor’s [14, 15], and Langmuir and
+Blodgett’s work [16] to quantitatively understand the be-
+haviour of the deposition efficiency ηd near the critical value
+of the Stokes number.
+Just above the critical value, ηd ∼
+exp(−1/δ 1/2). This unusual scaling comes from the fact that
+for an off-axis particle trajectory, the particle’s displacement
+from the cylinder’s axis increases exponentially with time,
+while the time to reach the cylinder scales as 1/δ 1/2. Thus
+unless the initial displacement from the axis is exponentially
+small, the particle misses the cylinder. The scaling follows di-
+rectly from the (inviscid) flow field we used; the behaviour of
+particles in Stokes flow with stick boundary conditions is very
+different [3, 20, 26]. The lesson here is that particle deposi-
+tion from flowing air is very sensitive to details of the flow
+field, especially near the stagnation point. The slow increase
+in deposition efficiency follows from the slip boundary con-
+ditions. Particle deposition has varied applications, from rime
+formation on aircraft wings v to pollination [8, 9]. The care-
+ful analysis of particle trajectories near the forward stagnation
+point will greatly contribute to the understanding of this issue.
+Here, we have studied a cylinder, but Langmuir and Blod-
+gett [16] also found a critical value of the Stokes number
+for spheres in inviscid, axisymmetric flow. This was at the
+slightly lower value of Stc = 1/12. It is straightforward to
+show that the scaling laws found here for a cylinder are the
+same (up to multiplicative constants) for a sphere. The details
+are in Appendix E. Our results for the scaling were obtained
+by expanding around the stagnation point, it is possible that
+they are general to at least most inviscid flows impinging on a
+locally parabolic surface in two or three dimensions.
+IX.
+SUPPLEMENTARY MATERIAL
+The supplementary material is a Python Jupyter notebook
+that performs all numerical calculations, and produces all fig-
+ures in this work.
+ACKNOWLEDGMENTS
+It is a pleasure to thank Josh Robinson and Patrick Warren
+for inspiring and useful discussions. The data that support the
+
+mono.
+1.5
+g= 1/2
+1.0
+M
+0.5
+0.5
+1.0
+1.5
+StMED7
+findings of this study are available within the article and in a
+Python Jupyter notebook in supplementary material.
+Appendix A: Langmuir and Blodgett expression for deposition
+efficiency
+The expression used by Langmuir and Blodgett [16] is
+ηd =
+�
+�
+�
+0
+St ≤ 1/8
+0.466[log10(8St)]2 1/8 < St < 1.1
+St/(St+π/2)
+1.1 < St
+(A1)
+Note that the functional form near St = 1/8 is incorrect but
+that it is quite close to numerical calculations. This expres-
+sion, and closely related ones, have been and are used exten-
+sively to model ice accretion on the leading edges of surfaces
+moving through air [7, 10, 17, 18]. Finstad et al.[17, 18] sum-
+marise work up to 1998 on improving the approximation of
+Eq. (A1). None of the improvements has the correct func-
+tional form for small δ = St − 1/8 but it should be said the
+work was focused on producing empirical expressions that are
+accurate over a wide range of values of the Stokes number, not
+on obtaining the correct value near the dynamical transition.
+Appendix B: Details of numerical calculations
+The numerical calculations were done with a Python
+Jupyter notebook available in the supplementary material.
+Particle trajectories were calculated as follows.
+Particles
+were started at Cartesian coordinates upstream of the cylin-
+der (xC0,yC0), i.e., a distance xC0 along the xC-axis (the flow
+direction), and displaced a perpendicular distance yC0 off axis.
+All calculations here are for xC0 = −10; we checked that in-
+creasing the distance upstream further to −20 had very lit-
+tle effect. Trajectories were obtained by integrating Newton’s
+equation of motion ((3)), and checking for collisions, which
+occur whenever r = 1.
+1.
+Verification of scaling found by analytical mathematics, by
+comparison to numerical calculations
+On axis, our analytical results predict that as the critical
+Stokes number is approached from above, the time to reach
+the surface of the cylinder and collide, scales as 1/δ 1/2. There
+is another contribution, which is the time for the particle to go
+from its initial position upstream to a point where x ≪ 1. That
+contribution is approximately equal to initial displacement/U.
+So, we fit a function of the form A + Bδ −1/2 to numerical
+results for the time to collision. The numerical results and fit
+are shown in Fig. 6. We see excellent agreement.
+We perform the same analysis for the trajectories that define
+ηd, at the edge of the strip of air over which collisions occur.
+The results are in Fig. 6, and again the agreement is excellent.
+FIG. 6. Plot of time to collision, as a function of δ = St − Stc. The
+points are numerical results, and the lines are fits of function A +
+Bδ −1/2. Blue points and cyan line are for on axis collisions, with
+best fit values A = 9.78 and B = 0.309. Orange points and red line
+are for collisions at the edge of the deposition zone, with A = 9.92
+and B = 0.309. Note that both values for B are close to our prediction
+of tCOLL = π/(8
+√
+2δ 1/2) = 0.278/δ 1/2.
+FIG. 7. Plot of ln(ηd) as a function of 1/δ 1/2, to show exp(−1/δ 1/2)
+scaling of the deposition efficiency. Points are numerical results, line
+is fit of function A+B(1/δ 1/2) with best fit values A = −0.193 and
+B = −1.118. Note that value for B is close to our prediction of α =
+−1.015/δ 1/2 (Eq. (20)).
+We also predict that the deposition efficiency has the scaling
+form
+lnηd = A+B 1
+δ 1/2
+δ ≪ 1.
+(B1)
+In Fig. 7 we compare the results of numerical calculations
+with a fit of this form. Again the agreement is excellent.
+A final check is on the match between the numerics and
+the analytics; the analytical mathematics relies on an expan-
+sion in x and y and so is only valid near the stagnation point
+where x,y ≪ 1. We are interested in the deposition efficiency
+ηd(δ). The deposition efficiency is set by the trajectory with
+the largest initial Cartesian yC coordinate, call it yC0, at large
+and negative Cartesian xC (far upstream of the cylinder) for
+
+13.0
+on axis
+time to collision
+on axis fit
+12.5
+off axis
+12.0
+off axis fit
+11.5
+11.0
+6
+10
+8
+4
+1/81/2-3
+fit
+-4
+off axis
+-5
+-6
+(pu)ul
+-7
+-8
+-9
+-10
+-11
+5
+3
+4
+6
+8
+9
+10
+7
+1/61/28
+FIG. 8. Plot of y, ˙y and their combination ˙y − µ2y which sets C0,
+as functions of the initial off-axis deviation in Cartesian coordinates.
+All trajectories start at xC = −10, and end at x = 0.1. The Stokes
+number is St = 1/4.
+which a collision occurs. There the particle velocity is taken
+to be that in flow field, which is close to U along the Cartesian
+xC axis.
+What the analytic mathematics uses is the initial boundary
+condition on y, as it appears as C0, which has the scaling
+C0 ∼ v0 − µ2y0 ∼ exp(α)
+(B2)
+and in effect shows that C0 ∼ exp(−1/δ 1/2). Here y0 and v0
+are the initial conditions on y and ˙y, respectively. To complete
+the link between the scaling we obtained for C0 and ηd we
+need to show that C0 varies smoothly with the value of yC0.
+We do this in Fig. 8. There we have plotted y, ˙y and ˙y − µ2y
+at x = 0.1, as functions of yC0. These are all obtained from
+numerical calculations. We see that they all vary smoothly
+with yC0, so the scaling found for C0 will translate ηd. In
+Fig. 8 we selected x = 0.1 as a value less than one but not
+≪ 1, plots with x = 0.05 and x = 0.2, are similar.
+Appendix C: Solution for on-axis case
+The solution of Eq. (8) is
+x(t) = A0 exp(λ1t)+B0 exp(λ2t)
+(C1)
+λ1,λ2 = −1±√1−8St
+2St
+,
+(C2)
+where A0 and B0 are fixed by the initial conditions. With x(t =
+0) = x0 and ˙x(t = 0) = u0, A0 and B0 are given by
+A0 = λ2x0 −u0
+λ2 −λ1
+and B0 = u0 −λ1x0
+λ2 −λ1
+.
+(C3)
+For St < 1/8 both λ are real (overdamped SHM solutions),
+and so the particle approaches the cylinder surface (x = 0) at
+a speed which decays as exp(−t), and hence there is only a
+collision in the t → ∞ limit, i.e., no collision at finite time. A
+similar phenomena occurs in the St = 1/8 case. But if St >
+1/8 then we have complex λ (underdamped SHM solution),
+and the collision will occur in finite time. In this case the
+solution can be written as
+x(t) = exp[−t/(2St)]
+�
+A0 cos
+�√8St−1
+2St
+t
+�
++B0 sin
+�√8St−1
+2St
+t
+��
+,
+(C4)
+or by using trigonometric identities
+x(t) = ψ exp[−t/(2St)]cos
+�√8St−1
+2St
+t +φ
+�
+,
+(C5)
+where ψ is an amplitude and φ is a phase which are related to
+the constants A0 and B0, which are given by
+A0 = x0
+and B0 = 2Stu0 +x0
+√8St−1 .
+(C6)
+Thus
+φ = tan−1
+�2Stu0/x0 +1
+√8St−1
+�
+.
+(C7)
+And rewriting using δ
+x(t) = ψ exp[−t/(2St)]cos
+�
+8
+√
+2δ 1/2
+1+8δ t +φ
+�
+.
+(C8)
+The collision occurs when x(tCOLL) = 0 and is set by the co-
+sine term, and so by the angular frequency, which scales as
+δ 1/2. In the δ → 0 limit, φ in Eq. (C7) tends to −π/2 (as
+u0 < 0) and the collision time is set by 8
+√
+2δ 1/2tCOLL = π,
+which leads to Eq. (12).
+Appendix D: Solution of Eq. (17)
+In order to derive the scaling argument on the coupling term
+in Eq. (17), we consider only the exponentially growing term
+on the RHS from the solution Eq. (14). Hence the differential
+equation becomes
+¨x+ ˙x+2x
+St
+= C2
+0µ2
+1 exp(2µ1t).
+(D1)
+The general solution to this equation consists of a com-
+plementary function which satisfies the homogeneous equa-
+tion Eq. (8) and a particular integral which satisfies the in-
+homogeneous RHS.
+The general solution is then
+x(t) = exp[−t/(2St)]
+�
+A0 cos
+�√8St−1
+2St
+t
+�
++ B0 sin
+�√8St−1
+2St
+t
+��
++E0 exp(2µ1t),
+(D2)
+
+0.20
+y-μ2y
+0.1
+0.15
+二
+X
+when
+0.10
+0.05
+0.00
+0.01
+0.00
+0.02
+when
+yc
+Xc=  109
+where
+E0 =
+µ2
+1C2
+0
+4µ2
+1 + 2µ1
+St + 2
+St
+,
+and
+A0 = x0 −E0,
+(D3)
+B0 = 2Stu0 +x0 −E0(1+4Stµ1)
+√8St−1
+.
+(D4)
+Appendix E: Inertia-driven collisions with a sphere
+For a sphere in an inviscid axisymmetric flow, the flow field
+is[25]
+usphere
+U
+=
+�
+1− R3
+r3
+�
+cos(θ)�r−
+�
+1+ R3
+2r3
+�
+sin(θ)�θ,
+(E1)
+which when expanded about the forward stagnation point with
+U = R = 1 gives
+usphere =
+�
+−3x+6x2 +O(cubic terms)
+� ˆr
++
+�
+−3
+2y+ 3
+2xy+O(cubic terms)
+�
+ˆθ.
+(E2)
+As Eq. (E2) is the same as the equivalent for a cylinder,
+Eq. (6), apart from numerical prefactors, there is also a critical
+value of the Stokes number for a sphere. The different numer-
+ical factors shift it to the lower value of Stc = 1/12, which
+agrees with the result of Langmuir and Blodgett [16]. The
+change from a −2x term in the flow field for a cylinder to a
+−3x term for a sphere just shifts the critical value down by a
+factor of 2/3. The equivalent of Eq. (8) for a sphere just has a
+−3x term in place of the −2x term. This means that we have
+the same 1/δ 1/2 scaling of the time to collide.
+In spherical polar coordinates, the acceleration terms along
+a constant azimuthal angle trajectory, in Eq. (3) are the same
+as those for the two-dimensional cylindrical polar coordinates.
+Hence the left-hand-sides of Eq. (4), and thus of Eq. (7a) and
+Eq. (7b), are the same as presented here, i.e. the ˙y2 term is the
+significant term, and the analysis presented for the cylinder
+can be repeated for the sphere. The result is that the scaling
+law for the deposition efficiency has the same δ dependence as
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+

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@@ -0,0 +1,1602 @@
+Discontinuous Jump Behavior of the Energy Conversion in Wind Energy Systems
+Pyei Phyo Lin,1, ∗ Matthias W¨achter,1 M. Reza Rahimi Tabar,2, 1 and Joachim Peinke1
+1ForWind, Institute of Physics, University of Oldenburg, Oldenburg, Germany
+2Department of Physics, Sharif University of Technology, Tehran 11155-9161, Iran
+(Dated: January 16, 2023)
+The power conversion process of a wind turbine can be characterized by a stochastic differential
+equation (SDE) of the power output conditioned to certain fixed wind speeds. An analogous ap-
+proach can also be applied to the mechanical loads on a wind turbine, such as generator torque. The
+constructed SDE consists of the deterministic and stochastic terms, the latter corresponding to the
+highly fluctuating behavior of the wind turbine. Here we show how advanced stochastic analysis of
+the noise contribution can be used to show different operating modes of the conversion process of
+a wind turbine. The parameters of the SDE, known as Kramers-Moyal (KM) coefficients, are esti-
+mated directly from the measurement data. Clear evidence is found that both, continuous diffusion
+noise and discontinuous jump noise are present. The difference in the noise contributions indicates
+different operational regions. In particular, we observe that the jump character or discontinuity in
+power production has a significant contribution in the regions where the control system switches
+strategies. We find that there is a high increase in jump amplitude near the transition to the rated
+region, and the switching strategies cannot result in a smooth transition. The proposed analysis
+provides new insights to the control strategies of the wind turbine.
+I.
+INTRODUCTION
+Wind energy is one of the most promising contributions
+to the global energy transition from fossil fuels to clean
+and sustainable energy.
+Europe could install around
+105 GW of new wind energy capacity in the period of
+2021–2025 as reported in [1]. However, the complex and
+intermittent nature of wind makes wind energy produc-
+tion difficult to predict, which is important for a stable
+energy supply [2–5].
+Furthermore this nature of wind
+may cause premature mechanical fatigue failure [6, 7].
+It is known that the commonly used industry standard
+by the International Electrotechnical Commission [8] is
+not describing properly the variability of wind and wind
+power [7, 9]. In particular, if one investigates time series
+of the power output of a wind turbine, one can find very
+rapid power fluctuations, which can become larger than
+50% of the rated power [10]. Such short time power fluc-
+tuations in the range of MW will represent special loads
+for the drive train and also for the power grid, as these
+fluctuations seem to some add up in a wind farm instead
+of being averaged out [9, 10].
+In this contribution, we focus on a statistically ad-
+vanced description of the power fluctuations of a wind
+turbine. In recent years, it has been shown that the power
+conversion process of a wind turbine can be modeled by
+a stochastic Langevin differential equation of the power
+output P conditioned to certain fixed wind speeds u [10–
+12]. An analogous approach has also been used to model
+the mechanical loads on a wind turbine such as genera-
+tor torque T [13]. The advantage of this approach is that
+the model equations (in form of the Langevin equation)
+can be extracted directly from given data. This model
+∗ pyei.phyo.lin@uol.de
+can reproduce the stochastic, turbulent and intermittent
+nature of wind power [10].
+In the Langevin modeling
+of conversion dynamics of a wind turbine, the focus was
+up-to now on the deterministic part of the power time se-
+ries, while the question remained open how to correctly
+capture the abrupt large power fluctuations mentioned
+above.
+The Langevin equation describes a diffusion process
+with continuous trajectory. It consists of the determin-
+istic term and the continuous stochastic term which is
+modeled by a Wiener process or a Brownian motion. Its
+two model parameters which are the drift and diffusion
+coefficients can be estimated directly from the measure-
+ment data [14–16]. These parameters are also known as
+Kramers-Moyal (KM) coefficients which are considered
+up to second order in the Langevin equation. For the
+continuous process, the coefficients higher than third or-
+der are negligible.
+Looking at the temporally high-resolved wind power
+data, one can see portions of time series, which look like a
+diffusive process (see Fig. 1 (a), but there are also periods
+where sudden big jumps of the delivered power become
+obvious, see Fig. 1 (b).
+In this contribution, we aim to investigate how far
+these jumps make it necessary to extend the stochastic
+description.
+If the higher order KM coefficients (> 3)
+are not negligible, they would be an indicator of non-
+continuity in the process [16, 17].
+One possibility to
+model this behavior is the extension of the Langevin dif-
+fusion process to a jump-diffusion process. An additional
+discontinuous stochastic term for the jump process is in-
+troduced for which we assume that it can be modeled by
+a Poisson process. In this more general stochastic ap-
+proach two more parameters arise which are the jump
+rate and the jump amplitude. We show how they can be
+estimated from the higher order KM coefficients. This
+analysis aims to give a more realistic stochastic descrip-
+arXiv:2301.05553v1  [eess.SY]  9 Jan 2023
+
+2
+25.00
+25.05
+25.10
+25.15
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+t [h]
+P P max
+25.00
+25.05
+25.10
+25.15
+−0.15
+−0.05
+0.05
+0.15
+t [h]
+∆P P max
+(a)
+(c)
+17.15
+17.20
+17.25
+17.30
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+t [h]
+P P max
+17.15
+17.20
+17.25
+17.30
+−0.15
+−0.05
+0.05
+0.15
+t [h]
+∆P P max
+(b)
+(d)
+FIG. 1. Wind power time series, spanning the period of ten minutes. (a) shows the period where the power changes are not
+very large.(b) shows the period where the power changes are very large, up to about 40% of the rated power. Increments
+∆P := P(t + τ) − P(t) emphasize the fluctuations and are shown for sampling period τ = 1s in (c) and (d).
+tion of the power output of a wind turbine. Relation to
+control strategies, or the use for improved modeling of the
+wind energy resource in a power grid, will be discussed
+in this paper.
+Our aim is to show in detail the procedure to es-
+timate the general stochastic jump-diffusion process
+with Wiener and Poisson noise to achieve an advanced
+stochastic characterization and modelling of the wind
+power conversion dynamics of a wind turbine. Here, we
+analyze the SCADA data from a wind turbine with res-
+olution of 1 Hz. The article is organized as follows. At
+first, we describe the analysed data. Next, the stochastic
+analysis method is summarized, and it is shown how it is
+possible to quantify and separate the contributions of dif-
+fusion and jump fluctuations. Finally, the results of the
+data analysis for power output conditioned to wind speed
+are presented. In addition, we investigate the stochastic
+relation between generator torque and generator rota-
+tional speed.
+II.
+STOCHASTIC DATA ANALYSIS OF WIND
+ENERGY SYSTEM
+A.
+Data description
+The measurement data are extracted from a wind tur-
+bine of a wind farm. The wind farm is installed onshore
+over an area covering roughly 4 km2 and is surrounded by
+flat rural terrain with 12 identical variable-speed, pitch-
+regulated wind turbines. The rated power of each turbine
+is in the order of 2 MW. The values were made anony-
+mous to keep the confidentiality of the data. Thus, all the
+data are normalized with their corresponding maximum
+for our analysis.
+The measured quantities are the net electrical power
+output, P, generated by the wind turbine, the wind
+speed, u, measured on the nacelle by a cup anemome-
+ter and the rotational speed or rpm, Ω, of the generator.
+The torque, T, on the generator is calculated from the
+power and rpm of the generator using the relation
+T = 60 s
+2π
+P
+Ω .
+(1)
+All measurements were performed at a sampling fre-
+
+3
+quency fs = 1 Hz. The measurement campaign was con-
+ducted over a period of eight months, from June 2009
+till February 2010. The same data were used also in the
+study of [18].
+B.
+Power Conversion Process Described by
+Stochastic Dynamics
+Assuming the validity of a diffusive process, the power
+conversion process of a wind turbine can be modelled as
+a stochastic Langevin equation of the power output P
+conditioned to certain fixed wind speed u [10–12],
+dP(t, u) = D(1)(P|u) dt +
+�
+D(2)(P|u) dWt ,
+(2)
+where Wt is a Wiener process, a scalar Brownian mo-
+tion.
+The general non-linear functions D(1)(P|u) and
+D(2)(P|u) are the drift and the diffusion functions, which
+in case of the Langevin equation (2) are identical to the
+first and second order Kramers-Moyal (KM) coefficients.
+In general, the j-th order KM coefficients, K(j)(P|u), can
+be directly determined from given data P for each wind
+speed u, using their definitions in terms of conditional
+incremental averaging, cf. [14, 16], as
+K(j)(P|u) = lim
+∆t→0
+1
+∆t
+�
+(P(t + ∆t) − P(t))j|P (t)=P, u(t)=u
+�
+.
+(3)
+The Langevin equation describes a continuous diffusion
+process where K(j)(P|u) = 0 for j ≥ 3 and D(j)(P|u) =
+K(j)(P|u) for j = 1, 2. Further details on methods of this
+estimation can be found in [14, 15].1 All the higher order
+KM coefficients vanish when the fourth order KM coef-
+ficient K(4)(P|u) is negligible according to the Pawula
+theorem [19].
+When the signal of a stochastic process
+has sharp changes, or discontinuities, at some instants,
+typically higher order Kramers-Moyal coefficients and es-
+pecially K(4)(P|u) are not negligible anymore. In this
+case, an extension of the Langevin-type modeling with
+an additional jump noise is needed, see [16, 17, 20–23].
+Such a jump-diffusion dynamics for a power conversion
+process is given by
+dP(t, u) = D(1)(P|u) dt +
+�
+D(2)(P|u) dWt + ξ dJt , (4)
+where again Wt is a Wiener process, D(1)(P|u) and
+D(2)(P|u) are the drift and the diffusion functions. In the
+1 KM
+coefficients
+of
+a
+Langevin
+process
+in
+x(t)
+are
+de-
+fined
+for
+j
+=
+1, 2
+as
+K(j)(x, t)
+=
+D(j)(x, t)
+=
+1
+j! lim∆t→0
+1
+∆t
+�
+(x(t + ∆t) − x(t))j |x(t)=x
+�
+in [14, 15]. In order
+to stay consistent with the jump-diffusion process, our defini-
+tion differs by a factor of
+1
+j! , and dWt =
+� t+dt
+t
+Γ(τ) · dτ where
+⟨Γ(t)⟩ = 0 and ⟨Γ(t)Γ(t′)⟩ = δ(t−t′). The corresponding Fokker-
+Planck equation will be
+∂
+∂t p(x, t) = − ∂
+∂x
+�
+D(1)(x, t) p(x, t)
+�
++
+1
+2
+∂2
+∂x2
+�
+D(2)(x, t) p(x, t)
+�
+.
+following we assume that ξ dJt is a Poisson jump process.
+The coefficient ξ is the jump size, which is assumed to be
+normally distributed, ξ ∼ N(0, σ2
+ξ), with zero mean and
+variance σ2
+ξ. ξ is also known as jump amplitude. Jt is a
+Poisson jump process which is a zero-one jump process
+with jump rate λ(P|u) [16, 24]. The drift and diffusion
+coefficients and the jump rate are now related to the KM
+coefficients K(j)(P|u) in the following way [17]:
+D(1)
+j
+(P|u) = K(1)(P|u),
+(5)
+D(2)
+j
+(P|u) + λ(P|u)⟨ξ2⟩ = K(2)(P|u),
+(6)
+λ(P|u)⟨ξj⟩ = K(j)(P|u)
+for j > 2.
+(7)
+Since ξ has zero mean, its second order moment is
+⟨ξ2⟩ = σ2
+ξ. From Eq. (5), it can be seen that the esti-
+mation of the drift coefficient is the same for the diffu-
+sion process, which obeys the Langevin equation, and the
+jump-diffusion process. Here we go into more details of
+the noise part and do not assume anymore a vanishing
+K(4) = 0.
+Jump amplitude σ2
+ξ and jump rate λ can be esti-
+mated by using Eq. (7) with j = 4 and 6 and Wick’s
+theorem [25, 26] for Gaussian random variables, i.e.,
+⟨ξ2n⟩ = (2n)!
+2nn! ⟨ξ2⟩n,
+σ2
+ξ(P|u) = K(6)(P|u)
+5K(4)(P|u),
+(8)
+λ(P|u) = K(4)(P|u)
+3σ4
+ξ(P|u) .
+(9)
+C.
+Results
+1.
+Results for Electrical Power Output
+First, we analyze the relation between wind speed and
+power. For chosen fixed wind speed values with bin sizes
+of 0.5 ms−1, the KM coefficients K(j)(P|u) are deter-
+mined with the assumption of stationarity within the
+corresponding wind speed bin.
+Firstly, the drift coef-
+ficients are determined. The zero-crossings of the drift
+coefficient, D(1)(P|u) = 0, correspond to the stable fixed
+points or equilibria of each wind speed bin if the slope
+of D(1) is negative [11, 27].
+Zero-crossings with posi-
+tive slope are unstable fixed points. Alternatively, this
+can be expressed by a drift potential, which is defined
+as Φ = −
+�
+P D(1)(P|u) dP. The zero-crossings with neg-
+ative slope of the drift correspond correspond to min-
+ima of the drift potential. An example of a drift coeffi-
+cient and corresponding potential for the wind speed of
+u = 0.41 umax is shown in Fig. 2.
+For each wind speed bin, there can be single or mul-
+tiple fixed points.
+With these stable fixed points, we
+can reconstruct the characteristic power curve, which we
+
+4
+G
+G
+G
+G
+GG
+G
+G
+GG
+GGGGGGGGGGGGGGGGGGGGGGGGG
+G
+G
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+−0.03
+0.00
+0.02
+0.04
+P P max
+D (1)
+(a)
+G
+G
+G
+G
+G
+G
+G
+G
+GGGGGGGGGGGGGGGGGGGGGGGGGGGG
+G
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+−0.30
+−0.20
+−0.10
+0.00
+P P max
+Φ
+(b)
+FIG. 2. Drift coefficient D1(P|u) (a) and corresponding potential Φ (b) for the wind speed of u = 0.41 umax. Zero-crossings
+of the drift coefficient D1(P|u) = 0 or local minima of drift potential Φ are stable fixed points which describe the equilibrium
+dynamics.
+call Langevin Power Curve (LPC) [28, 29], as shown in
+Fig. 3 (a). These stable fixed points can already be used
+for a definition of different operational states of the wind
+turbine. In our case, we mark three distinct states (P1,
+P2 and P3) which separate the operational regions by
+blue dotted lines. Multiple fixed points are found to be
+at these states. Near operation point P2 in Fig. 3 (a),
+we observe the shifting of fixed points in a discontinuous
+way. Such details cannot be detected by the standard
+averaging procedure of power curve defined by [8].
+The fixed point analysis and characterization of power
+output of a wind turbine by Langevin equation (2) or dif-
+fusion process have been studied by [11, 12, 18]. In their
+works, they extensively focused on the drift coefficient.
+Higher order KM coefficients were not considered. In our
+work here, we focus on the noisy part and evaluate the
+higher order KM coefficients.
+As explained in Sec. II B, first the fourth-order KM co-
+efficient K(4)(P|u) is estimated to see if jump noise mat-
+ters. An example of K(4)(P|u) and the jump amplitude
+σ2
+ξ(P|u) for the wind speed of u = 0.41 umax with their
+medians (solid black lines) is shown in Fig. 7. The fixed
+point for this wind speed bin is located at P = 0.7 Pmax,
+see Fig. 2.
+Statistically, we can obtain more accurate
+results near the fixed point due to the better coverage
+of data, whereas for regions with less data (farther away
+from the fixed point) the results become more noisy and
+outliers are seen. A robust method to estimate is to use
+the medians instead of the means [30]. Some examples
+are shown in Appendix A.
+In the following, we investigate details of the median
+values. Thus, we simplify the process to those with con-
+stant parameters σ2
+ξ and λ for the jump process. The
+P-dependence of these parameters can also be studied,
+which we do not do here to keep the discussion simpler,
+see Fig. 7.
+Fig. 4 (a) shows that there is an increase
+of �
+K(4)(P|u) near the state P3, the transition point to
+the rated power. This behavior of �
+K(4) shows that not
+only diffusive noise is present, thus we proceed to ana-
+lyze also the higher order KM-coefficients from which we
+can determine the jump amplitude �
+σ2
+ξ and jump rate �λ
+for each wind speed as shown in Fig. 5 (a) and (c). The
+jump amplitude σ2
+ξ is highest between the states P2 and
+P3 which is just below the transition to the rated power
+region where the switching of the control strategy play a
+major role. The jump rate �λ is highest in the region of
+rated power.
+In order to quantify the overall jump contribution, we
+determine the product �
+λσ2
+ξ as shown in Fig. 6 (c). Again
+we see that the jump contribution �
+λσ2
+ξ is highest around
+the state P3. Moreover, we also determine the median
+of the diffusion to jump ratio �
+D(2)
+λσ2
+ξ over the power bins
+for each wind speed bin obtained from the resolved D(2)
+λσ2
+ξ
+values as shown in Fig. 6 (e) to find out whether diffusive
+or jump noise is dominating. If the ratio is large, there
+is more diffusive noise and vice versa. The jump noise
+is dominant in the rated power region after the state
+
+5
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+u umax
+P P max
+G
+GGGG
+G
+G
+G
+G
+G
+G
+G
+G
+G
+GGG
+G
+G
+G
+G
+G
+G
+G
+G
+G
+GGGGGGGGGGGGGGGGGGG
+P1
+P2
+P3
+G Fixed Points
+(a)
+0.6
+0.7
+0.8
+0.9
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Ω Ωmax
+T T max
+G
+G GGG
+GG
+G G
+G
+G
+G
+G G G G G G G G G GG G G G
+G
+G
+G
+G
+G
+G G G G
+T1
+T2
+T3
+G Fixed Points
+(b)
+FIG. 3. Characteristic power curve, (a), determined from the zero crossings of the drift coefficients at each wind speed bin and
+characteristic torque curve, (b), at each rotational speed rpm bin, presented in red open circles. They are also called Langevin
+Power Curve (LPC) and Langevin Torque Curve (LTC), respectively. The blue background shows the density scatter plot of
+the measurement data and darker regions indicate more data points are available. The black dots are the outliers of the density
+scatter plot. Three distinct states (P1, P2 and P3) for LPC and (T1, T2 and T3) for LTC which separate the operational
+regions are marked by blue dotted lines.
+P3. The diffusive noise is dominant between the states
+P1 and P2, which also coincides with the lowest jump
+contribution λσ2
+ξ, Fig. 6 (c).
+The analysis in this subsection shows two important
+points. First, a jump process is present and should be
+included in an advanced stochastic description or, respec-
+tively, model. Second, below rated power, first a diffusive
+stochastic behavior is dominating, while jumpy noise be-
+comes important for the considered wind turbine close to
+the transition to rated power.
+2.
+Results for Generator Torque
+Apart from the dynamical dependence of power on the
+wind speed, another central characteristic of the wind
+turbine is the generator torque T vs. rotational speed or
+rpm Ω [13]. The generator torque was calculated from
+power and rpm data according to Eq. (1), whereas the
+rpm was measured independently. A similar stochastic
+approach like for the power and wind speed is applied to
+the torque and rpm dynamics T(t, Ω), next. Based on the
+drift coefficient, a characteristic curve like the Langevin
+Power Curve is calculated as shown in Fig. 3 (b). We
+refer to it as Langevin Torque Curve (LTC). We again
+mark three distinct states (T1, T2 and T3) with blue
+dotted lines which can be deduced from the fixed point
+analysis.
+Next, we also determine the jump contribution λσ2
+ξ
+and the diffusion to jump ratio D(2)
+λσ2
+ξ for each rpm bin, see
+Fig. 6 (d) and (f). In Fig. 6 (d), we see that the median
+of the jump contribution �
+λσ2
+ξ becomes significant in the
+region between T2 and T3, cf. Fig. 3 (b), at relatively
+high rpm values. Outside this regime, jump noise does
+not dominantly contribute to the dynamics. On the other
+hand, the median of the diffusion to jump ratio �
+D(2)
+λσ2
+ξ takes
+its largest values at low values of rotational speed rpm.
+At the rotational speeds rpm below the state T1, diffusive
+behavior is much more dominant.
+The conclusions from the previous subsection II C 1 ap-
+ply also to the torque analysis in an analogous way. Sim-
+ilar to the power characteristic, a jump process is also
+present for the torque characteristic, which could be ex-
+pected as we compute the torque from power and rota-
+tional speed. Diffusive stochastic behavior is dominating
+for low rpm, but in the region from T1 on (Ω/��max ≳ 0.7)
+diffusive and jumpy behavior seem to be more balanced
+for the torque case.
+
+6
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+u umax
+K (4)
+GG
+G
+G
+GGGGGGGGGG
+GG
+G
+G
+GG
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+GGGGGGGGGGGGGGG
+0
+0.00001
+0.00002
+GG
+G
+G
+GGGGGGGGGG
+GG
+G
+G
+GG
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+GGGGGGGGGGGGGGG
+P1
+P2
+P3
+(a)
+0.6
+0.7
+0.8
+0.9
+1.0
+Ω Ωmax
+K (4)
+GGGGGGG
+G
+GGGGGGGGGGGGG
+G
+G
+G
+GGGGGGG
+0
+0.0002
+0.0004
+GGGGGGG
+G
+GGGGGGGGGGGGG
+G
+G
+G
+GGGGGGG
+T1
+T2
+T3 (b)
+FIG. 4. The median of the fourth-order KM coefficient, �
+K(4)(u) over the power bins for each wind speeds bin (a) and over
+the torque bins for each rpm bin (b). There is an increase of �
+K(4)(u) near the transition to the rated region. Statistical
+uncertainties are shown as gray-shaded background. Blue dotted lines are the distinct states observed from the fixed point
+analysis, see Fig. 3.
+III.
+CONCLUSION AND OUTLOOK
+In our work, we investigate the contribution of the
+higher-order KM coefficients to the stochastic conversion
+dynamics of a wind turbine. As described in Sec. II B,
+these higher-order coefficients allow to quantify the con-
+tributions of diffusive behavior and jump noise, and in-
+dicate that discontinuities in the trajectory of the mea-
+surement data are due to the stochastic jump noise. The
+main results are that we can quantify with our proposed
+method how the amplitudes and the ratio of the two noise
+contributions change in different operating ranges of a
+wind turbine. The region below rated power seems to
+provide the highest values of the amplitudes (D(2) and
+σ2
+ξ).
+Sometimes the maximal values are found for the
+transition states defined by the fixed point characteris-
+tics. The ratio between the contributions of the diffusive
+and jumpy noise shows that at low wind speed and low
+power a diffusive noise is dominating whereas for higher
+power more jump noise is present, with some detailed
+differences for power and torque. All this indicates that
+it is the interplay between the stochastic driving wind
+speed and the reacting control system that determines
+the noise contribution. In particular, the jump contri-
+bution is closely linked to the control system as one can
+see, to our interpretation, in the rapid changes of σ2
+ξ in
+Fig. 5 (b). Interestingly, this is more prominent in the
+torque signal than in the power signal. It is well known
+that the control system is not operating directly with the
+wind signal but with toque T and the rotational speed
+Ω.
+Near the states T1, T2 and T3, there are three dis-
+tinct operational rotational speeds Ω which the control
+system prefers to approach. It is a common control strat-
+egy to avoid certain resonance frequencies of the struc-
+ture in order to mitigate excessive loads. We show this
+by evaluating the drift potential Φ(Ω) of the rotational
+speed. The minima of this potential correspond to the
+preferred rotational speeds, see Appendix B. Moreover,
+looking at Fig. 3 (b), between the states T2 and T3, there
+is a steep gradient which enforces a large change in gener-
+ator torque T at only a small regime in rotational speed
+Ω. In this range, we also observe that there is a huge
+increase in both diffusive and jump noise by more than
+two orders of magnitude, see Fig. 6 (b) and (d).
+So far we used the stochastic methods to character-
+ize the dynamics of the wind energy conversion process.
+It goes without saying that the characterization can be
+used to compare quantitatively different turbines.
+Po-
+tential failures in the control system should be detectable
+by comparison of the different stochastic terms. One may
+see how with time some noise contribution changes as the
+system gets old, or one may show how different wind tur-
+bines or different contril strategies perform differently in
+a dynamic sense. Together with detailed knowledge of a
+specific turbine this should also be useful for monitoring,
+e.g., performance or structural health.
+At last we would like to point out that besides this
+
+7
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.000
+0.002
+0.004
+u umax
+σξ
+2
+G
+G
+GGG
+G
+G
+G
+GG
+G
+G
+G
+G
+G
+G
+GG
+G
+G
+GG
+GG
+G
+G
+G
+GGG
+GGGGGG
+GG
+G
+G
+G
+G
+GGG
+G
+G
+G
+GG
+G
+G
+G
+G
+G
+G
+GG
+G
+G
+GG
+GG
+G
+G
+G
+GGG
+GGGGGG
+GG
+G
+G
+P1
+P2
+P3
+(a)
+0.6
+0.7
+0.8
+0.9
+1.0
+0.000
+0.004
+0.008
+Ω Ωmax
+σξ
+2
+GGGGGGG
+G
+GGGGGGGGGGGG
+G
+GGG
+G
+GGGGG
+GGGGGGG
+G
+GGGGGGGGGGGG
+G
+GGG
+G
+GGGGG
+T1
+T2
+T3 (b)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.4
+0.8
+u umax
+λ
+G
+G
+G
+G
+G
+GGGG
+GGGG
+G
+G
+G
+GGGG
+GG
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+GGGG
+GGGG
+G
+G
+G
+GGGG
+GG
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+P1
+P2
+P3
+(c)
+0.6
+0.7
+0.8
+0.9
+1.0
+0.0
+0.4
+0.8
+Ω Ωmax
+λ
+GGG
+G
+GG
+GGG
+GGGGGG
+G
+GG
+G
+G
+G
+G
+G
+G
+G
+G
+GGG
+G
+GG
+GGG
+GGGGGG
+G
+GG
+G
+G
+G
+G
+G
+G
+G
+G
+T1
+T2
+T3 (d)
+FIG. 5. The median of jump amplitude, �
+σ2
+ξ, over the power bins for each wind speed bin (a) and over the torque bins for each
+rpm bin (b). There is a significant increase in jump amplitude near the transition to the rated region. The median of jump
+rate, �λ, over the power bins for each wind speed bin (c) and over the torque bins for each rpm bin (d). There is a significant
+increase in jump rate after the transition to the rated region and a small increase near the cut-in region for power analysis. The
+jump rate for torque analysis is similar in between all three distinct states shown by blue dotted lines as in Fig. 3. Statistical
+uncertainties are shown as gray-shaded background.
+characterization the stochastic methods presented here
+also deliver the explicit form of the stochastic differential
+equations.
+Thus, it is also possible to use our results
+as very efficient dynamics models for power and torque.
+Long time simulations can be done easily. Such models
+are of interest for the simulation of the contribution of
+wind energy to the power grid and for the simulation of
+loads.
+ACKNOWLEDGMENTS
+The authors would like to thank Vlaho Petrovi´c
+and Christian Philipp for helpful discussions.
+We
+acknowledge financial support by the Federal Ministry
+for Economic Affairs and Climate Action of Germany in
+the framework of the projects “WEA-Doktor” (reference
+0324263A) and “WiSAbigdata” (reference 03EE3016A).
+Appendix A: Median as a Robust Estimator
+Statistically, we can obtain more accurate results near
+the fixed point due to the better coverage of data,
+whereas for regions with less data (farther away from
+the fixed point) the results become more noisy and out-
+liers are seen. A robust method to estimate the typical
+value of K(4)(P|u) and σ2
+ξ(P|u), as examples, is to use
+the medians �
+K(4)(u) and �
+σ2
+ξ(u) as shown by solid lines in
+Fig. 7.
+
+8
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+u umax
+D (2)
+G
+G
+GGG
+GG
+G
+GG
+G
+G
+G
+GG
+GGGGGGGG
+G
+G
+G
+G
+G
+G
+GG
+G
+G
+G
+0.00001
+0.0001
+0.001
+G
+G
+GGG
+GG
+G
+GG
+G
+G
+G
+GG
+GGGGGGGG
+G
+G
+G
+G
+G
+G
+GG
+G
+G
+G
+P1
+P2
+P3
+(a)
+0.6
+0.7
+0.8
+0.9
+1.0
+Ω Ωmax
+D (2)
+G
+G
+GGGGGGG
+GGGGGGGGGGG
+G
+G
+GG
+G
+G
+0.000001
+0.0001
+0.01
+G
+G
+GGGGGGG
+GGGGGGGGGGG
+G
+G
+GG
+G
+G
+T1
+T2
+T3 (b)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+u umax
+λ σξ
+2
+G
+G
+GGG
+G
+GG
+G
+G
+G
+GG
+GGG
+GGGGGGGG
+G
+GGG G
+GG
+GG
+G
+0.000001
+0.0001
+0.01
+G
+G
+GGG
+G
+GG
+G
+G
+G
+GG
+GGG
+GGGGGGGG
+G
+GGG G
+GG
+GG
+G
+P1
+P2
+P3
+(c)
+0.6
+0.7
+0.8
+0.9
+1.0
+Ω Ωmax
+λ σξ
+2
+G
+G
+G
+G
+GG
+GGG
+GGGGGGGGGGG
+G
+G
+GGG
+G
+0.000001
+0.0001
+0.01
+G
+G
+G
+G
+GG
+GGG
+GGGGGGGGGGG
+G
+G
+GGG
+G
+T1
+T2
+T3 (d)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+1
+2
+3
+4
+5
+u umax
+D (2) λ σξ
+2
+G
+G
+G
+GG
+G
+G
+G
+G
+G
+G
+G
+GG
+GGGGGG
+GGG
+GG
+G
+GG GGGGGG
+G
+G
+G
+GG
+G
+G
+G
+G
+G
+G
+G
+GG
+GGGGGG
+GGG
+GG
+G
+GG GGGGGG
+P1
+P2
+P3
+(e)
+0.6
+0.7
+0.8
+0.9
+1.0
+0.0
+1.0
+2.0
+Ω Ωmax
+D (2) λ σξ
+2
+G
+G
+G
+G
+G
+G
+GG
+GGGGGGGGGGGG G
+GGGG
+G
+G
+G
+G
+G
+G
+G
+GG
+GGGGGGGGGGGG G
+GGGG
+G
+T1
+T2
+T3 (f)
+FIG. 6. The median of diffusion coefficient, �
+D(2), (a), overall jump contribution, �
+λσ2
+ξ, (c), and the diffusion to jump ratio, �
+D(2)
+λσ2
+ξ ,
+(e), over the power bins for each wind speed bin, and respective quantities over the torque bins for each rpm bin in (b), (d),
+and (f). Sub-figures (a-d) are plotted in semi-logarithmic scale for better visualization. Statistical uncertainties are shown as
+gray-shaded background. Blue dotted lines are the distinct states observed from fixed point analysis, see Fig. 3.
+
+9
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+GGGGGGGGGGGGGGGGGGGGGGGGGGG
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+P P max
+K (4)
+0
+0.0002
+0.0005
+(a)
+G
+G
+G
+G
+GGG
+GG
+G
+G
+G
+G
+GGGG
+GGG
+GGGGGGGGGGGGGGG
+GG
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.00
+0.02
+0.04
+P P max
+σξ
+2
+(b)
+FIG. 7. Fourth-order KM coefficient K(4)(P|u) (a) and the jump amplitude σ2
+ξ(P|u) (b) for the wind speed of u = 0.41 umax.
+(For the corresponding drift term see Fig. 2). The solid black lines are their respective medians �
+K(4)(u) and �
+σ2
+ξ(u). The fixed
+point for this wind speed bin is P = 0.7 Pmax. Statistically, more accurate results can be obtained near the fixed point due to
+the better coverage of data. By using the median, our results are more robust to outliers far away from the fixed points.
+Appendix B: Drift Potential of Rotational Speed
+We analyzed all the data of rotational speed Ω in the
+range of 0.6 Ωmax and Ωmax without any conditioning
+or binning on other variables.
+We evaluated the drift
+coefficient D(1)(Ω) and then determined the drift poten-
+tial which is Φ(Ω) = −
+�
+Ω D(1)(Ω) dΩ which is plotted
+in Fig. 8. Minima of the drift potential corresponds to
+the stable fixed points or equilibria.
+Here we can ob-
+serve three minima around the three states T1, T2 and
+T3 which the control system prefers to approach. It is a
+common control strategy to avoid certain resonance fre-
+quencies of the structure in order to mitigate excessive
+loads as shown in Fig. 3.
+In Fig. 8 (b), we can clearly observe a minimum around
+the state T1. From our results in Fig. 4, 5 and 6, there is
+also a slight increase in noises around this state. This in-
+dicates that the control system of the wind turbine starts
+switching the strategies at this state around T1. As a
+remark, we calculated the deterministic potential only
+which reflects the mechanical and control mechanism of
+the wind turbine.
+[1] WindEurope,
+Wind
+energy
+in
+europe:
+2020
+statistics
+and
+the
+outlook
+for
+2021-
+2025,
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+platform/product/wind-energy-in-europe-2020-
+statistics-and-the-outlook-for-2021-2025/ (2021).
+[2] J.-F. m. c. Muzy, R. Ba¨ıle, and P. Poggi, Intermittency of
+surface-layer wind velocity series in the mesoscale range,
+Phys. Rev. E 81, 056308 (2010).
+[3] R. Calif and F. G. Schmitt, Multiscaling and joint mul-
+tiscaling description of the atmospheric wind speed and
+the aggregate power output from a wind farm, Nonlinear
+Processes in Geophysics 21, 379 (2014).
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+Anvari,
+G.
+Lohmann,
+M.
+W¨achter,
+P.
+Milan,
+E. Lorenz, D. Heinemann, M. R. R. Tabar, and J. Peinke,
+Short term fluctuations of wind and solar power systems,
+New Journal of Physics 18, 063027 (2016).
+[5] K. Schmietendorf, J. Peinke, and O. Kamps, The impact
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+stability and quality, The European Physical Journal B
+90, 10.1140/epjb/e2017-80352-8 (2017).
+[6] T.
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+

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+Speech Driven Video Editing via an Audio-Conditioned Diffusion Model
+Dan Bigioi
+University of Galway
+d.bigioi1@nuigalway.ie
+Shubhajit Basak
+University of Galway
+s.basak1@nuigalway.ie
+Hugh Jordan
+Trinity College Dublin
+jordanhu@tcd.ie
+Rachel McDonnell
+Trinity College Dublin
+ramcdonn@tcd.ie
+Peter Corcoran
+University of Galway
+peter.corcoran@universityofgalway.ie
+Abstract
+In this paper we propose a method for end-to-end
+speech driven video editing using a denoising diffusion
+model.
+Given a video of a person speaking, we aim to
+re-synchronise the lip and jaw motion of the person in re-
+sponse to a separate auditory speech recording without re-
+lying on intermediate structural representations such as fa-
+cial landmarks or a 3D face model. We show this is pos-
+sible by conditioning a denoising diffusion model with au-
+dio spectral features to generate synchronised facial mo-
+tion. We achieve convincing results on the task of unstruc-
+tured single-speaker video editing, achieving a word error
+rate of 45% using an off the shelf lip reading model. We
+further demonstrate how our approach can be extended to
+the multi-speaker domain. To our knowledge, this is the first
+work to explore the feasibility of applying denoising diffu-
+sion models to the task of audio-driven video editing. 1
+1. Introduction
+The idea behind audio-driven video editing is to be able
+to re-synchronise the lip and jaw movements of an actor
+in a video, in response to a new speech input signal. This
+new speech signal may come from the original, or different
+speaker in order to fix any mistakes captured in the orig-
+inal recording or dub over the original voice. Regardless
+of the source of the speech, it is imperative that the perfor-
+mance of the actor is never diminished. No matter how the
+lip and jaw movements change in response to the new au-
+dio, the facial expressions, and emotions portrayed by the
+actor should remain as close to the original performance as
+possible.
+Achieving such seamless audio-driven video editing is
+an exciting prospect for the entertainment industry, one with
+1All code, datasets, and models used as part of this work are made
+publically available here: https://www.github.com/anonymous
+the potential of being applied towards movies, TV shows,
+live streaming, and even home made content uploaded to
+platforms such as YouTube, TikTok, and others. Giving
+video content creators the ability and option to edit their
+work without having to go through time consuming, and
+expensive re-shoots, allows them to work with a greater
+tolerance for error during filming. Additionally, achieving
+true audio-driven video editing will enable its use within
+automatic dubbing pipelines. This will have a huge im-
+pact on the world of cinema and television, allowing for
+the further democratisation of video content, making it sig-
+nificantly easier, and more cost-effective to dub English-
+language movies/TV shows/videos into other languages and
+vice-versa. With the rapid advances in deep learning, and
+talking head generation techniques, this exciting prospect is
+getting closer and closer to reality.
+Generally, speech-driven video generation approaches
+can be grouped into two distinct types: structured, and un-
+structured generation. Structured generation refers to tech-
+niques that use the speech signal to first generate an inter-
+mediate structural representation of the face, before using
+said structure to aid in rendering the photo-realistic frame,
+an approach followed by works such as [8, 31, 69, 88, 91].
+Unstructured generation techniques on the other hand such
+as [20,30,73,90] , utilise image reconstruction techniques to
+generate the photo-realistic frame directly in an end-to-end
+manner.
+Diffusion models [60] are a relatively new class of gener-
+ative model that have been gaining traction in recent months
+due to their strong performance on image synthesis tasks,
+outperforming traditionally state of the art GAN (Genera-
+tive Adversarial Network) [22] based methods in some in-
+stances [17]. The last number of months in fact have seen
+diffusion models being applied to image to image transla-
+tion tasks [53, 57], as well as towards the video generation
+problem [27, 86], audio synthesis [11, 35], and many oth-
+ers [81].
+Utilising conditioning signals such as text and
+1
+arXiv:2301.04474v1  [cs.CV]  10 Jan 2023
+
+even images, diffusion models have shown that they can be
+trained and conditioned towards generating a specific de-
+sired output at inference time with relative ease [53]. They
+also achieve high mode coverage unlike GANs, and main-
+tain high sample quality. This ability makes them an ideal
+candidate for application towards the task of unstructured
+audio-driven video editing, a task that has thus far been
+dominated by GAN-based approaches [9,12,73]. As part of
+this work we make the following contributions to the field:
+• A novel unstructured end-to-end approach for audio-
+driven video editing based on the proposed archi-
+tecture by in Palette [57], a denoising U-Net model
+trained for image to image translation tasks. We in-
+stead condition the network on audio frames and train
+it to inpaint the lower half region of the face such
+that the lip and jaw movements are synchronised to
+the conditioning audio signal. We train on single and
+multi speaker versions of the GRID data set [14]. We
+demonstrate convincing results despite access to lim-
+ited data, and training hardware. All project code, and
+trained single speaker and multi speaker models are
+made available to the public.
+• We introduce a simple conditioning mechanism for the
+task of audio driven video editing with diffusion mod-
+els. We condition the network using mel-spectrogram
+features combined with the previously generated frame
+(to maintain temporal stability) to generate the next
+frame in the sequence. To our knowledge, this is the
+first attempt at using an audio signal to condition a dif-
+fusion model to generate an image (video frames in our
+case).
+2. Related Works
+2.1. Audio Driven Video Generation
+“Deep fakes”, as they have commonly been referred to
+in public discourse, are synthetic video or image content of
+a person(s) generated by a deep learning algorithm. Audio-
+driven video editing is a research topic that falls under the
+much broader scope of such “deep fake” research, specif-
+ically under audio-driven deep fakes which are what this
+section will focus most upon. For a broader review of the
+literature surrounding deep fakes and the various techniques
+used to generate them, we recommend [45, 70] for good
+overviews of the field.
+As touched upon a bit earlier, audio-driven deep fakes
+can be categorised by whether they are generated by lever-
+aging an audio driven structural representation of the face,
+or without.
+There have been numerous approaches over
+the years relating to the former, ranging from ones such
+as [2,7,10,16,19,31,39,56,66,68,74,75,79,91] which gen-
+erate a set of 2D facial landmark co-ordinates from audio,
+or [8,15,32,37,52,62,63,69,76,77,83–85,87] which predict
+expression parameters from audio to drive a 3D face model.
+What these approaches all have in common is that they use
+these intermediate structural representations as input to a
+separate neural rendering model which is typically trained
+as an image to image translation task to generate the final
+photo realistic image frame. As of the date of this submis-
+sion, GAN-based [22] approaches such as Pix2Pix [29], Cy-
+cleGAN [93], and other variations have proved immensely
+popular for this task, however it would not be surprising to
+see diffusion models being used for this in the very near fu-
+ture given their success so far on traditional image2image
+tasks [57].
+Non structural / End-to-end methods on the other hand
+utilise latent feature learning and image reconstruction tech-
+niques to generate a photo-realistic video sequence from an
+input speech signal and reference image/video in an end-
+to-end manner. Approaches such as [9, 20, 30, 36, 43, 49,
+65, 73, 89, 90, 92] have seen much success in recent times.
+Each of these approaches differ from the one used in this pa-
+per as they are all GAN/VAE (variational autencoder) [34]
+based probabilistic methods while ours leverages the power
+of a denoising diffusion model. While current end-to-end
+approaches suffer from low output resolution quality com-
+pared to structural methods, there is a lot of potential for im-
+provement, especially by exploiting diffusion models abil-
+ity to synthesise high quality samples while maintaining
+good mode coverage / diversity.
+2.2. Diffusion Models
+Denoising diffusion models [60,64] have seen great suc-
+cess on a wide variety of different challenges, ranging from
+image2image translation tasks like inpainting, colorisation,
+image upscaling, uncropping [6, 26, 41, 42, 50, 53, 57, 59],
+audio generation [11, 28, 33, 35, 38, 48, 67, 80], text-based
+image generation [4, 21, 23, 46, 51, 55, 58], video genera-
+tion [24,27,82,86], and many others. For a thorough review
+on diffusion models and all of their recent applications, we
+recommend [81].
+Diffusion models are a class of generative probabilistic
+models that consist of two steps: 1) the forward diffusion
+process that destroys data by steadily adding small amounts
+of random Gaussian noise over a series of time steps until
+it is destroyed. 2) The reverse diffusion process where a
+learning algorithm is trained to restore structure in the data
+by steadily removing noise over a series of time steps. The
+trained model can then sample information from a random
+distribution of Gaussian noise and steadily denoise it over a
+series of time steps to attain the desired output.
+Sohl-Dickstein et al. [60] developed the first diffusion
+model and coined the term. Ho et al. [25] combined de-
+noising score matching with Langevin dynamics [64] and
+diffusion models to synthesise images.
+This ignited a
+2
+
+steady interest in diffusion models, with Nichol et al. [47]
+building upon the work of [25] showing that by making
+small adjustments to the diffusion process, they could sam-
+ple data faster and achieve competitive log-likelihoods to
+GAN-based methods with minimal impact to sample qual-
+ity. They also found that training diffusion models with
+more computational power typically lead to better sample
+quality. Chen et al. [11] and Kong et al. [35] applied dif-
+fusion models to the task of audio synthesis, succeeding in
+generating high quality samples. Dhariwal and Nichol [17]
+demonstrated that diffusion models beat GANs on image
+synthesis, also introducing the concept of “classifier guid-
+ance” for conditional generation.
+As diffusion models are trained under a single loss, and
+do not rely on a discriminator, they are more stable dur-
+ing training and do not suffer from typical issues associated
+with training GANs such as mode collapse, and vanishing
+gradient. They produce high quality output samples, and
+display high mode coverage unlike GANs [78]. Despite
+these advantages, their sampling speed is very slow due to
+the need to run the backwards diffusion process many thou-
+sands of times on the same sample to denoise it completely.
+Xiao et al. [78] and Rombach et al [53] attempted at speed-
+ing up the sampling and training times associated with dif-
+fusion models with the former proposing a method to model
+the denoising distribution using a complex multi modal dis-
+tribution in order to facilitate larger diffusion steps, and the
+latter applying diffusion models in the latent space of a pre-
+trained autoencoder to reduce the complexity. This is an
+ongoing focus of research in the field, and it is a certainty
+that more works tackling the inference/training speed prob-
+lem will emerge.
+The work in this paper builds upon the work presented
+in Palette [57], a denoising diffusion model trained specifi-
+cally for the task of image2image translation, which in turn
+was heavily influenced by the 256x256 class conditional U-
+NET of [17]. We utilise the same overall architecture as
+discussed in Palette with a few variations in the hyperpa-
+rameters. Additionally, we modify the training procedure
+for the task of video editing, and introduce a feature con-
+catenation mechanism for conditioning the network using
+speech mel-spectrogram features, as well as information re-
+lated to the previously generated frame in the sequence so
+that the network can generate temporally coherent frames.
+3. Methodology
+3.1. Problem Formulation
+We frame the problem of audio-driven video editing as
+an inpainting task with a few key changes. Traditionally, in-
+painting is an image-to-image translation task where a neu-
+ral network must learn to fill in a masked out region of the
+image with realistic content. For video editing, we must
+Figure 1. Audio Conditioning Feature, Rectangular face mask
+provide the network with additional context, to help guide
+its generation process. As our approach works on a frame-
+by-frame basis, we must show the network the preceding
+frame in addition to the current masked frame. This is to
+ensure that there is temporal stability between consecutive
+frames. Additionally, audio information related to the previ-
+ous, current, and future frame must all be provided as well.
+Future audio frames are included to ensure that lip move-
+ments produced by plosives, sounds created by the letters
+“p, t, k, b, d, g”, are correctly generated. This is because
+the lip movements associated with plosives form before the
+sound is spoken. We concatenate all this information to the
+image channels of the current frame that is being edited,
+and pass this frame through the network. Fig. 2 depicts an
+overview of this process.
+3.2. Data Processing
+3.2.1
+Dataset
+We rely on the GRID [14] audio-visual speech data set to
+carry out the work in this paper. This is a multi speaker
+data set consisting of 34 speakers (18 male, 16 female), with
+each speaker uttering 1000 short 6-word sentences. We train
+three models: 1) A single speaker model trained with videos
+from speaker S1. 2) A multi-speaker model trained on 10%
+of the data set, where we keep speakers S1, S33, and S34
+unseen to the network. 3) A fine tuned single speaker model
+trained on top of the base multi speaker model to determine
+whether faster convergence was possible while achieving
+similar results to the base single speaker.
+3.2.2
+Audio Processing
+The audio from the GRID [14] data set is recorded with
+a sampling rate of 25 KHz. From the audio we compute
+mel-spectrogram features with non overlapping windows of
+length 40ms and 256 mel bands. We choose 256 so that
+when we concatenate the audio features to the image chan-
+nels as depicted in Fig. 2, the dimensions will be the same
+as the target image.
+Alternatively, one could use a lin-
+ear transformation on top of the standard spectrogram of
+3
+
+Single ID
+Multi-ID
+Image Size
+256x256
+256x256
+Total Frames
+73704
+222000
+Diffusion Steps
+2000
+2000
+Noise Schedule
+Linear
+Linear
+Linear Start
+1e − 06
+1e − 06
+Linear End
+0.01
+0.01
+Input Channels
+10
+10
+Inner Channels
+64
+64
+Channels Multiple
+1, 2, 4, 8
+1, 2, 4, 8
+Res Blocks
+2
+2
+Head Channels
+32
+32
+Drop Out
+0.2
+0.2
+Batch Size
+10
+10
+Training Epochs
+895
+185
+Learning Rate
+5e − 05
+5e − 05
+Table 1. Hyperparameter Set Up
+80/128 channels to transform it to the desired shape. For
+each video frame, we have a corresponding audio feature
+of shape [1x256]. Since we condition the network using
+3 audio frames per video frame (the past, present, and fu-
+ture one), we repeat the first two audio frames 85 times
+each, the third 86 times, and concatenate them to end up
+with an image of shape [256x256x1] that we can use to
+condition the network, as illustrate in Fig. 1. We find that
+conditioning with audio via concatenation is a rather sim-
+ple approach that works, however, in future work we plan
+to explore more complex techniques such as feeding audio
+embeddings through various layers of the U-net.
+3.2.3
+Video Processing
+First, we resize all videos in the data set to be 256 width x
+256 height from the default 360 x 288. This is done to de-
+crease the training time by decreasing the number of pixels,
+and to ensure that the image can be accepted as input by the
+U-net. For every frame in the videos we must compute the
+region that should be masked out. Using an off the shelf
+facial landmark extractor [40], we compute the facial land-
+mark co-ordinates to determine the position of the jaw. We
+then mask out the bottom portion of the face with these co-
+ordinates just below half of the nose, as depicted by Fig. 1.
+We apply the rectangular face mask to data samples at train
+time, before they are fed into the neural network.
+There is a very important reason for applying such a
+rectangular face mask rather than a mask in the shape of
+a face: to hide the jaw contour. There is a very strong cor-
+relation between lip movement, jaw movement, and overall
+head pose. Should the jaw contour be visible to the net-
+work, the network will learn to predict the lip movements
+from that alone, discarding the audio conditioning signal
+entirely, treating it as noise. This is a real problem when
+doing audio-driven video editing that is yet to be addressed
+in the literature, especially when working on relatively non-
+complex data sets such as GRID [14]. We discovered that
+applying such a rectangular face mask minimises this prob-
+lem significantly, however it still exists. When testing our
+single speaker model using silence as input, the lip move-
+ments are significantly diminished though movements do
+occasionally still occur. This is because for every individual
+speaker, there exists a slight correlation between head pose
+and lip movement that the model learns, especially when
+trained on a single speaker. We suspect that by training on
+a larger single speaker dataset, with more variety in both
+the head pose and background of the speaker, the network
+will accord even more attention to the conditioning speech
+signal in order to drive the mouth movements.
+3.2.4
+Audio Video Alignment
+Each video frame is aligned with the 40ms of audio preced-
+ing it, and 80ms after it, totalling a 120ms window of au-
+dio information that is used to condition the network when
+generating the lip/jaw movements for each frame.
+Care
+must be taken when choosing the audio window, too large
+and the network won’t use the most meaningful informa-
+tion available to it, too small and there may not be enough
+context for the network to generate more complex lip move-
+ments caused by plosives. We arrived at our 120ms window
+through empirical tests and observations. Note that image
+frame 0 does not have any audio preceding it. Instead of
+discarding it, we use it to commence the generation process,
+acting as an initial “identity” frame. At inference time, each
+generated frame is then re-input into the network serving as
+the “previous frame” to generate the next frame in the se-
+quence, maintaining temporal coherence. At train time, we
+use the previous real frame.
+3.3. Model Architecture
+We follow the general U-Net [54] architecture described
+by [57], which in turn is based off the model proposed
+by [25] with modifications inspired by the works of [17,59].
+For this work we use a lightweight version of the 256x256
+U-net architecture described by [17], minus the class condi-
+tioning mechanism. Like [57] we also introduce additional
+conditioning of the source image via the concatenation of
+our audio features and the previous frame. Tab. 1 displays
+the hyper-parameters we use to train our diffusion model
+for the task of audio-driven video editing. Notably, we omit
+the use of attention within the up/downsampling layers of
+the U-Net in an effort to speed up training. For all of our
+experiments, we train using a batch size of 10 per GPU on
+4 32gb V100 GPUs in parallel. For inference, we rely on a
+4
+
+Figure 2. High-Level Overview of Network Architecture including the forward and backward diffusion processes
+single GPU to generate the videos, as we use the previously
+generated frame as input to the network.
+A diffusion model is defined as having two steps, the
+forward diffusion process where the data is gradually de-
+stroyed, and the learned backwards diffusion process which
+reconstructs the data, and is used during training and infer-
+ence.
+3.3.1
+Forward diffusion process
+As defined by [60], the forward diffusion process is a
+Markov chain that adds small amounts of noise to the data
+y over a predefined number of time steps T, until the data
+is completely destroyed at time step t=T. This state is repre-
+sented as yT with y0 representing the data before any noise
+was added to it. The Markov chain is defined by:
+q (y1:T |y0) :=
+T
+�
+y=1
+q (yt|yt−1)
+(1)
+where at each step, Gaussian noise is added by:
+q (yt|yt−1) := N (yt; √αtyt−1, (1 − αt)I) ,
+(2)
+with αt := (1 − βt), representing the hyperparameters of
+our fixed noise scheduler. [25] show that it is possible to
+sample yt at any step t in closed form:
+q (yt|y0) := N
+�
+yt; √ ¯αty0, (1 − ¯αt)I
+�
+,
+(3)
+with ¯αt := �t
+s=1 αs. This is an important observation, as
+it significantly speeds up the forward diffusion process, and
+can be used to train the model on the fly with random noise
+levels at each forward step.
+3.3.2
+Backwards diffusion process
+Given a noisy image ¯y defined as:
+¯y :=
+√
+¯αy0 +
+√
+1 − ¯αϵ, ϵ ∼ N(0, I)
+(4)
+the goal of the backwards diffusion process is to learn an
+algorithm that can denoise and restore the noisy image to
+its original image Y0. Following the approach in [57], we
+train a neural network fθ(x, ¯y, ¯α) to predict the noise gen-
+erated at time t, optimising the Lsimple objective proposed
+by [25]:
+Et,y0,ϵ[
+���fθ(x,
+√
+¯αy0 +
+√
+1 − ¯αϵ, ¯α) − ϵ
+���
+2
+]
+(5)
+where x represents the conditioning audio and previous
+frame input to our network, ¯y the noisy image, and ¯α the
+noise level. During training, we only calculate the loss for
+the masked region of the face to save on compute, following
+the approach in [57].
+5
+
+Ground Truth Frame
+Previous Frame in
+Current Frame Being
+Past, Present, Future
+Y(0) 
+Video
+Denoised Y(T)
+audio frames
+q(ytlyt-1) := N(yti Vatyt-1,(1 - αt)I)
+Conditioning Via
+Concatenation
+Forward Diffusion Process
+Backwards Diffusion Process
+DownSample
+For a given image with noise level at timestep t, the U-
+Block
+Net is trained to predict the noise added at time step t.
+U-Net with
+UpSample Block
+A trained model is then able to start with a completely
+gradient descent
+noisy image, and subsequently denoise it over a series
+Global Attention
+step computed on
+Block
+of timesteps T.
+predicted noise vs
+Skip Connection
+actual noise
+Vefe(c, Vyo + V1-ae,a) -
+Generated Frame
+Denoised image
+Y(0)
+Y(T-1)
+Repeat T
+Times to
+Denoising image
+J1-αtEt
+Fully
+Y(T)
+Denoise
+ImageFollowing [25], to run inference, each step of the back-
+wards diffusion process can then be computed by:
+yt−1 ←
+1
+√αt
+�
+yt − 1 − αt
+√1 − ¯αt
+fθ(x, yt, ¯αt)
+�
++
+√
+1 − αtϵt,
+(6)
+where ϵ ∼ N(0, I). The backwards diffusion step is re-
+peated for as many times necessary to denoise the image
+fully. Please see Fig. 2 for a high level view of our network
+architecture, and to better understand where each equation
+is used. For a more detailed discussion behind these equa-
+tions, and how they are derived, please see [25,60,64].
+4. Experiments & Results
+We train and evaluate three versions of our video-editing
+diffusion model, a single speaker model, a multi speaker
+model, and a single speaker model we fine-tuned on top of
+the multi speaker one. We evaluate the videos generated by
+our models against the ground truth using a number of ob-
+jective metrics. We compare our results to recent methods
+for audio-driven video generation [13,30,36,65,72]. The re-
+sults we provide for each model are cited directly from their
+own research papers, with each model tested on a subset of
+the GRID [14] data set, unless explicitly stated otherwise.
+4.1. Evaluation Metrics
+We use a a number of objective metrics to measure the
+quality of our generated videos, allowing us to compare
+them directly to the ground truth, and other state of the art
+audio driven video generation methods from the literature.
+To make a fair comparison we calculate the Image Quality
+metrics(SSIM, PSNR) only on the masked portion of the
+image as shown in Fig. 1.
+SSIM (Structural Similarity Index): This is a perceptual
+metric to quantify the degradation of image quality.
+A
+larger SSIM signifies the better quality of the reconstructed
+image.
+PSNR (Peak Signal to Noise Ratio): We compute the
+peak signal to noise ratio between the ground truth and
+the generated image. The higher the PSNR the better the
+quality of the reconstructed image.
+CPBD (Cumulative Probability Blur Detection) [44]:
+This is a perceptual based no reference objective image
+sharpness metric.
+Similar to [30, 36, 72] we have used
+this metric to compare the CPBD results on the generated
+videos.
+WER (Word Error Rate): It evaluates the performance of
+a pre-trained speech recognition network on a given video.
+Similar to previous works [30,36,72] we use the LipNet [3]
+model which is pre-trained on GRID data set and achieves
+95.2 percent accuracy.
+Facial Action Unit (AU) [18] Recognition: Following the
+previous works [13, 65] we also evaluate our reconstructed
+images with respect to five facial action units (AU10:
+Upper Lip Raiser, AU14: Dimpler, AU20: Lip Stretcher,
+AU25: Lips Part, AU26: Jaw Drop). We use the Facial
+Behavior Analysis Toolkit [5] to detect the presence of
+these AUs (boolean true on activation) on each generated
+frames and compare them with the ground truth frames.
+Finally we calculate the average F1 score and the average
+accuracy based on the AU recognition.
+ACD (Average Content Distance) [71]: Similar to [72]
+we use Openface Face Recognizer [1] to calculate the
+Cosine(ACD-C) and Euclidean(ACD-E) distance between
+the generated frame and ground truth image. The smaller
+the distance between two images the similar the images.
+4.2. Single Speaker Model
+We train our single speaker model on identity S1 using
+data from the GRID audio visual corpus [14]. There are
+1000 videos in total, each of them roughly 3 seconds in
+length totalling about 50 minutes of audio-visual content
+for training. We train our model on 996 videos, withhold-
+ing 4 of them for testing purposes. We call this the “unseen”
+test set. The “seen” test set consists of videos that the model
+has seen during training, but with different speech inputs to
+the originals. Our unseen test set is relatively small with
+respect to the size of our data set as we wanted to give the
+model as much information as possible about the speaker it
+was training on given that there were only about 50 minutes
+worth of audio/visual content available.
+Tab. 2 depicts the results our model scores when tested
+on the unseen data set versus other approaches in the lit-
+erature. While the results we obtain are not state of the
+art, they demonstrate that using a denoising diffusion model
+to do audio-driven video editing, is indeed quite feasible,
+and produces reasonable results. Further time spent train-
+ing the model, and exposure to a larger data set should im-
+prove these scores further. Additionally, using our single
+speaker model, we edit a number of videos by introducing
+new speech inputs instead of the originals, and attach these
+videos to our supplementary materials section, encouraging
+readers to have a look.
+4.3. Multi-Speaker Training
+We train our multi speaker model on 30 different identi-
+ties using a subset of 100 random videos per speaker from
+the audio visual GRID corpus [14] for 185 epochs. Dur-
+ing training, we withheld identities S1, S33, and S34 en-
+tirely from the training set of the network so that we could
+6
+
+Method
+WER↓
+ACD-C↓
+ACD-E↓
+Avg. F1 AU
+Avg. Acc. AU
+SSIM↑
+PSNR↑
+CPBD
+OneShotA2V [36]
+27.5
+0.005
+0.09
+-
+-
+0.881
+28.571
+0.262
+OneShotA2V(lombard) [36]
+26.1
+0.002
+0.064
+-
+-
+0.922
+28.978
+0.453
+RSDGAN [72]
+23.1
+-
+1.47x10−4
+-
+-
+0.818
+27.100
+0.268
+Speech2Vid [30]
+58.2
+-
+1.48x10−4
+0.738
+78.97
+0.720
+22.662
+0.255
+CRAN [65]
+-
+-
+-
+0.710
+78.71
+0.694
+28.041
+-
+Chen et al. [13]
+-
+-
+-
+0.751
+80.92
+0.769
+29.838
+-
+Ours (Single Speaker)
+45.85
+0.018
+0.170
+0.502
+93.73
+0.896
+32.22
+0.341
+Ours(MultiSpeaker)
+76.3
+0.024
+0.196
+0.43
+88.21
+0.780
+32.214
+0.352
+Ours(Fine-Tuned)
+55.01
+0.021
+0.173
+0.47
+92.37
+0.85
+32.612
+0.34
+Table 2. Quantitative comparison with previous works on image quality, lip synchronization and facial feature metrics.
+Figure 3. Multi-speaker failure cases
+use them for additional testing purposes. We use approx-
+imately 10% of the entire data set to train our model for
+two major reasons: 1) To train on the entire data set with
+our current hardware would take approximately 15 hours
+per epoch. We intend to incorporate the findings of [53]
+into our future work to significantly speed up training time.
+2) We sought to train a “base” model using multiple iden-
+tities but with relatively few samples per speaker, and use
+it to fine tune a single-speaker model with more samples,
+investigating whether the fine-tuned model could be trained
+for less time than the single-speaker model while achieving
+similar performance.
+When testing on identities unseen to the model, it would
+struggle to maintain the identity of the speaker consistent
+throughout the generation process. We speculate that since
+our model relies on the previously generated frame alone to
+generate the next frame in the sequence, over time, infor-
+mation about the original identity is lost, as demonstrate in
+Fig. 3.
+When testing on identities previously seen by the net-
+work during training, we report relatively poor results when
+compared to other methods in the literature as depicted by
+Tab. 2. We speculate this is due to the much reduced data
+set size, and training time accorded to the model. As [25]
+state, diffusion model output quality typically scales up with
+additional training time and data, and we expect this to be
+the case as well here. Despite these issues, the results still
+look very promising Fig. 4, with certain identities perform-
+ing better than others. We encourage readers to view the
+provided multi speaker video samples for a mix of both fail-
+ure cases and successful videos.
+4.4. Single Speaker Fine Tuned
+We fine tune a single speaker model using identity S1
+on top of the pre-trained multi speaker model discussed
+above. We use the same training hyper-parameters as the
+base single speaker model, however we maintain 20 random
+videos unseen to the network for testing, and train it for only
+150 epochs instead of 895. We report reasonable results in
+Tab. 2 on the unseen test set . We show that by fine tuning
+on a pre-trained multi speaker model, we achieve slightly
+worse results to the base single speaker model while train-
+ing for significantly less time. It stands to reason that with
+further training, we would see even better results. We pro-
+vide videos demonstrating this models editing capabilities
+in the supplementary materials.
+5. Limitations & Future Work
+Training & Inference Speed: It is no secret that diffu-
+sion models are slow, both to train, and to sample from.
+Our model is no exception, taking approximately 30 min-
+utes/epoch to train the single speaker model, 90 min-
+utes/epoch for the multi speaker one, and approximately 1
+minute to generate 1 frame with 2000 diffusion steps on
+a single 32gb v100 GPU. We plan on updating our model
+with the approach proposed by [53], to facilitate training in
+the latent space, in addition to methodically shrinking the
+number of parameters our model has to determine the op-
+timum set up. We suspect our current model has too many
+parameters for the task at hand, and intend to reduce it. It
+remains to be seen how image quality will be impacted by
+these changes, however a faster model would allow for more
+in depth tests, and comparisons across a wider range of data
+sets, furthering the field.
+Multi Speaker Model: While we demonstrate that our sin-
+gle speaker models perform reasonably well (still a lot of
+room for improvement!), further work must be done to ac-
+7
+
+Figure 4. Sample frames generated by our base single speaker, fine tuned single speaker, and multi speaker models. Note that the single
+speaker model shows the best lip movement accuracy. Observe how the speakers lips close at the start of the word “blue”, and gradually
+open as the word progresses.
+complish reliable multi-speaker performance. We want to
+train the model longer, and with more data to improve the
+lip synchronization. To facilitate this we need to imple-
+ment the improvements mentioned above. Additionally, we
+noticed that our current multi speaker model struggled to
+keep the identity consistent throughout the generation pro-
+cess. To address this, we propose introducing an additional
+“identity” frame in the conditioning process, that the net-
+work may use as a reference for how the speaker should
+appear.
+Dataset Size: We train our networks on such small subsets
+of the GRID data set due to our limited hardware capacity.
+Despite these limitations however, we achieve convincing
+results, as shown in Tab. 2. We encourage readers with ac-
+cess to more powerful hardware to train on the full length
+data set, as well as other sources such as the Obama White
+House single speaker data set, or the BBC Lip Reading Data
+set [61].
+Talking head generation: The task of audio-driven video
+editing involves modifying a small portion of an already ex-
+isting video in response to a new audio signal. We demon-
+strate that diffusion models can be used successfully to-
+wards this goal. The next step is extending this functionality
+to talking head generation, where the network must learn to
+synthesize full frame videos from a driving audio signal and
+single image. This is a challenging task as the network must
+now also generate natural head movements, eye blinks, and
+facial expressions as well as maintaining accurate lip and
+jaw movements synchronised to the audio. We plan to ex-
+plore this task in our future work, and study various ways
+into conditioning the network to control the facial aspects
+mentioned above.
+6. Conclusion
+Throughout this work, we demonstrate the feasibility in
+applying denoising diffusion models to the task of end to
+8
+
+end audio-driven video editing. Although the slow sam-
+pling and training speeds associated with diffusion models
+hindered our approach in the multi-speaker domain, we still
+show reasonable results within the single speaker context,
+generating high quality videos. With our work, we take
+a promising first step forward towards achieving accurate
+audio-driven video editing with denoising diffusion models.
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+Minimizing Trajectory Curvature of ODE-based Generative Models
+Sangyun Lee 1 Beomsu Kim 2 Jong Chul Ye 2
+Abstract
+Recent ODE/SDE-based generative models, such
+as diffusion models and flow matching, define a
+generative process as a time reversal of a fixed for-
+ward process. Even though these models show im-
+pressive performance on large-scale datasets, nu-
+merical simulation requires multiple evaluations
+of a neural network, leading to a slow sampling
+speed. We attribute the reason to the high cur-
+vature of the learned generative trajectories, as
+it is directly related to the truncation error of a
+numerical solver. Based on the relationship be-
+tween the forward process and the curvature, here
+we present an efficient method of training the
+forward process to minimize the curvature of gen-
+erative trajectories without any ODE/SDE simula-
+tion. Experiments show that our method achieves
+a lower curvature than previous models and, there-
+fore, decreased sampling costs while maintaining
+competitive performance. Code is available at
+https://github.com/sangyun884/fast-ode
+1. Introduction
+Many machine learning problems can be formulated as dis-
+covering the underlying distribution from observations. Ow-
+ing to the development of deep neural networks, deep gen-
+erative models exhibit superb modeling capabilities.
+Classically, variational Autoencoders (VAE) (Kingma
+& Welling, 2013), Generative Adversarial Networks
+(GAN)
+(Goodfellow
+et
+al.,
+2014),
+and
+invertible
+flows (Rezende & Mohamed, 2015) have been extensively
+studied. However, each model has its drawback. GANs have
+dominated image synthesis for several years (Karras et al.,
+2019; Brock et al., 2018; Karras et al., 2020b), but carefully
+selected regularization techniques and hyperparameters are
+needed to stabilize training (Miyato et al., 2018; Brock et al.,
+2018), and their performance often does not transfer well to
+other datasets. Invertible flows enable exact maximum like-
+lihood training, but the invertibility constraint significantly
+1Soongsil University 2KAIST. Correspondence to: Jong Chul
+Ye <jong.ye@kaist.ac.kr>.
+restricts the architecture choice, which means that they can-
+not benefit from the development of scalable architectures.
+VAEs do not suffer from the invertibility constraint, but their
+sample quality is not as good as other models.
+Apart from the vanilla VAEs, recent studies utilize their hi-
+erarchical extensions (Child, 2020; Vahdat & Kautz, 2020)
+as they offer more expressivity to both inference and gen-
+erative components by assuming nonlinear dependencies
+between latent variables. However, they often have to rely
+on heuristics such as KL-annealing or gradient skipping due
+to training instabilities (Child, 2020; Vahdat & Kautz, 2020).
+Although continuous normalizing flows (Chen et al., 2018)
+do not suffer from the invertibility constraint and can be
+trained on a stationary objective function, training requires
+simulating ODEs, which prevents them from being applied
+to large-scale datasets.
+Recent ODE/SDE-based approaches attempt to settle these
+issues by defining the generative process as a time reversal of
+a fixed forward process. Diffusion models (Song & Ermon,
+2019; Song et al., 2020; Ho et al., 2020; Sohl-Dickstein
+et al., 2015) define the generative process as a time reversal
+of a forward diffusion process, where data is gradually trans-
+formed into noise. By doing so, they can be trained on a
+stationary loss function (Vincent, 2011) without ODE/SDE
+simulation. Moreover, they are not restricted by the in-
+vertibility constraint and can generate high-fidelity samples
+with great diversity, allowing them to be successfully ap-
+plied to various datasets of unprecedented scales (Saharia
+et al., 2022; Ramesh et al., 2022).
+Flow matching (Liu et al., 2022; Lipman et al., 2022) pro-
+vides a different perspective on this model class. From
+this viewpoint, the training of diffusion models can be seen
+as matching the forward and reverse vector fields. Since
+stochasticity is not a root of the success of these models (Kar-
+ras et al., 2022) and flow matching offers an alternative per-
+spective that is fully explained under the ODE scheme, we
+hereafter refer to these types of models as ODE-based gen-
+erative models. If necessary, a generative ODE can be easily
+converted to an SDE and vice versa (Song et al., 2020).
+However, drawing samples from ODE-based generative
+models requires multiple evaluations of a neural network
+for accurate numerical simulation, leading to slow sampling
+speed. While many studies have attempted to develop fast
+arXiv:2301.12003v1  [cs.LG]  27 Jan 2023
+
+Minimizing Trajectory Curvature of ODE-based Generative Models
+Figure 1. Forward and reverse trajectories of denoising diffusion model (Ho et al., 2020), flow matching (Liu et al., 2022), and our method
+on 2D dataset (left). The intersection between forward trajectories makes reverse trajectories collapse toward the average direction,
+resulting in increased curvature and suboptimal sample qualities with a limited number of function evaluations (NFE). In contrast, our
+approach successfully unties the crossover between forward trajectories, leading to low-curvature reverse trajectories. This phenomenon
+also holds true in high-dimensional spaces, as demonstrated by reverse process visualization on MNIST, CIFAR-10, and CelebAHQ (64 ×
+64) datasets (middle). As a result, our method makes less truncation error when the number of function evaluations (NFE) is small (right).
+samplers for pre-trained models (Lu et al., 2022; Zhang &
+Chen, 2022), there seems to be a limit to lowering the costs.
+We attribute the reason to the high curvature of the learned
+generative trajectories. The curvature is intriguing since it is
+directly related to the truncation error of a numerical solver.
+Intuitively, zero curvature means that generative ODEs can
+be accurately solved with only one function evaluation.
+Since a generative process is a time reversal of the forward
+process, it is evident that its curvature is also somehow de-
+termined by the forward process, but the exact mechanism is
+yet unexplored. We find that the flow matching perspective
+offers an interesting insight into the relationship between the
+forward process and the curvature. Based on our observa-
+tion, we propose an efficient method of training the forward
+process to reduce curvature. Specifically, our contributions
+are as follows:
+• We investigate the relationship between the forward
+process and curvature from a flow matching perspec-
+tive. We find that the degree of intersection between
+forward trajectories is positively related to the curva-
+ture of generative processes.
+• We propose an efficient method of learning the forward
+process to reduce the degree of intersection between
+forward trajectories without any ODE/SDE simulation.
+We show that our method can be seen as a β-VAE (Hig-
+gins et al., 2016) with a time-conditional decoder.
+• Experiments show that our method achieves lower cur-
+vature than previous models and, therefore, demon-
+strates decreased sampling costs while maintaining
+competitive performance.
+2. Background
+ODE/SDE-based generative models effectively model com-
+plex distributions by repeatedly composing a neural net-
+work, making trade-offs between execution time and sample
+quality. In this paper, we focus on ODE-based generative
+models since they yield the same marginal distribution as
+SDEs while being conceptually simpler and faster to sam-
+ple (Song et al., 2020).
+Different from continuous normalizing flows (CNF) (Chen
+et al., 2018), recent ODE-based models do not require ODE
+simulations during training and therefore are more scal-
+able. At a high level, they define a forward coupling q(x, z)
+between data distribution p(x) and prior distribution p(z)
+and subsequently an interpolation xt(x, z) for t ∈ [0, 1]
+between a pair (x, z) ∼ q(x, z) such that x0(x, z) = x
+and x1(x, z) = z. Training objectives are variants of the
+denoising autoencoder objective
+min
+θ Et∼U(0,1)Ex,z∼q(x,z)[λ(t)||x − xθ(xt(x, z), t)||2
+2],
+(1)
+where λ(t) is a weighting function. Here, a neural network
+xθ(xt, t) is trained to reconstruct the data x from the cor-
+rupted observation xt. In the following, we briefly review
+two popular instances of such models: the denoising dif-
+fusion model and flow matching. We refer the readers to
+Appendix A for a detailed background.
+
+MNIST
+0.08 
+Ours
+Flow matching
+Forwardtrajectory
+0.06
+0.04
+0.02
+(b) MNIST
+0.00-
+101
+102
+CIFAR-10
+0.12
+Reverse trajectory, NFEs = 20
+Ours
+0.10
+Flow matching
+0.08 -
+(c) CIFAR-10
+0.04
+0.02
+0.00 -
+101
+102
+Ours
+CelebAHQ 64
+0.06
+ NFEs = 3
+Ours
+— Flow matching
+0.05
+0.04
+low matching
+ 0.03 
+0.02
+0.01
+0.00
+Denoising diffusion
+Flow matching
+101
+Ours
+102
+t=1
+(d) CelebAHQ 64
+0=↑
+NFEs
+(a) 2D Mixture of GaussiansMinimizing Trajectory Curvature of ODE-based Generative Models
+Denoising diffusion models
+Denoising diffusion mod-
+els (Ho et al., 2020) employ the prior p(z) = N(0, I),
+forward coupling q(x, z) = p(x)p(z), and a nonlinear in-
+terpolation
+xt(x, z) = α(t)x +
+�
+1 − α(t)2z
+(2)
+with a predefined nonlinear function α(t). λ(t) is often
+adjusted to improve the perceptual quality for image syn-
+thesis (Ho et al., 2020). Sampling can be done by solving
+probability flow ODEs (Song et al., 2020).
+Flow matching
+However, the choice of Eq. (2) seems un-
+natural from a flow matching perspective as it unnecessarily
+increases the curvature of generative trajectories. In flow
+matching (Liu et al., 2022; Lipman et al., 2022), the inter-
+mediate sample xt is rather defined as a linear interpolation
+xt(x, z) = (1 − t)x + tz
+(3)
+since it has a constant velocity across t for given x and z.
+After training, sampling is done by solving the following
+ODE backward:
+dzt = zt − xθ(zt, t)
+t
+dt,
+(4)
+where dt is an infinitesimal timestep, and z1 is sampled
+from p(z). Instead of predicting x, Liu et al. (2022) directly
+learns the velocity zt−xθ(zt,t)
+t
+. The effectiveness of this
+sampler in reducing the sampling costs has been investigated
+in various contexts (Karras et al., 2022; Liu et al., 2022;
+Lipman et al., 2022). We build our method based on flow
+matching, using the linear interpolation and the ODE in
+Eq. (4) for sampling.
+3. Curvature Minimization
+3.1. Curvature
+For a generative process Z = {zt(z)} with the initial value
+z1(z) = z , we informally define curvature as the extent to
+which the trajectory deviates from a straight path:
+C(Z) = Et
+����z1(z) − z0(z) − ∂
+∂tzt(z)
+����
+2
+2
+(5)
+The average curvature Ez∼p(z)[C(Z)] should be the main
+concern in designing the ODE-based models since it is di-
+rectly related to the truncation error of numerical solvers.
+Zero curvature means the path is completely straight. There-
+fore, a single step of the Euler solver is sufficient to obtain
+an accurate solution.
+Since a generative process is a time reversal of the forward
+process, its curvature is determined by the forward process.
+As an illustrative example, consider a generative ODE that
+is trained on Eq. (1). In the optima, xθ(zt, t) is a minimum
+mean squared error estimator E[x|xt = zt], and the average
+curvature of the generative processes governed by Eq.(4)
+becomes
+Ez,t
+����z1(z) − z0(z) − 1
+t zt(z) + 1
+t E[x|xt = zt(z)]
+����
+2
+2
+,
+(6)
+which is a function of the posterior q(x|xt). Since we define
+xt as an interpolation between x and z, the posterior is de-
+termined by the forward coupling q(x, z). In previous work,
+q(x, z) is fixed, and so is the curvature of the generative
+process in optima. In the following, we further examine the
+relationship between forward coupling and curvature and
+show that we can improve the curvature by finding better
+q(x, z).
+3.2. Curvature and the degree of intersection
+Specifically, we observe that Eq. (6) is related to the degree
+of intersection of the forward trajectories
+I(q) = Et,x,z∼q(x,z)[||z−x−E[z−x|xt(x, z)]||2
+2], (7)
+which becomes zero when there is no intersection at any
+xt. As shown in Fig. 1, the intersection between forward
+trajectories makes the reverse vector field collapse toward
+the average direction, leading to high curvature. As the
+degree of intersection decreases, the reverse paths are grad-
+ually straightened. When I(q) = 0, the posterior q(x|xt)
+becomes a Dirac delta function, E[x|xt = zt(z)] = z0(z)
+for every t, and Eq. (6) becomes zero, i.e., the paths are com-
+pletely straight. Therefore, it is natural to seek a forward
+coupling q(x, z) that minimizes Eq. (7). We can estimate
+Eq. (7) by minimizing the following upper bound with re-
+spect to θ.
+Proposition 1. Let xt(x, z) be the linear interpolation de-
+fined as Eq. (3). Then, we have
+I(q) ≤ Et,x,z∼q(x,z)
+� 1
+t2 ||x − xθ(xt, t)||2
+2
+�
+.
+(8)
+The bound is tight when xθ(xt, t) = E[x|xt].
+Sketch of Proof. Using Eq. (3), we obtain z − x = (xt −
+x)/t. Plugging it into Eq. (7), we have
+I(q) = Et,x,z∼q(x,z)[||1
+t (x − E[x|xt])||2
+2],
+(9)
+which is bounded by Eq. (8).
+For the independent coupling q(x, z) = p(x)p(z), the up-
+per bound of I(q) coincides with Eq. (1) with λ(t) = 1/t2,
+which is the training loss of Liu et al. (2022). In this sense,
+
+Minimizing Trajectory Curvature of ODE-based Generative Models
+Liu et al. (2022) estimates the upper bound of the degree of
+intersection of the independent coupling but does not really
+minimize it. Intuitively, the degree of intersection can be
+measured by the reconstruction error of an optimal decoder
+since the decoding is more difficult when multiple inputs are
+encoded into a single point. See Fig. 2 for an illustration.
+Figure 2. Reconstruction error is (a) high when forward trajectories
+intersect (b) and low when they do not.
+3.3. Parameterizing q(x, z)
+After estimating I(q) by updating θ, we search for q that
+minimizes I(q). Although there are many ways to solve
+this optimization problem, there are two practical con-
+siderations. First, the optimization needs to be efficient.
+Moreover, q(z|x) should define a smooth map from x to
+z since we have to approximate E[x|xt] using a neural
+network with finite capacity in practice. Therefore, we
+propose to parameterize the coupling as a neural network
+qφ(x, z) = qφ(z|x)p(x), where we define qφ(z|x) as a
+Gaussian distribution. With qφ(z) =
+�
+qφ(x, z)dx and a
+weight β, we optimize
+min
+φ I(qφ) + βDKL(qφ(z)||p(z)).
+(10)
+The second KL term ensures qφ(x, z) is a valid coupling
+between p(x) and p(z). See Appendix B for more details.
+Joint training
+In practice, we jointly minimize Eqs. (8)
+and (10) with respect to both θ and φ. This leads to our loss
+function
+min
+θ,φ Et,x,z∼qφ(x,z)[ 1
+t2 ||x − xθ(xt(x, z), t)||2
+2
++βDKL(qφ(z|x)||p(z))],
+(11)
+which resembles the β-VAE objective (Higgins et al., 2016)
+in that Eq. (11) reduces to the β-VAE loss if we fix t to
+1. Since the decoder xθ is conditioned on time step, ODE-
+based models can synthesize higher quality samples than
+β-VAEs by iteratively refining the blurry initial predictions.
+From this viewpoint, previous methods (Liu et al., 2022;
+Ho et al., 2020) can be seen as degenerate cases where the
+encoder qφ(z|x) collapses into the prior by setting β → ∞.
+See Fig. 3 for a visual schematic of our method.
+Figure 3. A visual schematic of the proposed method.
+4. Related Works
+Alternative forward processes
+There have been several
+approaches to finding alternative forward processes for dif-
+fusion models. It has been demonstrated that other types of
+degradation, such as blurring, masking, or pre-trained neural
+encoding, can be used for the forward process (Rissanen
+et al., 2022; Lee et al., 2022; Hoogeboom & Salimans, 2022;
+Daras et al., 2022; Gu et al., 2022; Bansal et al., 2022). How-
+ever, they are either purely heuristic or rely on an inductive
+bias that is not necessarily well-supported by theory.
+Learning forward process
+A few studies attempted to
+learn the forward process. Kingma et al. (2021) proposed
+to learn the signal-to-ratio function of the forward process
+jointly with generative components. However, the inference
+model of Kingma et al. (2021) is linear and thus has lim-
+ited expressivity. Zhang & Chen (2021) proposed nonlinear
+diffusion models, where the drift function of the forward
+SDEs are neural networks. Although they introduce more
+flexibility in inference models, training requires the simu-
+lation of forward/reverse SDEs, which causes a significant
+computational overhead. Our method possesses the advan-
+tages of both methods. Our inference model is expressive
+since we set q(z|x) as a neural network. Since we define
+xt as an interpolation between x and z and qφ(xt|x) as a
+Gaussian distribution, sampling is done with one forward
+pass for an arbitrary t, enabling efficient training as in previ-
+ous methods (Ho et al., 2020; Liu et al., 2022). Moreover,
+even though the nonlinear forward process appeared to im-
+prove the sampling efficiency of diffusion models (Zhang
+& Chen, 2021), the exact mechanism of the improved sam-
+pling speed was vague. In this paper, we convey a clear
+motivation for learning the forward process by revealing the
+relationship between the forward process and the curvature
+of the generative trajectories.
+Fast samplers
+Accelerating the sampling speed of dif-
+fusion models is an active research topic, which is often
+
+(a) High reconstruction error
+(b) Low reconstruction error
+Data
+Noise
+Interpolated sample
+ → Encoding
+Decoding
+→ Reconstruction errorKL loss
+p(z)
+Encoder
+(1 -t) +tz
+Decoder
+Reconstruction lossMinimizing Trajectory Curvature of ODE-based Generative Models
+tackled by developing fast solvers (Lu et al., 2022; Zhang &
+Chen, 2022). Our work is in an orthogonal direction since
+they focus on taming the high curvature ODEs while we
+aim to minimize the curvature itself. We expect the effect of
+these methods to be additive to ours and leave the detailed
+investigation for future work.
+Straightness of Neural ODEs
+The importance of the
+straightness of neural ODEs in reducing the sampling cost
+has been previously discussed. Based on Benamou-Brenier
+formulation of the optimal transport problem (Benamou &
+Brenier, 2000), Finlay et al. (2020) regularized the norm of
+the vector field of the CNFs to encourage the straightness,
+which is later generalized in Kelly et al. (2020) where the
+norm of K-th order derivative is minimized. Since CNFs
+are trained on the maximum likelihood objective, any vec-
+tor field that defines the transport map from p(z) to p(x)
+is optimal, and it is therefore possible to narrow down the
+search space by utilizing the additional constraint without
+drastically compromising the performance. In contrast, re-
+cent ODE-based generative models (Song & Ermon, 2019;
+Ho et al., 2020; Liu et al., 2022; Lipman et al., 2022) train
+the neural ODE to match a pre-defined forward flow using
+Eq. (1). Thus the solution is unique, and any additional
+regularization makes models deviate from optima.
+Figure 4. The relationship between the degree of intersection be-
+tween forward trajectories and curvature of reverse trajectories.
+The first column shows forward and reverse trajectories induced by
+the independent coupling q(x, z) = p(x)p(z). As the degree of
+intersection between forward trajectories is decreased by lowering
+β, reverse paths are gradually straightened.
+5. Experiment
+5.1. 2D dataset
+Fig. 4 demonstrates the visual results and the estimated
+upper bounds of the degree of intersection on the 2D toy
+dataset. The leftmost column shows the forward trajectories
+induced by the independent coupling q(x, z) = p(x)p(z)
+used in previous work. Since a data point can be mapped
+to any noise, the forward trajectories largely intersect with
+each other, and as a result, the reverse trajectories collapse
+toward the average direction where the actual density is low,
+and the curvature increases as the reverse trajectories need
+to bend toward the modes. As β decreases, qφ(z|x) tries to
+untie the tangled trajectories, leading to low curvature.
+Figure 5. Effects of β on curvature. The results on MNIST, CIFAR-
+10, and CelebAHQ (64 × 64) are indicated by red, green, and gray
+colors. Dashed lines indicate the curvatures of the independent
+coupling baselines.
+5.2. Image generation
+We further conduct an experiment on the image dataset to
+investigate the relationship between I(q) and E[C(Z)] in
+the high-dimensional space. We estimate E[C(Z)] by simu-
+lating 10, 000 generative trajectories using the Euler solver
+with 128 steps and then divide by the number of pixels. As
+shown in Fig. 5, the average curvature is the highest when us-
+ing independent forward coupling q(x, z) = p(x)p(z) and
+lowered as β decreases. Fig. 6 shows that the generative vec-
+tor field induced by independent coupling q(x, z) = p(x, z)
+initially predicts the blurry images and then bends toward
+the mode, resulting in the high curvature. Lower β yields
+more consistent results across the time steps.
+Fig. 7 shows that the model trained with the lower β per-
+forms better with the limited NFEs and asymptotically ap-
+proaches the performance of baseline indicated by a dashed
+line. When β is as low as 1, the generative process is almost
+straight, but the sample quality is degraded because of high
+DKL(qφ(z)||p(z). As shown in Fig. 8, the gap between
+reconstruction FID (rFID) and FID is large when β = 1
+and gradually becomes smaller as β increases. Moreover,
+the distribution of the norm of latent vectors gradually ap-
+proaches p(z) as β increases. From this observation, we can
+see that β is an important hyperparameter that determines
+the trade-off between sample quality and computational
+cost, and there are little advantages of setting β to ∞ as in
+previous work (Liu et al., 2022; Ho et al., 2020) in most
+cases.
+
+Forward trajectory
+Reverse trajectory
+independent, I(g) ≤ 32
+β = 100, I(q) ≤ 15.23
+β = 30, I(q) ≤ 5.3
+β = 10, I(g) < 2.190.08
+0.07
+0.065
+0.06
+0.053
+0.05
+0.046
+Curvature
+0.038
+0.04
+0.033
+0.03
+0.02
+-0.016
+0.014
+0.01
+0.012
+0.010
+0.00
+1
+5
+10
+15
+20
+βMinimizing Trajectory Curvature of ODE-based Generative Models
+Figure 6. Visualization of intermediate samples xθ(zt, t) with
+varying β. Lower β allows for sharper initial predictions, as indi-
+cated by red boxes.
+(a) Quantitative results
+(b) Qualitative results
+Figure 7. Trade-off between FID10K and the number of function
+evaluations (NFE) with varying β. The low curvature generative
+process produces more high-quality samples than the baseline with
+limited NFEs.
+5.3. Distillation
+Even though distillation is an effective way to train the
+one-step student models from the teacher diffusion models,
+(a) FID gap with respect to β
+(b) Distribution of ||z||2
+Figure 8. The gap between reconstruction FID (rFID) and FID
+values (a), and distribution of the norm of z ∼ qφ(z) (b). rFID is
+measured using samples reconstructed from qφ(z).
+the performance of the student model is suboptimal due to
+the distillation error (Liu et al., 2022; Luhman & Luhman,
+2021; Salimans & Ho, 2022). Given that the teacher tra-
+jectories with higher NFEs are more difficult to distill, our
+low-curvature generative ODEs would make less distillation
+error since they achieve the same level of sample quality
+using relatively lower NFEs. Based on this intuition, we
+investigate the effect of our method on reducing the distil-
+lation error. As shown in Table 1, the teacher ODE with
+β = 10 achieves a similar FID score using half as many
+NFEs compared to the baseline model. This resulted in
+a smaller distillation error and an improved FID score of
+the one-step model while reducing the cost of generating
+paired data by half. Fig. 9 demonstrates that our method
+with β = 10 obtains a superior one-step model than baseline
+with independent coupling in terms of sample fidelity.
+Table 1. Effects of curvature on distillation performance. The re-
+sults of independent coupling and our method with β = 10 are
+reported. Distillation error is measured as a mean-squared error on
+the test set.
+Independent
+β = 10
+FID / Error
+NFEs
+FID / Error
+NFEs
+Teacher
+3.60 / -
+20
+3.52 / -
+10
+Distilled
+6.25 / 0.0208
+1
+4.41 / 0.0157
+1
+(a) β = 10
+(b) Independent
+Figure 9. Synthesis results of one-step models.
+
+B
+β= 10
+β=
+: 20
+p(z)
+25
+26
+27
+28
+29
+30
+I/z||2SSSSS
+222
+38888
+C
+:20
+88
+000
+Independent
+Independent20.0
+20
+18.8
+independent
+17.5
+learned
+15.0
+15
+3.0
+12.5
+FID10K
+2 :5.
+10.0
+B
+-10
+8.5
+2.0
+7.5
+1.5
+64
+128
+5.0
+5
+2.5
+0.0
+1
+5
+10
+20
+32
+64
+128
+NFEs7
+S
+6
+6
+子
+NFEs=4 
+8
+7A
+7
+?子
+0.
+6
+6
+NFEs=128
+？子
+3
+S
+3
+s
+β= 5
+Independent12 -
+11.36
+10:
+8
+FID
+rFID
+6
+4
+2
+0.68
+0.28
+0.23
+0
+1
+5
+10
+20
+βMinimizing Trajectory Curvature of ODE-based Generative Models
+5.4. Size of encoder
+Since we train qφ(z|x), a natural question is how much
+additional computational cost is needed for training our
+model. We experiment with two settings of the encoder,
+same and small. In same setting, we use the identical
+architecture with a generative component except that the
+number of output channels is twice for predicting a diagonal
+covariance. In small setting, we use roughly 20 times
+smaller architecture for the encoder model. See Appendix C
+for a detailed configuration. As shown in Table 2, a small
+encoder performs just as well or even better than a larger
+encoder, and part of the reason is that we can use a larger
+batch size in the small setting. The additional cost of our
+method is negligible with the use of a lightweight architec-
+ture for qφ(z|x), so we stick to small setting throughout
+our experiments.
+Table 2. Performance comparison of same and small encoder
+settings on CIFAR-10 dataset, measured by FID10K.
+Setting \ NFEs
+128
+40
+20
+10
+5
+4
+Same Encoder
+5.52
+6.23
+7.74
+11.49
+22.90
+30.97
+Small Encoder
+5.39
+6.07
+7.51
+11.19
+22.33
+30.16
+5.5. Comparison with state-of-the-arts
+Table 3 shows the unconditional synthesis results of our
+approach on the CIFAR-10 dataset. Results of recent meth-
+ods are also provided as a reference. We experiment with
+two configurations, config A and config B, which we detail
+in Appendix C. We try three solvers, Euler solver, Heun’s
+2nd order method, and the black-box RK45 method from
+Scipy (Virtanen et al., 2020), and find that RK45 works
+well when we are able to fully simulate ODEs while Heun’s
+2nd order method performs better than other solvers with
+small NFEs. As shown in the table, we can see that the
+performance gap between our method and the baseline is
+huge when the sampling budget is limited. For instance, our
+method with β = 10 achieved an FID score of 18.74, which
+is significantly better than the baseline’s score of 37.19 when
+NFEs is 5. Surprisingly, our method with β = 20 exhibits
+superior sample qualities across all NFEs, even in the case
+of full sampling using the RK45 solver. See Fig. 10 for vi-
+sual comparison. Additional qualitative results are provided
+in Appendix D.
+6. Discussion and Limitations
+One limitation of our method is that for our encoding dis-
+tribution qφ(z|x), we use a Gaussian distribution for the-
+oretical and practical conveniences: sampling is easily im-
+plemented in a differentiable manner, and KL divergence
+is tractable. However, our simple Gaussian encoder cannot
+eliminate the intersection completely. We believe that it
+would be beneficial to use a more flexible encoding distribu-
+tion, for instance, using the hierarchical latent variable as in
+Child (2020); Vahdat & Kautz (2020).
+Additionally, the trade-off between sample quality and com-
+putational cost is determined by the value of β, which must
+be manually selected by a practitioner. This is problematic
+as one has to train a model from scratch for each value of
+β, which would potentially lead to excessive energy con-
+sumption. However, using a reasonably high value of β
+consistently outperforms the baseline regardless of the sam-
+pling budget, as shown in our experiments. Therefore, one
+could reduce the sampling cost without compromising per-
+formance by conservatively setting β to a high value in most
+cases.
+7. Conclusion
+In this paper, we mainly discussed the curvature of the
+ODE-based generative models, which is crucial for sam-
+pling efficiency. We revealed the relationship between the
+degree of intersection between forward trajectories and the
+curvature and presented an efficient algorithm to reduce the
+intersection by training a forward coupling. We demon-
+strated that our method successfully reduces the trajectory
+curvature, thereby enabling accurate ODE simulation with
+significantly less sampling budget. Furthermore, we showed
+our method effectively decreases the distillation error, im-
+proving the performance of one-step student models. Our
+approach is unique and complementary to other acceleration
+methods, and we believe it can be used in conjunction with
+other techniques to further decrease the sampling cost of
+ODE-based generative models.
+8. Societal Impacts and Reproducibility
+We anticipate this work will have positive effects, as it can
+significantly decrease the computational resources required
+during sampling. However, it is important to note that the
+same technology can also be used to create malicious con-
+tent, and thus, proper regulations need to be put in place to
+ensure that this technology is used responsibly and ethically.
+We make the code and model checkpoints for our method
+publicly available at the following GitHub remote reposi-
+tory: https://github.com/sangyun884/fast-ode. Implementa-
+tion details can also be found in Appendix. C.
+References
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+Table 3. Comparison with state-of-the-arts on CIFAR-10 dataset. * Our reimplementation.
+Method
+NFEs(↓)
+IS (↑)
+FID (↓)
+Recall (↑)
+GANs
+StyleGAN2 (Karras et al., 2020a)
+1
+9.18
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+0.41
+StyleGAN2 + ADA (Karras et al., 2020a)
+1
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+-
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+-
+DFNO (Zheng et al., 2022)
+1
+-
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+-
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+1
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+-
+9.12
+-
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+0.57
+2-Rectified Flow (RK45)
+110
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+0.54
+3-Rectified Flow (RK45)
+104
+9.01
+3.96
+0.53
+2-Rectified Flow Distillation
+1
+9.01
+4.85
+0.50
+Our results
+Rectified Flow* (config A, RK45)
+134
+9.18
+2.87
+-
+Rectified Flow* (config B, RK45)
+132
+9.48
+2.66
+-
+Rectified Flow* (config B, Heun’s 2nd order method)
+9
+8.48
+12.92
+-
+Rectified Flow* (config B, Heun’s 2nd order method)
+5
+7.04
+37.19
+-
+Ours (β = 20, config B, RK45)
+118
+9.55
+2.45
+-
+Ours (β = 20, config B, Heun’s 2nd order method)
+9
+8.75
+9.96
+-
+Ours (β = 20, config B, Heun’s 2nd order method)
+5
+7.83
+24.40
+-
+Ours (β = 10, config A, RK45)
+110
+9.32
+3.37
+-
+Ours (β = 10, config A, Heun’s 2nd order method)
+9
+8.67
+8.66
+-
+Ours (β = 10, config A, Heun’s 2nd order method)
+5
+8.09
+18.74
+-
+Figure 10. Qualitative comparison between our method (β = 10) and baseline.
+
+CIFAR-10
+CelebAHQ 64
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+
+Minimizing Trajectory Curvature of ODE-based Generative Models
+A. Flow Matching
+Diffusion models have been interpreted as the variational approaches (Sohl-Dickstein et al., 2015; Ho et al., 2020) or
+score-based models (Song & Ermon, 2019; Song et al., 2020), and their deterministic samplers are derived post hoc.
+However, stochasticity is not a key factor in the success of these models. The state-of-the-art performance can be achieved
+without stochasticity (Karras et al., 2022), and the incorporation of stochasticity makes sampling slow and complicates the
+theoretical understanding. Flow matching (Liu et al., 2022; Lipman et al., 2022) provides a useful viewpoint for explaining
+the recent iterative methods (Ho et al., 2020; Song & Ermon, 2019) from a pure ODE perspective. For the purpose of brevity,
+we only consider the variance-preserving diffusion models (Song et al., 2020). Readers are encouraged to refer to Liu et al.
+(2022) for a more comprehensive explanation.
+The variance-preserving diffusion models define the following noise distribution
+q(xt|x) = N(α(t)x, (1 − α(t)2)I),
+(12)
+where α(t) is set to exp(− 1
+2
+� t
+0(as + b) ds) with a = 19.9 and b = 0.1. Another way to see this is to consider the nonlinear
+interpolation between x and z sampled independently from q(x, z) = p(x)p(z):
+xt(x, z) = α(t)x +
+�
+1 − α(t)2z
+(13)
+This forward flow represents the dynamics of particles that move from p(x) to p(z). Note that this cannot be used for
+generative modeling as it requires x to compute the velocity. To estimate the velocity without having x, a neural network
+xθ(xt, t) is trained by optimizing
+min
+θ Ex,z,t[λ(t)||x − xθ(xt(x, z), t)||2
+2].
+(14)
+From this viewpoint, the choice of nonlinear interpolation is unnatural since it unnecessarily increases the curvature of both
+forward and reverse (generative) trajectories. For this reason, Liu et al. (2022) defines the following constant-velocity flow
+with an initial value x and endpoint z:
+dxt(x, z) = (z − x)dt
+(15)
+x0(x, z) = x
+(16)
+Instead of predicting x, they directly train a vector field vθ(xt, t) to match the velocity of forward flow by minimizing the
+flow matching loss
+LF M =
+� 1
+0
+E[||(z − x) − vθ(xt, t)||2
+2] dt,
+(17)
+where vθ(xt, t) = E[z − x|xt] in the optima. Samples are drawn by solving the following ODE backward:
+dxt = vθ(xt, t)dt
+(18)
+It is shown that Eq. (18) yields the same marginal distribution as the forward flow at every t (see Theorem 3.3 in Liu et al.
+(2022)).
+Given xt = (1 − t)x + tz and z − x = (xt − x)/t, we can further find the connection with diffusion models by
+reparameterizing vθ(xt, t) = (xt − xθ(xt, t))/t and writing Eq. (17) as
+� 1
+0
+E[||(z − x) − vθ(xt, t)||2
+2] dt =
+� 1
+0
+E[||(xt − x)/t − vθ(xt, t)||2
+2] dt
+(19)
+=
+� 1
+0
+E[||(xt − x)/t − (xt − xθ(xt, t))/t||2
+2] dt
+(20)
+=
+� 1
+0
+E[ 1
+t2 ||x − xθ(xt, t))||2
+2] dt.
+(21)
+This is equivalent to Eq. (1) with λ(t) = 1/t2, and Eq. (18) is equal to Eq. (4). To our knowledge, the effectiveness of
+Eq. (18) in reducing the truncation error is first examined in Karras et al. (2022) under the variance-exploding scheme and
+later in Liu et al. (2022) and Lipman et al. (2022) (OT conditional vector fields in their work) concurrently.
+
+Minimizing Trajectory Curvature of ODE-based Generative Models
+B. Derivation of Loss Function
+B.1. Estimating DKL(qφ(z)||p(z))
+We factorize DKL(qφ(z)||p(z)) via following algebraic manipulation:
+DKL(qφ(z)||p(z)) = Ep(x)Eqφ(z|x)
+�
+log qφ(z)
+p(z)
+�
+(22)
+= Ep(x)Eqφ(z|x)
+�
+log qφ(z)
+qφ(z|x) + log qφ(z|x)
+p(z)
+�
+(23)
+= Ep(x)Eqφ(z|x)
+�
+log qφ(z)p(x)
+qφ(z|x)p(x)
+�
++ Ep(x)[DKL(qφ(z|x)||p(z))]
+(24)
+= −Iqφ(x,z)(x, z) + Ep(x)[DKL(qφ(z|x)||p(z))]
+(25)
+We can derive the variational lower bound of the mutual information as
+Iqφ(x,z)(x, z) = H(x) − H(x|z)
+(26)
+= H(x) + Eqφ(x,z)[log qφ(x|z)]
+(27)
+= H(x) + Eqφ(x,z)
+�
+log pψ(x|z) + log qφ(x|z)
+pψ(x|z)
+�
+(28)
+= H(x) + Eqφ(x,z)[log pψ(x|z)] + Eqφ(z)[DKL(qφ(x|z)||pψ(x|z))]
+(29)
+≥ H(x) + Eqφ(x,z)[log pψ(x|z)],
+(30)
+where the bound is tight when the variational distribution pψ(x|z) is equal to qφ(x|z). For that, we need to optimize
+minψ Eqφ(z)[− log pψ(x|z)], which becomes the reconstruction loss Ep(x)Eqφ(z|x)
+�
+||xψ(z)−x||2
+2
+2σ2
+�
+if we set pψ(x|z) =
+N(x; xψ(z), σ2I). Consequently, we arrive at
+DKL(qφ(z)||p(z)) ≤ inf
+ψ Ep(x)Eqφ(z|x)
+�||xψ(z) − x||2
+2
+2σ2
+�
++ Ep(x)[DKL(qφ(z|x)||p(z))] + const.
+(31)
+B.2. Our loss function
+We further set xψ(z) = xθ(z, 1) for parameter sharing. Then, our loss function is
+min
+θ,φ I(q) + βDKL(qφ(z)||p(z))
+(32)
+≤ Et,x,z∼qφ(x,z)
+� 1
+t2 ||x − xθ(xt(x, z), t)||2
+2 + β ||xθ(z, 1) − x||2
+2
+2σ2
++ βDKL(qφ(z|x)||p(z))
+�
++ const
+(33)
+= Et,x,z∼qφ(x,z)
+�¯λ(t)||x − xθ(xt(x, z), t)||2
+2 + βDKL(qφ(z|x)||p(z))
+�
++ const,
+(34)
+where ¯λ(t) is 1/t2 if t ̸= 1 and βδ(0) in t = 1 with Dirac delta function δ(·). Empirically, we observe that setting ¯λ(t) to
+1/t2 for every t leads to better performance.
+C. Implementation Details
+Table. 4 shows the training and architecture configuration we use in our experiments. In our experiment, we directly
+parameterize the vector field vθ(xt, t) following Liu et al. (2022). For MNIST and CIFAR-10 datasets, we employ DDPM++
+architecture (Song et al., 2020) in the codebase of Karras et al. (2022)1. For the CelebAHQ dataset, we use the U-Net
+architecture of (Dhariwal & Nichol, 2021)2. We evaluate FID using the code of (Karras et al., 2022). We fix the random
+seed to 0 throughout all experiments. We linearly increase the learning rate as in previous studies (Karras et al., 2022; Song
+1https://github.com/NVlabs/edm
+2https://github.com/openai/guided-diffusion
+
+Minimizing Trajectory Curvature of ODE-based Generative Models
+Table 4. Architecture and training configurations. 1We use 200K and 300K iterations for β = 10 and independent coupling, respectively.
+2We use 500K and 600K iterations for β = 20 and independent coupling, respectively.
+CIFAR-10 (A)
+CIFAR-10 (B)
+MNIST
+Iterations
+varies1
+varies2
+60K
+Batch size
+128
+128
+256
+Learning rate
+3e − 4
+2e − 4
+3e − 4
+LR warm-up steps
+78125
+5000
+8000
+EMA decay rate
+0.9999
+0.9999
+0.9999
+EMA start steps
+300
+1
+300
+Dropout probability
+0.13
+0.13
+0.13
+Channel multiplier
+128
+128
+32
+Channels per resolution
+[2, 2, 2]
+[2, 2, 2]
+[2, 2, 2]
+Xflip augmentation
+X
+O
+X
+# of params (generator)
+55.73M
+55.73M
+2.15M
+# of params (encoder)
+2.2M
+2.2M
+2.2M
+# of ResBlocks
+4
+4
+2
+t range
+[0, 1]
+[1e − 5, 1]
+[0, 1]
+et al., 2020). We use Adam optimizer with β1 = 0.9, β2 = 0.999, and eps = 1e − 8 for MNIST and CIFAR-10 datasets.
+For the CelebAHQ dataset, we use AdamW with β1 = 0.9, β2 = 0.9999, and weight decay = 0.1.
+In small setting for encoder architecture, we use the MNIST generator architecture in Tab. 4, which is more than 20 times
+smaller than CIFAR-10 models. For the distillation experiment, we use 500K pairs sampled from teacher ODEs. We find
+that student models overfit if the number of pairs is less than 500K.
+For unconditional CIFAR-10 generation, we use two solvers – RK45 and Heun’s 2nd order method. We set both atol and
+rtol to 1e − 5 for RK45 as in previous work (Song et al., 2020; Liu et al., 2022). We experiment with two configurations,
+config A and config B, and find that config A converges faster than config B at the expense of performance. Overall, our
+method converges faster than the independent coupling baseline.
+D. Additional Results
+We further provide additional synthesis results of our method in Figs. 11 and 12.
+
+Minimizing Trajectory Curvature of ODE-based Generative Models
+β = 10
+β = 20
+independent
+NFEs = 5
+NFEs = 10
+NFEs = 128
+Figure 11. Uncurated MNIST samples.
+
+Minimizing Trajectory Curvature of ODE-based Generative Models
+β = 10
+β = 20
+independent
+NFEs = 5
+NFEs = 9
+full sampling
+Figure 12. Uncurated CIFAR-10 samples.
+

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@@ -0,0 +1,1510 @@
+Astronomy & Astrophysics manuscript no. paper3
+©ESO 2023
+January 9, 2023
+On the degree of stochastic asymmetry in the tidal tails of star
+clusters
+J. Pflamm-Altenburg1, P. Kroupa1, 2, I. Thies1, Tereza Jerabkova3, Giacomo Beccari3, Timo Prusti4, and Henri M. J.
+Boffin3
+1 Helmholtz-Institut für Strahlen- und Kernphysik (HISKP), Universität Bonn, Nussallee 14–16, D-53115 Bonn, Germany
+e-mail: jpa@hiskp.uni-bonn.de
+2 Charles University in Prague, Faculty of Mathematics and Physics, Astronomical Institute, V Holešoviˇckách 2, CZ-180 00 Praha
+8, Czech Republic
+3 European Southern Observatory, Karl-Schwarzschild-Strasse 2, D-85748 Garching bei München, Germany
+4 European Space Research and Technology Centre (ESA ESTEC), Keplerlaan 1, 2201 AZ Nordwijk, Netherlands
+Received . . . ; accepted ...
+ABSTRACT
+Context. Tidal tails of star clusters are commonly understood to be populated symmetrically. Recently, the analysis of Gaia data
+revealed large asymmetries between the leading and trailing tidal tail arms of the four open star clusters Hyades, Praesepe, Coma
+Berenices and NGC 752.
+Aims. As the evaporation of stars from star clusters into the tidal tails is a stochastic process, the degree of stochastic asymmetry is
+quantified in this work.
+Methods. For each star cluster 1000 configurations of test particles are integrated in the combined potential of a Plummer sphere and
+the Galactic tidal field over the life time of the particular star cluster. For each of the four star clusters the distribution function of the
+stochastic asymmetry is determined and compared with the observed asymmetry.
+Results. The probabilities for a stochastic origin of the observed asymmetry of the four star clusters are: Praesepe ≈1.7 σ, Coma
+Berenices ≈2.4 σ, Hyades ≈6.7 σ, NGC 752 ≈1.6 σ.
+Conclusions. In the case of Praesepe, Coma Berenices and NGC 752 the observed asymmetry can be interpreted as a stochastic
+evaporation event. However, for the formation of the asymmetric tidal tails of the Hyades additional dynamical processes beyond a
+pure statistical evaporation effect are required.
+Key words. open clusters and associations: general - open clusters and associations: individual: Hyades, Praesepe, Coma Berenices,
+NGC 752 - stars: kinematics and dynamics
+1. Introduction
+In general, stars form spatially confined in the densest regions
+of molecular clouds (Lada & Lada 2003; Allen et al. 2007). Af-
+ter their formation different processes lead to the loss of stellar
+members:
+Early gas expulsion: The gas in the central part of the newly
+formed star cluster has not been completely converted into stars.
+Once the most massive stars have ignited the ionising radiation
+heats up the remaining gas leading to its removal from the star
+cluster. The initially virialised mixture of stars and gas turns into
+a dynamically hot and expanding star cluster. After loosing a
+substantial amount of members the remaining star cluster reviri-
+alises now having a larger diameter than at its birth. This process
+only takes a few crossing times which are typically of the order
+of a few Myr for open star clusters (e.g. Baumgardt & Kroupa
+2007).
+Stellar ejections: In the Galactic field OB-stars are observed
+moving with much higher velocities than the velocity dispersion
+of the young stellar component in the Galactic field. They are
+assumed to be ejected form young star clusters either by the dis-
+integration of massive binaries where one component explodes
+in a supernova (supernova ejection) leaving the second compo-
+nent with its high orbital velocity, or by close dynamical in-
+teractions of multiple stellar systems (e.g. in binary-binary en-
+counters) with energy transfer between the components. By or-
+bital shrinkage of one binary potential energy is transferred to
+the other binary leading to its disintegration and leaving the two
+components with high kinetic energy behind (e.g. Poveda et al.
+1967; Pflamm-Altenburg & Kroupa 2006; Oh & Kroupa 2016).
+In both cases, the velocity of the star is higher than the escape
+velocity of the star cluster allowing the particular star to escape
+from the star cluster into the Galactic field.
+Stellar evolution: Due to stellar evolution, the total mass of
+the star cluster decreases continuously, reducing the binding en-
+ergy and the tidal radius of the star cluster. The negative binding
+energy of stars which are only slightly bound may turn to a pos-
+itive value.
+Evaporation and tidal loss of stars: Gravitationally bound
+stellar systems embedded in an external gravitational field lose
+members due to the tidal forces. In the frame of a star cluster
+its potential well is lowered by the tidal forces at two opposite
+points known as Lagrange points. The velocities of stars in the
+Maxwell tail of the velocity distribution are sufficiently high to
+escape through these two Lagrange points. Each of both streams
+of escaping stars forms an elongated structure pointing in oppo-
+site directions, these being the tidal tails. If the size of the star
+cluster is small enough compared to the spatial change of the
+Article number, page 1 of 8
+arXiv:2301.02251v1  [astro-ph.GA]  5 Jan 2023
+
+A&A proofs: manuscript no. paper3
+external force field then both depressions of the star cluster po-
+tential at the Lagrange points are equal. This is typically the case
+for star clusters in the solar vicinity orbiting around the Galactic
+centre. Thus, both streams of stars through the Lagrange points
+are equal and the tidal pattern is expected to be symmetric with
+respect to the star cluster (e.g. Küpper et al. 2010). Therefore,
+both tidal arms should be equal.
+Recently, the detailed analysis of Gaia data revealed asym-
+metries in the tidal tails of the four open star clusters Hyades
+(Jerabkova et al. 2021), Preasepe, Coma Berenices (Jerabkova
+et al. in prep.) and NGC 752 (Boffin et al. 2022), challenging the
+common assumption of symmetric tidal tails. However, as it can
+not be predicted through which Lagrange point a star will es-
+cape from the star cluster into one of the tidal arms, evaporation
+can be interpreted as a stochastic process. Thus, different num-
+bers of members in both tidal tails of an individual star cluster
+are expected. It is the aim of this work to quantify the degree of
+asymmetry in tidal tails due to the stochastic population of tidal
+tails and to compare with the observations.
+In Sect. 2 we describe how (a)symmetry in tidal tails of star
+clusters is quantified throughout this work. The data of the ob-
+served star clusters and the derived asymmetries are presented in
+Sect. 3. The evaporation process of the numerical Monte Carlo
+model is outlined in Sect. 4 including the determination of statis-
+tical asymmetries due to stochastic evaporation. Section 5 com-
+pares the numerical asymmetries with a theoretical model which
+interprets the evaporation of stars through both Lagrange points
+into the tidal tails as a Bernoulli experiment.
+2. Measuring the (a)symmetry of tidal tails
+In order to explore the degree of (a)symmetry in the tidal tails
+the distance distribution of tail members is quantified. Küpper
+et al. (2010) performed N-body simulations and calculated the
+stellar number density as a function of the distance to the star
+cluster along the star cluster orbit. In their simulations 21 mod-
+els covering a range of different initial conditions such as the
+galactocentic orbital radius or the inclination of the orbit were
+computed with initially 65536 particles using the Nnody4 code
+(Aarseth 1999, 2003) which performs a full force summation
+over all particles. This high number of particles leads to a smooth
+population of the tidal arms. Thus, only small statistical fluctua-
+tions are expected. In order to explore the statistical fluctuations
+of the member number of tidal tails of open star clusters with
+only a few hundred stars in the tidal tails multiple simulations of
+star clusters resembling the observed ones are required.
+In order to avoid effects by uncertainties of the Galactic tidal
+field, here the actual velocity vector of the star cluster is used as
+the reference (Fig.1) and not the distance to the cluster along the
+orbit of the star cluster. The membership and distance of each
+star is determined as follows: For each star its distance is given
+by its position vector, ri = |ri|, in the star cluster reference frame.
+If its distance is larger than the tidal radius, rt, it is a tidal tail star.
+The orientation angle, ϕi, is the angle enclosed by the position
+vector, ri, of the star and the actual velocity vector of the star
+cluster, vcl. A star is considered to be a member of the leading
+arm if 0 ≤ ϕi ≤ 90◦, and to be a member of the trailing arm if
+ϕi > 90◦. Here, the membership of a star to belong to either the
+leading or the trailing arm is a sharp criterium without any prob-
+ability. For those stars located very close to the border separating
+the leading from the trailing arm the errors of the observed data
+of the positions of the stars are not considered.
+The normalised asymmetry, ϵ, is given by the difference of
+the number of members in the leading tail, nl, and the number
+ϕi
+vcl
+ri
+rt
+Fig. 1. Sketch of the method to quantify the (a)symmetry in tidal tails.
+The angle of the orientation, ϕi, of each star is the angle enclosed by the
+position vector, ri, of the particular star with respect to the centre of the
+star cluster and the actual velocity vector, vcl, of the star cluster.
+of members in the trailing tail, nt, divided by the total number of
+tidal tail stars, n,
+ϵ = nl − nt
+nl + nt
+= nl − nt
+n
+.
+(1)
+3. Observed open star clusters
+In this section the structure of the tidal tails of the Hyades,
+Praesepe, Coma Berenices and NGC 752 are analysed using the
+method described in Sect. 2. The present-day stellar distributions
+have been obtained by the Jerabkova et al. (2021) compact con-
+vergent point (CCP) method which allows the tidal tails to be
+mapped to their tips.
+Table 1 summarises the observed values of the star clusters
+which are of interest for this work (see Kroupa et al. 2022 for
+details).
+The results of the analysis of the observational data are
+shown in Figs. 2–8. The data for the Hyades are published in
+Jerabkova et al. (2021) and the detailed analysis of its tidal tails
+can be seen in Fig. 2. The upper left panel shows the spatial dis-
+tribution of all stars associated with the Hyades. These are all
+stars, which are identified to be cluster members, that is, have a
+distance to the cluster centre less than the tidal radius (Table 1),
+and those stars, which are identified to be tidal tail members, that
+is, have a distance to the cluster centre larger than the tidal ra-
+dius. Cluster members are marked by black dots, members of the
+leading tidal arm by green dots, members of the trailing tidal arm
+by red dots. The positive x-axis points towards the Galactic cen-
+tre, the positive y-axis towards the Galactic rotation. The black
+arrow indicates the actual direction of motion of the Hyades clus-
+ter with respect to the Galactic rest frame. It can be clearly seen
+that the leading arm is more populated than the trailing arm. Fur-
+thermore, the leading tidal arm has also a higher surface density
+of tidal tail members as can be seen in Fig. 3.
+In the upper right panel of Fig. 2 the orientation angle and
+the distance of all stars are shown using the method described
+in Sect. 2. Stars with an orientation angle smaller than 90◦ be-
+long formally to the leading arm, stars with an angle larger than
+90◦ to the trailing arm. Within the tidal radius the cluster is well
+represented by a Plummer model (Röser et al. 2011). Thus, in
+Article number, page 2 of 8
+
+J. Pflamm-Altenburg et al.: On the degree of stochastic asymmetry in the tidal tails of star clusters
+star cluster
+Hyades
+Coma
+Berenices Praesepe
+NGC 752
+d / pc
+47.5
+85.9
+186.2
+438.4
+M / M⊙
+275
+112
+311
+379
+bPl / pc
+3.1
+2.7
+3.7
+4.1
+rt / pc
+9.0
+6.9
+10.8
+9.4 (1.2285◦)
+t / Myr
+680
+750
+770
+1750
+n
+541
+640
+833
+298
+nl
+351
+348
+384
+163
+nt
+190
+292
+449
+135
+ϵ
+0.298
+0.088
+-0.078
+0.094
+nl,50−200
+162
+133
+87
+56
+nt,50−200
+64
+111
+140
+43
+x / pc
+-8344.61
+-8305.78
+-8439.78
+-8294.05
+y / pc
+-0.23
+-5.84682
+-69.384
+275.07
+z / pc
+10.69
+112.516
+128.552
+-158.408
+vx / km s−1
+-32.01
+8.69065
+-33.3617
+-8.19
+vy / km s−1
+212.37
+226.515
+210.711
+216.262
+vz / km s−1
+6.44
+6.16155
+-1.83814
+-12.81
+Table 1. Data of the star clusters. (upper section:) d: distance Sun–star
+cluster, M: current stellar mass within tidal radius rt, bPl: Plummer pa-
+rameter obtained from the observed half mass radius, rt: observed tidal
+radius, t: average of published ages of the star clusters, n: total number
+of tidal tail members, nl: number of members in the leading tidal tail, nt:
+number of members in the trailing tidal tail, nl,50−200: number of mem-
+bers in the leading tidal tail at a distance of 50–200 pc from the cluster
+centre, nt,50−200: number of members in the trailing tidal tail at a distance
+of 50–200 pc from the cluster centre. (lower section:) current position
+and velocity of the star clusters in the Galactic inertial rest frame used
+for the Monte Carlo simulations for an assumed Solar distance of 8.3
+kpc to the Galactic centre, and 27 pc above the Galactic plane and a
+local rotational velocity of 225 km/s as in Jerabkova et al. (2021). The
+velocity components of the Sun in the Galactic rest frame are 11.1 km/s
+towards to the Galactic centre, 232.24 km/s into the direction of Galactic
+rotation and 7.25 km/s in positive vertical direction. Thus, the peculiar
+velocity of the Sun is [11.1, 7.24, 7.25] km/s. See Kroupa et al. (2022)
+for details.
+the ϕ-r-diagram the region between 0 and 180◦ is fully popu-
+lated. The region with distances between the tidal radius of 9 pc
+and ≈ 50 pc is still uniformly populated but much sparser than
+within the bound cluster. This corresponds to a relatively spher-
+ically shaped region around the Hyades cluster, which is some-
+times called a stellar corona of a star cluster. At distances larger
+than ≈ 50 pc two distinct, sharply confined tidal arms are visible.
+The lower left panel of Fig. 2 shows the cumulative number
+of stars in both tidal arms separately as a function of the dis-
+tance to the cluster centre. Up to ≈20-30 pc both distributions
+are nearly equal. Beyond ≈30 pc the distributions start to deviate
+from each other. Up to ≈100 pc the cumulative distributions in-
+crease nearly constantly. The total number of stars in the leading
+arm increases faster than the number of stars in the trailing arm.
+At ≈100 pc both distributions flatten and have again a constant
+but shallower slope.
+The lower right panel in Fig. 2 shows the cumulative nor-
+malised asymmetry, ϵ(≤ r), calculated by Eq. (1).
+Figure 4 shows the same data analysis for the Praesepe star
+cluster (data are taken from Jerabkova et al. in prep.). It can be
+seen in the lower left panel that the radial cumulative distance
+distribution at small distances to the star cluster centre shows the
+same functional behaviour as in the case of the Hyades. Within
+the tidal radius the cumulative distributions rise rapidly and be-
+come abruptly shallower in slope at the tidal radius. However,
+the cumulative distributions of both arms are identical up to a
+-600
+-400
+-200
+ 0
+ 200
+ 400
+ 600
+-600
+-400
+-200
+ 0
+ 200
+ 400
+ 600
+xMW [pc]
+yMW [pc]
+ 0
+ 50
+ 100
+ 150
+ 0
+ 100
+ 200
+ 300
+ 400
+ 500
+ 600
+leading
+trailing
+ϕ [°]
+r [pc]
+ 0
+ 50
+ 100
+ 150
+ 200
+ 250
+ 300
+ 350
+ 400
+ 0
+ 100
+ 200
+ 300
+ 400
+ 500
+ 600
+leading
+trailing
+cum. N
+r [pc]
+-0.4
+-0.2
+ 0
+ 0.2
+ 0.4
+ 0
+ 100
+ 200
+ 300
+ 400
+ 500
+ 600
+ǫ(≤r)
+r [pc]
+Fig. 2. Hyades: (upper left:) Shown is the spatial distribution of the
+members of the Hyades (black), the leading tidal arm (green) and the
+trailing tidal arm (red) from Jerabkova et al. (2021). The coordinates
+xMW and yMW refer to a non-rotating rest frame of the Milky Way with
+a shifted origin where the position of the Sun lies at (0,0). The positive
+x-axis points towards the Galactic centre, the positive y-axis towards
+Galactic rotation. The arrow indicates the motion of the star cluster
+centre in the Galactic rest frame. (upper right:) Shown are the posi-
+tions of cluster and tidal tidal members in polar coordinates calculated
+as described in Sect. 2. (lower left:) Radial cumulative distribution of
+the number of stars in the leading and trailing tidal arm of the Hyades
+(lower right:) Radial cumulative evolution of the normalised asymme-
+try (Eq. 1) considering tidal tail stars with a distance to the cluster centre
+smaller or equal to r.
+distance of ≈50–70 pc. Beyond the stellar corona the distribu-
+tions diverge continuously from each other, whereas the trailing
+arm contains more members than the leading arm, contrary to
+the Hyades. The surface density can be seen in Fig. 5.
+Figure 6 shows the analysis of the Coma Berenices star clus-
+ter (data are taken from Jerabkova et al. in prep.). The radial
+cumulative distributions have the same qualitative trend as those
+of the Hyades. Within the star cluster and the stellar corona both
+distribution functions are identical. Again, beyond a distance of
+≈50–70 pc to the cluster centre the distribution diverge continu-
+ously from each other. In this case the leading arm contains more
+members. The surface density can be seen in Fig. 7.
+Figure 8 shows the analysis of the NGC 752 star cluster (data
+are taken from Boffin et al. 2022). Due to the increasing errors
+of the parallaxes with increasing distance of the star cluster the
+analysis of the tidal tails in three dimensions is effected by a dis-
+tortion of the sample in the x-y-plane (Boffin et al. 2022). There-
+fore, the stellar sample is analysed in projection on the sky.
+4. Monte Carlo simulations
+In order to compare the observed asymmetries of tidal tails with
+the degree of asymmetry due to the stochastic evaporation of
+stars from star clusters embedded in a Galactic tidal field, a large
+number of test particle integrations are performed in the Galactic
+gravitational potential.
+4.1. Numerical model
+In the simulations a star cluster is set up as a Plummer phase-
+space distribution (Plummer 1911; Aarseth et al. 1974) with ini-
+tial parameters bPl being the Plummer radius and the total mass,
+Article number, page 3 of 8
+
+A&A proofs: manuscript no. paper3
+-600
+-400
+-200
+ 0
+ 200
+ 400
+ 600
+-600
+-400
+-200
+ 0
+ 200
+ 400
+ 600
+xMW [pc]
+yMW [pc]
+-4
+-3
+-2
+-1
+ 0
+ 1
+ 2
+log10(ρ [pc-2] )
+Fig. 3. Surface number density of stars of the Hyades (cluster plus both
+tidal arms). Each dot represents one star. The local number density is
+calculated with the 6th-nearest neighbour method (Casertano & Hut
+1985) and colour-coded. The black arrow indicates the velocity vec-
+tor of the Hyades star cluster in the Galactic rest frame. The centre of
+the star cluster is located at the intersection of the two thin dashed lines.
+-800
+-600
+-400
+-200
+ 0
+ 200
+ 400
+ 600
+ 800
+-800-600-400-200  0  200 400 600 800
+xMW [pc]
+yMW [pc]
+ 0
+ 50
+ 100
+ 150
+ 0  100  200  300  400  500  600  700  800  900
+leading
+trailing
+ϕ [°]
+r [pc]
+ 0
+ 100
+ 200
+ 300
+ 400
+ 500
+ 0
+ 100
+ 200
+ 300
+ 400
+ 500
+ 600
+trailing
+leading
+cum. N 
+r [pc]
+-0.4
+-0.2
+ 0
+ 0.2
+ 0.4
+ 0
+ 100
+ 200
+ 300
+ 400
+ 500
+ 600
+ǫ(≤r)
+r [pc]
+Fig. 4. Praesepe from Jerabkova et al. (in prep.): Similar to Fig. 2.
+MPl. The centre of mass of the model, rPl, moves in a Galactic
+potential as given in Allen & Santillan (1991). The orbit of each
+stellar test particle is integrated in both gravitational fields, the
+Plummer potential as the gravitational proxy of the star cluster
+and the full Galactic gravitational potential. Thus, the equations
+of motion of the star i and of the cluster model are
+ai = −∇ΦPl − ∇ΦMW ,
+(2)
+aPl = −∇ΦMW .
+(3)
+The test particles are distributed in the model cluster potential
+according to the Plummer phase-space distribution function and
+are integrated in time in the static Galactic potential and a static
+Plummer potential, whose origin is integrated as a test particle
+in the Galactic potential. In a self-gravitating system those par-
+ticles evaporate from the cluster which gained sufficient energy
+by energy redistribution between the gravitationally interacting
+particles to exceed the binding energy to the cluster. In the model
+cluster all particles are treated as test particles and are integrated
+-800
+-600
+-400
+-200
+ 0
+ 200
+ 400
+ 600
+ 800
+-800
+-600
+-400
+-200
+ 0
+ 200
+ 400
+ 600
+ 800
+xMW [pc]
+yMW [pc]
+-4
+-3
+-2
+-1
+ 0
+ 1
+ 2
+log10(ρ [pc-2] )
+Fig. 5. Praesepe: Similar to Fig. 3.
+-600
+-400
+-200
+ 0
+ 200
+ 400
+ 600
+-600
+-400
+-200
+ 0
+ 200
+ 400
+ 600
+xMW [pc]
+yMW [pc]
+ 0
+ 50
+ 100
+ 150
+ 0
+ 100
+ 200
+ 300
+ 400
+ 500
+ 600
+leading
+trailing
+ϕ [°]
+r [pc]
+ 0
+ 50
+ 100
+ 150
+ 200
+ 250
+ 300
+ 350
+ 400
+ 0
+ 100
+ 200
+ 300
+ 400
+ 500
+ 600
+leading
+trailing
+cum. N 
+r [pc]
+-0.4
+-0.2
+ 0
+ 0.2
+ 0.4
+ 0
+ 100
+ 200
+ 300
+ 400
+ 500
+ 600
+ǫ(≤r)
+r [pc]
+Fig. 6. Coma Berenices from Jerabkova et al. (in prep.): Similar to
+Fig. 2.
+-600
+-400
+-200
+ 0
+ 200
+ 400
+ 600
+-600
+-400
+-200
+ 0
+ 200
+ 400
+ 600
+xMW [pc]
+yMW [pc]
+-4
+-3
+-2
+-1
+ 0
+ 1
+ 2
+log10(ρ [pc-2] )
+Fig. 7. Coma Berenices: Similar to Fig. 3.
+in a static potential without gravitational interaction between the
+particles. As energy redistribution does not occur in this model
+the evaporation process is simulated as follows: If the Plummer
+sphere were set up in isolation all particles had negative energy
+and were bound to the cluster. As the model cluster is positioned
+Article number, page 4 of 8
+
+J. Pflamm-Altenburg et al.: On the degree of stochastic asymmetry in the tidal tails of star clusters
+ 10
+ 15
+ 20
+ 25
+ 30
+ 35
+ 40
+ 45
+ 50
+ 55
+ 0
+ 10
+ 20
+ 30
+ 40
+ 50
+ 60
+DE [°]
+RA [°]
+ 0
+ 50
+ 100
+ 150
+ 0
+ 5
+ 10
+ 15
+ 20
+leading
+trailing
+ϕ [°]
+r [°]
+ 0
+ 50
+ 100
+ 150
+ 200
+ 0
+ 5
+ 10
+ 15
+ 20
+leading
+trailing
+cum. N
+r [°]
+-0.4
+-0.2
+ 0
+ 0.2
+ 0.4
+ 0
+ 5
+ 10
+ 15
+ 20
+ǫ(≤r)
+r [°]
+Fig. 8. NGC 752 from Beccari et al. (2022): Similar to Fig. 2.
+in the Galactic external potential the combined potential is low-
+ered in two opposite points on the intersecting line defined by the
+Galactic origin and the cluster centre, being the Lagrange points.
+As a consequence, a fraction of the initial set of stars have pos-
+itive energy with respect to the Lagrange points and lead to a
+continuous stream of escaping stars across the clusters’s tidal
+threshold (or práh according to Kroupa et al. 2022), with indi-
+vidual escape time scales up to a Hubble time. This method has
+been successfully tested in Fukushige & Heggie (2000).
+The aim in this work is to quantify the expected distribu-
+tion of the asymmetry between both tidal tails for the observed
+total number of tidal tail members due to stochastic evaporation
+through the Lagrange points. As not all test particles escape from
+the star cluster into the tidal arms a few test runs are required to
+calibrate the total number of test particles, such that the number
+of escaped stars is nearly equal to the observed number of tidal
+tail members.
+As the stellar test particles are performing many revolutions
+around the star cluster centre before evaporating into the Galac-
+tic tidal field, the equations of motion are integrated with a time-
+symmetric Hermite method (Kokubo et al. 1998). The expres-
+sions of the accelerations and the corresponding time derivatives
+are listed in App. A for completeness.
+As the total mass of the Plummer model is constant during
+the simulation, models with minimum and maximum star cluster
+mass are calculated in order to enclose the mass loss of real star
+clusters. Röser et al. (2011) estimated an initial stellar mass of
+the Hyades of 1100 M⊙ and determined a current stellar mass
+gravitationally bound within the tidal radius of 275 M⊙. There-
+fore, in the minimum model the mass of the Plummer sphere is
+given by the current stellar mass of the star cluster (Table 1), in
+the maximum model the mass of the Plummer sphere is set to
+four times the current stellar mass. The Plummer parameter, bPl,
+is identical in both models.
+Furthermore, these two models also take into account the
+possible range of tidal radii: a minimum model (current mass)
+with the smallest tidal radius and a maximum model (here 4x
+the current mass) with a plausible maximum tidal radius.
+The orbits of the four star clusters are integrated backwards
+in time over their assumed life time from their current position
+(Table 1) to their location of formation. At this position 1000
+randomly created Plummer models are set up for each maximum
+and minimum cluster. Each configuration is integrated forward
+in time over the assumed age of the respective star cluster. Fi-
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+ 12
+ 14
+ 16
+-0.4
+-0.2
+ 0
+ 0.2
+ 0.4
+Hyades
+normalised distribution
+(nl-nt)/(nl+nt)
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+ 12
+ 14
+ 16
+-0.4
+-0.2
+ 0
+ 0.2
+ 0.4
+Praesepe
+normalised distribution
+(nl-nt)/(nl+nt)
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+ 12
+ 14
+ 16
+-0.4
+-0.2
+ 0
+ 0.2
+ 0.4
+Coma Berenices
+normalised distribution
+(nl-nt)/(nl+nt)
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+ 12
+ 14
+ 16
+-0.4
+-0.2
+ 0
+ 0.2
+ 0.4
+NGC 752
+normalised distribution
+(nl-nt)/(nl+nt)
+Fig. 9. Monte Carlo results for the minimum model: Shown is the nor-
+malised distribution of the asymmetry in the Monte Carlo simulations
+for each star cluster model. The solid curve is the best-fitting Gaussian
+distribution function. The vertical solid line marks the observed value.
+nally all particles outside the tidal radius are assigned to their
+corresponding tidal arm using the method described in Sect. 2
+and the normalised asymmetry, ϵ, is calculated.
+4.2. Statistical asymmetry
+The numerically obtained distribution of the normalised asym-
+metry of all four star clusters is shown by the histograms in Fig. 9
+for the minimum models and in Fig. 10 for the maximum mod-
+els. The solid curves show the best-fitting Gaussian function.
+The vertical line marks the observed asymmetry (Table 1). The
+resulting mean asymmetry, µ, and the dispersion, σ, of the Gaus-
+sian fits are listed in Table 2. In the last column in Table 2 the
+observed asymmetry is tabulated in units of the respective model
+dispersion. For example, in the case of the maximum model the
+observed asymmetry of the Praesepe cluster is a 1.7 σ event.
+Dispersion and mean value of the minimum and maximum
+are in good agreement. It can be concluded that the mass of the
+Plummer model has only a small influence. In the case of the
+Praesepe, the Coma Berenices and the NGC 752 star cluster the
+observed asymmetries have a probability less than 3 σ and can be
+interpreted as pure statistical events. But, if the observed asym-
+metric tidal tails of the Hyades were solely the result of stochas-
+tic evaporation, then the asymmetry would be at least a 6.7 σ
+event.
+5. Theoretical considerations
+In this section a theoretical stochastic evaporation model is de-
+veloped and compared with the results of the numerical models
+of Sect. 4.
+5.1. Theoretical distribution function, f(ϵ), of the asymmetry
+The evaporation of a star into one of the tidal tails can be treated
+as a Bernoulli-experiment. Let pl be the probability for a star to
+end up in the leading tail, (1 − pl) the probability to end up in
+the trailing tail. For a total number n of tidal tail members the
+probability that nl stars are located in the leading arm is given by
+Article number, page 5 of 8
+
+A&A proofs: manuscript no. paper3
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+ 12
+ 14
+ 16
+-0.4
+-0.2
+ 0
+ 0.2
+ 0.4
+Hyades
+normalised distribution
+(nl-nt)/(nl+nt)
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+ 12
+ 14
+ 16
+-0.4
+-0.2
+ 0
+ 0.2
+ 0.4
+Praesepe
+normalised distribution
+(nl-nt)/(nl+nt)
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+ 12
+ 14
+ 16
+-0.4
+-0.2
+ 0
+ 0.2
+ 0.4
+Coma Berenices
+normalised distribution
+(nl-nt)/(nl+nt)
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+ 12
+ 14
+ 16
+-0.4
+-0.2
+ 0
+ 0.2
+ 0.4
+NGC 752
+normalised distribution
+(nl-nt)/(nl+nt)
+Fig. 10. Similar to analysis as in Fig. 9 but for the maximum model.
+Cluster
+µ
+σ
+|xobs − µ|
+Hyades (min)
+-0.0088
+0.042
+7.3 σ
+Hyades (max)
+-0.0086
+0.046
+6.7 σ
+Hyades (theo.)
+0
+0.043
+6.9 σ
+Praesepe (min)
+-0.0163
+0.036
+-1.7 σ
+Praesepe (max)
+-0.0142
+0.037
+-1.7 σ
+Praesepe (theo.)
+0
+0.035
+-2.2 σ
+Coma Berenices (min)
+-0.0091
+0.041
+2.4 σ
+Coma Berenices (max)
+-0.0092
+0.040
+2.4 σ
+Coma Berenices (theo.)
+0
+0.040
+2.2 σ
+NGC 752 (min)
+-0.0052
+0.055
+1.6 σ
+NGC 752 (max)
+-0.0046
+0.056
+1.6 σ
+NGC 752 (theo.)
+0
+0.058
+1.6 σ
+Table 2. Parameter of Gaussian fits of the asymmetry distribution of the
+different Monte Carlo models.
+a binomial distribution
+bn(nl) =
+�n
+nl
+�
+pnl
+l (1 − pl)n−nl ,
+(4)
+with an expectation value of E(nl) = npl and a variance Var(nl) =
+npl(1 − pl).
+According to the theorem by de Moivre-Laplace, the bino-
+mial distribution converges against the Gaussian distribution,
+g(nl) =
+1
+�
+2πσ2
+l
+e
+−
+(nl−µl)2
+2σ2
+l
+,
+(5)
+for increasing nl with an expectation value E(nl) = µl = npl and
+a variance Var(nl) = σ2
+l = npl(1 − pl).
+The normalised asymmetry, ϵ, is related to the number of
+members of the leading tail, nl, by
+ϵ = nl − nt
+n
+= 2nl
+n − 1 .
+(6)
+The relation between the distribution function of the normalised
+asymmetry, f(ϵ), and the distribution of the member number of
+stars in the leading tail, g(nl), is given by
+f(ϵ) dϵ = g(nl) dnl .
+(7)
+The distribution function of the asymmetry can then be calcu-
+lated by
+f(ϵ) = g(nl(ϵ))
+�����
+dnl
+dϵ (ϵ)
+����� =
+1
+�
+2πσ2ϵ
+e
+− (ϵ−µϵ )2
+2σ2ϵ
+,
+(8)
+with dnl/dϵ following from Eq. (6) and
+σ2
+ϵ = 4σ2
+l
+n2 = 4pl(1 − pl)
+n
+,
+(9)
+and
+µϵ = 2µl
+n − 1 = 2pl − 1 .
+(10)
+5.2. Symmetric evaporation
+In the case of a symmetric population of the tidal tails, the evap-
+oration probabilities into both arms are identical, pl = pt = 1
+2.
+The expectation value and the variance are
+µϵ = 0
+,
+σϵ =
+1√n .
+(11)
+For all four clusters the theoretically expected asymmetry due to
+stochastic evaporation, if the tidal tails are symmetrically popu-
+lated, is listed in Table 2. It can be seen that in all four cases the
+theoretical values are close to the numerically obtained ones.
+5.3. Asymmetric evaporation
+Now, consider the case that the evaporation and the distribution
+processes within the vicinity of the star cluster into the tidal arms
+were asymmetric. According to Eq. 10 the expectation value of
+the asymmetry, µϵ, increases, that is, more stars evaporate into
+the leading arm than into the trailing arm, if the population prob-
+ability, pl, of the leading arm increases.
+For a given observed asymmetry, ϵobs, the probability of this
+event can be calculated as multiples, k, of the dispersion for the
+assumed evaporation probability pl,
+k = ϵobs − µϵ(pl)
+σϵ(pl)
+=
+√n(ϵobs − 2pl + 1)
+2
+�
+pl(1 − pl)
+.
+(12)
+Fig. 11 shows this function for all four star clusters. The vertical
+dashed line marks the case of symmetrically populated tidal tails.
+For example, the Hyades have an observed asymmetry of ϵobs =
+0.298 (Table 1). In the case of a symmetric evaporation, pl = 1
+2,
+the theoretical dispersion is σ = 0.043. Thus, the probabillity of
+this event is 0.298/0.043 = 6.9 σ. This point is marked in Fig. 11
+by the intersection of the solid line labeled with Hyades and the
+vertical dashed line.
+On the other hand, in order to increase the event probability
+(decreasing k) the probability of evaporation into the leading arm
+needs to be increased. For given k, Eq. (12) can be solved for the
+required evaporation probability, pl, into the leading arm. The
+emerging quadratic equation leads to two solutions of pl,
+p1,2 = −a
+2 ±
+��a
+2
+�2
+− b ,
+(13)
+where
+a = −k2 + n(ϵobs + 1)
+k2 + n
+and
+b = n(ϵobs + 1)2
+4k2 + 4n
+.
+(14)
+If the observed asymmetry of the Hyades should be a 3 σ event
+then an evaporation probability into the leading arm of approxi-
+mately pl = 0.585 is required.
+Article number, page 6 of 8
+
+J. Pflamm-Altenburg et al.: On the degree of stochastic asymmetry in the tidal tails of star clusters
+-10
+-5
+ 0
+ 5
+ 10
+ 0.3
+ 0.4
+ 0.5
+ 0.6
+ 0.7
+ 0.8
+Hyades
+Coma Berenices
+Praesepe
+NGC 752
+k
+pl
+Fig. 11. Shown is the probability of the observed asymmetry of all four
+star clusters as a multiple of the dispersion in dependence of the as-
+sumed evaporation probability into the leading arm, pl. See Sect. 5.3
+for details.
+6. Discussion and Conclusions
+The common assumption that both tidal tails of star clusters,
+moving on nearly circular obits around the Galactic centre,
+evolve equally has recently faced a challenge as the analysis of
+Gaia data reveal asymmetries in the tidal tails of four nearby
+open star clusters.
+Because the evaporation of stars can be treated as a stochastic
+process the normalised difference of the number of member stars
+of both tails should follow a distribution function. This distribu-
+tion has been quantified here by use of Monte Carlo simulations
+of test particle configurations integrated in the full Galactic po-
+tential and compared with a theoretical approach. It emerges that
+the theoretical and numerical results agree with each other.
+Comparing the individual distribution functions of the asym-
+metry with the observed ones, it can be concluded that the ob-
+served asymmetry of Praesepe, Coma Berenices and NGC 752
+might be the result of the stochastic nature of the evaporation
+of stars through both Lagrange points. On the other hand, the
+asymmetry of the Hyades is a 6.7 σ event. In order to interpret
+the asymmetry as a 3 σ event, asymmetric evaporation probabil-
+ities into the leading arm of 58.5% and into the trailing arm of
+41.5% are required.
+It might be speculated that external effects might lead to an
+additional broadening of the distribution function of the asym-
+metry. Assuming a different value for the Galactic rotational ve-
+locity or a different position of the Sun than used in this work
+might not lead to a larger scatter as the 50/50% evaporation prob-
+abilities at both Lagrange points are not expected to vary in an
+almost flat rotation curve.
+A stronger effect on the asymmetry of the tidal tails might be
+due to local deviations from a logarithmic potential (as required
+in the case of a flat rotation curve). Such local variations can be
+a result of an interaction with a Galactic bar or spiral arms (cf.
+Bonaca et al. 2020; Pearson et al. 2017). How strong this effect
+will be can be hardly estimated and will be explored in further
+numerical studies. However, the main result here, that the pure
+evaporation of stars through the Lagrange points is basically a
+simple Bernoulli process, remains solid.
+If the asymmetric evaporation probabilities have an internal
+origin, then larger asymmetries in the star cluster potential and
+the kinematics of the evaporation process is required than New-
+tonian dynamics can provide (Kroupa et al. 2022). On the other
+hand the asymmetry might be due to an external perturbation,
+for example through the encounter with a molecular cloud (Jer-
+abkova et al. 2021). However, the detailed analysis of the asym-
+metry in the tidal tails of the Hyades reveals that the process
+leading to the asymmetry must affect both arms equally in terms
+of the qualitative structure. The bending of the radial number dis-
+tributions occur at the same distance to the star cluster but with
+different strength (Fig. 2, lower left panel). If an encounter with
+an external object had occurred on one side of the star cluster
+then it is expected that more than only the total number of tidal
+tail members is reduced. Instead, the functional form of the cu-
+mulative number distribution in both arms would be completely
+different and rapid changes of the slopes of the cumulative num-
+ber distribution would not be expected to occur at the same dis-
+tance from the star cluster centre in both tidal tails on opposite
+sites of the star cluster as is observed in the tidal tails of the
+Hyades (Fig. 2, lower left panel).
+References
+Aarseth, S. J. 1999, PASP, 111, 1333
+Aarseth, S. J. 2003, Gravitational N-Body Simulations (Gravitational N-Body
+Simulations, by Sverre J. Aarseth, pp. 430. ISBN 0521432723. Cambridge,
+UK: Cambridge University Press, November 2003.)
+Aarseth, S. J., Henon, M., & Wielen, R. 1974, A&A, 37, 183
+Allen, C. & Santillan, A. 1991, Revista Mexicana de Astronomia y Astrofisica,
+22, 255
+Allen, L., Megeath, S. T., Gutermuth, R., et al. 2007, Protostars and Planets V,
+361
+Baumgardt, H. & Kroupa, P. 2007, MNRAS, 380, 1589
+Beccari, G., Jerabkova, T., Boffin, H. M. J., et al. 2022, in preparation
+Boffin, H. M. J., Jerabkova, T., Beccari, G., & Wang, L. 2022, MNRAS, 514,
+3579
+Bonaca, A., Pearson, S., Price-Whelan, A. M., et al. 2020, ApJ, 889, 70
+Casertano, S. & Hut, P. 1985, ApJ, 298, 80
+Fukushige, T. & Heggie, D. C. 2000, MNRAS, 318, 753
+Jerabkova, T., Boffin, H. M. J., Beccari, G., et al. 2021, A&A, 647, A137
+Jerabkova, T., Boffin, H. M. J., Beccari, G., et al. in prep., in preparation
+Kokubo, E., Yoshinaga, K., & Makino, J. 1998, MNRAS, 297, 1067
+Kroupa, P., Jerabkova, T., Thies, I., et al. 2022, MNRAS, 517, 3613
+Küpper, A. H. W., Kroupa, P., Baumgardt, H., & Heggie, D. C. 2010, MNRAS,
+401, 105
+Lada, C. J. & Lada, E. A. 2003, ARA&A, 41, 57
+Oh, S. & Kroupa, P. 2016, A&A, 590, A107
+Pearson, S., Price-Whelan, A. M., & Johnston, K. V. 2017, Nature Astronomy,
+1, 633
+Pflamm-Altenburg, J. & Kroupa, P. 2006, MNRAS, 373, 295
+Plummer, H. C. 1911, MNRAS, 71, 460
+Poveda, A., Ruiz, J., & Allen, C. 1967, Boletin de los Observatorios Tonantzintla
+y Tacubaya, 4, 86
+Röser, S., Schilbach, E., Piskunov, A. E., Kharchenko, N. V., & Scholz, R. D.
+2011, A&A, 531, A92
+Article number, page 7 of 8
+
+A&A proofs: manuscript no. paper3
+Appendix A: Hermite formulae for the
+Allen-Santillan potential
+The total Galactic gravitational field has contributions from
+three components: Galactic bulge, disk and halo. The rotation-
+ally symmetric gravitational potentials are taken from Allen &
+Santillan (1991). The time-symmetric Hermite integrator from
+Kokubo et al. (1998) requires the accelerations a and their time
+derivatives j = ˙a and are listed below. The following notations
+are used: r = (x, y, z), ρ = (x, y, 0), z = (0, 0, z), ˙r = v = (˙x, ˙y, ˙z),
+˙ρ = (˙x, ˙y, 0), ˙z = (0, 0, ˙z).
+Φ(ρ, z) = Φbulge(ρ, z) + Φdisk(ρ, z) + Φhalo(ρ, z)
+(A.1)
+a = abulge + adisk + ahalo = −∇Φbulge − ∇Φdisk − ∇Φhalo
+(A.2)
+j = jbulge + jdisk + jhalo
+(A.3)
+Appendix A.1: bulge
+Φbulge(ρ, z) = −GM1
+1
+�
+ρ2 + z2 + b2
+1
+(A.4)
+abulge = −GM1
+�
+ρ2 + z2 + b2
+1
+�− 3
+2 r
+(A.5)
+jbulge = −GM1
+�
+ρ2 + z2 + b2
+1
+�− 3
+2
+������v − 3
+˙r • r
+ρ2 + z2 + b2
+1
+r
+������
+(A.6)
+Appendix A.2: disk
+Φdisk(ρ, z) = −GM2
+1
+�
+ρ2 +
+�
+a2 +
+�
+z2 + b2
+2
+�2
+(A.7)
+a = −GM2
+�
+ρ2 +
+�
+a2 +
+�
+z2 + b2
+2
+� 1
+2 �2�− 3
+2
+(A.8)
+������������
+ρ +
+������������
+1 +
+a2
+�
+z2 + b2
+2
+������������
+z
+������������
+(A.9)
+j = 3M2G
+�
+ρ2 +
+�
+a2 +
+�
+z2 + b2
+2
+� 1
+2 �2�− 5
+2
+·
+(A.10)
+������������
+ρ • ˙ρ +
+������������
+1 +
+a2
+�
+z2 + b2
+2
+������������
+z • ˙z
+������������
+������������
+ρ +
+������������
+1 +
+a2
+�
+z2 + b2
+2
+������������
+z
+������������
+(A.11)
+−M2G
+�
+ρ2 +
+�
+a2 +
+�
+z2 + b2
+2
+� 1
+2 �2�− 3
+2
+·
+(A.12)
+������������
+˙ρ −
+a2z • ˙z
+�
+z2 + b2
+2
+� 3
+2 z +
+������������
+1 +
+a2
+�
+z2 + b2
+2
+������������
+˙z
+��������������
+(A.13)
+component
+parameter
+bulge
+M1 = 606.0
+bulge
+b1 = 0.3873
+disk
+M2 = 3690.0
+disk
+a2 = 5.3178
+disk
+b2 = 0.2500
+halo
+M3 = 4615.0
+halo
+a3 = 12.0
+halo
+γ = 2.02
+Table A.1. Parameters of the potential used in this work.
+Appendix A.3: halo
+Φhalo = −GMhalo(≤ r)
+r
+−
+GM3
+(γ − 1)a3
+����������−
+γ − 1
+1 +
+� r
+a3
+�γ−1 + ln
+�������1 +
+� r
+a3
+�γ−1�������
+����������
+100
+r
+,
+(A.14)
+where
+M(≤ r) =
+M3
+� r
+a3
+�γ
+1 +
+� r
+a3
+�γ−1
+(A.15)
+is the total halo mass within the Galactocentric radius r.
+ahalo = −GM(≤ r)
+r3
+r
+(A.16)
+jhalo = −GM(r)
+r3
+����������v − r • v
+r2
+����������γ − 3 −
+(γ − 1)
+� r
+a3
+�γ−1
+1 +
+� r
+a3
+�γ−1
+���������� r
+����������
+(A.17)
+Appendix A.4: Units and parameters
+In Allen & Santillan (1991) the spatial variables of the poten-
+tial have the unit kpc, the velocity the unit 10 km s−1 and the
+potential the unit 100 km2 s−2. Thus, the accelerations needs to
+be multiplied by 0.104572 in order to have the unit pc Myr−2
+and their time derivatives, j, by 1.06936 × 10−3 to have the unit
+pc Myr−3. The mass has the unit 2.3262 × 107 M⊙ and is scaled
+such that the gravitational constant is G = 1. The parameters of
+the potentials are summarised in Table A.1.
+Appendix A.5: Hermite formulae for the Plummer sphere
+ΦPl(r) = −GMPl
+1
+�
+r2 + b2
+Pl
+(A.18)
+aPl = −GMPl
+�
+r2 + b2
+Pl
+�− 3
+2 r
+(A.19)
+jPl = −GMPl
+�
+r2 + b2
+Pl
+�− 3
+2
+������v − 3 ˙r • r
+r2 + b2
+Pl
+r
+������
+(A.20)
+Article number, page 8 of 8
+

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+arXiv:2301.12955v1  [math.AC]  30 Jan 2023
+Invertible bases and root vectors for analytic
+matrix-valued functions
+Vanni Noferini∗
+Abstract
+We revisit the concept of a minimal basis through the lens of the theory of modules
+over a commutative ring R. We first review the conditions for the existence of a basis
+for submodules of Rn where R is a B´ezout domain. Then, we define the concept of
+invertible basis of a submodule of Rn and, when R is an elementary divisor domain,
+link it to the Main Theorem of [G. D. Forney Jr., SIAM J. Control 13, 493–520, 1975].
+As an application, we let Ω ⊆ C be either a connected compact set or a connected open
+set, and we specialize to R = A(Ω), the ring of functions that are analytic on Ω. We
+show that, for any matrix A(z) ∈ A(Ω)m×n, ker A(z) ∩ A(Ω)n is a free A(Ω)-module
+and admits an invertible basis, or equivalently a basis that is full rank upon evaluation
+at any λ ∈ Ω. Finally, given λ ∈ Ω, we use invertible bases to define and study maximal
+sets of root vectors at λ for A(z). This in particular allows us to define eigenvectors
+also for analytic matrices that do not have full column rank.
+Keywords: Analytic function, eigenvector, free module, minimal basis, invertible basis,
+root vector, maximal set
+2020 MSC: 15B33, 15A18, 65H17, 15B99
+1
+Introduction
+In the context of the theory of polynomial matrices [9], the concept of a minimal basis was
+introduced by D. Forney Jr. in [6]. Let F be a field, and denote by F[x] (resp. F(x)) the ring
+of univariate polynomials over F (resp. the field of fractions of F[x]). If V is any subspace of
+F(x)n, it is clear that a polynomial basis for V exists. Forney gave the following definition.
+Definition 1.1 Let B = {b1(x), . . . , bp(x)} be a polynomial basis for V ⊆ F(x)n. Then, the
+quantity Ω = �p
+i=1 deg bi(x) is called the order of B. If B has minimal order among all
+possible polynomial bases of V , it is called a minimal basis of V .
+∗Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100, FI-00076, Aalto,
+Finland (vanni.noferini@aalto.fi.
+Supported by an Academy of Finland grant (Suomen Akatemian
+p¨a¨at¨os 331240).
+1
+
+Then, Forney showed that the degrees of a minimal basis, called the minimal indices of
+V , are uniquely determined by V . Moreover, he gave some equivalent characterization of a
+minimal basis in [6, Main Theorem]. To state it, we must define the high coefficient matrix
+{P} of a polynomial matrix P(x) as the matrix whose columns retain the coefficients of the
+highest order monomials in each column of P(x). For example, over F = Q,
+
+
+
+
+
+1
+2x2
+0
+x + 1
+−1
+x2 + x + 1
+
+
+
+
+ =
+
+
+1
+2
+0
+0
+−1
+1
+
+ .
+Theorem 1.2 (Forney) Let M(x) ∈ F[x]n×p. The following are equivalent:
+1. The set of the columns of M(x) is a minimal basis for the subspace they span;
+2. (a) For every prime polynomial p(x) ∈ F[x], M(x) is nonsingular modulo p(x) (b) {M}
+is nonsingular;
+3. (a) The GCD of all p × p minors of M(x) is 1 (b) The largest degree of all the p × p
+minors of M(x) is the sum of the degrees of the columns of G(x);
+4. If b(x) = M(x)a(x) is a polynomial vector then (a) a(x) must be polynomial (b) The
+degree of b(x) is the maximum, for all 1 ≤ i ≤ p, of the sum of the degree of the i-th
+column of M(x) and the degree of the i-th coefficient of a(x).
+(For conciseness, we have omitted a fifth equivalent condition from Forney’s original theorem,
+that relates (for all d) the minimal indices of V with the dimension over F of Vd, the vector
+space of all polynomial n-uples in V of degree ≤ d.)
+Two more remarks on minimal bases are in order. First, Forney observed already in
+[6] that property 4(a) in his Main Theorem is equivalent to the fact that the F[x]-module
+M = V ∩F[x]n is free and that a minimal basis is a basis for M. However, Forney mentioned
+this (correct) fact in passing, without explaining it in detail; a more thorough treatement
+can be found in [8, Section 2.4]. Moreover, Forney showed that properties (a) are equivalent
+to each other (and to the fact that a minimal basis of V is a basis of M) and properties (b)
+are equivalent to each other.
+Second, root polynomials and root vectors at λ ∈ C for a (possibly not full column rank)
+polynomial matrix P(x) were defined in [4, 20], extending the original notion for regular
+polynomial matrices [9] by building on the concept of the vector space kerλ P(x); the latter
+was defined as the span of a minimal basis of ker P(x), evaluated at x = λ. A so-called
+maximal set of root polynomials is a useful tool as it provides all the local positive partial
+multiplicities at λ: see [4, 20] as well as [22] for an extension to rational matrices. The idea
+of root vectors has proved useful for theoretical analysis [4, 5, 20, 22] as well as to analyze
+conditioning [17, 21] and to design practical algorithms for singualr eigenvalue problems
+[12, 14]. However, a careful inspections of the arguments in [4, 20, 22] shows that to define
+kerλ P(x) it suffices in fact to start from a, not necessarily minimal, basis satisfying properties
+(a) in Forney’s Main Theorem 1.2. In other words, properties (b) in Theorem 1.2 are not
+2
+
+necessary to correctly define root polynomials at a finite point. In this paper, we propose to
+call a polynomial basis that satifies properties (a) in Forney’s Main Theorem 1.2 invertible,
+since a matrix whose columns are the vectors of such basis is left invertible (over F[x]).
+Equivalent characterizations of left invertibility are that the matrix has trivial Smith form,
+i.e., all the invariant factors are units, or that it is a submatrix of a unimodular matrix; see
+also [1, Theorem 3.3].
+The main goals of this paper are (1) To extend the notion of an invertible basis to certain
+submodules of Rn when R is a B´ezout domain (but not necessarily a principal ideal domain)
+and (2) To use invertible bases to define a maximal set of root vectors for matrices over the
+ring of analytic functions on a connected (open or compact) set. We expect that the resulting
+theory may be useful, for example, for applications to the nonlinear eigenvalue problem [10].
+We will take a module-theoretical approach, but keeping the exposition accessible for an
+audience with expertise in numerical linear algebra. In particular, in Section 2 we explicitly
+recall the algebraic concepts about modules that we will need. In Section 3, we show that
+when R is a B´ezout domain with field of fractions G, then a submodule of Rn is free if
+and only if it is finitely generated (Theorem 3.6), and in particular that for any suspace
+V ⊆ Gn then V ∩ Rn is free and has an invertible basis. This result is certainly not new:
+see for example [7, Theorem 1.13.3]; however, our presentation and in particular the link to
+properties (a) in Theorem 1.2 (see Theorem 3.9) seem to not be well known in the linear
+algebra community. Finally, in Section 4 we specialize to the ring A(Ω) of analytic functions
+over a connected, and either open or compact, set Ω, and we use the existence of invertible
+basis to define and characterize maximal sets of root vectors for matrices over A(Ω). This
+allows us in particular to define eigenvectors even for analytic matrices that do not have full
+column rank: in that case, an eigenvector can be seen as a certain equivalence class.
+2
+Commutative rings, modules, and free modules
+In this section we refer to some basic algebra material: the notions that we recall, and
+more, can be found for example in [3, 7, 8, 11, 13]. Throughout, we consider a commutative
+ring R with unity, and we assume 1 ̸= 0 so that R is not the trivial ring {0}, because
+it contains at least two distinct elements. A module over the commutative ring R, or R-
+module, is a non-empty set M which is a commutative group with respect to the addition
++ (in particular, there is a zero element 0 ∈ M and every x ∈ M has an inverse −x ∈ M
+such that x + (−x) = 0) and is endowed with a scalar multiplication · : R × M → M that
+satisfies the following axioms: (A1) a · (x + y) = a · x + a · y for all x, y ∈ M, a ∈ R; (A2)
+(a+b)·x = a·x+b·x for all x ∈ M, a, b ∈ R; (A3) (ab)·x = a·(b·x) for all x ∈ M, a, b ∈ R;
+(A4) 1 · x = x for all x ∈ M.
+More informally, axioms (A1)–(A4) say that a module is to a commutative ring what
+a vector space is to a field.
+(In fact, a vector space is an R-module in the case of R
+being a field.) In particular, the concepts of linear independence and of linear span both
+still make sense. Namely, we say that x1, . . . , xn ∈ M are linearly independent if, given
+a1, . . . , an ∈ R, �n
+i=1 ai · xi = 0 ⇒ ai = 0 ∀ i. Moreover, we say that the module M is
+3
+
+(finitely) generated by the elements y1, . . . , ym ∈ M (called generators) if, for every x ∈ M,
+there exist b1, . . . , bm ∈ R such that x = �m
+i=1 bi · yi. Linear independence can be applied
+also to infinite sets of elements of M: in this case, one requires that every possible finite
+subset is linearly independent. Similarly, a module M may also be infinitely generated: in
+this case, it is necessary to have an infinite number of generators but every element of M is
+still required to be an R-linear combination of finitely many generators. If S ⊂ M is a set
+of generators of the R-module M, we write M = spanRS.
+However, when combining these two concepts, a striking difference between modules and
+vectors spaces emerges. Indeed, define a basis of a module as a linearly independent set of
+generators. Every vector space has a basis1 but the same is not true for modules.
+Example 2.1 Take R = Z. Then, it is easy to verify that M = R is a Z-module. Assume
+for a contradiction that it has a basis B and let b0 ∈ B ⊆ R. Suppose that 2 ≤ k ∈ Z, then
+by assumption there exist b1, . . . , bn ∈ B such that, for some z0, z1, . . . , zn ∈ Z,
+b
+k =
+n
+�
+i=0
+zibi ⇒ b −
+n
+�
+i=0
+(kzi)bi = 0,
+contradicting the linear independence of the basis. (To deal with the case where z0 ̸= 0, note
+that 1 − kzi ̸= 0 as otherwise |k| = 1, contradicting k ≥ 2.)
+Nevertheless, some modules do have a basis.
+A module that has a basis is called free.
+Moreover, for convenience of exposition, we will conventionally agree that the trivial module
+M = {0} is also free, with its basis being the empty set; this excentricity actually makes
+sense if we formally agree that the sum of an empty set of elements of M yields 0, and in the
+following it will allow us to avoid the clumsy need to repetitively exclude the trivial module
+when making formal statements about free modules.
+Proposition 2.2 Let R be a commutative ring. Rn (the set of all possible n-uples of ele-
+ments of R) is a free R-module.
+Proof. It is clear that the canonical basis {ei}n
+i=1, where (ei)j = 1 if i = j and (ei)j = 0
+if i ̸= j, is a basis of Rn.
+The example of Proposition 2.2 is in fact unique up to isomorphism, in the sense that
+every finitely generated free R-module with a basis of cardinality n is isomorphic to Rn, by
+the first isomorphism theorem.
+We conclude this section by recalling some further nomenclature and basic properties
+[3, 7, 8, 11, 13]. In the special case M ⊆ R, the R-module M is called an ideal. Recall also
+that a commutative ring R is called an integral domain if, for all a, b ∈ R, ab = 0 and b ̸= 0
+imply a = 0. An integral domain is called a principal ideal domain (PID) if every ideal of R
+is principal, i.e., generated by a single element of R; and it is called a B´ezout domain (BD) if
+every finitely generated ideal of R is principal. Obviously, every PID is a BD by definition,
+but the converse is not true. For example, the ring A(C) of entire functions is a BD [7] but
+1Of course, in the infinite dimensional case, this result is equivalent to the axiom of choice.
+4
+
+not a PID. To see that A(C) is not a PID, consider define J ⊂ A(C) as the set of entire
+functions f(z) that are zero at all but finitely many Gaussian integers: it is easy to verify
+that J is an ideal which is not finitely generated (and in particular not principal). Finally,
+an integral domain R is called an elementary divisor domain (EDD) if the following theorem
+holds.
+Theorem 2.3 (Smith) Let R be an elementary divisor domain and A ∈ Rm×n. Then there
+exist two unimodular (that is, invertible over R) matrices U ∈ Rm×m and V ∈ Rn×n such
+that A = USV where S ∈ Rm×n is diagonal and such that the (i, i) element of S divides the
+(i + 1, i + 1) element of S, for all i < min{m, n}. Such a matrix S is called a Smith form
+of A, and it is uniquely determined by A up to multiplication of each diagonal element by a
+unit of R. The diagonal elements of A are called invariant factors of A, and the product of
+the first k invariant factors is called the kth determinantal divisor of A and it is equal to the
+GCD of all the k × k minors of A. Both invariant factors and determinantal divisors are
+uniquely determined by A up to multiplication by a unit of R.
+It can be proved that every PID is an EDD and that every EDD is a BD [7], while to our
+knowledge it is still an open problem to decide whether there is a BD which is not an EDD
+or the definitions of EDD and BD are equivalent [16].
+3
+Submodules of Rn when R is a PID, an elementary
+divisor domain, or a B´ezout domain
+In this section, we study submodules of Rn where R is a commutative ring; we will often
+make further assumptions on R such as being a PID, EDD or BD. Most of the results in
+this section can be traced elsewhere, e.g., in [2, 6, 7, 8, 11], although our derivation does
+not strictly follow those sources, and thus the form in which we present some statements, or
+their proofs, may be original.
+Let R be a commutative ring and A ∈ Rm×n. The range, or column space, of A is defined
+as cs(A) := {x ∈ Rm : ∃ y ∈ Rn s.t. x = Ay}. It is a simple exercise to verify that cs(A) ⊆ Rm
+is an R-module. The null space (over R) of A is denoted by null(A) := {y ∈ Rn : Ay = 0};
+when R is an integral domain we notationally distinguish null(A) from ker A, that in this
+paper denotes the null space, or kernel, over G, the field of fractions of R. (For example,
+if R = F[x] is the ring of polyomials then G = F(x) is the field of rational functions.) It is
+clear that, when R is an integral domain, null(A) = ker A ∩ Rn.
+The following result [2, Theorem 5.10] is very general as no assumption on R, beyond
+being a commutative ring, is needed.
+Theorem 3.1 (Brown) Let R be a commutative ring and let M be a finitely generated R-
+module. Suppose that {m1, . . . , mk} ⊆ M is a linearly independent set and {p1, . . . , pn} ⊆ M
+is a set of generators of M. Then k ≤ n. Moreover, if k = n, then M is free and {p1, . . . , pn}
+is a basis of M.
+5
+
+An immediate corollary of Theorem 3.1 is that, if a matrix A ∈ Rn×p over a commutative
+ring has linearly independent (over R) columns, then cs(A) is a free module and the set
+of the columns of A is a basis. (This also follows from the first isomorphism theorem for
+modules since in this case cs(A) ∼= Rp/null(A) = Rp.) Generally, however, neither the range
+nor the null space of a matrix need to be free modules.
+Example 3.2 Let A = 2 ∈ Z4. Clearly, null(A) = {0, 2} is generated by 2 and 2 is the
+unique possible generator.
+However, {2} is not a basis for ker A since it is not a Z4-
+independent set (indeed in Z4 it holds 2 · 2 = 0). Moreover, cs(A) = null(A) is also not
+free.
+Luckily, though, over principal ideal domains the situation is much simpler.
+Theorem 3.3 Let R be a principal ideal domain. Then, every submodule of Rn is free.
+Proof. By Proposition 2.2, Rn is a free R-module. On the other hand, every submodule
+of a free R-module is free if R is a PID [11, Theorem 5.1].
+Corollary 3.4 Let R be a principal ideal domain and G its field of fractions. For every
+subspace V ⊆ Gn, the set M = V ∩ Rn is a free R-module.
+Proof. In view of Theorem 3.3, and since M ⊆ Rn, it is enough to show that M is a
+module. It is clear that M is a group with respect for addition, for x, y ∈ M implies both
+x + y ∈ V and x + y ∈ Rn; and it is also clear that if x ∈ M and p ∈ R then px ∈ Rn and
+px ∈ V . Finally, properties (A1)–(A4) are also immediate because M ⊆ Rn.
+Corollary 3.5 Let R be a principal ideal domain and let S be a subset of Rn. Then, M =
+spanRS is a free R-module.
+Proof. Similarly to Corollary 3.4, it is easy to verify that M is a module. Hence, M is a
+free module by Theorem 3.3.
+Either Corollary 3.4 or Corollary 3.5 (or both) provide, for example, a somewhat more
+natural enviroment for the results related to the filtration approach to minimal bases [18]:
+in that context, one can consider more directly a nested sequence of submodules of F[x]n
+rather than of subspaces of F(x)n.
+We now turn to B´ezout domains, where there are some complications with respect to
+principal ideal domain. Nevertheless, the complications are not too hard to overcome.
+Theorem 3.6 Let R be a B´ezout domain and let M ⊆ Rn be a submodule. Then, M is a
+free R-module if and only if it is finitely generated.
+Proof. We first prove necessity. Suppose that M is free and let B be a basis of M. If
+#B > n, let the columns of A ∈ Rm×n be any finite subset of B of cardinality m > n: then it
+suffices to take any nonzero vector c ∈ null(A) to immediately show that B must be linearly
+dependent. Hence, #B ≤ n, and in particular B is a finite set thus showing that M is finitely
+generated.
+6
+
+For sufficiency, we argue2 by induction on n. For the base case n = 1, note that if M ⊆ R
+is an R-module then M is in particular an ideal. But since M is finitely generated, then
+it is principal: let g be a generator of M = ⟨g⟩, then either {g} is a basis of M (if g ̸= 0)
+or M = {0} is trivial so ∅ is a basis (if g = 0). If n > 1, let M ⊆ Rn be an R-module
+generated by the finite set S = {si}i∈I ⊆ M, where I is a finite set of indces, and define
+J := {r ∈ R : ∃ b ∈ Rn−1 s.t. b ⊕ r ∈ M} and N := {a ∈ Rn−1 : a ⊕ 0 ∈ M}. By the first
+isomorphism theorem applied to the linear map that sends v ∈ M to its bottom component,
+both J and N are R-modules. (Note that both are obviously non-empty as 0 ∈ M.) First
+fix r ∈ J.
+Then there is b ∈ Rn−1 such that b ⊕ r = �
+i∈I cisi for some ci ∈ R.
+But
+this implies r = �
+i∈I ciσi where, for each i, σi is the bottom component of si. Hence, J
+is (finitely) generated by the set of the distinct bottom components of the elements of S.
+Next, fix a ∈ N. Then a ⊕ 0 = �
+i∈I disi, where again di ∈ R. But in turn this implies that
+N is (finitely) generated by the set of the distinct (n − 1)-uples that contain the top n − 1
+components of the elements in S. Hence, using the inductive assumption, both J ⊆ R and
+N ⊆ Rn−1 are free. Let BJ and BN be, respectively, a basis of J and N; in particular since
+J ⊆ R is an ideal we have #BJ ≤ 1 by the first part of the proof. If J = {0}, M = N ⊕ 0
+and hence BN ⊕ 0 is a basis of M. If J ̸= {0}, let {g} be a basis of J and take b0 to be any
+element of Rn−1 such that b0 ⊕ g ∈ M. Manifestly, given v ∈ M, there are a unique c ∈ R
+and a unique w ∈ N such that v = c(b0 ⊕ g) + (w ⊕ 0). It follows that M ∼= N ⊕ J and in
+particular (BN ⊕ 0) ∪ {b0 ⊕ g} is a basis of M.
+We now formalize the definition of an invertible basis anticipated in the introduction.
+Definition 3.7 Let R be a commutative ring and M ⊆ Rn be a free R-module with basis B.
+We say that B is an invertible basis of M if there is a matrix A ∈ Rn×p such that (1) the
+columns of A are the elements of B (2) A is left invertible over R, i.e., there exists L ∈ Rp×n
+such that LA = Ip.
+Proposition 3.8 Let R be an elementary divisor domain and G its field of fractions. For
+every subspace V ⊆ Gn, the set M = V ∩ Rn is a free R-module. Moreover, it admits an
+invertible basis.
+Proof. That M is a module can be easily proved as in Corollary 3.4. Since every EDD
+is a BD, and in view of Theorem 3.6, we must show that M is finitely generated: we will do
+so constructively, and the procedure will yield an invertible basis as a bonus. Let us start
+with a basis (over G) of V with elements in Rn: this can be easily constructed starting by
+any basis and scaling each element by least common denominators3. Let A ∈ Rn×p be the
+matrix whose columns are the elements of such a basis. Then, A = QSZ where Q ∈ Rn×n
+and Z ∈ Rp×p are unimodular while S ∈ Rn×p is a Smith form (see Theorem 2.3). Letting
+2Our proof for sufficiency follows the lead of [11, Theorem 5.1], which is written for the case of a PID,
+with two modifications: (1) For a BD we must prove “by hand” that the sets N and J, defined in our proof,
+are finitely generated (2) We have rephrased the proof in a language that we hope is more accessible for the
+linear algebra research community.
+3This is possible because every BD is a GCD domain, and it is known [15, Theorem 2] that in a GCD
+domain every pair of elements have both a GCD and a LCM.
+7
+
+Qp ∈ Rn×p be the matrix containing the leftmost p columns of Q and Sp ∈ Rp×p be the
+matrix containing the top p rows of S, we also have A = QpSpZ. We claim that the set
+of the columns of Qp is an invertible basis of M. Linear independence is clear, as Qp is a
+submatrix of a unimodular matrix. It remains to show that M is generated over R by the
+columns of Qp. To this goal, note that the columns of A span V over G, and hence for all
+v ∈ M ⊆ V there is c ∈ Gp such that v = Ac. Thus, v = Qp(SpZc). But Qp is left invertible
+(over R) by construction, so by letting L ∈ Rp×n be a left inverse it holds SpZc = Lv ∈ Rp:
+and since v is a generic element of M, the columns of Qp are a set of generators of M.
+Invertible bases retain (in a generalized sense) properties (a) in Forney’s Theorem 1.2.
+This in particular includes property 4(a) that, in the case R = F[x], was labelled “polynomial
+linear combination property” in [18]. To state Theorem 3.9, recall that in a commutative
+ring R, given a, b, r ∈ R, we write a ≡ b mod r to mean that r divides a − b, i.e., that there
+is d ∈ R such that a − b = rd. This notation extends elementwise to matrices over R: for
+A, B ∈ Rm×n we write A ≡ B mod r if there exists D ∈ Rm×n such that A − B = rD.
+Theorem 3.9 Let R be an elementary divisor domain and G its field of fractions.
+Let
+V ⊆ Gn be a suspace and suppose that set of the columns of the matrix Q ∈ Rn×p is a basis
+for V . Then the following are equivalent:
+1. The set of the columns of Q is an invertible basis for the module V ∩ Rn;
+2. For all non-units r ∈ R and for all Q′ ∈ Rn×p such that Q ≡ Q′ mod r, Q′ has full
+column rank;
+3. The GCD of all p × p minors in Q is 1;
+4. If b = Qa ∈ Rn for some a ∈ Gp, then a ∈ Rp;
+5. Q has trivial Smith form, i.e., a Smith form of Q is S =
+�
+Ip
+0
+�
+.
+Proof.
+1 ⇒ 2 By assumption there is L ∈ Rp×n such that LQ = Ip. Let r ∈ R and let D, Q′ ∈ Rp×n
+satisfy rD = Q′ − Q, then LQ′ = Ip + rLD ⇒ det(LQ′) ≡ 1 mod r. Therefore, if
+det(LQ′) = 0 then 1 = rd for some d ∈ R implying that r is a unit. Hence, if r is not
+a unit then det(LQ′) ̸= 0 and thus Q′ has full rank.
+2 ⇒ 3 Write Q = USV with U, V unimdular and S in Smith form. Let g ∈ R be the GCD
+of all p × p minors in Q, and suppose g is not a unit. Then, by Theorem 2.3, the p-th
+invariant factor of Q, or equivalently the (p, p) element of S, cannot be a unit, say,
+Spp = r for some non-unit r ∈ R. If S′ = S − repeT
+p is constructed from S by replacing
+Spp by 0, define Q′ = US′V . Clearly Q′ is rank deficient and Q − Q′ = rUepeT
+p V ≡ 0
+mod r.
+8
+
+3 ⇒ 5 If the GCD of all p × p minors of Q, that is the p-th determinantal divisor, is 1 then
+by Theorem 2.3 all the invariant factors of Q must be units of R, and hence
+�
+Ip
+0
+�
+is a
+Smith form of Q.
+5 ⇒ 1 Since STS = Ip and Q = USV for some unimodular U, V , a left inverse L can be
+consctructed as L = V −1STU−1.
+1 ⇒ 4 Suppose Q is an invertible basis; then, by definition, there exists L ∈ Rp×n such that
+LQ = Ip. Hence, a = Lb ∈ Rp.
+4 ⇒ 2 Let r ∈ R be a non-unit and suppose that there exists Q′ ∈ Rn×p which is rank
+deficient and satisfies Q′ ≡ Q mod r. Let c ∈ null(Q′) be such that the GCD of the
+elements of c is 1. (It is clear that such c exists, by taking any nonzero element of
+null(Q′) and dividing it by the GCD of its entries.) Let D ∈ Rp×n satisfy rD = Q−Q′,
+then Qc = Q′c + (rD)c = D(rc). Hence, if a = c/r ∈ Gp, we have that a ̸∈ Rp but
+Qa = Dc ∈ Rn.
+Remark 3.10 An important subtlety is that property 2 in Theorem 3.9 is being full rank
+modulo every non-unit, while property 2(a) in Theorem 1.2 is being full rank modulo every
+prime. Since every prime is a non-unit, it is clear that the former implies the latter.
+Over a PID, such as the ring of univariate polynomials over a field, then every element
+of the ring admits a unique (up to permutations and multiplications by units) prime decom-
+position; hence, over a PID the converse implication is also true and the two properties are
+thus equivalent, because every element of a PID is either a unit or divisible by a prime. For
+example, Theorem 3.9 implies that every subspace of Qn has an integer basis that is full rank
+modulo p, for every prime number p.
+In general, however, property 2 in Theorem 3.9 is not equivalent to being full rank modulo
+every prime if R is not a PID. For example let A ⊂ C be the ring of algebraic integers (roots
+of monic polynomials in Z[x]) which is a BD [13, Theorem 102] and an EDD [19, Theorem
+5] so Theorem 3.9 holds. On the other hand, A has no irreducible elements (and hence no
+prime elements), because the square root of an algebraic integer is an algebraic integer. It
+follows that A contains non-units (say, 2 ∈ A but 1
+2 ̸∈ A so 2 is not a unit of A) that are not
+divisible by any prime elements. We conclude that, for an EDD that is not a PID, property
+2 as stated in Theorem 3.9, and hence being an invertible basis, is possibly stronger than (the
+analogue of) property 2(a) as stated in Theorem 1.2; indeed there are EDDs in which being
+full rank modulo every prime is not equivalent to being an invertible basis. For example {2}
+is not an invertible basis of A although 2 is (vacuously) full rank modulo any prime.
+4
+Invertible bases and root vectors over A(Ω)
+Throughout this section, we fix a connected set Ω ⊆ C, assuming that Ω is either compact
+or open, and we denote by A(Ω) the ring of functions that are analytic on Ω. The spectral
+9
+
+theory of analytic matrices is relevant in the applications, and in particular in the context
+of nonlinear eigenvalue problems or in the context of signal processing: see [10] for a recent
+survey on nonlinear eigenvalue problems or [23] for some applications in signal processing.
+We recall that, under the stated assumptions, A(Ω) is an elementary divisor domain (the
+Smith theorem holds) and hence a BD: see for example [7]. Thus, the analysis of Section 3 is
+relevant. In fact, in the case where Ω is a connected compact set, even more strongly A(Ω)
+is actually a PID and a Euclidean domain [7, Lemma 1.3.7]. We consider A(Ω)m×n, the
+set of m × n matrices over A(Ω). By obvious extension of the properties of scalar analytic
+functions, every element of A(Ω)m×n admits a convergent power series at any point λ ∈ Ω,
+say, A(z) = �∞
+j=0 Aj(z − λ)j, with (Aj)j ⊂ Cm×n. A nonzero analytic matrix A(z) ̸= 0 has
+a root of order k at λ if A0 = · · · = Ak−1 = 0 ̸= Ak; or equivalently if A(z) = (z − λ)kB(z)
+for some B(z) ∈ A(Ω)m×n such that B(λ) ̸= 0.
+Let us now fix a matrix A(z) ∈ A(Ω)m×n.
+The most basic definition is that of an
+eigenvalue and of the multiplicities associated with it.
+Definition 4.1 (Eigenvalues and multiplicities) We say that λ ∈ Ω is an eigenvalue of
+the matrix A(z) ∈ A(Ω)m×n if rankA(λ) < rankA(z). Moreover, the partial multiplicities of
+an eigenvalue of A(z) are the orders of λ as a root of the nonzero invariant factors of A(z).
+The number of the nonzero partial multiplicities of λ is called the geometric multiplicity of
+λ, and the sum of the partial multiplicities is called the algebraic multiciplicity of λ.
+The free module M = null(A(z)) admits an invertible basis by Theorem 3.6. Let us take
+one such basis, arrange its elements as the columns of a matrix and denote the latter by
+Q(z). We first show that the set of the columns of Q(z) is an invertible basis if and only if
+no λ ∈ Ω is an eigenvalue of Q(z).
+Corollary 4.2 The set of the columns of the matrix Q(z) ∈ A(Ω)n×p is an invertible basis
+for the module they span if and only if Q(λ) has full column rank for all λ ∈ Ω.
+Proof. The statement follows from Theorem 3.9, but for completeness let us give an
+argument more specific to A(Ω). Suppose that the set of the columns of Q(z) is an invertible
+basis for the module cs(Q(z)), then there exists L(z) ∈ A(Ω)p×n such that L(z)Q(z) = Ip,
+implying L(λ)Q(λ) = Ip and hence Q(λ) has full column rank. Conversely, if Q(λ) has full
+colum rank for all λ ∈ Ω then, noting that the units of A(Ω) are precisely the analytic
+functions with no zeros in Ω, the Smith form of Q(z) is trivial, and hence Q(z) is left
+invertible.
+For brevity, and coherently with common practice in the literature on minimal bases,
+from now on we may write, e.g., “Q(z) is an invertible basis of the module M” as shorthand
+to mean “the set of the columns of Q(z) is an invertible basis for the module M”.
+Definition 4.3 Let A(z) ∈ A(Ω)m×n and suppose that Q(z) ∈ A(Ω)n×p is an invertible
+basis of null(A(z)). Given λ ∈ Ω, we define kerλ A(z) := span Q(λ) ⊆ Cn.
+10
+
+It is clear that kerλ A(z) is well defined, as if Q(z) and T(z) are two invertible bases
+of null(A(z)) then it is easy to prove that Q(z) = T(z)U(z) for some unimodular U(z) ∈
+A(Ω)p×p. It follows that kerλ A(z) is a subspace of Cn. The next result generalizes [4, Lemma
+2.9].
+Lemma 4.4 v ∈ kerλ A(z) ⇔ ∃w(z) ∈ A(Ω)n : A(z)w(z) = 0 and w(λ) = v.
+Proof. Let Q(z) be an invertible basis of null(A(z)). Then from the existence of w(z) we
+deduce that w(z) = Q(z)c(z) for some c(z) ∈ Ap, and hence v = Q(λ)c(λ). Conversely, if
+v = Q(λ)c for some constant c ∈ Cp then set w(z) := Q(z)c.
+We are now in the position to extend some relevant definitions from [4, 20] to the case
+of analytic matrices. (See also [22].)
+Definition 4.5 (Root vectors) Given A(z) ∈ A(Ω)m×n and λ ∈ Ω, we say that r(z) ∈
+A(Ω)n is a root vector of order k ≥ 1 at λ for A(z) if r(λ) ̸∈ kerλ A(z) and A(z)r(z) has a
+root of order k at λ, i.e., A(z)r(z) = (z − λ)kv(z) for some v(z) ∈ A(Ω)m, v(λ) ̸= 0.
+Definition 4.6 (λ-independent, complete, and maximal sets of root vectors) Given
+A(z) ∈ A(Ω)m×n and λ ∈ Ω, suppose that Q(z) ∈ A(Ω)n×p is an invertible basis of null(A(z))
+and that {ri(z)}s
+i=1 ⊂ A(Ω)n is a set of root vectors at λ for A(z) having orders {ki}s
+i=1.
+Then:
+• Such a set is λ-independent if
+�
+Q(λ)
+r1(λ)
+. . . rs(λ)
+�
+∈ Cn×(p+s) has full column rank;
+• A λ-independent set is complete if there is no λ-independent set of strictly larger car-
+dinality;
+• Such a complete set is ordered if k1 ≥ · · · ≥ ks > 0;
+• A complete ordered set is maximal if, for all j, there is no root vector v(z) at λ of order
+k > kj such that
+�
+Q(λ)
+x1(λ)
+. . .
+xj−1(λ)
+v(λ)
+�
+has full column rank.
+At this point, we are well positioned to generalize from polynomials (or rational functions)
+to analytic functions several useful results found in [4, Sections 3 and 4] and in [5, 22]. We
+state them below as a number of propositions and theorems; in many cases, however, we
+omit the proofs as they are essentially the same as the proofs in [4, 22] for the polynomial
+case. Here, “essentially” points to minor modifications that, once made, allow us to follow
+the same main ideas of the original proofs. Typical examples of these minor modifications
+are: one may need to replace minimal bases with invertible bases, the ring of polynomials
+with the ring A(Ω), the field of rational functions with the field of functions meromorphic on
+Ω, or the expansion of a polynomial round a point to the Taylor series of an analytic function
+about a point in its domain of analyticity, etc. When, instead, a proof must be significantly
+different than its analogue appeared in [4, 5, 22] for polynomial or rational matrices, we have
+included it.
+11
+
+Theorem 4.7 (Generalization of Theorem 3.1 in [4]) Let S(z) ∈ A(Ω)m×n be in Smith
+form, and suppose that the rank of S(x) is r and that the geometric multiplicity of λ as an
+eigenvalue of S(z) is s. Then {er, er−1, . . . , er−s+1}, where ei ∈ Cn is the i-th vector of the
+canonical basis, is a maximal set of root vectors at λ for S(x); moreover, their orders are
+the nonzero partial multiplicities of λ as an eigenvalue of S(x).
+We note that Theorem 4.7 was stated in [4] referring to the local Smith form at λ rather
+than to the global Smith form. The same could be done for analytic matrices, but for the
+sake of conciseness – and since the results that follow can be equivalently deduced from
+either the local or global version of Theorem 4.7 – we have decided here to state a global
+version rather than having to introduce the definition of local Smith form; interested readers
+can find it for instance in [4].
+Lemma 4.8 (Generalization of Lemma 3.3 in [4]) Let P(z), Q(z) ∈ A(Ω)m×n and λ ∈
+Ω. Suppose that Q(z) = A(z)P(z)B(z) for some A(z) ∈ A(Ω)m×m, B(z) ∈ A(Ω)n×n and that
+det A(λ) det B(λ) ̸= 0; moreover, let M(z), N(z) be invertible bases for null(P(z)), null(Q(z))
+respectively. Then kerλ P(z) = span M(λ) = span B(λ)N(λ) and kerλ Q(z) = span N(λ) =
+span adj B(λ)M(λ).
+Theorem 4.9 (Generalization of Theorem 3.4 in [4]) Let P(z), Q(z) ∈ A(Ω)m×n and
+λ ∈ Ω. Suppose that Q(z) = A(z)P(z)B(z) for some A(z) ∈ A(Ω)m×m, B(z) ∈ A(Ω)n×n
+and that det A(λ) det B(λ) ̸= 0. Then:
+• If {vi(z)}s
+i=1 are a maximal (resp. complete, λ-independent) set of root vectors at λ for
+Q(z) with orders ℓ1 ≥ . . . ℓs > 0, then {B(z)ri(z)}s
+i=1 are a maximal (resp. complete,
+λ-independent) set of root vectors at λ for P(z), with the same orders;
+• If {wi(z)}s
+i=1 are a maximal (resp. complete, λ-independent) set of root vectors at λ
+for P(z) with orders ℓ1 ≥ . . . ℓs > 0, then {adj B(z)ri(z)}s
+i=1 are a maximal (resp.
+complete, λ-independent) set of root vectors at λ for Q(z), with the same orders.
+Note that Theorem 4.7 and Theorem 4.9 imply that every analytic matrix A(z) has
+a maximal set of root vectors at λ, whose cardinality is precisely equal to the geometric
+multiplicity of λ as an eigenvalue of A(z). (To make full sense of the latter statement, we
+must formally agree that, if λ is not an eigenvalue, then the empty set is a set, and in fact
+the only possible set, of root vectors at λ for A(z).)
+Theorem 4.10 (Generalization of Theorem 3.10 in [22]) Let A(z) ∈ A(Ω)m×n and
+suppose that Q(z) ∈ A(Ω)n×p is an invertible basis for null(A(z)). Suppose that {vi(z)}s
+i=1
+are root vectors at λ ∈ Ω for A(z). Then, they are a complete set if and only if the set of
+the columns of the matrix
+B :=
+�Q(λ)
+v1(λ)
+. . . vs(λ)�
+is a basis for ker A(λ).
+12
+
+Proof. By definition, A(z)Q(z) = 0 and A(z)vi(z) = (z − λ)kwi(z), implying A(λ)B = 0.
+If {vi(z)}s
+i=1 is complete, then it is in particular λ-independent so B has full column rank.
+If the set of the columns of B is not a a basis for ker A(λ), we can complete it to a basis by
+adding some (constant) vectors {xi}c
+i=1. But then xi are root vectors at λ for A(z), and by
+construction {vi(z)}s
+i=1 ∪ {xi}c
+i=1 are a λ-independent set, contradicting completeness of the
+original set. Conversely, if the set of root vectors is not complete then the set of the columns
+of B cannot be a basis of ker A(λ) for dimensional reasons, as by starting from a complete
+set of root vectors we can construct t > p + s linearly independent vectors in ker A(λ).
+Theorem 4.11 (Generalization of Theorem 4.1 and Theorem 4.2 in [4]) Suppose that
+A(z) ∈ A(Ω)m×n has partial multiplicities m1 ≥ · · · ≥ ms at λ ∈ Ω. Then:
+1. All complete sets of root vectors of A(z) at λ have the same cardinality, equal to s;
+2. All maximal sets of root vectors of A(z) at λ have the same ordered list of orders
+m1 ≥ · · · ≥ ms, equal in turn to the partial multiplicities of λ as an eigenvalue of
+A(z);
+3. If {vi(z)}s
+i=1 is an ordered complete set of root vectors at λ for A(z) having orders
+ℓ1 ≥ · · · ≥ ℓs, then
+3.1 mi ≥ ℓi for all i = 1, . . . , s;
+3.2 {vi(z)}s
+i=1 is a maximal set of root vectors at λ for A(z) if and only if ℓi = mi for
+all i = 1, . . . , s;
+3.3 {vi(z)}s
+i=1 is a maximal set of root vectors at λ for A(z) if and only if �s
+i=1 mi =
+�s
+i=1 ℓi.
+Theorem 4.12 (Generalization of Lemma 5.2 in [5]) Suppose that A(z) ∈ A(Ω)m×n
+has partial multiplicities m1 ≥ · · · ≥ ms at λ ∈ Ω.
+Moreover, let {vi(z)}c
+i=1 be a λ-
+independent set of root vectors at λ for A(z) having orders ℓ1 ≥ · · · ≥ ℓc. Then c ≤ s
+and ℓi ≤ mi for all i = 1, . . . , c.
+The concept of root vectors allows us also to give appropriate definitions of eigenvectors,
+even for rectangular or square but singular analytic matrices. To this goal, we give below
+some relevant definitions that generalize analogous ones given (for polynomial or rational
+matrices) in [4, 22]. Fix an analytic matrix A(z) ∈ A(Ω)m×n and a scalar λ ∈ Ω, and note
+that generally kerλ A(z) ⊆ ker A(λ). However, the inclusion is strict if and only if λ is an
+eigenvalue of A(z) if and only if the cardinality of any maximal set of root vectors at λ for
+A(z) is positive. This motivates the following definition.
+Definition 4.13 An equivalence class [v] ∈ ker A(λ)/ kerλ A(z) is called a right eigenvector
+of A(z) associated with λ if [v] ̸= [0].
+13
+
+Following the same arguments as in [4, 22], we see that Definition 4.13 implies that
+eigenvectors associated with λ exists if and only if λ is an eigenvalue. Theorem 4.10 implies
+moreover that [v] is an eigenvector if and only if
+[0] ̸= [v] ∈ span{[x1(λ)], . . . , [xs(λ)]}
+where {xi(λ)}s
+i=1 is any complete set of root vectors at λ for A(z). Equivalently, [v] is an
+eigenvector if and only if v = �s
+i=1 xi(λ)ci + w for some coefficients ci ∈ C and some vector
+w ∈ kerλ A(z). When λ is a simple eigenvalue, it is possible to make v unique up to a phase,
+by asking that v ∈ kerλ A(z)⊥ and that ∥v∥2 = 1; in the polynomial case, this was useful for
+example in [14, 17] in the context of conditioning analysis and in [12, 14] to devise practical
+algorithms. We expect that similar developments may be possible also in the analytic case.
+We conclude by noting that, if A(z) has full column rank, then kerλ A(z) = {0} and thus we
+recover the familiar definition of an eigenvector being a nonzero vector v such that A(λ)v = 0.
+Remark 4.14 In this paper, we focused on extending the theory of root vectors from polyno-
+mial [4] to analytic matrices, by using invertible bases and the theory of modules. In [22], the
+theory of root vectors was extended from polynomial to rational matrices by using the theory
+of discrete valuations. Adding valuation theory to the tools of this paper makes it similarly
+possible to state a theory of root vectors for meromorphic matrices. We opted to not do it in
+this paper for two reasons: (1) because meromoprhic matrices do not appear to be as central,
+in the current linear algebra literature, as polynomial, analytic or rational matrices (2) for
+simplicity of exposition, i.e., to avoid an excessive density of tools that are possibly not too
+familiar to every matrix theorist or numerical linear algebraist. We thus leave the task to
+future research, but for now we comment for an interested reader that it is not difficult to
+generalize the theory in [22] from rational to meromorphic functions, and then to combine
+it with the theory of this paper, thus producing a theory of root vectors for meromorphic
+functions.
+4.1
+Example
+In this subsection, we work out an example to illustrate the results of Section 4. Let Ω = C
+so that A(Ω) is the ring of entire functions, and consider the analytic matrix
+A(z) =
+
+
+2ze2z
+zez
+z sinh(z)
+e2z(sin(z)2 − 2z)
+ez(sin(z)2 − z)
+ez cos(z) sin(z)2 − z sinh(z)
+ez(2zez − sin(z)2)
+zez − sin(z)2
+cos(z)3 − cos(z) + z sinh(z)
+
+ .
+We are going to study the spectral properties of A(z) at its eigenvalue λ = 0: in particular
+we will exhibit a maximal set of root vectors, thus obtaining the partial multiplicities of 0 as
+well as corresponding eigenvectors (according to Definition 4.13). As a preliminary step, note
+that a straightforward computation shows that det A(z) = 0, and that the leading principal
+2 × 2 minor of A(z) is e3zz sin(z)2 ̸= 0. Hence, rank A(z) = 2. It can be readily verified that
+14
+
+an invertible basis of null(A(z)) is
+Q(z) =
+
+
+ez cos(z) − sinh(z)
+ez(sinh(z) − 2ez cos(z))
+e2z
+
+ ⇒ ker0 A(z) = span
+
+
+
+
+1
+−2
+1
+
+
+
+ .
+Let now
+v1(z) =
+
+
+1
+−ez
+0
+
+ ,
+v2(z) =
+
+
+1
+−2ez
+0
+
+ .
+We claim that {v1(z), v2(z)} is a maximal set of root vectors at 0 for A(z). Note first that
+A(z)v1(z) = ze2z
+
+
+1
+−1
+1
+
+ ,
+A(z)v2(z) = ez sin(z)2
+
+
+0
+−ez
+1
+
+ ,
+and that
+v1(0) =
+
+
+1
+−1
+0
+
+ ̸∈ ker0 A(z),
+v2(0) =
+
+
+1
+−2
+0
+
+ ̸∈ ker0 A(z),
+and hence v1(z), v2(z) are root vectors at 0 for A(z) of order 1 and 2 respectively: note
+indeed the Taylor expansions zez = z + o(z) and ez sin(z)2 = z2 + o(z). The set {vi(z)}2
+i=1
+is 0-independent because
+rank
+�
+Q(0)
+v1(0)
+v2(0)
+�
+= rank
+
+
+1
+1
+1
+−2
+−1
+−2
+1
+0
+0
+
+ = 3.
+Moroever, we have A(0) = 0 and hence, by Theorem 4.10, the set {vi(z)}2
+i=1 is complete.
+Finally, one can compute a Smith form of A(z), for example by computing determinantal
+divisors (GCDs of minors of a given size); indeed, recall from the introduction that the
+invariant factors are the ratios of the determinantal divisor. We thus obtain that the invariant
+factors of A(z) are z, sin(z)2, 0. This implies that the partial multiplicities of 0 are 1, 2, and
+by Theorem 4.11 the set {vi(z)}2
+i=1 is therefore maximal.
+Moreover, [v1(0)] and [v2(0)],
+seen as equivalence classes
+mod ker0 A(z), are linearly independent eigenvectors of A(z)
+associated with the eigenvalue 0.
+Acknowledgements
+I thank Froil´an Dopico for reading a preliminary version of this paper and sharing very useful
+remarks.
+15
+
+References
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+17
+

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+Unified simulation methods for quantum acoustic devices
+Hugo Banderier,∗ Maxwell Drimmer,† and Yiwen Chu
+Department of Physics, Eidgen¨ossiche Technische Hochschule Z¨urich, 8093 Z¨urich, Switzerland
+and
+Quantum Center, Eidgen¨ossiche Technische Hochschule Z¨urich, 8093 Z¨urich, Switzerland
+(Dated: January 13, 2023)
+In circuit quantum acoustodynamics (cQAD), superconducting circuits are combined with acous-
+tic resonators to create and control non-classical states of mechanical motion.
+Simulating these
+systems is challenging due to the extreme difference in scale between the microwave and mechanical
+wavelengths. All existing techniques simulate the electromagnetic and mechanical subsystems sep-
+arately. However, this approach may not be adequate for all cQAD devices. Here, we demonstrate
+a single simulation of a superconducting qubit coupled to an acoustic and a microwave resonator
+and introduce two methods for using this simulation to predict the frequencies, coupling rates, and
+energy-participation ratios of the electromechanical modes of the hybrid system. We also discuss
+how these methods can be used to investigate important dissipation channels and quantify the non-
+trivial effects of mode hybridization in our device. Our methodology is flexible and can be extended
+to other acoustic resonators and quantum degrees of freedom, providing a valuable new tool for
+designing hybrid quantum systems.
+I.
+INTRODUCTION
+Circuit quantum acoustodynamics (cQAD) provides
+the opportunity to combine the unique advantages
+of superconducting (SC) circuits and mechanical res-
+onators [1, 2].
+A device that integrates the numerous
+long-lived modes of compact mechanical resonators [3–
+5] with the strong quantum nonlinearity of SC qubits is
+potentially useful for quantum information processing [6–
+8] and tests of fundamental physics [9]. SC qubits have
+already been combined with a wide variety of mechani-
+cal elements, including membranes [10, 11], bulk acous-
+tic wave (BAW) resonators [12–14], surface acoustic wave
+resonators [15–17], and phononic crystals [18].
+Finite element (FE) simulations are a crucial de-
+sign tool in both circuit quantum electrodynamics
+(cQED) [19–24] and solid mechanics [25] for predicting
+the behavior of complex solid-state structures.
+Both
+fields have independently developed mature techniques
+which rely on different strategies and software. For exam-
+ple, quantum circuits are modeled using electromagnetic
+simulation software like Ansys HFSS [26], Microwave
+Office [27], or Sonnet [28] while COMSOL Multiphysics
+(COMSOL) [29] is the preferred choice for performing
+FE simulations of acoustic resonators.
+Techniques from cQED and solid mechanics can be
+combined in order to simulate cQAD devices. If an elec-
+tromechanical system can be modeled as a lumped ele-
+ment, it is possible to simulate the electromechanical re-
+sponse in isolation from the Josephson circuit [30, 31].
+This response is represented by an equivalent circuit
+which can be quantized to determine the system’s Hamil-
+tonian [32]. In certain cases, however, isolating an elec-
+∗ Current affiliation : Oeschger Centre for Climate Change Re-
+search and Institute of Geography, Universit¨at Bern, 3012 Bern,
+Switzerland
+† Corresponding author : max.drimmer@phys.ethz.ch
+trically small subsystem may not be possible meaning
+that the system cannot be described by simple circuit.
+The ℏBAR [33, 34], a cQAD device that features a 3-D
+transmon qubit [35] piezoelectrically coupled to a high-
+overtone BAW resonator (HBAR), falls into this cate-
+gory.
+In this paper, we demonstrate two simulation ap-
+proaches that unify FE techniques from cQED and solid
+mechanics using a single COMSOL model without the
+need for an equivalent circuit.
+In the first approach,
+which we call the “unhybridized eigenmode approach,”
+we solve for the eigenmodes of the electric and displace-
+ment fields separately (i.e.
+without any piezoelectric
+coupling).
+This yields the mode structure of the un-
+hybridized electrical and mechanical subsystems.
+One
+can then evaluate the electromechanical coupling rate
+between an electrical and a mechanical eigenmode us-
+ing an overlap integral inside the piezoelectric material
+(Eq. 5). In the second approach, which we call the “hy-
+bridized eigenmode approach”, we simultaneously solve
+for the coupled electric and displacement fields to find
+the dressed eigenmodes of the entire system. We demon-
+strate this approach by first presenting a Hamiltonian
+formulation that extends the energy-participation ratio
+(EPR) method [23] to mechanical degrees of freedom.
+We then use it to extract important Hamiltonian param-
+eters that arise from electromechanical coupling, such as
+cross-Kerr nonlinearities, anharmonicities, and mechani-
+cal EPRs in the dispersive regime. While simulating the
+coupled fields is more computationally intensive, the re-
+sults reflect that hybridization of the electromagnetic and
+mechanical subsystems affects the frequency and shape
+of the modes, which can in turn impact the predicted
+coupling and loss rates.
+arXiv:2301.05172v1  [quant-ph]  12 Jan 2023
+
+2
+II.
+HYBRID QUANTUM INTERACTIONS
+We consider a general system consisting of a SC cir-
+cuit with J transmons and N − J linear electromagnetic
+modes interacting with a mechanical resonator support-
+ing M modes. Even though the N linear electromagnetic
+modes and M mechanical modes are identically described
+as modes of a bosonic resonator, we use different letters
+to distinguish them for clarity in the rest of this sec-
+tion. Each of the J transmons can be described using a
+Hamiltonian that is a sum of a linear resonator term and
+a nonlinear term [32]. When the interactions are written
+under the rotating wave approximation, the Hamiltonian
+is
+ˆH
+ℏ =
+N
+�
+n
+˜ωnˆ˜a†
+nˆ˜an+
+M
+�
+m
+Ωmˆb†
+mˆbm+
+N,M
+�
+n,m
+˜gnm
+�
+ˆ˜a†
+nˆbm + ˆb†
+mˆ˜aj
+�
+−
+N,N
+�
+n̸=n′
+ςnn′
+�
+ˆ˜a†
+nˆ˜an′ + ˆ˜a†
+n′ˆ˜an
+�
+−
+J
+�
+j
+Ej
+ℏ
+�
+cos ˆθj + 1
+2
+ˆθ2
+j
+�
+(1)
+where we have introduced ˜ωn and ˆ˜an (Ωm and ˆbm) as the
+frequencies and bosonic ladder operators of the nth (mth)
+linear electromagnetic (mechanical) resonator mode, the
+Josephson energy of the jth junction Ej and its flux
+ˆθj = θZPF,j
+�
+ˆ˜aj + ˆ˜a†
+j
+�
+where θZPF,j are the associated
+zero-point fluctuations (ZPFs), and ςnn′ (˜gnm) are the
+electromagnetic (electromechanical) two-mode coupling
+rates. The detuning between two modes are defined as
+∆nn′ = ˜ωn′ − ˜ωn and ∆nm = Ωm − ˜ωn.
+The cou-
+pling between two modes are said to be dispersive if
+|∆nn′| ≫ |ςnn′| or |∆nm| ≫ |gnm|.
+Eq. 1 is a nonlinear Hamiltonian that cannot be di-
+rectly modeled using standard FE simulation techniques.
+Instead, we follow Ref. [19] by rewriting the Hamiltonian
+as a sum of linear and nonlinear terms. Then we can
+use FE simulations to find the linear eigenmodes, from
+which we extract the relevant parameters that describe
+the nonlinear terms using the EPR method.
+A.
+Unhybridized eigenmode approach
+If we ignore the coupling between the electromagnetic
+and mechanical degrees of freedom (third term in Eq. 1),
+we can simulate the two subsystems individually. The
+results of electromagnetics-only simulations (i.e. simu-
+lations with solid mechanics and piezoelectricity turned
+off) are dressed eigenstates of the linear electromagnetic
+part of the Hamiltonian (Terms 1 and 4 of Eq. 1). Thus
+the Hamiltonian can be partially diagonalized
+ˆH
+ℏ =
+N
+�
+n=1
+ωnˆa†
+nˆan+
+M
+�
+m
+Ωmˆb†
+mˆbm+
+N,M
+�
+nm
+gnm
+�
+ˆa†
+nˆbm + ˆb†
+mˆan
+�
+−
+J
+�
+j
+Ej
+ℏ
+∞
+�
+p=4
+cp
+�
+�
+N
+�
+n=1
+ϕnjˆan + H.c.
+�
+�
+p
+(2)
+In this expression, ˆa are the dressed electromagnetic lad-
+der operators, cp are the cosine expansion coefficients,
+and ϕnj are the ZPFs of the flux in the j-th junction
+when only dressed mode n is excited and Ej are the j-th
+junction’s Josephson energies.
+The solutions of the mechanical simulations are ex-
+actly the modes described Term 2 of Eq. 1. The elec-
+tromechanical interaction is now written in terms of the
+dressed electromagnetic eigenmodes. The exact expres-
+sion of gnm depends on the nature of the interaction.
+Here, we will consider the piezoelectric coupling which
+we describe in detail in Appendix A. From now on we
+will refer to the dressed modes ˆan,ˆbm of this picture as
+“unhybridized” because the next approach will further
+hybridize these.
+B.
+Hybridized eigenmode approach
+As there is no conceptual difference between an elec-
+tromagnetic and a mechanical mode in Eq. 1, we can
+equivalently choose to express the Hamiltonian in terms
+of N + M hybrid electromechanical modes with eigen-
+values ξk, bosonic ladder operators ˆck, and associated
+junction flux ZPFs φkj
+ˆH
+ℏ =
+N+M
+�
+k=1
+ξkˆc†
+kˆck −
+J
+�
+j
+Ej
+ℏ
+∞
+�
+p=4
+cp
+�
+�
+N+M
+�
+k=1
+φkjˆck + H.c.
+�
+�
+p
+(3)
+Under the dispersive and the perturbative assumptions,
+detailed in Appendix B, we can limit the expansion of
+the second term to p = 4 and only keep only excitation
+number-preserving interactions to obtain
+ˆHp=4
+ℏ
+=
+�
+k
+−∆kˆc†
+kˆck−1
+2αkˆc†2
+k ˆc2
+k−
+�
+l<k
+1
+2χklˆc†
+kˆckˆc†
+l ˆcl (4)
+This expression highlights several experimentally rel-
+evant quantities:
+∆k
+=
+1
+2
+�
+l χkl are the effective
+Lamb shifts, αk are the anharmonicities, and χkl =
+ℏ−1 �
+j Ejφ2
+kjφ2
+lj are the total cross-Kerr shifts induced
+between modes k and l. Under these approximations all
+the parameters depend on the zero-point fluctuations of
+the junctions’ fluxes in each mode, φkj. We note that in
+cQAD, the perturbative assumption is not always valid in
+the dispersive regime. However, as shown in Appendix B,
+the correction to these quantities can still be expressed
+using the same ZPFs.
+
+3
+In the two approaches described above, we have devel-
+oped Hamiltonians with key unknowns that can be ob-
+tained from simulations. In the unhybridized eigenmode
+approach, they are the bare mode frequencies ωn and
+Ωm, the junction flux ZPFs in the bare electromagnetic
+modes ϕnj, as well as the piezoelectric pairwise coupling
+rates gnm.
+In the hybridized eigenmode approach, we
+only need to solve for the hybridized mode frequencies ξk
+and the junction flux ZPFs φkj.
+III.
+SIMULATING HYBRID QUANTUM
+DEVICES
+A.
+Physics interfaces
+We begin each simulation by choosing physics inter-
+faces in COMSOL, each of which defines a vector field
+along with its equations of motion to be solved. We used
+the Electromagnetic Waves, Frequency Domain in-
+terface (emw) in the RF Module for modeling SC circuits
+and microwave cavities and used the Solid Mechanics
+interface (solid) in the Structural Mechanics Module for
+the acoustic resonator.
+The second step is dynamically combining these inter-
+faces together. In general, linking interfaces A and B is
+simply done by calling the field defined in A in a domain
+or boundary condition on B and vice versa.
+In many
+cases, COMSOL has built-in multiphysics interfaces that
+perform this step. However, no such interface exists be-
+tween solid and emw.
+Therefore, the coupling must
+be defined manually. In this work, electromagnetic and
+mechanical objects are coupled by the piezoelectric ef-
+fect [36]. In a piezoelectric medium, the wave equations
+for the electric and displacement fields are modified to
+become Eqs. A23 and A24 as derived in Appendix A.
+These modifications are implemented in our simulation
+using three additional domain conditions, represented as
+nodes of the physics interfaces. Specifically, the effec-
+tive medium and external current density nodes
+are added to the emw interface and an external stress
+node is added to the solid interface. A detailed descrip-
+tion is provided in Appendix C.
+B.
+Model of the device
+We use the method presented in this work to simu-
+late an ℏBAR similar to those used in Refs. [33, 34].
+This device is comprised of two chips: The first has one
+single-junction transmon qubit with one pad extended to
+form an antenna, and the second has a piezoelectric dome
+which transduces the qubit’s electric field into mechan-
+ical modes of the HBAR (Fig. 1a). The two substrates
+are bonded together such that the antenna is aligned un-
+derneath the dome [34]. Then the assembly is situated
+in a 3-D microwave cavity (Fig. 1b).
+cQED elements
+are simulated by applying a perfect electric conductor
+z
+2 µm
+AlN
+Al
+40 µm
+a)
+b)
+d)
+x
+y
+100 μm
+c)
+z
+x
+y
+x
+y
+5 mm
+200 μm
+~1 μm
+FIG. 1. Simulating the ℏBAR. a) A schematic of the ℏBAR
+device. The bottom chip is a 3-D transmon qubit, modified
+with an antenna (the extension at the right), on a dielectric
+substrate. The top chip hosts a HBAR, formed by a piezo-
+electric dome, positioned above the qubit antenna. The pur-
+ple arrows represent the electric field of the qubit-like mode.
+b) Full simulation space defined by half of a rectangular SC
+microwave cavity with with cylindrical ends (lavender vol-
+ume). The simulation is symmetric about the x-z plane clos-
+est to the viewer. The cyan structure in the red rectangle is
+the ℏBAR. c) Displacement field of a simulated fundamental
+HBAR mode in a 2-D slice through the symmetry axis. This
+image is not to scale and is cropped to show the bottom 20%
+of the HBAR so that the curvature of the dome and displace-
+ment profile of the mode are clearly visible. The diameter of
+the HBAR and half wavelength of the acoustic mode are in-
+dicated by scale bars. d) Electric field of the qubit-like mode
+centered in a region around the 3-D transmon.
+boundary condition to the superconductors (the surfaces
+of the microwave cavity and the leads of the qubit) and
+representing the Josephson junction as a lumped element
+inductor [19]. In our simulation, the substrate of both the
+qubit and HBAR chips is c-axis oriented sapphire and the
+piezoelectric dome is made of c-axis oriented aluminum
+nitride (AlN).
+To perform a 3-D full-wave eigenmode simulation of
+such a device, several simplifications have to be made.
+The whole device is symmetric about the x − z plane at
+y = 0, so we can use symmetry boundary conditions to
+reduce the simulation space. We observe that the long-
+lived modes of the HBAR are confined inside a cylindrical
+
+广x4
+volume with a small transverse area in the x − y plane
+(Fig. 1c).
+We therefore only simulate the mechanical
+fields inside a cylindrical volume with the radius of the
+piezoelectric dome. Finally, we simulate an HBAR with
+a substrate thickness of 40 µm, an order of magnitude
+smaller than the 420 µm thick devices in Refs. [33, 34], in
+order to speed up the simulations. A detailed description
+of the device model can be found in Appendix C.
+C.
+Extracting Hamiltonian parameters
+An eigenmode simulation returns a set of field distri-
+butions which are labeled by their eigenfrequencies. In
+the unhybridized approach, these are En and um. We
+use an overlap integral to extract the coupling rates be-
+tween the unhybridized electromagnetic and mechanical
+modes [37]
+gnm = Anm
+�
+V,piezo
+En · eT : εmdV
+(5)
+where eT is the transpose of the piezoelectric tensor, εm
+is the strain tensor derived from um, and the proportion-
+ality constant Anm comes from normalizing the fields to
+that of a single photon and phonon. We justify this for-
+mula by deriving the piezoelectric Hamiltonian in a mul-
+timode Jaynes-Cummings form in Appendix A, where
+Eq. A17 is the full expression for gnm.
+For the hybridized eigenmode simulations, we obtain
+the electric and displacement fields Ek and uk, respec-
+tively, for each ξk. COMSOL also computes several de-
+rived quantities; in this work, we make use of the elec-
+tric displacement Dk, strain εk, and stress Sk fields, the
+current through the jth junction element Ikj, and the
+time-averaged global electrical and strain energies Eelec,k
+and Estrain,k. In order to use the EPR method, one must
+calculate the energy-participation ratio of the kth mode
+in the energy of jth junction. In Appendix D, we show
+that the EPRs of the hybridized qubit-HBAR modes can
+be written as
+pkj =
+Wkj
+Eelec,k + Estrain,k
+(6)
+where the average inductive energy Wkj
+=
+1
+2LjI2
+kj
+and the time-averaged electrical and mechanical energy
+stored in the system for mode k Eelec,k, Estrain,k are de-
+fined by Eqs. D6 and D7 in terms of the fields Ek and
+εk. These quantities are easily obtained from the sim-
+ulation solutions: Lj is the lumped element inductance
+that we define, while Ikj, the electric current through
+the jth lumped element, is calculated from the solution
+as Ikj = 1
+w
+�
+elem j Jk ·tdS, where w is the junction width,
+Jk is the surface current, and t a unit direction vector.
+In Appendix D, we show that the ZPF of the junction’s
+flux can be written in terms of the extracted EPR
+pkj =
+Ejφ2
+kj
+1
+2ℏξk
+,
+(7)
+which in turn defines the Hamiltonian of Eq. 3.
+Fur-
+thermore, the EPR can be used to calculate various loss
+mechanisms that arise from or are modified by the hy-
+bridization of the modes, as shown in Appendix E.
+D.
+Finite element considerations
+In finite element method (FEM) simulations, space
+is divided into polyhedra whose vertices define a mesh.
+Building this mesh, or “meshing,” is done automatically
+in modern FE software, but some user input is almost
+always needed in the case of more involved geometries.
+Finding a mesh that is coarse enough so that the simula-
+tion runs in a reasonable time but fine enough to capture
+all the important physical phenomena can be challenging.
+A rule of thumb for meshing FE simulations is to use
+at least five meshing elements per wavelength [38]. We
+can immediately see the issue for hybridized eigenmode
+simulations: the software has to simultaneously solve for
+E and u, which have wavelengths that differ by five orders
+of magnitude at the same frequency. In the case of u, for
+which λ ∼ 1 µm in typical GHz frequency cQAD devices,
+a fine mesh can quickly make the simulation intractable
+if defined over a too big volume. However, we show that
+even in the case a HBAR, which has a relatively large
+volume compared to most mechanical resonators used in
+cQAD systems, a meshing procedure can be found to
+keep the simulation at a reasonable size while resolving
+all of the relevant physics.
+The mesh additionally needs to be optimized to re-
+duce the number of so-called spurious modes. These un-
+physical modes are a source of inaccuracy in many FEM
+applications [39–41]. A handmade mesh was created in
+order to minimize the number of elements and spurious
+solutions (the meshing procedure and parameters are re-
+ported in Appendix C). Our simulations were able to
+solve for 150 eigenmodes in under 2 hours on a com-
+puter with 64 GB of memory. Our meshing procedure
+drastically reduced, but could not completely eradicate,
+spurious modes.
+IV.
+RESULTS
+A.
+Unhybridized eigenmode approach
+We choose to solve the unhybridized solid mechanics
+eigenmode simulations near a frequency corresponding
+to λ ≈ 1800 nm in both sapphire and AlN. At this fre-
+quency, the mode has half a wavelength in the piezo-
+electric dome which is expected to maximize the over-
+lap between the acoustic mode’s strain field with the
+piezoelectrically-induced external stress resulting from
+the qubit mode’s electric field. With our geometry, this
+corresponds to modes with a longitudinal mode number
+q = 49.
+A cross-section of such a mode is shown in
+Fig. 1c. Because the resonator is a 3-D object, it supports
+
+5
+many modes with this longitudinal number with differ-
+ent patterns in the transverse plane, as well as different
+polarizations. We observe both Laguerre-Gaussian (LG)
+and Hermite-Gaussian (HG) modes in our results. The
+2-D profiles of these modes are represented in Fig. 2a.
+Note that we always show the 2-D pro��le correspond-
+ing to uz irrespective of the mode’s polarization, see
+also Appendix C 6. While we are mainly interested in
+longitudinal-like polarized modes (as defined in Eq. C1),
+the anisotropy of the material’s piezoelectric tensor may
+give shear-like polarized modes nontrivial coupling to the
+qubit, making them interesting to study as well. We also
+perform an unhybridized EM simulation to compute the
+coupling rates, as described in the next paragraph. An
+example electric field distribution can be seen in Fig. 1d,
+and an example mechanical displacement field distribu-
+tion can be seen in Fig. 1c in 2-D and Fig. 4b in 3-D.
+6.440
+6.442
+6.444
+6.446
+6.448
+6.450
+f [GHz]
+104
+106
+g/2  [Hz]
+b)
+LG(0, 0)L
+LG(0, 1)L
+LG(0, 1)SH
+HG(2, 0)L
+LG(0, 3)L
+HG(3, 0)L
+HG(0, 4)L
+HG(2, 2)L
+6.446
+6.448
+6.450
+6.452
+6.454
+6.456
+f [GHz]
+104
+106
+g/2  [Hz]
+c)
+LG(0, 0)L
+LG(0, 1)L
+LG(0, 1)SH
+HG(2, 0)L
+LG(0, 3)L
+HG(3, 0)L
+HG(0, 4)L
+HG(2, 2)L
+104
+106
+g/2  [Hz]
+d)
+Unhybridized
+Hybridized
+a)
+LG (0, 0)
+LG (0, 1)
+HG (2, 0)
+LG (0, 3)
+HG (3, 0)
+HG (0, 4)
+HG (2, 2)
+FIG. 2. Acoustic modes and electromechanical coupling rates.
+a) 2-D longitudinal displacement (uz) profiles of acoustic
+modes observed in our simulations.
+b) An acoustic spec-
+trum with electromechanical coupling rates calculated using
+the unhybridized eigenmode approach. c) An electro-acoustic
+spectrum with electromechanical coupling rates calculated us-
+ing the hybridized eigenmode approach. d) A mode-by-mode
+comparison of the electromechanical coupling rates extracted
+from unhybridized (green) and hybridized (red) simulations.
+The coupling rates in b), c), and d) refer to a qubit mode at
+ω = 2π × 6.424 GHz.
+Using these results, overlap integrals are performed be-
+tween the EM qubit mode (n = q in Eq. 5) and the me-
+chanical modes in order to compute the two-mode cou-
+pling rates gqm (hereafter referred to as g for simplicity)
+and plot them in Fig. 2b against the mechanical modes’
+frequencies.
+We find that the zeroth transverse order mode of a 40
+µm HBAR has a qubit-phonon coupling rate g ≈ 2π × 1
+MHz. The higher-order transverse modes have lower cou-
+pling rates due to their overlap mismatch with the qubit’s
+electric field profile. We note that an antenna radius of
+r = 20 µm was chosen, but it could be optimized in shape
+to increase the coupling rate to any of the observed acous-
+tic modes.
+B.
+Hybridized eigenmode approach
+Next, we turn on the piezoelectric coupling between
+the solid and emw interfaces. To test our implemen-
+tation and demonstrate the capabilities of this simula-
+tion framework, we perform eigenfrequency simulations
+while sweeping the qubit’s inductance L such that the fre-
+quency of the qubit-like mode (defined as the mode with
+highest EPR) intersects the set of high-overtone HBAR
+modes mentioned in the previous section. Note that the
+acoustic mode frequencies shift up by about 5 MHz com-
+pared to the uncoupled modes because the piezoelectric
+effect increases the stiffness of the AlN material [42].
+The eigenfrequencies found in the simulations are
+shown in Fig. 3a and are plotted against the frequency of
+the unhybridized (UH) qubit mode. We observe avoided
+crossings in good agreement with the values of g com-
+puted in the previous section. This comparison between
+the unhybridized and hybridized approaches provides a
+sanity check for the physics modeling. The EPRs calcu-
+lated from this frequency sweep, shown in Fig. 3b, are
+another way to visualize the hybridization of the qubit
+with the acoustic modes. When the qubit hybridizes with
+a mechanical mode, a significant fraction of the qubit-like
+mode EPR is allocated to the mechanical-like mode.
+One challenge of analyzing these simulations is the
+presence of a significant number of spurious modes in
+the results, which are a source of inaccuracy in the de-
+rived quantities. We find them when looking for GHz-
+frequency eigenmodes of the displacement field in both
+unhybridized and hybridized simulations. These modes
+appear as several point-like defects on mesh edges and
+nodes (see Appendix C 4). We observed that simulations
+with finer transverse meshing tended to have more spu-
+rious modes.
+To address this issue, we identify the so-called ”physi-
+cal modes” in post-processing by finding the modes whose
+displacement fields best match those of the LG and HG
+modes. These, together with the qubit mode, are used
+for further analysis, while the rest are discarded as spu-
+rious modes (see Appendix C). This process is not ideal,
+however, since the physical modes can be hybridized with
+the spurious modes through their coupling to the qubit.
+This causes an issue which we refer to as EPR dilution,
+where part of the physical mode’s EPR is distributed
+to spurious modes with nearby frequencies. The effect of
+EPR dilution is evident for certain bare qubit frequencies
+in Fig. 3b where the EPRs of all the modes sum to less
+than one (eg. around 6.451 GHz). Evidently, however,
+the effect of EPR dilution is on the 20% level.
+We now focus on the case of a large detuning be-
+
+6
+Bare qubit frequency [GHz]
+6.442
+6.444
+6.446
+6.448
+6.450
+6.452
+6.454
+6.456
+6.458
+Mode frequency [GHz]
+a)
+Legend
+qubit
+LG(0, 0)L
+LG(0, 1)L
+LG(0, 1)SH
+HG(0, 2)SH
+HG(2, 0)L
+LG(0, 3)L
+HG(3, 0)L
+HG(0, 4)L
+HG(2, 2)SH
+HG(4, 0)L
+HG(1, 4)SH
+HG(2, 2)L
+HG(0, 2)L
+6.442
+6.444
+6.446
+6.448
+6.450
+6.452
+6.454
+6.456
+Bare qubit frequency [GHz]
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Energy participation ratio (p)
+b)
+FIG. 3. Results of hybridized electromechanical simulations.
+a) The mode spectrum near the acoustic fundamental longi-
+tudinal mode at ω = 2π × 6.445 GHz show several avoided
+crossings. The size of each avoided crossing indicates the cou-
+pling between the mechanical mode and the qubit. The leg-
+end above this plot applies to both subfigures. b) Energy-
+participation ratios of each mode shown in Fig. 3a to the
+junction. Both longitudinal-like and shear-like modes are la-
+beled according to their uz profiles. The green line indicates
+the sum of the EPRs of all of the modes in the figure.
+Mode
+Qubit
+LG(0,0)
+HG(2,0)
+f [GHz]
+6.424
+6.445
+6.451
+p
+0.95
+1.4 × 10−3 5.4 × 10−5
+χk,l/2π [Hz]
+Qubit
+(0,0)
+(1,0)
+Qubit
+5.2 × 108
+4.4 × 104
+2.3 × 103
+(0,0)
+-
+1.1 × 103
+1.7 × 102
+(1,0)
+-
+-
+1.7 × 100
+TABLE I. Top: Frequencies and EPRs of the qubit-like mode
+and the two most strongly coupled acoustic modes. Bottom:
+Pair-wise (cross-) Kerr couplings χk,l/2π between the modes.
+tween the qubit mode and the same family of LG acous-
+tic modes as in Figs. 2b, 3a and 3b in order to demon-
+strate the extraction of design quantities in the dispersive
+regime. We choose a value for the junction inductance
+such that the unhybridized qubit is ∆UH
+q,(0,0) ≈ 2π × 21
+MHz below the LG(0, 0) mode. We compute the self- and
+cross-Kerr couplings using the EPRs, from which we can
+further compute the mode’s anharmonicity αk = 1
+2χk,k
+and its Lamb shift ∆k = 1
+2
+�
+l χk,l.
+The qubit’s anharmonicity can be compared with its
+capacitive energy EC = e2
+hC , since αq ≈ EC
+2h [43] for trans-
+mons. We can infer the capacitance C of the qubit from
+the slope of its inverse squared frequency using ω−2
+q
+= LC
+and obtain C = 67 fF, finally giving an expected anhar-
+monicity of EC
+2h = 288 MHz. This is in reasonably good
+agreement with the EPR result of αq = 261 MHz (see
+Table I), considering that EC
+h
+overestimates αq [44].
+We further verify the results in Table I by using them
+to compute the coupling rate g between the correspond-
+ing unhybridized modes with the approximate relation-
+ship [44]
+g2
+q,l ≈ ∆UH
+q,l χq,l
+∆UH
+q,l + αq
+2αq
+(8)
+The mode numbers k, l either refer to unhybridized
+modes (on g and ∆UH) or to their hybridized counter-
+parts (on χ and α). The results are shown in Fig. 2c.
+Figure 2d shows good agreement between the g’s com-
+puted using the two methods we described. The discrep-
+ancies may be in part explained by EPR dilution and the
+approximate nature of Eq. 8.
+V.
+OUTLOOK
+We have demonstrated a technique for the simulation
+of cQAD devices that unifies electromagnetic and me-
+chanical degrees of freedom under the EPR method. Im-
+portantly, we showed that quantum circuits, including
+SC qubits, can be combined with solid mechanics within
+the powerful multiphysics framework of COMSOL. By
+combining the necessary physics interfaces, we can ex-
+tract the hybridized eigenmodes of the entire system.
+We showcased this technique by simulating an ℏBAR,
+and showed that it gives results in good agreement with
+other methods for estimating device parameters in the
+dispersive regime.
+The example shown in this work may be the most com-
+putationally intensive out of the existing cQAD systems
+due to the large size of the acoustic resonator. There-
+fore, applying our methodology to other types of devices
+should prove relatively straightforward. More generally,
+our technique may be extended to other types of hybrid
+quantum systems such as a SC qubit coupled to magnonic
+resonators [2, 45] or devices where acoustic resonators
+interact with other qubits such as color centers [46] or
+quantum dots [47].
+The simulation files used in this paper will be made
+available in the supplementary material.
+
+7
+ACKNOWLEDGEMENTS
+We thank S. Marti, Y. Dahmani, V. Jain, G. Steele,
+J. Franse, N. Egli, R. Benevides, and U. von L¨upke for
+valuable discussions. This project has received funding
+from the European Research Council (ERC) under the
+European Union’s Horizon 2020 research and innovation
+programme (Grant agreement No. 948047)
+
+8
+Appendix A: Piezoelectric Hamiltonian
+In this appendix, we derive the quantum Hamiltonian
+for a piezoelectric solid from first principles.
+1.
+Field quantization
+We begin by quantizing the electric and displacement
+fields. Following the procedure in [48], we will explicitly
+quantize the displacement field.
+Following Chapters 3
+and 4 of [49], we define the displacement field u(x, t),
+strain tensor ε(x, t) =
+1
+2
+�
+∇u(x, t) + (∇u)T (x, t)
+�
+and
+stress tensor S(x, t).
+In an anisotropic linear material
+with stiffness tensor c and density ρ, Newton’s second
+law implies the following wave equation (neglecting body
+forces)
+∇ · S = ρ∂2u
+∂t2
+cijlm
+∂2ul
+∂xj∂xm
+= ρ∂2ui
+∂t2
+(A1)
+where step 2 is written in index notation using Einstein’s
+convention, and Hooke’s law is used since we assumed a
+linear elastic material. We split time and space depen-
+dencies of a displacement mode assuming a frequency Ω,
+keeping the convention for naming frequencies from the
+main text (ω for electromagnetics and Ω for solid me-
+chanics)
+u(x, t) = u0e−iΩth(x) + c.c.
+(A2)
+where u0 is a constant and h(x) is a normalized function
+that contains the mode shape and polarization of the
+mode. With u0(t) = u0e−iΩt, we write the Hamiltonian
+of the system
+H = T + V
+=
+�
+dV 1
+2ρ∂ui
+∂t
+∂u∗
+i
+∂t +
+�
+dV 1
+2cijlm
+∂ui
+∂xj
+∂ul
+∂xm
+= 1
+2ρΩ2��iu0(t) − iu∗
+0(t)
+��2 �
+dV
+��h(x)
+��2
++ 1
+2
+�
+dSnjcijlmui
+∂ul
+∂xm
+− 1
+2
+�
+dV uicijlm
+∂2ul
+∂xj∂xm
+= 1
+2ρΩ2��iu0(t) − iu∗
+0(t)
+��2 + 0 − 1
+2
+�
+dV uiρ∂2ui
+∂t2
+= 1
+2ρΩ2 ���iu0(t) − iu∗
+0(t)
+��2 +
+��u0(t) + u∗
+0(t)
+��2�
+= 2ρΩ2��u0(t)
+��2
+(A3)
+where the fourth step assumes no energy leaves the sys-
+tem’s volume and uses the wave Eq. A1 and the spatial
+normalization of h. nj is the j-th element of a unit nor-
+mal vector to the surface in the second term of step 3.
+We define the following conjugate variables
+p = −iΩρ(u0(t) + c.c)
+q = (u0(t) + c.c)
+(A4)
+which gives the following familiar Hamiltonian
+H = p2
+2ρ + 1
+2ρΩ2q2
+(A5)
+that we quantize using mechanical ladder operators
+ˆp = −i
+�
+ρℏΩ
+2
+�
+ˆb − ˆb†�
+ˆq =
+�
+ℏ
+2ρΩ
+�
+ˆb + ˆb†�
+. (A6)
+Identifying ˆu0(t) =
+�
+ℏ
+2ρΩˆb lets us express the quantum
+displacement field as
+ˆu(x, t) =
+�
+ℏ
+2ρΩh(x)ˆb(t) + H.c.
+(A7)
+and the single mode quantum strain tensor as
+ˆε(x, t) =
+�
+ℏ
+2ρΩ∇h(x)ˆb(t) + H.c.
+(A8)
+The multimode extension of this derivation is straight-
+forward and also follows the recipe from [48]
+ˆu(x, t) =
+M
+�
+m=1
+�
+ℏ
+2ρΩm
+hm(x)ˆbm(t) + H.c.
+=
+M
+�
+m=1
+�
+um(x)ˆbm(t) + H.c.
+�
+ˆε(x, t) =
+M
+�
+m=1
+�
+ℏ
+2ρΩm
+∇hm(x)ˆbm(t) + H.c.
+=
+M
+�
+m=1
+�
+εmˆbm(t) + H.c.
+�
+where mode m has frequency ωm, normalized shape
+hm(x) and is created (resp.
+annihilated) by operator
+ˆbm(t) (resp. ˆb†
+m(t)).
+The same derivation for the electric field gives
+ˆE(x, t) =
+N
+�
+n=1
+�
+−
+�
+ℏωn
+2ϵ0
+f n(x)ˆan(t) + H.c.
+�
+=
+N
+�
+n=1
+�
+En(x)ˆan(t) + H.c.
+�
+with ϵ0 the vacuum permittivity and normalized shape
+functions
+�
+V ϵr(x)f ∗
+n(x)f m(x)dV = δnm where ϵr(x) is
+the relative permittivity in the medium.
+
+9
+2.
+Piezoelectricity
+Before moving on to the piezoelectric Hamiltonian, we
+first specify the classical piezoelectric relations we will
+use.
+In a piezoelectric medium, we define the perme-
+ability µ, permittivity at constant strain ϵε, density ρ,
+stiffness tensor at constant electric field cE, its inverse
+sE and the strain-charge form piezoelectric coupling ten-
+sor d = sE : e. We write the piezoelectric constitutive
+relations in stress-charge from
+�
+S
+D
+�
+=
+�
+�
+cE −eT
+e
+ϵε
+�
+�
+�
+ε
+E
+�
+(A9)
+In this equation, the product is either a simple matrix
+product or a double dot product and the sum is always
+done on the last index(ices). For example,
+�
+cEε
+�
+ij
+≡
+�
+cE : ε
+�
+ij
+=
+�
+k,l
+cE,ijklεkl
+(A10)
+�
+ϵεE
+�
+i =
+�
+j
+ϵε,ijEj
+(A11)
+3.
+Electromechanical Coupling
+The stored electric and mechanical energies in a system
+at any time can simply be written as
+Eelec(t) = 1
+2
+�
+E(x, t) · D∗(x, t)dV
+(A12)
+Estrain(t) = 1
+2
+�
+S(x, t) : ε∗(x, t)dV
+(A13)
+From these and the piezoelectric constitutive relations,
+we have that the added energy due to piezoelectricity is
+given by
+Epiezo(t) = −
+�
+dV E(x, t) · eT : ε(x, t)
+(A14)
+which can finally be expanded in terms of the ladder op-
+erators
+ˆHpiezo = −ℏ
+�
+nm
+gnm
+�
+ˆan + ˆa†
+n
+� �
+ˆbm + ˆb†
+m
+�
+(A15)
+where we have switched to Schr¨odinger’s picture (ˆa(t) →
+ˆa, ˆb(t) → ˆb) to remain consistent with the main text and
+the EPR method, and with
+gnm = 1
+2
+�
+ωn
+ρϵ0Ωm
+�
+dV
+�
+f i
+n
+�∗
+(x)eijk (∇hm)jk (x)
+(A16)
+In terms quantities that can be easily extracted from
+simulation results, the coupling rate between the n-th
+electromagnetic and the m-th mechanical mode can be
+expressed as
+gnm
+2π =
+�
+ωn
+Ωm
+1
+ρϵ0
+�
+V , piezo E∗
+n(x) · eT : εm(x)dV
+4π
+��
+V E∗T
+n (x)ϵrEn(x)dV
+��
+V , SM u∗m(x) · um(x)dV
+(A17)
+We compute Eq. A17 for all solutions of the pure SM
+simulation using COMSOL’s feature Volume Integra-
+tion in the piezoelectric medium, calling the E-field from
+a chosen solution of the pure EM simulation using COM-
+SOL’s operator withsol.
+For example, one term con-
+tributing to the integrand in the numerator of Eq. A17
+therefore looks like
+conj(withsol(’sol2’, sext11)) * solid.eXX
+(A18)
+where sol2 refers to the solution of the pure EM simu-
+lation, sext11 is the 11 component of the external stress
+Sext = −eT · E and solid.eXX is the 11 component of
+the strain tensor.
+4.
+Electromagnetic-elastomechanical wave equation
+in a piezoelectric medium
+We can now derive the full wave equation by using the
+piezoelectric relations A9 along with Maxwell equations,
+where J, H and B = µH are the electric current, mag-
+netic and magnetic flux fields, respectively, and ρe is the
+electric charge density
+∇ × E = −iωB
+(A19)
+∇ × H = J + iωD
+(A20)
+∇ · B = 0
+(A21)
+∇ · D = ρe,
+(A22)
+We combine the two first Maxwell equations together
+with A9 to write a first equation of motion for the electric
+field
+∇ × µ−1∇ × E − ω2ϵεE = ω2e : ε
+(A23)
+While combining A1 with A9 gives
+∇ ·
+�
+cE : ε
+�
++ ρω2u = ∇ ·
+�
+eT · E
+�
+(A24)
+Appendix B: cQAD dispersive regime considerations
+The form of the Hamiltonian in Eq. 4 requires the dis-
+persive regime assumption for all interacting pairs of two
+electromagnetic modes (n, n′); |∆nn′| ≫ |ςnn′| and for
+all pairs of an electromagnetic and a mechanical mode
+(n, m); |∆nm| ≫ |gnm|. It also requires the perturbative
+assumption
+∆H
+kl ≫ Ej
+ℏ
+�
+ˆφj
+�p
+∀ k, l, j and ∀ p ≥ 4
+
+10
+expressed in the hybridized eigenmode approach (“H”) ,
+with ∆H
+kl := ξl − ξk.
+This condition is usually satisfied in cQED, but not
+always in cQAD [34, 50, 51]. For example, in the case
+analyzed in the main text, with a single junction la-
+beled j = 0, the second assumption is not respected,
+as ∆H
+q,(0,0) = 21 MHz < E0
+ℏ φ4
+q0 = 310 MHz where q refers
+to the qubit mode.
+This intermediate regime requires
+an additional transformation to go from the fourth order
+expansion of Eq. 3 to Eq. 4, namely a Schrieffer-Wolff
+transformation that removes the term proportional to
+�
+l̸=q ˆc†
+qˆcqˆc†
+l ˆcq + H.c., which has a time dependence of
+frequency ∆H
+ql and therefore cannot be neglected in the
+rotating wave approximation. Using [43], in the simple
+case of a single acoustic-like mode l with φl0 ≪ φq0, this
+changes the expression for the cross-Kerr coupling rates
+from E0
+ℏ φ2
+q0φ2
+l0 to E0
+ℏ φ2
+q0φ2
+l0
+1
+1+
+αq
+∆H
+ql
+. We can see this correc-
+tion has a significant effect for large
+αq
+∆H
+ql ratios, which is
+the case in our results. In terms of the EPRs, the qubit’s
+anharmonicity does not change, but the cross-Kerr cou-
+plings becomes
+χql = ℏ
+E0
+ξqξlpqpl
+4
+1
+1 +
+ℏ
+E0
+ξ2qp2q
+8∆H
+ql
+(B1)
+Appendix C: COMSOL simulation setup
+In COMSOL, all the information needed for a simu-
+lation is stored in a single file which, in the software,
+presents itself as a tree with several levels of nodes. At
+top level there is the file node, which consists in global
+definitions, components, studies and results. In a com-
+ponent node, we can define a model’s geometry, mesh-
+ing options and most importantly the physics interfaces,
+defining which fields will be solved in the model along
+with their equations of motion.
+Each of these nodes
+also feature subnodes which give additional details. In
+physics interfaces, the subnodes can be domain condi-
+tions, such as the main equation of motion or initial val-
+ues, or boundary conditions like fixed or free.
+1.
+Geometry in detail
+The geometry consists of a 5 × 30.5 × 17.8 mm3 3-D
+microwave cavity with two sapphire substrates: a bottom
+one (the qubit substrate, 5 × 2.6 × 0.420 mm3) on top of
+which the transmon and its antenna sit and a top one (the
+HBAR substrate, 5 × 2.6 × 0.04 mm3) which features the
+piezoelectric dome, made of AlN, looking down above the
+antenna. The two substrates are 3.0 µm apart in the z
+direction and the piezoelectric dome has a 100 µm radius
+and a 900 nm maximum height.
+These are typical values for the devices used in [13, 33,
+34]. To keep the simulations light, top substrates of only
+40 µm are used in the simulations used to produce the re-
+sults of the main text. This approximation is necessary to
+make the computations tractable, making some numer-
+ical results incomparable to actual experiments.
+They
+are still useful as proofs of concept and may actually be
+experimentally relevant in the coming years as one path
+being explored in the future experiments is the use of
+thinner HBARs, like the one presented by Bl´esin et. al in
+a recent proposal for microwave-optical transduction [52].
+As a further simplification, since the whole geometry
+is symmetric about the x−z plane at y = 0, the model is
+cut in half there (axes definitions can be seen in Fig. 1)
+and symmetry boundary conditions are used.
+Since we observed that the elastic waves excited in the
+HBAR by this configuration where well confined to a
+small region in the x − y plane, we define a cylinder cut
+of the HBAR substrate of the same radius as the piezo-
+electric dome. We will only solve for the displacement
+field inside of this cylinder instead of the whole HBAR
+substrate to avoid extending the required fine mesh any
+more than necessary. The boundaries of this cylinder cut
+are modeled as low reflecting boundaries to avoid any
+unphysical reflections.
+2.
+Physics modeling
+A 3-D microwave cavity is simply empty space sur-
+rounded by a perfect electric conductor (PEC). In most
+FE software, modeling the PEC is done using a bound-
+ary condition of the same name, which avoids having to
+model an actual layer of metal which would need to be
+meshed and would make the simulation more complex.
+Losses can be modeled using either scattering boundary
+conditions or perfectly matched layers (PML) on certain
+parts of the exterior boundary which correspond to phys-
+ical objects such as input and output ports.
+The transmon qubit is drawn as a 2-D object. The
+superconducting aluminum can simply be modeled as a
+PEC while the Josephson junction is modeled as a linear
+inductance using a lumped element boundary condition.
+This recipe without the lumped element can be used to
+model coplanar waveguides or 2-D resonators.
+All the nodes used in the Electromagnetic waves
+physics interface are detailed here, where an item with
+a ■ is a domain condition while an item with a □ is a
+boundary condition:
+■ The Wave equation, electric node is applied to
+all domains. It defines the EOM for the electromag-
+netic part of the simulation; the Helmholtz equa-
+tion in the frequency domain. Without any addi-
+tional node, this equation reads
+∇ × µ−1∇ × E − ω2ϵE = 0
+■ The effective medium 1 subnode defines a mod-
+ified relative susceptibility ˜ϵr = ϵr,AlN − e : dT /ϵ0
+and applies it to the domain corresponding to the
+
+11
+piezoelectric dome. This is the first of three steps
+for piezoelectric implementation.
+■ The External current density node adds a
+source term of the form −iωJext to the right-hand-
+side of the Helmholtz equation for the domain cor-
+responding to the piezoelectric dome.
+We define
+Jext = iωe : ε. This is the second step for piezo-
+electric implementation.
+□ The Perfect electric conductor 1 & 2 nodes
+define perfectly reflecting boundaries for the elec-
+tromagnetic fields. The first node is applied to the
+exterior boundaries (sides of the cavity) while the
+second one is applied to the transmon’s geometry
+parts since it is implemented as a 2-D object. The
+boundary equation is simply n × E = 0, where n is
+a normal vector to the boundary element.
+□ The Lumped element node acts like a linear cir-
+cuit containing at most a resistor, a capacitor and
+an inductor.
+It has to be connected to conduc-
+tors (perfect electric conductors in our case) on two
+sides. We use it to act like the linear part of our
+Josephson junction, so we define it as an inductor.
+□ The Perfect magnetic conductor node is sim-
+ply a symmetry boundary condition for the electric
+field. This allows us to cut the whole system in half
+along the x − z plane at y = 0 since the system is
+the same on both sides.
+And for the Solid Mechanics interface, which is only
+solved for in the piezoelectric dome and a cylinder cut
+of the HBAR above it, with the same height as the top
+substrate and the same radius as the dome:
+■ The Linear elastic material node defines the
+EOM for the solid mechanics part of the simula-
+tion in all selected domains (piezoelectric dome and
+HBAR cylinder cut). The equation is the standard
+elastic wave equation: ∇ · S + ρω2u = 0. Without
+additional nodes we have S = cE : ε
+■ The subnode External Stress adds the following
+external stress Sext = −eT · E to S in the domain
+corresponding to the piezoelectric slab. This is the
+third and final step for piezoelectric implementa-
+tion.
+■ The Prescribed displacement node can be used
+to simplify the simulation to only include the z
+component of the displacement field uz, leaving ux
+and uy at 0 everywhere. This has been shown to
+be a very good approximation while drastically re-
+ducing the number of spurious modes in the results
+of the simulation.
+□ The Free boundary condition is applied on the
+sides and bottom of the piezoelectric dome and on
+top of the HBAR.
+□ The Low reflecting boundary node is used to
+avoid reflection on the unphysical boundaries on
+the side of the HBAR’s cylinder cut.
+Note that
+a perfectly matched layer (PML) is usually pre-
+ferred to this kind of boundary conditions, but un-
+fortunately in our case it can’t be implemented (see
+note).
+□ The Symmetry boundary condition node is
+used on the the boundaries on the x − z plane at
+y = 0 to cut the system in half as well.
+3.
+Meshing procedure
+To properly mesh the HBAR, we need to respect the
+rule of thumb of 5 elements per wavelength in the longi-
+tudinal direction while also resolving higher-transverse-
+order modes since we expect non-negligible coupling to
+them.
+In addition to increasing the simulation size, a
+finer transverse mesh was also observed to increase the
+number of spurious modes (see next section). Because
+no simple metric characterizing the spurious modes is di-
+rectly available in COMSOL’s results, a mesh refinement
+study could not be used to limit their presence.
+For our model, a handmade mesh was created for the
+part of the geometry where solid mechanics are solved
+using a mapped and a swept node, which allow us to
+control the number of meshing points in all three cylin-
+drical directions using distribution subnodes. For all
+simulations whose results are reported in this work, the
+cylindrical region is divided into a shell with inner radius
+30 µm and a mapped mesh with 10 azimuthal and 6
+radial elements and center region, which is a free quad
+surface mesh with maximum element size 5 µm.
+The
+rest of the simulation space can then be meshed with au-
+tomatically generated tetrahedrons, where the only user
+input are size specifications. We observed that the pa-
+rameters with the most impact were the maximum el-
+ement size and z-stretching ratio. A light convergence
+analysis was performed to ensure the meshing was suf-
+ficiently dense near the junction where the electric has
+a strong gradient. Using this meshing procedure, a hy-
+bridized eigenmode simulation finds 150 modes in 2 hours
+on a computer with 64 GB of memory.
+4.
+Spurious modes
+FE eigenmode simulations can converge to modes that
+are not physical, referred to as spurious modes [39–41].
+They appear in solid mechanics simulations at GHz fre-
+quency and with fine mesh features, yielding field dis-
+tributions made out of point defects that can be seen
+in Fig. 4. In our situation, these modes can appear in
+greater numbers than physical modes. They cause several
+problems. First, if one wants to find a certain number
+of modes (higher order transverse modes of the HBAR
+
+12
+in our case), one typically has to ask the solver for many
+more modes than this number. Thus, the presence of spu-
+rious modes in the results artificially increases the solve
+time.
+Another detrimental effect of the spurious modes is
+“EPR dilution,” where the EPR of a physical mode will
+be shared among several spurious modes that are nearby
+in frequency. The coupling of the qubit mode to a spuri-
+ous mode is typically not higher than 5×103 kHz, but in
+certain cases the frequency difference between a spurious
+mode and a physical or qubit mode can be lower, creat-
+ing a significant hybridization in the hybrid simulations.
+This reduces the value of the physical or qubit mode’s
+EPR and the quantities obtained through it, such as the
+cross-Kerr coupling rate.
+No method was found to entirely remove spurious
+modes from the results of solid mechanics eigenmode
+simulations of an HBAR. We also could not find any
+one-number metric that distinguishes them from phys-
+ical modes, and their estimated convergence error (using
+COMSOL’s error estimates for example) is lower than
+that of the physical modes, meaning stronger convergence
+requirements make this issue worse. The only two things
+one can do to mitigate this problem is optimize the mesh-
+ing (previous section) and post-process the data.
+FIG. 4. Illustration of two modes with neighboring frequen-
+cies in an unhybridized solid mechanics simulation, a) a spu-
+rious mode and b) a physical LG(1, 1) mode.
+5.
+Solver settings and convergence
+All simulations used in this work are done using ”eigen-
+mode” COMSOL studies that uses a direct MUMPS
+solver, with most settings kept as default. However, in
+the case of hybridized simulations, one change needs to
+be done in order for the solver to converge to sensible re-
+sults [53]. The COMSOL settings ”Scaling” and ”Resid-
+ual Scaling” should be set to 102 for the electric field ⃗E
+and to 10−20 for the displacement field ⃗u. This is done
+in the nodes found under Study ▶ Solver Configura-
+tions ▶ Solution ▶ Dependant variables.
+6.
+Acoustic Polarization and Post-Processing
+After the results are computed by COMSOL, we apply
+a post-processing procedure in order to extract quantities
+of interest from the simulation. We expect the physical
+eigenmodes of the HBAR to include Laguerre-Gaussian
+or Hermite-Gaussian modes with longitudinal (compo-
+nent 33 of ε) or shear (components 13 or 23) polariza-
+tion.
+These modes admit an analytical expression for
+their longitudinal mode profile (uz at the top surface).
+By computing the pointwise distance between the mode
+profiles of each eigenmode in the results and these ana-
+lytical mode profiles, we can find the best match in the
+results, and simply assign all results that aren’t good fits
+for any of the reference modes as spurious modes. Ad-
+ditionally, the longitudinal or shear nature of a physical
+eigenmode can be simply extracted using, for example,
+the weight of a tensor component in the strain energy
+of the mode. Formally, the polarization is attributed to
+component c, with c in {11, 12, 13, 21, 22, 23, 31, 32, 33
+}, if c respects
+1
+4Re
+�
+V,SM Sc(x) : ε∗
+c(x)dV.
+Estrain,k
+>
+1
+4Re
+�
+V,SM S˜c(x) : ε∗
+˜c(x)dV.
+Estrain,k
+(C1)
+for all other components ˜c. To keep things simple, all
+physical modes are recognized and labeled according to
+their uz profiles. For other mechanical resonator geome-
+tries, physical modes can usually also be visually dis-
+tinguished from spurious ones, but a similar automated
+method for distinguishing them from spurious modes may
+need to be developed.
+Appendix D: Hybrid EPR method
+The goal of the EPR method, developed by Minev et.
+al [23], is to compute the coefficients of the cQED Hamil-
+tonian from the so-called energy-participation ratios. We
+will mirror a simplified version of the derivation from this
+paper but in the case of a hybrid Hamiltonian (Eq. 3).
+The energy of classical mechanical resonator’s eigen-
+mode oscillates in time between strain and kinetic energy.
+Analogously, for a classical electronic circuit, it oscillates
+between inductive and capacitive energy. In the unhy-
+bridized eigenmode approach, for each mechanical (elec-
+tromagnetic) oscillator in the system, the time-averaged
+strain (linear inductive) energy is equal to the time aver-
+aged kinetic (capacitive) energy. Equivalently, each form
+of energy’s time average is equal to half the time-averaged
+
+b)
+a)13
+total linear [54] energy for this mode
+Elin. ind,n = Eelec,n = 1
+2Etotal, EM,n
+Estrain,m = Ekin,m = 1
+2Etotal, mech,m.
+(D1)
+Elin. ind,n is the sum of the energy stored in the magnetic
+field Emag,n as well as the energies of the lumped element
+inductances �
+j
+1
+2LjI2
+nj. The EPR is then defined as [23]
+pkj := linear inductive energy in junction j in mode k
+total linear inductive energy in mode k
+(D2)
+In the hybridized eigenmode approach, the definition
+of the total linear inductive energy in mode k is extended
+to include the strain energy:
+Elin. ind,k + Estrain,k = Eelec,k + Estrain,k
+= 1
+2
+�
+ˆHlin
+�
+k = ℏ
+2
+N+M
+�
+l=1
+ξl
+�
+ˆc†
+l ˆcl
+�
+k
+(D3)
+We have introduced • as the time average and ⟨•⟩k as
+the expectation value of an operator over a state with
+excitations in a single mode. The numerator of Eq. D2
+is unchanged since a junction’s inductive energy does not
+include any mechanical part. It is defined as the the time-
+averaged linear inductive excitation energy (as opposed
+to absolute energy) at junction j when only mode k is
+excited
+�1
+2Ej ˆφ2
+j
+�
+k
+−
+�1
+2Ej ˆφ2
+j
+�
+0
+(D4)
+Defining a general Fock state for this system as
+|µ1, . . . , µN+M⟩
+we see that writing the EPR in terms of a single-mode
+Fock state makes it independent of the excitation number
+and links it to the ZPF of the junction’s flux
+pkj = ⟨µk| 1
+2Ej ˆφ2
+j |µk⟩ − ⟨0| 1
+2Ej ˆφ2
+j |0⟩
+1
+2
+�N+M
+l
+ℏξl ⟨µk| ˆc†
+l ˆcl |µk⟩
+=
+Ejφ2
+kj
+1
+2ℏξk
+(D5)
+In practice, we extract the denominator of Eq. D2 by
+performing finite-sum integrals over the volumes where
+the fields E and u are defined
+Eelec,k = 1
+4Re
+�
+V
+Ek(x) · D∗
+k(x)dV
+(D6)
+Estrain,k = 1
+4Re
+�
+V,SM
+Sk(x) : ε∗
+k(x)dV
+(D7)
+Appendix E: Modeling dissipation
+Our simulation framework can also be used to study
+dissipation in cQAD devices. In this Appendix, we in-
+troduce the basics for this next step in the method. The
+relevant loss mechanisms in a ℏBAR-like device can be
+separated into two different categories based on how they
+can be estimated using simulations.
+In the first case,
+which we call semi-analytical loss, a lossy element (sur-
+face or volume) has an intrinsic quality factor that is
+taken from the literature. Then, its participation in the
+overall quality factor is weighted by the element’s energy-
+participation ratio which is computed from the results of
+the simulation. These ratios are referred to as “lossy”
+EPRs to distinguish them from the junction EPRs dis-
+cussed in the rest of the paper, even though the principle
+is the same. In this section, we illustrate three examples
+of such dissipation mechanisms: bulk dielectric and sur-
+face inductive losses [23, 55, 56] as well as losses due to
+surface roughness in the acoustic resonator. The second
+category of losses includes mechanisms that can be fully
+characterized numerically, so we call it numerical loss.
+One such mechanism present in our system is so-called
+phonon diffraction loss, where we consider all phonons
+leaving the center region of the HBAR as lost and quan-
+tify this using a numerical flux integration.
+a.
+Semi-analytical loss calculations
+For a given EPR-based loss mechanism L, its overall
+contribution to a mode’s quality factor is a weighted in-
+verse sum with contributions from all lossy elements
+1
+QL
+k
+=
+�
+l
+pL
+kl
+QL
+l
+(E1)
+Bulk dielectric losses of the electromagnetic field are
+characterized by the loss tangent δl of a lossy solid l.
+This loss tangent is the inverse of an intrinsic quality
+factor Qdiel
+l
+, and the contribution to the overall quality
+factor of a mode from one such solid is weighted by its
+energy-participation ratio
+pdiel, bulk
+kl
+= 1
+Ek
+1
+4Re
+�
+Vl
+E∗
+m · ϵEmdV.
+(E2)
+We have defined the total energy of mode k as Ek =
+2Eelec,k + 2Estrain,k (see Eq. D1).
+Surface inductive losses are caused by surface currents
+and result in Ohmic loss. These are characterized by an
+intrinsic quality factor estimated at unity for metals such
+as the copper of the microwave cavity, and higher than
+105 for SC aluminum [23]. The contribution of a lossy
+surface l is computed using
+pind, surf
+kl
+= 1
+Ek
+λlµl
+4 Re
+�
+surfl
+H∗
+k,∥ · Hk,∥ds
+(E3)
+where λl is the skin depth of the surface’s material and
+µl its permeability.
+Acoustic losses due to surface roughness are estimated
+using a method from Ref. [57]. Surface roughness limits
+
+14
+the quality factor to
+Qrough
+k
+=
+h2
+2nkσ2 ,
+where nk is the longitudinal mode number, σ2 =
+�
+z2�
+is
+the height variance of the surface assuming a Gaussian
+distributed roughness, and h is the height of the HBAR
+such that
+���qk
+���h = nkπ, where qk is the mode’s wave
+vector. This quality factor only applies to the HBAR, so
+it has to be weighted by the fraction of energy stored in
+the mechanics 2Estrain,k
+Ek
+.
+b.
+Numerical loss calculations
+The plano-convex shape of the HBAR was chosen to
+provide both longitudinal and transverse confinement to
+the acoustic modes. However, to study the effect of im-
+perfections in this geometry, such as the finite size of the
+dome, we can use simulations to calculate the acoustic
+energy leaving the Fabry-P´erot cavity (the region of the
+sapphire substrate above the piezoelectric dome). This
+can be treated as loss because, even if the substrate has
+a finite size and reflecting boundaries, the timescale on
+which the energy is reflected back into the mode region
+is much longer than the typical timescale of operations
+we’re interested in [33, 34]. We compute a quality fac-
+tor due to diffraction loss using a flux integral of the
+mechanical Poynting vector P a
+Qdiff
+k
+= ωk
+Ek
+�
+Sd P a · dσ
+(E4)
+.
+Here Sd is a cylindrical surface defines the bound-
+ary of the acoustic cavity, and is parametrized by (x0 +
+R cos θ, R sin θ, z), where x0 = 1 mm is the position of
+the center of the antenna, R = 90 µm, θ ∈ [−π/2, π/2]
+and z ∈ [−0.9, 40] µm, which includes the entire height
+of the substrate.
+c.
+Effects of hybridization on losses
+An interesting new feature that arises from our simula-
+tion framework is the ability to study mechanical losses in
+hybrid qubit- or cavity-like modes, and electromagnetic
+losses in mechanical-like modes. These new effects can
+only be studied once one has access to the full dynam-
+ics of the hybridized eigenmodes and are thus a unique
+feature of the hybridized approach.
+As an example, we show how the qubit mode, once hy-
+bridized with the HBAR in the same dispersive regime
+as in the main text, acquires a new loss channel through
+phonon diffraction. Fig. 5 shows the LG(0, 0) mode of
+the unhybridized and hybridized simulations as well as
+the qubit-like mode in the hybridized simulation. The
+FIG. 5.
+Acoustic displacement along a line defined by the
+interface between the piezoelectric dome and the sapphire and
+the symmetry axis of the simulation (y = 0).
+The sharp
+features on the qubit mode are the result of hybridization
+with spurious modes.
+displacement profile of the bare mechanical mode (black)
+and the hybridized mechanical-like mode (green) are al-
+most identical. However, we see that for the qubit-like
+mode (blue), the piezoelectric coupling to the qubit elec-
+tric field, which is asymmetric due to the thin lead of
+the antenna, results in an asymmetric displacement field.
+This asymmetry is not captured in the unhybridized ap-
+proach. Such a modification of the acoustic mode shape
+could lead to additional loss through imperfect mode con-
+finement. Studying these effects using the techniques de-
+scribed in the previous section will be the subject of fu-
+ture work.
+
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@@ -0,0 +1,2276 @@
+arXiv:2301.03623v1  [quant-ph]  9 Jan 2023
+Weyl conjecture and thermal radiation of finite systems
+M. C. Baldiotti,1, ∗ M. A. Jaraba,1, † L. F. Santos,2, ‡ and C. Molina3, §
+1Departamento de Física, Universidade Estadual de Londrina, CEP 86051-990, Londrina-PR, Brazil.
+2Instituto de Física, Universidade de São Paulo,
+Caixa Postal 66318, CEP 05315-970, São Paulo-SP, Brazil.
+3Escola de Artes, Ciências e Humanidades, Universidade de São Paulo,
+Avenida Arlindo Bettio 1000, CEP 03828-000, São Paulo-SP, Brazil.
+In this work, corrections for the Weyl law and Weyl conjecture in d dimensions are obtained and
+effects related to the polarization and area term are analyzed. The derived formalism is applied on
+the quasithermodynamics of the electromagnetic field in a finite d-dimensional box within a semi-
+classical treatment. In this context, corrections to the Stefan-Boltzmann law are obtained. Special
+attention is given to the two-dimensional scenario, since it can be used in the characterization of
+experimental setups. Another application concerns acoustic perturbations in a quasithermodynamic
+generalization of Debye model for a finite solid in d dimensions. Extensions and corrections for
+known results and usual formulas, such as the Debye frequency and Dulong-Petit law, are calculated.
+Keywords:
+Weyl law, Weyl conjecture, quasithermodynamics of the electromagnetic field,
+quasithermodynamics of acoustic perturbations, generalized Debye model
+I.
+INTRODUCTION
+The analysis of thermal radiation is widespread in a large variety of finite-temperature systems. Theoretical research
+includes treatments based on fluid dynamics and/or quantum field theory. Laboratory and observational applications
+involve particle phenomenology in colliders, properties of solids in laboratories, characteristics of the cosmic microwave
+background in dedicated observatories, among many more setups. From lower-dimensional settings to models with
+arbitrary number of dimensions, thermal radiation frequently plays a significant role.
+A common procedure for the investigation of thermal phenomena is based on the definition of a suitable thermody-
+namic limit within a statistical mechanics model. Usually, the transition from statistical mechanics to thermodynamics
+implies that the boundary conditions are neglected. In this way, the dependence of the obtained results with the actual
+volume and shape of the physical system is not considered. Although this approach is useful for many purposes, the
+strict thermodynamic limit disregards many interesting insights about the actual system of interest. One way to mit-
+igate this problem is to define an intermediary regime between the pure statistical-mechanic treatment and the strict
+thermodynamic regime, where the volume of the system is large, but finite. This is the so-called quasithermodynamic
+limit [1].
+On a very general level, the characteristics of thermal radiation in a given setup are directly linked to the asymptotic
+distribution of eigenvalues of a suitable wave equation. One of the first investigators to explore this connection was
+Rayleigh, studying the problem of stationary acoustic waves in a cubic room [2, 3]. Rayleigh’s analysis showed the
+importance of a term proportional to the volume of the room and to the cube of the mode frequency (the V · ν3
+term).
+This result appeared in the (incorrect) description of the thermal radiation with the Rayleigh-Jeans law.
+Eventually Planck improved this purely classical analysis, but even within the quantum description the eigenvalue
+distribution remains unaltered. The same problem, and hence with the same V · ν3 term as the result, emerges in the
+investigation of the vibration modes in a solid, with the so-called Debye model. In this treatment, Debye proposed
+that the asymptotic behavior of the eigenvalues do not depend on the shape of the solid. This proposal was rigorously
+proved by Weyl, and today it is known as the Weyl law. An overview of this development can be seen in [4, 5].
+A central question considering the strict thermodynamic regime is when the approximation of infinite volume
+adequately describes a real physical system. For the treatment of this issue, it is necessary an estimate of the terms
+that are being dropped in the thermodynamic limit. A first step in this direction is given by Weyl conjecture, which
+predicts corrections proportional to the area of the body. This conjecture has been proven in a variety of domains
+and in this process, several methods can be applied. For instance, asymptotically expanding the solutions of the
+Helmholtz equation, using the Neumann-Poincaré construction for the Brownell Green’s function and considering the
+decomposition of the mode density via multiple reflection expansion [6, 7].
+∗ baldiotti@uel.br
+† jaraba.marcosrod@uel.br
+‡ luisf@usp.br
+§ cmolina@usp.br
+
+2
+For many important problems, involving for example the electromagnetic field, vector solutions and polarization
+effects must be considered. For this purpose, a possible approach is to decompose the vector fields into solutions of
+the suitable scalar wave equation [6, 8–10]. Following this program, many works in the pertinent literature indicate
+that the area term (i.e., the term of Weyl conjecture) does not participate in describing the behavior of the thermal
+electromagnetic radiation.
+Previous comments refer mainly to usual scenarios with three dimensions. But the relevance of the proposed treat-
+ment appears in models with different dimensionalities. For instance, the thermodynamic and quasithermodynamic
+analyses of two-dimensional systems have practical applications in the so-called single-layer materials [11], with high-
+lights to the graphene [12]. As a more theoretical application, we can mention the thermodynamic properties of
+photon spheres [13], thermal radiation of the two-dimensional bosonic and fermionic modes of black holes [14], as
+well the description of thermodynamics properties of BTZ black holes [15]. Considering three-dimensional systems,
+corrections in thermal radiation play an important role in the analysis of the background microwave radiation [16].
+The development is also relevant in the thermodynamic description of the phenomenon of sonoluminescence (hot
+spot theory), in which pulses of light are created by means of the insertion of sound waves into liquids or gases [17].
+Systems with more than three spatial dimensions are also explored. For instance, Hawking emission of a black hole
+is altered in brane-world scenarios [18–20]. Thermal radiation phenomena associated with gravitational emission in
+brane-world models could be significant in the early Universe. Similar effects could be expected in colliders and near
+active astrophysical objects, within models involving extra dimensions (see for example [21] and references therein).
+Proposals linking anti-de Sitter geometries and conformal field theories (AdS/CFT correspondences) offer a great
+variety of applications for d-dimensional thermodynamic results [22–25].
+In the present work, Weyl law and its extensions are explored through an intuitive approach. Quasithermodynamic
+analysis and its description of systems with a finite volume and relevant boundary conditions are a central issue in
+this article. The developed formalism is applied on the quasithermodynamics of the electromagnetic field in a cavity
+and acoustic perturbations in a solid. Generalizations for d-dimensional setups are derived. One of the contributions
+of this work is to incorporate polarization effects directly into the asymptotic expansions of the mode distributions,
+using mixed boundary conditions. We emphasize the role of the area term, showing that, under certain conditions, a
+distinct quasithermodynamic behavior emerges.
+In addition to the correction coming from the eigenvalue distribution, an expected characteristic of the phenomenol-
+ogy of black-body radiation in finite cavities is the existence of a minimum energy.1 For the two-dimensional case,
+this issue was studied in [26]. Generalizing this development for arbitrary dimensions, our treatment allows us to
+compare the effects associated to the minimum energy with those related to corrections of the spectral distribution.
+This work is organized as follows. In section II, the proposed formalism associated to the Weyl law is established.
+In section III, with the techniques introduced, higher-order corrections to the Weyl law are obtained. More complex
+setups are considered in section IV, where mixed boundary conditions and degeneracies are treated. In section V
+we turn to physics applications, linking the results previously obtained with thermodynamics and quasithermody-
+namics. In section VI, the thermodynamic treatment of the electromagnetic field in a finite cavity is conducted.
+The quasithermodynamics of acoustic perturbations is considered in section VII, where Debye model is analyzed and
+extended. In section VIII final comments and future perspectives are presented. Further details on the calculation of
+the hypervolume associated to the axes and counting functions are discussed in appendices A and B.
+II.
+SCALAR FIELD AND WEYL LAW
+Electromagnetic and mechanic perturbations in cavities and solids are the main interest in the present work.
+However, as we will see later on, the thermodynamics of those systems can be described in terms of a simpler scalar
+perturbation. Let us consider a scalar field ψ (x1, . . . , xd) in d dimensions, confined in a cubic cavity of size L, which
+respects the Helmholtz equation,
+∇2
+dψ (x1, . . . , xd) + k2ψ (x1, . . . , xd) = 0 , xi ∈ Ωd , k ∈ R .
+(1)
+In Eq. (1), ∇2
+d denotes the d-dimensional Laplacian. The hypervolumes of domain Ωd and its boundary ∂Ωd are given
+respectively by
+|Ωd| = Ld , |∂Ωd| = 2dLd−1 .
+(2)
+1 This lower bound on the energy of the system would be associated to the quantum vacuum, according to [26].
+
+3
+Two different boundary conditions for Eq. (1) will be initially explored, namely the Neumann condition (∂nψ = 0 at
+∂Ωd) and the Dirichlet condition (ψ = 0 at ∂Ωd).
+Solutions of the d-dimensional Helmholtz equation (1) can be constructed using the one-dimensional version of (1),
+which are
+φ±
+n (x) = e−iπ/4 ± eiπ/4
+√
+4L
+�
+eiknx ± e−iknx�
+, kn = π
+Ln , n =
+�
+0, 1, 2, . . .(+)
+1, 2, 3, . . .(−) .
+(3)
+The label (+) indicates solutions satisfying Neumann conditions, while (−) denotes solutions obeying Dirichlet con-
+ditions. From the one-dimensional solutions, d-dimensional generalizations can be constructed as
+ψ± (x1, . . . , xd) = φ±
+n1(x1)φ±
+n2(x2) · · · φ±
+nd(xd) ,
+(4)
+with
+k2 = π2
+L2 (n2
+1 + · · · + n2
+d) .
+(5)
+It should be noted that, for the Dirichlet case, the solutions with ni = 0 should be excluded in order to avoid the null
+solution.
+Considering Eq. (5), the set of solutions of the Helmholtz equation (1) can be labeled by points in a discrete lattice.
+Specifically, we consider the d-dimensional Cartesian lattice Ld
+ǫ,
+Ld
+ǫ ≡
+�
+(ǫ1, ǫ2, . . . , ǫd) ∈ Rd : ǫi = ǫni , ni = 0, 1, 2, . . .
+�
+,
+(6)
+where the (real and positive) ǫ is the lattice parameter. We denote the region ˜Ωd ⊂ Rd as one of the 2d sections of
+the d-dimensional sphere of radius k, i.e.,
+˜Ωd ≡
+�
+(ǫ1, ǫ2, . . . , ǫd) ∈ Rd : ǫ2
+1 + ǫ2
+2 + · · · + ǫ2
+d < k2 , ǫi ⩾ 0
+�
+.
+(7)
+The main questions addressed in the present work can be formulated as counting problems. In this direction, let
+us define a counting function N (d)(k) as
+N (d) (k) =
+�
+# (n1, . . . , nd) : π2
+L2
+�
+n2
+1 + · · · + n2
+d
+�
+< k2
+�
+.
+(8)
+The function N (d)(k) can be expressed as
+N (d)(k) = card
+�
+Ld
+ǫ ∩ ˜Ωd
+�
+, ǫ = π
+L ,
+(9)
+where “card” denotes the cardinality of the set [27]. It should be noted that the complete dependence of N (d)(k) on
+k is described by ˜Ωd, since the lattice is (for now) fixed. We are interested in the asymptotic behavior of N (d)(k) as
+k → ∞.
+One method for the analysis is to adopt a “coarse grained” version of the lattice, where the counting of the discrete
+points is substituted by the calculation of volumes. More specifically, the function N (d)(k) is approximated by the
+volume Vd(k) generated by hypercubes of side length ǫ, centered on the points of the lattice belonging to Ld
+ǫ ∩ ˜Ωd(k).
+We illustrate this coarse graining in Figs. 1 and 2 of the next section. The volume of each hypercube is ǫd, and hence
+Vd (k) = ǫdN (d)(k). The total volume Vd (k) tends to infinity as k → ∞, N (d)(k) → ∞ and ǫ is kept fixed.
+Expression (9) and the coarse graining introduced are a well-known algorithm for the counting process, in which
+the lattice is kept fixed. In the present work, we propose an alternative approach. Instead of fixing ǫ (that is, the
+lattice), we fix the volume Vd (k). In other words, we propose to maintain the domain fixed and adjust the lattice.
+This can be accomplished rescaling ǫi,
+ǫi −→ ǫi = k−1ǫi = ǫni , ǫ = π
+kL .
+(10)
+With the rescaling, both the counting function N (d) (k) and the volume Vd become dependent only on the lattice,
+that is,
+Vd (ǫ) = ǫdN (d)(ǫ) .
+(11)
+
+4
+Intuitively, Vd(ǫ) should tend to the volume |˜Ωd| as the lattice becomes more dense (ǫ → 0+), where
+���˜Ωd
+��� = 2−dωd , ωd =
+πd/2
+Γ
+� d
+2 + 1
+� ,
+(12)
+and Γ denotes the usual gamma function. The precise development of this relation will lead to the so-called Weyl
+law, which will be explicitly shown with the approach proposed in this work.
+To clarify the notation, let us denote the counting function for the Neumann case as N (d)
++ , and analogously N (d)
+−
+for
+the Dirichlet case. We focus on the difference in the counting problem considering the Neumann and Dirichlet setup.
+This difference is a result of the inclusion of the points belonging to the ǫi-axis for the calculation of N (d)
++ , and the
+exclusion for N (d)
+− .
+Let us treat the Neumann boundary condition first, since (as will be shown in later sections) N (d)
+−
+can be expressed
+in terms of N (d)
++ . The counting function N (d)
++
+can be written as
+N (d)
++ (ǫ) =
+M(ǫ)
+�
+n1=0
+· · ·
+M(ǫ)
+�
+nd−1=0
+
+�
+1 − ǫ2(n2
+1 + · · · + n2
+d−1)
+ǫ
+ , M(ǫ) =
+�1
+ǫ
+�
+,
+(13)
+with ⌊x⌋ representing the integer function (or floor function) of x [28]. Using the sawtooth function η(x), η(x) ≡ x−⌊x⌋,
+Eq. (11) is written as
+V +
+d (ǫ) = ǫd−1
+M(ǫ)
+�
+n1=0
+· · ·
+M(ǫ)
+�
+nd−1=0
+�
+1 − ǫ2(n2
+1 + · · · + n2
+d−1) − ǫd
+M(ǫ)
+�
+n1=0
+· · ·
+M(ǫ)
+�
+nd−1=0
+η
+�1
+ǫ
+�
+.
+(14)
+Since 0 ≤ η < 1, the second term of Eq. (14) obeys the following property:
+0 ≤ ǫd
+M(ǫ)
+�
+n1=0
+· · ·
+M(ǫ)
+�
+nd−1=0
+η < ǫ [ǫ + ǫM (ǫ)]d−1 ,
+(15)
+implying that
+lim
+ǫ→0+ ǫd
+M(ǫ)
+�
+n1=0
+· · ·
+M(ǫ)
+�
+nd−1=0
+η
+�1
+ǫ
+�
+= 0 .
+(16)
+Previous results lead to the Weyl law, as we will show in the following. Considering the first term in Eq. (14), we
+observe that
+ǫ =
+M −1
+1 + ηM −1 ⇒ lim
+ǫ→0+ ǫ =
+lim
+M→∞
+M −1
+1 + ηM −1 =
+lim
+M→∞ M −1 ,
+(17)
+and we can write
+lim
+ǫ→0+ ǫd−1
+M(ǫ)
+�
+n1=0
+· · ·
+M(ǫ)
+�
+nd−1=0
+�
+1 − ǫ2 �
+n2
+1 + · · · + n2
+d−1
+�
+=
+lim
+M→∞
+1
+M d−1
+M(ǫ)
+�
+n1=0
+· · ·
+M(ǫ)
+�
+nd−1=0
+�
+1 − n2
+1 + · · · + n2
+d−1
+M 2
+.
+(18)
+The last terms in Eq. (18) is the Riemann sum which gives the volume of ˜Ωd. Hence, using Eq. (16), we conclude that
+lim
+ǫ→0+ V +
+d (ǫ) = 2−dωd .
+(19)
+Result (19) is the Weyl law for the scalar field in a cubic cavity, with Neumann boundary conditions.
+Although the Weyl law is a well-known result, the procedure adopted here (that is, a coarse graining with a scaling
+which fixes the domain and modifies the lattice) can be employed in the improvement of the result (19). Also, this
+approach will allow us to consider different boundary conditions and polarization effects.
+
+5
+III.
+BEYOND THE WEYL CONJECTURE
+Weyl law can be considered as the zero-order correction of the counting function. The first-order correction was
+conjectured by Weyl and later proven by Ivrii [31]. In our notation, the Weyl conjecture can be written as
+ǫdN (d)
+± (ǫ) = 2−dωd ± 2−ddωd−1ǫ + O(ǫwd) ,
+(20)
+where wd =
+�
+d2 − d + 1 + 1/d
+�
+/ (d − 1) [32]. In the “big-O” sense, no higher-order corrections are known2 [31, 32].
+Although for the scalar case the above correction is always the most relevant, as we will see, when considering the
+vector problem this correction may cancels. Therefore, its important to go beyond the first correction. For this goal
+a special type of “averaged corrections” can be considered. It is defined using Gaussian logarithmic averages, or the
+called Brownell’s ˜O formalism [32]. One says that f = ˜O (g) if, for some x0,
+����
+� ∞
+x0
+e− 1
+2 ρ2(ln
+y
+x′ )
+2
+df(x)
+dx
+����
+x′ dx′
+���� ≤ δρg (y) ,
+(21)
+for every ρ > 0 and some δρ < ∞. For the case d = 2, 3, it is found that
+ǫ2N (2)
+± (ǫ) = π
+4 ± ǫ + ǫ2
+4 + ˜O(ǫ ˜
+w2) ,
+(22)
+ǫ3N (3)
+± (ǫ) = π
+6 ± 3π
+8 ǫ + 3
+4ǫ2 ± ǫ3
+8 + ˜O(ǫ ˜
+w3) ,
+(23)
+where ˜w2 > 2 and ˜w3 > 3 [30, 32]. As will be seen below, the corrections (22)-(23) are given by a polynomial expansion
+whose coefficients are related to the geometric “shape” of the ˜Ωd boundary. Based on this geometric argument, it will
+be seen that we can always decompose ǫdN (d)
++ (ǫ) as the sum of two contributions: one continuous component, that
+we will denote by F(d)(ǫ), and other non-continuous.
+Our goal in this section is to explore the relation between the functional form of F(d)(ǫ) and the corrections derived
+from the Brownell formalism. We will employ a “bottom-up approach”, considering in detail particular values for d,
+and then extrapolating the results for general d.
+The one-dimensional case (d = 1) is trivial, however it is crucial to construct the bottom-up approach. For this
+case, the Brownell corrections coincide with the exact corrections. From Eq. (13), the counting function for Neumann
+boundary condition is given by
+ǫN (1)
++ (ǫ) = 1 − ǫη
+�
+ǫ−1�
+∼ 1 ⇒ F(1)(ǫ) = 1 .
+(24)
+For the Dirichlet boundary condition the result is similar:
+ǫN (1)
+− (ǫ) = ǫ[N (1)
++ (ǫ) − 1] = 1 − ǫ[1 + η
+�
+ǫ−1�
+] ∼ 1 .
+(25)
+For the two-dimensional Neumann scenario, given the shape of ˜Ω2 boundary, we can decompose ǫ2N (2)
++
+as
+ǫ2N (2)
++ (ǫ) = π
+4 + axes-boundary area + area of the boundary curve.
+(26)
+This result is illustrated in Fig. 1. More precisely, the “axes-boundary area” is the excess area localized around the
+axes (which are not included in ˜Ω2). As seen in Fig. 1, this quantity is
+axes-boundary area = ǫ + ǫ2
+4 .
+(27)
+Notice that we are considering the contribution of each axis in the interval [0, 1], implying that the contribution of
+the central square is ǫ2/4. The contribution of the area of the boundary curve is unknown.3 However we know that it
+2 One writes f(x) = O(g(x)) if there exists a real δ > 0 and x0 such that |f (x)| ≤ δg (x) for all x > x0. That is, f is smaller than g as
+x → ∞, and the asymptotic behavior of f is bounded by the function g.
+3 This is related to the famous, and still open, Gauss circle problem [29].
+
+6
+Figure 1. Two-dimensional coarse graining and lattice considered in the counting process. The solid gray region contributes
+with an area equals to π/4 (one fourth of the circle) and each axis contributes with an area equals to ǫ/2.
+is described by a non-continuous function, otherwise N (2)
++ (ǫ) would be a continuous function and we know that this
+is not true. Hence, it follows that F(2)(ǫ) in this case is
+F(2)(ǫ) = π
+4 + ǫ + ǫ2
+4 ,
+(28)
+which agrees with the Brownell corrections given in Eq. (22) for the Neumann case.
+For the two-dimensional Dirichlet scenario, note that we can relate N (2)
+−
+and N (2)
++
+as
+N (2)
++ (ǫ) =
+N (2)
+− (ǫ)
+� �� �
+points outside the axes
++ 2N (1)
+− (ǫ) + 1
+�
+��
+�
+points at the axes
+.
+(29)
+Isolating N (2)
+−
+and using results (24) and (28), we obtain the Brownell corrections given in Eq. (22) for Dirichlet.
+The analysis for the three-dimensional case can be conducted in an analogous manner. To illustrate the development,
+let us consider Fig. 2, the generalization of two-dimensional lattice diagram (one eighth of the three-dimensional sphere
+instead of one fourth of the two-dimensional circle).
+With the three-dimensional lattice with Neumann boundary conditions, we get that F(3)(ǫ) for this case is
+F(3)(ǫ) = π
+6 + 3π
+8 ǫ + 3
+4ǫ2 + ǫ3
+8 .
+(30)
+Again, the Brownell corrections for Neumann given in Eq. (23), are obtained.
+For the three-dimensional case with Dirichlet boundary conditions, the counting function can be decomposed as
+N (3)
++ (ǫ) =
+N (3)
+− (ǫ)
+� �� �
+points outside the axes
++
+3N (2)
+− (ǫ)
+�
+��
+�
+points at the faces
++ 3N (1)
+− (ǫ) + 1
+�
+��
+�
+points at the axes
+.
+(31)
+Using previous results, we obtain expression (23).
+After studying the cases d = 2 and d = 3 in detail, we can generalize these results to higher dimensions. Let
+us initially consider the d-dimensional scenario with Neumann boundary conditions. As in Eq. (26), the counting
+function ǫdN (d)
++
+can be expressed as the sum of three contributions: the term |˜Ωd|, the hypervolume associated to the
+axes and the hypervolume of the boundary hypersurface. In appendix A, we show that the hypervolume associated
+to the axes is given by
+Hypervolume of the axes =
+d
+�
+n=1
+�d
+n
+�
+2−dωd−nǫn ,
+(32)
+
+6916 y16bnuod-29xA
+4
+H
+个
+E7
+Figure 2. Three-dimensional coarse graining and axes-boundary area. The dotted dark lines represent the 1/8 of the sphere,
+that contributes with π/6. The solids part of the cubes represents the axes-boundary area. The center cube contributes with
+ǫ3/8. The cubes in each axis contributes with ǫ2/4 . The cubes in the faces represents 3 times 1/4 of the area of the circles,
+i.e., 3π/4 times ǫ. But only half this value, i.e., 3πǫ/8, contributes.
+where
+�d
+n
+�
+=
+d!
+n! (d − n)! .
+(33)
+Hence,
+F(d) (ǫ) =
+d
+�
+n=0
+�d
+n
+�
+2−dωd−nǫn .
+(34)
+The result obtained above generalizes the corrections (22)-(23) to an arbitrary dimension. By repeating the procedure
+realized in [32] for d = 2, 3, and the using the Theorem (8.17) on this reference, we can find
+ǫdN (d)
++ (ǫ) = F(d) (ǫ) + ˜O(ǫ ˜
+wd) , ˜wd > d .
+(35)
+The Dirichlet case can be obtained using a generalization of expressions (29) and (31).
+This generalization is
+constructed in Appendix B, and the final expression can be condensed into
+ǫdN (d)
+± (ǫ) =
+d
+�
+n=0
+�
+(−1)
+d−n
+2
+�1∓1 �d
+n
+�
+2−dωd−nǫn + ˜O
+�
+ǫ ˜
+wd�
+.
+(36)
+Expression (36) is the main result of this section. Note that this equation also maintains the “alternating symmetry”
+observed for the cases d = 2, 3, when we go from Neumann to Dirichlet. If the coefficients do not follow this binomial
+pattern, the alternating symmetry in the signs of coefficients is broken.
+Rewriting expression (36) in terms of the variable k, we have the results provided by Brownell’s formalism for higher
+orders,
+N (d)
+± (k) = 1
+2d
+d
+�
+n=0
+�
+(−1)
+d−n
+2
+�1∓1 �d
+n
+�π(n−d)/2Ld−n
+Γ
+� d−n
+2
++ 1
+� kd−n + ˜O
+�
+kd− ˜wd�
+,
+(37)
+where ˜wd > d.
+
+T
+X
++
++
+H
++
+18
+IV.
+MIXED BOUNDARY CONDITIONS AND DEGENERACIES
+From the results involving Neumann and Dirichlet cases discussed, it is possible to treat more complex boundary
+conditions, dubbed here as “mixed boundary conditions”. Also, in the present section, we will consider the effects
+of degeneracies in the spectra. Those setups model physical systems described by Helmholtz equation which will be
+considered in this work. Further interesting examples (not directly addressed here) can be found in [33, 34].
+In the present development, mixed boundary conditions are constructed imposing Neumann and Dirichlet conditions
+in different axes. The solutions of Helmholtz equation (1) under these conditions can be written as
+ψM (x1, . . . , xd) = φ± (x1) · · · φ± (xd) .
+(38)
+In d dimensions, the number of possible scenarios with mixed boundary conditions is 2d, including the ones which
+are entirely Dirichlet or Neumann. In any case, the counting function (referred generally as N (d)
+M
+for an arbitrary
+boundary condition) can be decomposed as a linear combination in the form
+N (d)
+M (ǫ) =
+d
+�
+n=0
+AnN (n)
+−
+(ǫ) ,
+(39)
+where the coefficients {An} are related to the boundary conditions and possible degeneracies.
+For example, re-
+sult (B1) for the Neumann case without degeneracies is recovered with An =
+�d
+n
+�
+(see Appendix B for details). This
+expression (B1) can be generalized with the consideration of vector fields associated to the Helmholtz equation. This
+development will be necessary to the treatment of degeneracies. In the vector-field problem, beside the d modes
+{n1, n2, . . . , nd}, there are also d components for the fields, {F i
+n1···nd , i = 1, . . . , d}. However, contrary to the scalar
+field, the internal degrees of freedom of the vector fields must be taken into account. There is, the polarization of the
+vector field is an issue.
+Possible polarizations can be transverse and longitudinal. Indeed, the vector fields can be expressed as
+F i
+n1···nd = F 0i
+χ
+�
+k=1
+φ+
+nk
+d
+�
+j=χ+1
+φ−
+nj , i = 1, . . . , d ,
+(40)
+with constants F 0i ∈ C. Following the development which leaded to Eq. (B1), non-trivial solutions can be separated
+into (χ + 1) classes: those in which only one ni is null, those in which two ni are null, and so on, up to those which
+χ of the ni are equal to zero. Let us label these classes as “class zero”, “class one”, and so on, respectively.
+The solutions where χ of the possible nk are null (forming the class χ) have only one non-null component F i
+n1···nd.
+These solutions are polarized in the direction of this component. In the same way, solutions with (χ − 1) of the
+possible ni being null have two non-null components, and consequently two possible polarizations. Hence, the class n
+has a degenerescence equal to
+ξ(d,χ)
+χ−i = i + 1 , i = 0, . . . , χ − 1 , χ ≤ d .
+(41)
+For the class zero, where none of the ni is null, all the components F i
+n1···nd can be non-null and we have d degrees
+of freedom, ξ(d,χ)
+0
+= d . This counting of the degrees of freedom considers only the effect of the boundary conditions.
+In addition, one can also impose the orthogonality condition, which prevents the longitudinal modes. This condition
+requires that the amplitude vector to be perpendicular to the wave vector,
+d
+�
+n=1
+knF 0n = 0 ,
+(42)
+which reduces by one the number of degrees of freedom of the zero class configuration. Therefore, we can write
+ξ(d,χ)
+0
+= ξ(d)
+0
+= d − ˜ξ ,
+(43)
+where ˜ξ = 1 for systems which admit only transverse perturbations and ˜ξ = 0 for cases where longitudinal perturba-
+tions are considered.
+As commented in Appendix B, each class n of the (χ + 1) vector solutions behaves as the Dirichlet problem in
+(d − n) dimensions. The class with only one of the ni equal to zero (degenerescence equal to ξ(d,χ)
+1
+) can be divided
+
+9
+in d sub-classes (d cases where nj = 0). If two of the solutions have nj = 0 (degenerescence ξ(d,χ)
+2
+), the subclass can
+then be further separated in
+�d
+2
+�
+sets. The process continues, until eventually new sub-classes can not be produced.
+Given the counting process presented, the total number of modes can be expressed as the composition of the classes
+and sub-classes thus constructed,
+N (d)
+χ
+=
+χ
+�
+n=0
+(#n-class) × (n-degenerescence) × N (d−n)
+−
+,
+(44)
+or
+N (d)
+χ
+(ǫ) =
+χ
+�
+n=0
+�d
+n
+�
+ξ(d,χ)
+n
+N (d−n)
+−
+(ǫ) ,
+(45)
+with N (m)
+−
+given by Eq. (37), ξ(d,χ)
+m
+in Eq. (41) for m > 0 and ξ(d,χ)
+0
+in Eq. (43). We emphasize that expression (45)
+is one of the main results of the present work. For χ = d and no polarization (ξ(d)
+i
+= 1 , ∀i = 0, . . . , d), Eq. (B1) in
+Appendix B is recovered.
+In the following sections we will apply the formalism developed in concrete scenarios. Namely, we will discuss the
+thermodynamics of the electromagnetic field in a hypercubic cavity and acoustic perturbations in a generalized version
+of Debye model.
+V.
+THERMODYNAMIC AND QUASITHERMODYNAMIC LIMITS
+From the Weyl law and its extensions, we turn to physics applications. The goal of this section is to connect the
+derived expansions for the counting functions with thermodynamic analyses. In this context, corrections of the Weyl
+law will correspond to the transition from the strict thermodynamic limit to the quasithermodynamic approach.
+Let us assume a semi-classical treatment, with the physical system of interest being described by solutions of the
+Helmholtz equation (1). In the treatment, we consider a d-dimensional cubic box with side length L populated by
+a thermal gas. The gas is composed by effective massless particles (actually modes associated to electromagnetic
+or acoustic perturbations) with a well-defined energy and subjected to the Bose-Einstein statistics. The system is
+supposed to be in thermal equilibrium with constant temperature T . The number of modes is not conserved, and
+hence the chemical potential µ is null. Since the temperature and chemical potential are fixed, the grand canonical
+ensemble is assumed.
+A macroscopic treatment is established if a thermodynamic limit can be achieved. Since that we are employing a
+grand canonical ensemble, the strict thermodynamic limit is defined as [35]
+|Ωd| → ∞ with T fixed and µ = 0 ,
+(46)
+where |Ωd| is the hypervolume of the domain Ωd. As seen from Eq. (2), this condition implies that
+LT → ∞ with T ̸= 0 fixed and µ = 0 .
+(47)
+It is also important to consider not only the thermodynamic limit, but also how this limit is approached. Hence,
+we establish the quasithermodynamic limit [1] as
+|Ωd| large but finite, with T fixed and µ = 0 .
+(48)
+The quasithermodynamic is particularly relevant in the present work, where finite cavities or solids are considered.
+Given that the system of interest is compatible with thermodynamic and quasithermodynamic limits, the associated
+partition function can be written as
+ln Z = −
+� ∞
+0
+D(ω) ln
+�
+1 − exp
+�
+− ℏω
+kBT
+��
+dω ,
+(49)
+where D(ω) is the spectral density function. Expression (49) can be seen as the Thomas-Fermi approximation for
+the exact grand canonical partition function. From Z, all the associated thermodynamic and quasithermodynamic
+quantities can be readily calculated. For example, the internal energy is given by
+U = kBT 2 ∂
+∂T (ln Z) .
+(50)
+
+10
+In a practical implementation of the quasithermodynamic limit, combined with the assumption that the temperature
+of the system is fixed, the box length L should be large enough so that [24]
+kB
+ℏc LT ≫ 1 , with LT kept finite.
+(51)
+As reference, kB/(ℏc) ≈ 436.7 K−1 m−1 in SI units. The relation between the spectral density D in the partition
+function (49) and the counting functions studied in the previous sections (collectively denoted by N (d)) is given by
+D(ω) dω = dN (d)(ω)
+dω
+dω .
+(52)
+It follows that the Weyl law (19) is connected with the strict thermodynamic limit (47) when only the term with highest
+power of LT is considered in the asymptotic expansion of the counting function. For the link between extensions of
+Weyl law and the quasithermodynamic limit (51), subdominant corrections on powers of LT are also considered in
+this expansion. It should be noted that, although result (45) for the counting function is valid for any values of L, it
+is only when condition (51) is satisfied that the integral form (49) of the partition function can be employed.
+VI.
+QUASITHERMODYNAMICS OF THE ELECTROMAGNETIC FIELD
+A first application of the developed formalism involves quasithermodynamics of the electromagnetic field in a
+(hyper)cubic cavity. Let us consider a cubical cavity in d dimensions with side length L. Its faces are supposed to
+be perfect conductors, surrounding a vacuum region. The electric field inside the cavity satisfies a wave equation
+characterized by a velocity equals to c. Also, the conductivity of the walls guaranties that the tangential components
+of the electric field at the walls are null.
+The components of the electric field inside the cavity are given by [36]
+Ei
+n1···nd (x1, . . . , xd) = E0i × φ−
+n1 (x1) · · · φ+
+ni (xi) · · · φ−
+nd (xd) ,
+(53)
+where each component satisfies a mixed boundary condition (40) with χ = 1. Furthermore, as only transverse modes
+are present, we have ˜ξ = 1 in Eq. (43). Hence, using result (45), we find that the number of independent modes
+N (d)
+em = N (d)
+1
+is given by
+N (d)
+em (ω) = 1
+2d
+�
+(d − 1)
+cdπd/2
+(ωL)d
+Γ
+� d
+2 + 1
+� +
+d (3 − d)
+cd−1π(d−1)/2
+(ωL)d−1
+Γ
+� d−1
+2
++ 1
+�
+�
+.
+(54)
+In Eq. (54), it was assumed the vacuum dispersion relation k = ω/c for the electromagnetic field.
+For the three-dimensional case we have the well-known quadratic term cancellation in ω [9, 10]. In this case, we
+consider the next correction term in Eq. (45), furnishing
+N (3)
+em =
+L3
+3π2c3 ω3 − 3L
+2πcω .
+(55)
+Result (55) agrees with [9, 10, 30, 37]. However, Eq. (54) shows that this cancellation of the “area” term for the
+electromagnetic field only occurs in the three-dimensional case. This implies a drastic difference in d = 3 scenario
+when comparing with other dimensionalities.
+The spectral density D(d)
+em can be calculated deriving expression (54) with respect to ω,
+D(d)
+em = dN (d)
+em
+dω
+= 1
+2d
+�
+d (d − 1) Ldωd−1
+πd/2Γ
+� d
+2 + 1
+�
+cd + d (d − 1) (3 − d) Ld−1ωd−2
+π(d−1)/2Γ
+� d−1
+2
++ 1
+�
+cd−1
+�
+.
+(56)
+Hence, the internal energy of the system can be written as
+U (d) =
+� ∞
+0
+D(d)
+em (ω)
+ℏω
+exp
+�
+ℏω
+kBT
+�
+− 1
+dω = kB
+�
+(d − 1) θdLdT d+1 + (3 − d) d
+2θd−1Ld−1T d
+�
+,
+(57)
+where θm is defined as
+θm ≡ ζ (m + 1) Γ (m + 1) m
+2mπm/2Γ
+� m
+2 + 1
+�
+km
+B
+ℏmcm .
+(58)
+
+11
+The term ζ (z) in Eq. (58) denotes the Riemann zeta function. Expression (57) can be seen as an improved version of
+the Stefan-Boltzmann law for the electromagnetic field in a hypercubic cavity, in a quasithermodynamic treatment.
+In the strict thermodynamic limit, that is, when the correction is not considered, we recover the Stefan-Boltzmann
+law in d dimensions. In the strict thermodynamic regime, the (improved) result (57) can be compared with [38, 39].
+However, in those references the authors impose that ξ(d)
+em = ξ(d)
+0
+= 2 for the polarization of the d-dimensional electric
+field (as done for the three-dimensional scenario). The observation that ξ(d)
+em = d − 1 is the correct factor was made in
+[40].
+It is important to stress that, while the enforcement that ξ(d)
+em = 2 for any dimension corresponds to a simple factor
+for the main term of several thermodynamic quantities, this choice has a very drastic effect for the correction terms in
+the quasithermodynamic analysis. For instance, when considering the quasithermodynamics, the general enforcement
+of ξ(d)
+em = 2 would imply the cancellation of the first correction for any value of d (not appropriate), and not for d = 3,
+as we have obtained.
+The quasithermodynamic Stefan-Boltzmann law (57) can also be rewritten as
+U (d)
+Ld
+=
+� ∞
+0
+B(d) (ω, T, L) dω ,
+(59)
+implying that
+B(d) (ω, T, L) =
+�
+ω
+Γ
+� d
+2 + 1
+� − c (d − 3) √π
+Γ
+� d−1
+2
++ 1
+� 1
+L
+�
+d (d − 1) ℏωd−1
+2dπd/2cd
+�
+exp
+�
+ℏω
+kBT
+�
+− 1
+� .
+(60)
+Expression (60) is a quasithermodynamic version of the Planck formula.
+This result describes the effects of the
+boundedness of the cavity on the spectral density of the electromagnetic field.
+For the two-dimensional case, a more detailed discussion is in order. Our results for d = 2 can be compared with
+the analysis presented in [26]. An interesting remark suggested in [26] is that size effects prevent arbitrarily low
+frequencies in the system. With this observation, particularized here for a square system of area L2, the authors
+propose an internal energy U with the form
+U = 2
+� ∞
+ωmin
+D (ω)
+ℏω
+exp
+�
+ℏω
+kBT
+�
+− 1
+dω , D (ω) = L2ω
+2πc2 , ωmin =
+√
+2πc
+L
+.
+(61)
+The integral in Eq. (61) can be solved by making the change x = t + xmin, where
+x ≡ ℏω
+kBT , xmin ≡ ℏωmin
+kBT
+=
+√
+2πℏc
+kBT L .
+(62)
+One obtains
+U (xmin) =
+k3
+B
+πℏ2c2 L2T 3S (xmin) ,
+(63)
+S (xmin) = 2Li3
+�
+e−xmin�
++ 2xminLi2
+�
+e−xmin�
++ x2
+minLi1
+�
+e−xmin�
+,
+(64)
+where the polylogarithm function Lis (z) is defined as
+Lis+1 (z) =
+1
+Γ (s + 1)
+� ∞
+0
+ts
+exp (t) /z − 1 dt .
+(65)
+Following the development in [26], the term S (xmin) is expanded around xmin = 0,
+S(xmin) = 2ζ(3) − x2
+min
+2
++ x3
+min
+6
+− x4
+min
+48
+− x6
+min
+4320 + O
+�
+x7
+min
+�
+.
+(66)
+Observe that the expansion close to xmin = 0 is equivalent to consider [kB/(ℏc)] LT ≫ 1, as one sees from the
+definition of xmin in Eq. (62). Finally, using the expansion (66), and keeping only the first-order term x2
+min, the
+approach from [26] furnishes4
+U = 2ζ (3)
+π
+k3
+B
+ℏ2c2 L2T 3 − πkBT .
+(67)
+4 It should be remarked that Eq. (67) is not actually derived in [26].
+
+12
+We note that the first term in Eq. (67) differs from the result (57) presented in this work by a factor of 2. This
+discrepancy is just a consequence of the authors in [26] using the developments of [38] which, as previously commented,
+assume that ξ(d)
+em = 2 for any value of d. However, as a more important remark, it is possible to see from Eq. (61)
+that, while considering the size effects on the minimum frequency, these effects on the density of the modes D (ω) are
+not taken into account. In other words, the integrand of U in Eq. (61) is correct only in the thermodynamic limit.
+We propose an improvement of the results in [26] by considering size effects for both the minimal frequency and
+the density of the modes. Following the procedure that leads to Eq. (64), but using D(2)
+em of Eq. (56) instead of D(ω)
+in the integrand of the internal energy U in expression (61), we get
+U (2)
+em = πkBT
+�
+x−2
+minS (xmin) +
+√
+2
+π x−1
+min ˜S (xmin)
+�
+,
+(68)
+with S(xmin) presented in Eq. (64) and
+˜S (xmin) = Li2
+�
+e−xmin�
++ xminLi1
+�
+e−xmin�
+.
+(69)
+The expansion of ˜S(xmin) around xmin = 0 gives
+˜S (xmin) = π2
+6 − xmin + x2
+min
+4
+− x3
+min
+36
++ x5
+min
+3600 + O
+�
+x6
+min
+�
+.
+(70)
+Substituting the results (66) and (70) in Eq. (68), and keeping only terms of order up to x2
+min, we obtain
+U (2)
+em = ζ (3)
+π
+k3
+B
+ℏ2c2 L2T 3 + πk2
+B
+6ℏc LT 2 −
+�
+1
+2 +
+√
+2
+π
+�
+πkBT .
+(71)
+In addition to the already mentioned factor of 2, and a correction of
+√
+2/π in the last term in Eq. (67), we highlight the
+appearance of an “area” term proportional to T 2. This new term is the leading correction on the quasithermodynamics
+limit.
+It is important to notice that the second term in Eq. (71), that is, the leading correction, is precisely the last term
+presented in Eq. (57) for d = 2. This is a consequence of the fact that S in Eq. (66) does not have a linear term (i.e.,
+a term proportional to xmin). The existence of such linear term would change the second term in Eq. (57).
+The above procedure for the two-dimensional scenario can be generalized to arbitrary dimensions. In this case, to
+consider a minimal energy implies that
+U (d)
+em = d (d − 1) kBT
+��
+x(d)
+min
+�−d
+CdSd
+�
+x(d)
+min
+�
++
+�
+x(d)
+min
+�−1 ˜Cd ˜S
+�
+x(d)
+min
+��
+,
+(72)
+where
+x(d)
+min =
+ℏcπ
+kBLT
+�
+3d
+2 − 1 , Cd =
+�π
+8
+� d
+2 (3d − 2)
+d
+2
+Γ
+� d
+2 + 1
+� , ˜Cd = (−1)d
+2d−1π C1 ,
+Sd
+�
+x(d)
+min
+�
+=
+d
+�
+k=0
+�
+d
+k
+�
+Γ (k + 1)
+�
+x(d)
+min
+�d−k
+Lik+1
+�
+e−x(d)
+min
+�
+,
+(73)
+and ˜S is given by Eq. (69) for any value of d. Similarly to the two-dimensional case, it is straightforward to see that
+the limit of Sd around x(d)
+min = 0 does not have a linear term.
+It should be noticed that the power of x(d)
+min in the second term in the brackets of Eq. (72) does not depend on
+dimension and is equal to −1. Thus, the higher contribution of this term is proportional to LT 2, in the same way as in
+d = 2. Therefore, the leading terms in U (d)
+em of Eq. (72) are precisely those presented in expression (57). We conclude
+that, for d ̸= 3, the existence of a minimal frequency would not affect the quasithermodynamics behavior, exactly as
+in the thermodynamic limit, and the expression (57) is correct even if the minimal frequency is considered.5
+5 For the three-dimensional scenario, the absence of the area term could imply higher-order corrections.
+In this case, the minimum
+frequency would affect the first correction term, which is proportional to LT 2 .
+
+13
+VII.
+QUASITHERMODYNAMICS OF ACOUSTIC PERTURBATIONS
+A.
+General considerations
+We turn to the quasithermodynamics of acoustic perturbations, focusing on the d-dimensional version of the Debye
+model. Let us consider a harmonic solid, there is, an isotropic, elastic and continuous body. The solid is assumed
+to be a hypercube of dimension d and length size L. In this hypercube, a number of χ opposed faces are free, and
+hence the oscillations in these directions respect Neumann conditions. The remaining d − χ opposing faces are fixed,
+respecting Dirichlet conditions. The propagation of vibrations in the solid is associated to acoustic waves.
+In this setup, the components of the displacement field ui(x) of the particles that form the solid (atoms, ions,
+molecules, etc.) will be solutions of Helmholtz equation with the form (40). Specifically, ui(x) are given by
+ui
+n1···nd (x1, . . . , xd) = u0i × φ+
+n1 (x1) · · · φ+
+nχ (xχ)
+�
+��
+�
+free faces
+φ−
+nχ+1 (xχ+1) · · · φ−
+nd (xd)
+�
+��
+�
+fixed faces
+.
+(74)
+Contrary to the description of the electromagnetic field in a cavity, acoustic perturbations are free to oscillate in
+the longitudinal direction. Therefore, a distinct quasithermodynamic behavior, comparing to the thermodynamic of
+electromagnetic perturbations, should be expected in the present setup. In particular, for d = 3, we should expect an
+important role of the area term, the first term of quasithermodynamic correction.6
+Let us apply the results derived in the present work to generalize the well-known expressions in the usual three-
+dimensional Debye model and investigate the quasithermodynamic regime. From Eq. (45), with ˜ξ = 0 in (43), we
+obtain
+N (d)
+s
+(ω) =
+d
+πd/22d
+
+
+1
+Γ
+� d
+2 + 1
+�
+�
+ωL
+c(d)
+s0
+�d
++ (2χ − d) π1/2
+Γ
+� d−1
+2
++ 1
+�
+�
+ωL
+c(d)
+sχ
+�d−1
+ .
+(75)
+The quantities c(d)
+s0 and c(d)
+sχ represent an effective bulk sound velocities and will be discussed in the next section.
+Expression (75) can be interpreted as the number of modes associated to acoustic waves whose frequencies are lower
+than ω. Unlike the electromagnetic case, in the treatment of Debye model the area term is absent only for d even
+and when there is a precise balance between Neumann and Dirichlet boundary conditions: half the faces of the solid
+are free and half are fixed. In particular, for the three-dimensional scenario, there is always the influence of the area
+term.
+B.
+Influence of the area term
+Let us focus on the effects of the area term, considering the extreme scenarios, where all the walls are free (χ = d),
+or all the walls are fixed (χ = 0). In these cases,
+N (d)
+s± (ω) =
+1
+πd/2
+d
+2d
+
+
+1
+Γ
+� d
+2 + 1
+�
+�
+ωL
+c(d)
+s0
+�d
+±
+dπ1/2
+Γ
+� d−1
+2
++ 1
+�
+�
+ωL
+c(d)
+s±
+�d−1
+ .
+(76)
+In Eq. (76), the plus and minus signs are associated to the “all free” and “all fixed” types of solid, respectively. Also, c(d)
+s0
+can be interpreted as an effective velocity given by the linear superposition of velocities cl and ct, of the longitudinal
+mode and of the (d − 1) transverse modes,
+�
+c(d)
+s0
+�d
+= dcd
+t
+�
+d − 1 + cd
+t
+cd
+l
+�−1
+.
+(77)
+It is important to notice that this linear superposition can only be applied to the main term [30]. Indeed, the reflection
+of the modes in the walls of the solid produce a mixture of the modes. Hence, a purely transverse (or longitudinal)
+perturbation can be reflected as a superposition of transverse and longitudinal waves. The phenomenon generates an
+effective velocity c(d)
+s± which is different from c(d)
+s0 .
+6 Effects associated to the area term in a three-dimensional setup with low temperature are considered in [41, 42].
+
+14
+For the three-dimensional case, one approach to study the wave reflection in a specific wall is to consider a slab
+instead of a cube, and hence to ignore border effects in this wall.
+That is the approach followed in [41], where
+appropriate boundary conditions (Neumann or Dirichlet) are imposed on the faces parallel to the plane of the slab
+(the planes of reflection). However, periodic boundary conditions are enforced on the other faces (i.e., on the slab
+thickness direction), in an approach which captures border effects.
+Note that the number of faces with periodic
+boundary condition is equal to the dimension of the incident plane (two-dimensional in the three-dimensional case).
+Periodic conditions at the borders are justified if the borders are far enough from the region of interest. Within this
+simplified model, it was determined in [41] that
+�
+c(3)
+s+
+�2
+= 3
+�
+2
+�
+c2
+t
+�2 − 3c2
+tc2
+l + 3
+�
+c2
+l
+�2
+c2
+tc2
+l (c2
+l − c2
+t)
+�−1
+,
+(78)
+�
+c(3)
+s−
+�2
+= 3
+�
+2
+c2
+t
++ 1
+c2
+l
++
+�
+c2
+l − c2
+t
+�2
+c2
+l c2
+t (c2
+l + c2
+t)
+�−1
+.
+(79)
+Following the development presented in [41], it is observed that the presence of the power 2 in cl,t is a consequence
+of the number of periodic boundary conditions considered. That, in turn, is a result of the fact that the plates have
+two dimensions.
+In d dimensions, we can consider the reflection of the wave by any one of the 2d faces. In this case, a slab is
+defined as the set of all points lying between two (d − 1)-dimensional hyperplanes7 in Rd. In this way, the system
+is approximated by two infinite plates. Appropriate boundary conditions (Neumann or Dirichlet) are imposed on
+these plates, with periodic boundary conditions enforced on the other (d − 1) directions. In this way, considering the
+relation with the dimensionality of the system and the power series on ct and cl, the proposed generalization of the
+three-dimensional results in Eqs. (78)-(79) to the more general d-dimensional scenario is
+�
+c(d)
+s±
+�d−1
+= dcd−1
+t
+�
+d − 1 +
+�
+cd−1
+l
+�2 − cd−1
+l
+cd−1
+t
++ 2
+�
+cd−1
+t
+�2
+cd−1
+l
+�
+cd−1
+l
+∓ cd−1
+t
+�
+�−1
+.
+(80)
+C.
+Debye frequency
+One important parameter of the Debye model is the so-called Debye frequency. This quantity refers to the cutoff
+angular frequency of the waves propagating in the solid, resulting from the existence of a minimum distance between
+the particles that form the solid lattice. From Eq. (76), it is possible to determine the Debye frequency ω(d)
+D±, analyzing
+the number of degrees of freedom of the system. That is,
+nd + n∂l ≈ nd = N (d)
+s±
+�
+ω(d)
+D±
+�
+,
+(81)
+where n is the number of particles inside the cavity (bulk) and n∂ is the number of particles in the borders (edge).
+Considering the edge, the particles have l degrees of freedom. In expression (81), we assumed that n ≫ n∂ since we
+are performing a macroscopic (quasithermodynamic) treatment, and hence we can consider n as an approximation to
+the total number of particles of the system. In fact, we are interested in the border effects on the modes propagating
+on the bulk, not considering the superficial modes (Rayleigh modes). The contribution of the superficial modes can
+be disregarded in the quasithermodynamic regime.
+One relevant issue is the influence of the borders in Debye frequency. For the analysis of this point, we rewrite
+expression (81) in the form
+�
+ω(d)
+D±
+�d
++ Bd±
+�
+ω(d)
+D±
+�d−1
+−
+�
+ω(d)
+D0
+�d
+= 0 ,
+(82)
+7 See section 3.4 of [43].
+
+15
+with
+Bd± ≡ ±dπ1/2Γ
+� d
+2 + 1
+�
+Γ
+� d−1
+2
++ 1
+�
+�
+c(d)
+s0
+�d
+�
+c(d)
+s±
+�d−1
+L
+,
+(83)
+ω(d)
+D0 ≡ 2π1/2c(d)
+s0
+�
+Γ
+�d
+2 + 1
+�
+ρd
+� 1
+d
+.
+(84)
+In Eq. (84), ρd denotes the (hyper)volumetric density of the cube, defined as
+ρd ≡ n
+Ld .
+(85)
+The most physically relevant scenario is the three-dimensional solid. For d = 3, an exact solution for Eq. (82) can
+be obtained:
+ω(3)
+D± = πc(3)
+s0
+
+
+3�
+f0± +
+3�
+f1± ∓
+3
+�
+c(3)
+s0
+�2
+4
+�
+c(3)
+s±
+�2
+1
+L
+
+ ,
+(86)
+where
+fp± ≡ 6ρ3
+π
+∓
+27
+�
+c(3)
+s0
+�6
+32
+�
+c(3)
+s±
+�6
+1
+L3 + (−1)p
+
+9ρ2
+3
+π2 ±
+81
+�
+c(3)
+0
+�6
+32π2
+�
+c(3)
+s±
+�6
+ρ3
+L3
+
+
+1/2
+.
+(87)
+Let us consider other scenarios besides the most usual one. In the two-dimensional case, again an exact expression
+for the Debye frequency ω(2)
+D± can be produced,
+2ω(2)
+D± =
+�
+B2
+2± +
+�
+2ω(2)
+D0
+�2
+− B2± .
+(88)
+For d > 3, we use the approximation
+�
+ω(d)
+D±
+�d
++ Bd±
+�
+ω(d)
+D±
+�d−1
+=
+�
+ω(d)
+D± + Bd±
+d
+�d
++ O
+� 1
+L2
+�
+,
+(89)
+which is justified since we are considering the quasithermodynamic regime of the theory. Therefore, keeping only
+terms of order 1/L and employing result (82), we obtain
+ω(d)
+D± = ω(d)
+D0 ∓ π1/2Γ
+� d
+2 + 1
+�
+Γ
+� d−1
+2
++ 1
+�
+�
+c(d)
+s0
+�d
+�
+c(d)
+s±
+�d−1
+1
+L .
+(90)
+It should be remarked that, for the two- and three-dimensional cases, exact expressions (88) and (86) are available.
+With these results, higher-order contributions in 1/L are considered [when compared to Eq. (90)].
+In the strict thermodynamic limit (46) we observe that Bd± → 0, and
+ω(d)
+D± (L → ∞) = ω(d)
+D0 .
+(91)
+With d = 3, expression (91) reduces to the well-known Debye frequency. Summarizing, we have obtained in Eq. (90)
+a correction in Debye frequency for a harmonic solid in d dimensions, with free (+) or fixed (−) walls.
+It is interesting to notice that result (90) could offer a possible experimental test for the developments presented in
+this work. Indeed, from Eq. (90) we observe that
+ω(d)
+D+ ≤ ω(d)
+D0 ≤ ω(d)
+D− .
+(92)
+That is, a solid with free walls should have lower Debye frequency when compared with the standard result in the
+strict thermodynamic regime. In a similar way, a solid with fixed walls will have larger Debye frequency. These
+differences are, in principle, measurable. Given that in an electric conductor material the main contribution for the
+thermal capacity comes from the electrons that are not strongly bound to the lattice, we expect that the effect (92)
+will be more relevant in an electric insulator (non-metallic crystal).
+
+16
+D.
+Heat capacity
+We now consider the heat capacity of the solid in the quasithermodynamic regime. To evaluate the heat capacity
+at constant volume C(d)
+± , let us examine the internal energy U (d)
+±
+of the system for both boundary conditions studied.
+Using Eq. (76), we obtain that the density of modes is given by
+D(d)
+s± = dN (d)
+s±
+dω
+= κ(d)
+0 ωd−1 + κ(d)
+± ωd−2 ,
+(93)
+where the following quantities are defined:
+κ(d)
+0
+≡
+d2
+2dπd/2
+Ld
+Γ
+� d
+2 + 1
+� �
+c(d)
+s0
+�d ,
+(94)
+κ(d)
+± ≡ ±d2 (d − 1)
+2dπ
+d−1
+2
+Ld−1
+Γ
+� d−1
+2
++ 1
+�
+�
+c(d)
+s±
+�1−d
+.
+(95)
+Hence, the internal energy can be written as
+U (d)
+±
+= k2
+BT 2
+ℏ
+� θ(d)
+± /T
+0
+D(d)
+s±
+�kBT
+ℏ
+x
+� x dx
+ex − 1 ,
+(96)
+with
+θ(d)
+± ≡ ℏ
+kB
+ω(d)
+D±
+(97)
+denoting the Debye temperature in d dimensions. In general, the integral in Eq. (96) does not have an analytic
+solution. Let us investigate the regimes of low temperature (T ≪ θ(d)
+± ) and high temperature (T ≫ θ(d)
+± ).
+The low-temperature limit is more commonly treated in the pertinent literature. In three dimensions, this regime
+is captured by Debye law: the specific heat of a solid at constant volume varies as the cube of the absolute tem-
+perature T . We aim to improve Debye law in d dimensions considering solids with finite size, and hence adopting a
+quasithermodynamic description.
+It should be remarked that the quasithermodynamic limit (51) implies [kB/(ℏc)] LT ≫ 1. Therefore, low values for
+the temperature T should be compensated by corresponding large values for L. For low enough temperatures, the
+internal energy of the system can be written as
+U (d)
+±
+�
+T ≪ θ(d)
+±
+�
+= d!κ(d)
+0
+ℏd
+ζ (d + 1) kd+1
+B
+T d+1 + (d − 1)!κ(d)
+±
+ℏd−1
+ζ (d) kd
+BT d .
+(98)
+From expression (98) for U (d)
+± , corrections in Debye law can be obtained:
+C(d)
+±
+�
+T ≪ θ(d)
+±
+�
+= d! (d + 1) κ(d)
+0
+ℏd
+ζ (d + 1) kd+1
+B
+T d + d (d − 1)!κ(d)
+±
+ℏd−1
+ζ (d) kd
+BT d−1 .
+(99)
+In the strict thermodynamic limit (46) we recover the Debye law in d dimensions [44]. For the three-dimensional case,
+expression (99) reduces to the result previously discussed in [41].
+The high-temperature regime is less explored, and we will consider it in the present work. This scenario is actually
+more suitable for the quasithermodynamic treatment, because if T is high, solids with low size L can be more
+accurately analyzed. In three dimensions, assuming the thermodynamic limit, the behavior of the system in this
+regime is captured by Dulong-Petit law: the heat capacity of a solid with a mol of particles is approximated constant
+for high enough temperature. Corrections for the Dulong-Petit law in d dimensions considering a finite-size solid will
+be derived.
+In the high-temperature regime θ(d)
+± /T → 0, and the approximation ex − 1 ≈ x can be employed in Eq. (96). With
+this approach, the internal energy U (d)
+± (T ≫ θ(d)
+± ) can be explicitly written and the thermal capacity C(d)
+±
+determined:
+C(d)
+±
+�
+T ≫ θ(d)
+±
+�
+=
+�kB
+ℏ
+�d �
+κ(d)
+0
+d
+�
+θ(d)
+±
+�d
++
+ℏκ(d)
+±
+kB (d − 1)
+�
+θ(d)
+±
+�d−1
+�
+kB .
+(100)
+
+17
+In the strict thermodynamic limit (46) we observe that ωD± → ω(d)
+D0 and hence
+C(d)
+±
+�
+T ≫ θ(d)
+± , L → ∞
+�
+= κ(d)
+0 ω(d)
+0
+d
+kB .
+(101)
+For the three-dimensional case, result (101) is reduced to the usual Dulong-Petit law. For general values of d, Eq. (100)
+furnishes the correction of the Dulong-Petit law for a finite solid of arbitrary dimensionality.
+VIII.
+FINAL COMMENTS
+In the present work, we explored Weyl law and its extensions with an intuitive formalism, based on the association
+between point counting and volumes of sections of the sphere. In several published developments, the counting function
+depended strongly on the pertinent domain, with the associated lattice kept fixed. In our approach, the domain is
+kept fixed and the lattice is rescaled. Known results were rederived, showing the robustness of the method. Moreover,
+new results were obtained, including corrections for the Weyl conjecture in d dimensions, effects of the polarization,
+and an exploration of the role of the area term in the three-dimensional scenario. Applications of the previous results
+were investigated, with the quasithermodynamic analyses of the electromagnetic field in a finite cavity and acoustic
+perturbations in a finite solid.
+Applying the formalism to the thermodynamics and quasithermodynamics of the electromagnetic perturbations in
+a finite box within a semi-classical treatment, corrections to the d-dimensional Stefan-Boltzmann law were obtained
+and polarization and border effects were treated. In particular, we showed that the well-known cancellation of the
+area term only occurs in three dimensions. This effect turns the thermodynamics of the system distinct for d ̸= 3.
+The correction due to a minimum energy of the system is treated. In all scenarios except the three-dimensional case,
+the quasithermodynamic corrections suppress an eventual effect of a minimal energy.
+Two-dimensional results for the quasithermodynamics of the electromagnetic field can be linked to experimental
+setups. Indeed, the analysis of thermal radiation in d = 2 has applications in the description of single-layer materials
+(also known as “2D materials”), which include graphene, single layers of various dichalcogenides and complex oxides
+[11]. The results presented in [26], describing two-dimensional thermal radiation and improved in the present work,
+were used in [12] to study the emission spectra of a graphene transistor.
+Concerning the electromagnetic field in a three-dimensional cavity, there is some discussion in the literature involving
+the cancellation of area term. We believe this controversy is the result of an inadequate treatment of how polarization
+affects each term of the expansion (34). For instance, in [1] polarization is assumed to have the same effect on all
+terms of the expansion (34), and as a result the cancellation of the area term does not occur. Our result in Eq. (45)
+indicates that this approach is inadequate. Experimentally, the cancellation of the area term in a three-dimensional
+setup implies in a very small correction [9]. Therefore, other effects (such as scattering and diffraction) can supplant
+this correction. In fact, while the results in [45] are compatible with the negative value correction presented in Eq. (55),
+other experiments point to a positive correction term [46].
+Moreover, we believe that three-dimensional systems are not the most suitable setups for the observation of effects
+associated to the Weyl conjecture on electromagnetic radiation. Our results suggest that a better approach to this goal
+is the use of two-dimensional effective systems. Such systems could be constructed using single-layer or nanostructures
+graphene devices simulating two-dimensional blackbodies [12, 47].
+Another main application of the developed formalism concerns the thermodynamics of acoustic perturbations. An
+improved version of Debye model for a finite solid is treated. We reproduce the known results for the three-dimensional
+case in the limit of low temperatures and extend those results to arbitrary dimensions. New developments in the high-
+temperature scenario and the influence of the area term are explored. Extensions of the known formulas are obtained
+for Debye temperature and Dulong-Petit law.
+The presented development captures some effects associated to internal degrees of freedom, such as spin. Indeed,
+macroscopic effects associated to spin manifest itself in the polarization of the fields. This is more directly seen when
+the electromagnetic perturbation is considered. In this case, the two values of the photon spin (or, more precisely,
+of the photon helicity) can be related to the two polarizations of the classical field. For acoustic perturbations the
+problem is more subtle [48]. In an ideal isotropic medium, considering modes with long wavelength, it is possible to
+associate longitudinally polarized modes with spin-0 phonons, and transversely polarized modes with spin-1 phonons.
+In general, new internal degrees of freedom can be incorporated into the presented approach, as long as they do not
+modify the eigenvalue problem [i.e., the Helmholtz equation (1)]. This can be done by increasing the multiplicity
+count of the modes performed in section IV.
+
+18
+The analysis of arbitrary boundary conditions in the extensions of Weyl law, related with all self-adjoint extensions
+of the d-dimensional Laplacian operator, is a work in progress. In fact, as far as we know, there is no Weyl conjecture
+for such general scenario. We believe that the approach introduced in the present development, based on a counting
+function with the rescaling of the lattice, might be an important step in the problem. Further developments of the
+present work might include the application of the proposed formalism on gravitational systems. Also, a possible
+exploration of the influence of the area term in the Casimir effect could be conducted, considering three-dimensional
+systems with finite temperature. Analyses along those lines should appear in forthcoming presentations.
+Appendix A: Hypervolume associated to the axes
+The calculation of the hypervolume associated to the axes in the d-dimensional case will be presented. Before the
+actual calculation, let us fix the notation and definitions that are used in the development.
+We denote ˜Ωd as one of the 2d partitions of the d-dimensional unit sphere, that is,
+˜Ωd ≡
+�
+(ǫ1, ǫ2, . . . , ǫd) ∈ Rd : 0 ⩽ ǫi ⩽ 1 and ǫ2
+1 + ǫ2
+2 + · · · + ǫ2
+d ⩽ 1
+�
+.
+(A1)
+Also, the hyperplanes Hi are defined as
+Hi ≡
+�
+(ǫ1, ǫ2, . . . , ǫd) ∈ Rd : ǫi = 0
+�
+, i = 1, 2, . . ., d .
+(A2)
+The volume of a given subset S ⊂ Rd is indicated by |S|. For example,
+���˜Ωd
+��� = 2−dωd .
+(A3)
+Using the sets ˜Ωd and Hi defined in Eqs. (A1) and (A2), the subsets {Πi1,...,in} of ˜Ωd are constructed:
+Πi1,...,in = ˜Ωd ∩ Hi1 ∩ · · · ∩ Hin , 1 ⩽ in ⩽ d , 1 ⩽ n ⩽ d .
+(A4)
+The number of subsets {Πi1,...,in} is nd. However, some of this subsets are equal, because: (1) Πi1,...,in is symmetric
+under the switch of indexes; (2) and the idempotent property Hj ∩ Hi = Hi, for i = j. So, the number of distinct
+sets is
+#Πi1,...,in =
+�d
+n
+�
+=
+d!
+n! (d − n)! .
+(A5)
+The subsets {Πi1,...,in}, each one labeled by n indexes, can be interpreted as sections of the sphere in Rd−n, generated
+by different axes. For example, if d = 3,
+(n = 1)
+
+
+
+
+
+Π1 = Ω2 generated by ǫ2 and ǫ3
+Π2 = Ω2 generated by ǫ1 and ǫ3
+Π3 = Ω2 generated by ǫ1 and ǫ2
+, (n = 2)
+
+
+
+
+
+Π12 = Ω1 generated by ǫ3
+Π13 = Ω1 generated by ǫ2
+Π23 = Ω1 generated by ǫ1
+, (n = 3) {Π123 = (0, 0, 0) .
+(A6)
+With the symbology Πi1,...,in, n indicates the number of axes not included in the subset generation.
+From a given section Πi1,...,in, a cylinder CI can be defined, adding to Πi1,...,in an interval with the form I = [a, b]
+for each not included axes,
+CI
+i1,...,in = Πi1,...,in × I × I × · · · × I
+�
+��
+�
+n
+.
+(A7)
+It should be observed that, for each value of n, there are
+�d
+n
+�
+different cylinders with the same volume V n
+d , where
+V n
+d ≡
+��CI
+i1,...,in
+�� = |I|n |Πi1,...,in| = |I|n2n−dωd−n .
+(A8)
+Using previous results, the contribution of the axes to the total volume can be determined. Considering the volumes
+of the cylinders CI constructed with the interval I =
+�
+− ǫ
+2, 0
+�
+, we obtain
+Hypervolume of the axes =
+d
+�
+n=1
+�d
+n
+�
+V n
+d =
+d
+�
+n=1
+�d
+n
+� � ǫ
+2
+�n
+2n−dωd−n =
+d
+�
+n=1
+�d
+n
+�
+2−dωd−nǫn .
+(A9)
+
+19
+Appendix B: Counting functions
+Let us see how Neumann counting function can be constructed from the counting function for Dirichlet case and
+vice versa. We start by arranging the Neumann modes in d + 1 classes, labeled by j = 0, 1, . . . , d. The first class is
+composed by modes where all ni are non-null. The second class is composed by modes where only one of the ni is
+zero. In the third class two of the ni are zero. The process is continued in this fashion, up to d + 1 sets.
+Given the proposed partition of the Neumann modes, we observe that the class where none of the ni is zero is the
+solution for the Dirichlet problem in d dimensions, where all modes with ni = 0 must be excluded. The class with only
+one of the ni is zero can be further divided into d sub-classes where n1 = 0, or n2 = 0, etc. Hence, in each one of these
+sub-classes, disregarding the null ni, we obtain sub-classes composed by Dirichlet solutions in d − 1 dimensions. The
+class where two ni are null can be separated into
+�d
+2
+�
+sub-classes composed of modes which satisfy Dirichlet boundary
+conditions in d − 2 dimensions. Carrying on with this procedure, solutions of the Neumann problem can be written
+as a combination of the elements in the constructed sub-classes:
+N (d)
++ (ǫ) =
+d
+�
+j=0
+�d
+j
+�
+N (d−j)
+−
+(ǫ) .
+(B1)
+On the other hand, we know that the solutions for Dirichlet, unlike those for Neumann, exclude the mode ni = 0,
+and therefore the difference between the number of solutions of both will be given by
+N (d)
++ − N (d)
+−
+=
+�
+n
+(# Dirichlet solutions formed by n different axes) .
+(B2)
+Since there are d axes, for each set of n axes there will be a degeneracy of
+�d
+n
+�
+in the Dirichlet solutions, and hence
+N (d)
++
+− N (d)
+−
+=
+d−1
+�
+n=0
+�d
+n
+�
+N (n)
+−
+.
+(B3)
+Substituting N (d)
+−
+in the sum (B3), we get the relation (B1).
+The inverse relation is obtained writing Eq. (B1) as
+N (i)
++ =
+d
+�
+j=0
+P i
+jN (j)
+− , P i
+j =
+� �i
+j
+�
+,
+j ≤ i
+0
+j > i
+, i, j = 0, 1, 2, ..., d .
+(B4)
+The triangular matrix P is known as the Pascal’s matrix. It is invertible, since det P = 1, with an inverse given by
+�
+P −1�i
+j =
+�
+(−1)i−j �i
+j
+�
+,
+j ≤ i
+0,
+j > i
+.
+(B5)
+Therefore we can solve Eq. (B4) for N (d)
+− , obtaining
+N (d)
+−
+=
+d
+�
+n=0
+(−1)d−n
+�d
+n
+�
+N (n)
++ .
+(B6)
+ACKNOWLEDGMENTS
+L. F. S. acknowledges the support of Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) –
+Brazil, Finance Code 001; and the Fundação Araucária (Foundation in Support of the Scientific and Technological
+Development of the State of Paraná, Brazil). M. A. J. thanks Coordenação de Aperfeiçoamento de Pessoal de Nível
+Superior (CAPES) – Brazil, Finance Code 001, for financial support. This is the version of the article before peer
+review or editing, as submitted by to Journal of Physics A. IOP Publishing Ltd is not responsible for any errors or
+omissions in this version of the manuscript or any version derived from it. The Version of Record is available online
+at https://doi.org/10.1088/1751-8121/acb09b.
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+

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@@ -0,0 +1,1413 @@
+CFG2VEC: Hierarchical Graph Neural Network for
+Cross-Architectural Software Reverse Engineering
+Shih-Yuan Yu1, Yonatan Gizachew Achamyeleh1, Chonghan Wang1, Anton Kocheturov2, Patrick Eisen2,
+Mohammad Abdullah Al Faruque1
+1Dept. of Electrical Engineering and Computer Science, University of California, Irvine, CA, USA
+{shihyuay, yachamye, chonghaw, alfaruqu}@uci.edu
+2Siemens Technology, Princeton, NJ, USA, {anton.kocheturov, patrick.eisen}@siemens.com
+Abstract—Mission-critical embedded software is critical to
+our society’s infrastructure but can be subject to new security
+vulnerabilities as technology advances. When security issues
+arise, Reverse Engineers (REs) use Software Reverse Engineering
+(SRE) tools to analyze vulnerable binaries. However, existing tools
+have limited support, and REs undergo a time-consuming, costly,
+and error-prone process that requires experience and expertise
+to understand the behaviors of software and vulnerabilities. To
+improve these tools, we propose cfg2vec, a Hierarchical Graph
+Neural Network (GNN) based approach. To represent binary, we
+propose a novel Graph-of-Graph (GoG) representation, combining
+the information of control-flow and function-call graphs. Our
+cfg2vec learns how to represent each binary function compiled
+from various CPU architectures, utilizing hierarchical GNN and
+the siamese network-based supervised learning architecture. We
+evaluate cfg2vec’s capability of predicting function names from
+stripped binaries. Our results show that cfg2vec outperforms the
+state-of-the-art by 24.54% in predicting function names and can
+even achieve 51.84% better given more training data. Addition-
+ally, cfg2vec consistently outperforms the state-of-the-art for all
+CPU architectures, while the baseline requires multiple training
+to achieve similar performance. More importantly, our results
+demonstrate that our cfg2vec could tackle binaries built from
+unseen CPU architectures, thus indicating that our approach
+can generalize the learned knowledge. Lastly, we demonstrate its
+practicability by implementing it as a Ghidra plugin used during
+resolving DARPA Assured MicroPatching (AMP) challenges.
+Index Terms—Software Reverse Engineering; Binary Analysis;
+Cross-Architecture; Machine Learning; Graph Neural Network;
+I. INTRODUCTION
+In mission-critical systems, embedded software is vital in
+manipulating physical processes and executing missions that
+could pose risks to human operators. Recently, the Internet of
+Things (IoT) has created a market valued at 19 trillion dollars
+and drastically grown the number of connected devices to
+approximately 35 billion in 2025 [1]–[3]. However, while IoT
+brings technological growth, it unintendedly exposes mission-
+critical systems to novel vulnerabilities [4]–[6]. The reported
+This material is based upon work supported by the Defense Advanced
+Research Projects Agency (DARPA) and Naval Information Warfare Center
+Pacific (NIWC Pacific) under Contract Number N66001-20-C-4024 The
+views, opinions, and/or findings expressed are those of the author(s) and
+should not be interpreted as representing the official views or policies of
+the Department of Defense or the U.S. Government.
+Distribution Statement ”A” (Approved for Public Release, Distribution
+Unlimited).
+number of IoT cyberattacks increased by 300% in 2019 [7],
+while the discovered software vulnerabilities rose from 1.6k to
+100k [8]. The consequence can be detrimental, as indicated in
+[9], the Heartbleed bug [10] can lead to a leakage of up to 64K
+memory, threatening not only personal but also organizational
+information security. Besides, Shellshock is a bash command-
+line interface shell bug, but it has existed for 30 years and
+remains a threat to enterprises today [11], [12]. For mission-
+critical systems, unexpected disruptions can incur millions of
+dollars even if they only last for a few hours or minutes [13].
+As a result, timely analyzing these impacted software and
+patching vulnerabilities becomes critical.
+Fig. 1. Legacy software life cycle.
+However, mission-critical systems usually use software that
+can last for decades due to the criticality of the missions.
+Over time, these systems become legacy, and the number
+of newly-discovered threats can increase (as illustrated in
+Figure 1). Typically, for legacy software, the original devel-
+opment environment, maintenance support, or source code
+might no longer exist. To address vulnerabilities, vendors offer
+patches in the form of source code changes based on the
+current software version (e.g., ver 0.9). However, the only
+available data in the legacy system is binary based on its
+source code (e.g., ver 0.1). Such a version gap poses challenges
+in applying patches to the legacy binaries, leaving the only
+solution for applying patches to legacy software as direct
+binary analysis. Today, as Figure 2 shows, Reverse Engineers
+(REs) have to leverage Software Reverse Engineering (SRE)
+arXiv:2301.02723v1  [cs.SE]  6 Jan 2023
+
+田
+Deploy
+日日日
+[
+</>
+Developer
+Source Code
+Binary
+50 years ago
+MissionCriticalEmbeddedSystems
+Present
+-
+ Developing tools missing.
+Extract
+ Source code missing
+ Developer missing
+Legacy Binary
+MissionCriticalEmbeddedSystemstools such as Ghidra [14], HexRays [15], and radare2 [16]
+to first disassemble and decompile binaries into higher-level
+representations (e.g., C or C++). Typically, these tools take
+the debugging information, strings, and the symbol-table and
+binary to reconstruct function names and variable names,
+allowing REs to rebuild a software’s structure and functionality
+without access to source code [17]. For REs, these symbols
+encode the context of the source code and provide invaluable
+information that could help them to understand the program’s
+logic as they work to patch vulnerable binaries. However, sym-
+bols are often excluded for optimizing the binary’s footprint
+in mission-critical legacy systems where memory is limited.
+Because recovering symbols from stripped binaries is not
+straightforward, most decompilers assign meaningless symbol
+names to coding elements. As for understanding the software
+semantics, REs have to leverage their experience and expertise
+to consume the information and then interpret the semantics
+of each coding element.
+Recent works tackle these challenges with Machine Learn-
+ing (ML), aiming to recover the program’s information from
+raw binaries. For example, [18], and [19] associate code fea-
+tures to function names and model the relationships between
+such code features and the corresponding source-level infor-
+mation (variable names in [19], variable & function names
+in [18]). Meanwhile, [20] and [21] use an encoder-decoder
+network structure to predict function names from stripped
+binary functions based on instruction sequences and control
+flows. However, none of them support cross-architectural
+debug information reconstruction. On the other side, there
+exist works focusing on the cross-platform in their ML mod-
+els [22]–[24]. These works focus on modeling the binary code
+similarity, extracting a real-valued vector from each control-
+flow graph (CFG) with attributed features, and then computing
+the Structural Similarity between the feature vectors of binary
+functions built from different CPU architectures.
+In this paper, as part of a multi-industry-academia joint
+initiative between Siemens, the Johns Hopkins University
+Applied Physics Laboratory (JHU/APL), BAE Systems (BAE),
+and UCI, we propose cfg2vec, which utilizes a hierarchical
+Graph Neural Network (GNN) for reconstructing the name
+of each binary function, aiming to develop the capacity for
+quick patching of legacy binaries in mission-critical systems.
+Our Cfg2vec forms a Graph-of-Graph (GoG) representation,
+combining CFG and FCG to model the relationship between
+binary functions’ representation and their semantic names.
+Besides, cfg2vec can tackle cross-architectural binaries thanks
+to the design of Siamese-based network architecture, as shown
+in Figure 3. One crucial use case of cross-architectural de-
+compilation is patching, where the goal is to identify a known
+vulnerability or a bug and apply a patch. However, there can be
+architecture gaps when software with a bug can be compiled
+into many devices with diverse hardware architectures. For
+example, it is challenging to patch a stripped binary from an
+exotic embedded architecture compiled ten years ago that is
+vulnerable to a known attack such as Heartbleed [10]. While
+the reference patch is available in software, the reference
+architecture may not be readily available or documented, or
+the vendor may no longer exist. Under such circumstances,
+mapping code features across architectures is very helpful.
+It would allow for identifying similarities in code between a
+stripped binary that is vulnerable and its reference patch, even
+if the patch were built for a different type of CPU architecture.
+For cfg2vec, our targeted contributions are as follows:
+• We propose representing binary functions in Graph-of-
+Graph (GoG) and demonstrate its usefulness in recon-
+structing function names from stripped binaries.
+• We propose a novel methodology, cfg2vec that uses a
+hierarchical Graph Neural Network (GNN) to model
+control-flow and function-calling relations in binaries.
+• We propose using cross-architectural loss when training,
+allowing cfg2vec to capture the architecture-agnostic rep-
+resentations of binaries.
+• We release cfg2vec in a GitHub repository: https://github.
+com/AICPS/mindsight cfg2vec.
+• We integrate our cfg2vec into an experimental Ghidra plu-
+gin, assisting the realistic scenarios of patching DARPA
+Assured MicroPatching (AMP) challenge binaries.
+The paper is structured as follows: Section II discusses
+related works and fundamentals to provide a better under-
+standing of the paper. Section III describes cfg2vec, includ-
+ing problem formulation, data preprocessing, and our main
+pipeline introduction. Section IV shows our experimental
+results. Lastly, we conclude the paper in Section V.
+II. RELATED WORK
+This section introduces software reverse engineering back-
+grounds, discusses the related works using machine learning
+to improve reverse engineering, and ultimately covers graph
+learning for binary analysis.
+A. Software Reverse Engineering
+Software Reverse Engineering (SRE) aims at understanding
+the behavior of a program without having access to its source
+code, often being used in many applications such as detecting
+malware [25], [26], discovering vulnerabilities, and patching
+bugs in legacy software [27], [28]. One primary tool that
+Reverse Engineers (REs) use to inspect programs is disassem-
+bler which translates a binary into low-level assembly code.
+Examples of such tools include GNU Binutils’ objdump [29],
+IDA [15], Binary Ninja [30], and Hopper [31]. However, even
+with these tools, reasoning at the assembly level still requires
+considerable cognitive effort from RE experts.
+More recently, REs use decompilers such as Hex-Rays [32],
+or Ghidra [14] to reverse the compiling process by further
+translating the output of disassemblers into the code that
+ensembles high-level programming languages such as C or
+C++ to reduce the burden of understanding assembly code.
+From assembly instructions, these decompilers can use pro-
+gram analysis and heuristics to reconstruct variables, types,
+functions, and control flow structure of a binary. However,
+the decompilation is incomplete even if these decompilers
+generate a higher-level output for better code understanding.
+
+Fig. 2. The RE flow to solve security issues.
+The reason is that the compilation process discards the source-
+level information and lowers its abstraction level in exchange
+for a smaller footprint size, faster execution time, or even
+security considerations. The source-level information such as
+comments, variable names, function names, and idiomatic
+structure can be essential for understanding a program but is
+typically unavailable in the output of these decompilers.
+As Figure 2 demonstrated, REs use disassemblers or de-
+compilers to generate high-level source code. Besides, [33]
+indicates REs will take notes and grant a name to those
+critical functions related to the vulnerabilities. This will create
+an annotated source code based on the high-level machine-
+generated source code. While annotating the source code, REs
+also analyze the significant part related to the vulnerability
+and ignore those general instructions or unrelated codes. At
+the same time, understanding the logic flow among functions
+is another major task they must focus on resolving their
+tasks. After classification, annotation, and understanding, REs
+experiment with several viable remedies to find the correct
+patch to fix the vulnerability.
+B. Machine Learning for Reverse Engineering
+Software binary analysis is a straightforward first step to
+enhance security as developers usually deploy software in
+binaries [34]. Usually, experts conduct the patching process
+or vulnerability analysis by understanding the compilation
+source, function signatures, and variable information. How-
+ever, after the compilation, such information is usually stripped
+or confuscated deliberately (e.g., obfuscation). Software binary
+analysis becomes more challenging in this case as developers
+have to recover the source-level information based on their
+experience and expertise. The early recovery work for binaries
+focuses on manual completion but suffers from low efficiency,
+high cost, and the error-prone nature of reverse engineering.
+As Machine Learning (ML) has significantly advanced in its
+reasoning capability, applying ML and reconstructing higher-
+level source code information as an alternative to manual-
+based approaches has attracted considerable research attention.
+For example, [35] was the first approach that used neural
+network-based and graph-based models, predicting the func-
+tion types to assist the reverse engineer in understanding
+the binary. [36] also predicted function names with neural
+networks, aggregating the related features of sections of bi-
+nary vectors. Then, it analyzes the connections between each
+function in the source code (e.g., Java) and their corresponding
+function names for function name prediction. [18], on the other
+hand, did not use a neural network. It combined a decision-
+tree-based classification algorithm and a structured prediction
+with a probabilistic graphical model, then matched the func-
+tion name by analyzing symbol names, types, and locations.
+However, [18] can only predict from a predetermined closed
+set, incapable of generalizing to new names.
+As the languages for naming functions are similar to nat-
+ural language, recent research works start leaning toward the
+use of Natural Language Processing (NLP) [20], [21], [37].
+Precisely, these models predict semantic tokens based on the
+function names in the library, comprising the function name
+during inference. The underlying premise is that each token
+corresponds in some way to the attributes and functionality of
+the function. [20] uses Control-Flow Graph (CFG) to predict
+function names. It combined static analysis with LSTM and
+transformer neural model to realize the name of functions.
+However, the dataset that consisted of unbalanced data and in-
+sufficient features was limited and hindered utter performance.
+[37] was designed to solve the limitation of the dataset. It
+provided UbuntuDataset that contained more than 9 million
+functions in 22K software. [21] demonstrated the framework’s
+effectiveness by building a large dataset. It considers the
+fine-grained sequence and structure information of assembly
+code when modeling and realizing function name prediction.
+Meanwhile, [21] reduced the diversity of data (instructions or
+words) while keeping the basic semantics unchanged, similar
+to word stemming and semantics in NLP. However, these
+works have low precision scores for prediction tasks, exampled
+by [21], only achieving around 41% in correctly predicting
+the function name subtokens. Moreover, the metrics for the
+inference of unknown functions are substantially lower [21],
+making it difficult for REs to find it helpful in practice.
+Although many existing works can reconstruct source-
+level information, none of them supports reconstructing cross-
+platform debug information. Cross-compilation is becoming
+more popular in the development of software. Hardware
+manufacturers, for instance, often reuse the same firmware
+code base across several devices running on various archi-
+tectures [38]. A tool that performs cross-architecture function
+name prediction/matching would be beneficial if we have a
+stripped binary compiled for one architecture and a binary
+of a comparable program compiled for another architecture
+with debug symbols. We may use the binary with the debug
+symbols to predict the names of functions in the stripped
+binary, which significantly aids debugging. A tool that could
+capture the architecture-agnostic characteristics of binaries
+
+IDA/GNU
+IDA/GNU
+Dismiss unrelated BBs
+Hypothesis
+disprove
+ASM
+<>
+c
+approved
+Reconstruct code logic
+Reverse
+Verifying
+Engineers
+Vulnerabili
+high-level C
+binary
+annotated C
+patching
+Assembly
+Understand program
+code
+code
+SRE Tools
+Problem Solving from REswould also help in malware detection as the source code of
+malware can be compiled in different architectures [38], [39].
+Comparing two binaries of different architectures becomes
+more complicated because they will have different instruction
+sets, calling conventions, register sets, etc. Furthermore, as-
+sembly instructions from different architectures cannot often
+be compared directly due to the slightly different behavior of
+different architectures [40]. Cross-architecture function name
+prediction will assist in finding a malicious function in a
+program compiled for different architectures by learning its
+features from a binary compiled for just one architecture. The
+tools mentioned above are not architecture-agnostic; thus, we
+cannot utilize them for such applications. To address the flaws
+mentioned above, aid in creating more efficient decompilers,
+and make reverse engineering more accessible, we propose
+cfg2vec. Incorporating the cross-architectural siamese network
+architecture, our cfg2vec can learn to extract robust features
+that encompass platform-independent features, enhancing the
+state-of-the-art by achieving function name reconstruction
+across cross-architectural binaries.
+C. Graph Learning for Binary Analysis
+Graph learning has become a practical approach across
+fields [41]–[44]. Although conventional ML can effectively
+capture the features hidden in Euclidean data, such as images,
+text, or videos, our work focuses more on the application
+where the core data is graph-structured. Graphs can be ir-
+regular, and a graph may contain a variable size of unordered
+nodes; moreover, nodes can have a varying number of neigh-
+boring nodes, making deep learning mathematical operations
+(e.g., 2D Convolution) challenging to apply. The operations in
+conventional ML methods can only be applied by projecting
+non-Euclidean data into low-dimensional embedding space.
+In graph learning, Graph Embeddings (GE) can transform a
+graph into a vector (embedding of a graph) or a set of vectors
+(embedding of nodes or edges) while preserving the relevant
+and structural information about the graph [41]. Graph Neural
+Network (GNN) is a model aiming at addressing graph-related
+tasks in an end-to-end manner, where the main idea is to
+generate a node’s representation by aggregating its representa-
+tion and the representations of its neighbors [42]. GNN stacks
+multiple graph convolution layers, graph pooling layers, and a
+graph readout to generate a low-dimensional graph embedding
+from high-dimensional graph-structured data.
+In software binary analysis, many approaches use Control-
+Flow Graphs (CFGs) as the primary representations. For
+example, Genius forms an Attributed Control-Flow Graph
+(ACFG) representation for each binary function by extracting
+the raw attributes from each Basic Block (BB), a straight-line
+code sequence with no branching in or out except at the entry
+and exit, in an ACFG [22]. Genius measures the similarity
+of a pair of ACFGs through a bipartite graph matching algo-
+rithm, and the ACFGs are then clustered based on similarity.
+Genius leverages a codebook for retrieving the embedding
+of an ACFG based on similarity. Another approach, Gemini,
+proposes a deep neural network-based model along with a
+Siamese architecture for modeling binary similarities with
+greater efficiency and accuracy than other state-of-the-art mod-
+els of the time [23]. Gemini takes in a pair of ACFGs extracted
+from raw binary functions generated from known vulnerability
+in code and then embeds them with a shared Structure2vec
+model in their network architecture. Once embedded, Gemini
+trains its model with a loss function that calculates the cosine
+similarities between two embedded representations. Gemini
+outperforms models like Genius or other approaches such as
+bipartite graph matching. In literature, there exist other works
+that consider the Function Call Graph (FCG) as their primary
+data structures in binary analysis for malware detection [45].
+Our cfg2vec extracts relevant platform-independent features by
+combining the usage of CFG and FCG, resulting in a Graph-
+of-Graph (GoG) representation for cross-architectural high-
+level information reconstruction tasks (e.g., function name).
+III. CFG2VEC ARCHITECTURE
+This section begins with problem formulation. Next, as
+Figure 4 shows, we depict how our cfg2vec extracts the Graph-
+of-Graph (GoG) representation from each software binary.
+Lastly, we describe the network architecture in cfg2vec.
+A. Problem Formulation
+In our work, given a binary code, denoted as p, compiled
+from different CPU architectures, we extract a graph-of-graph
+(GoG) representation, G = (V , A) where V is the set of
+nodes and A is the adjacency matrix (As Figure 3 shows). The
+nodes in V represent functions and the edges in A indicate
+their cross-referencing relationships. That says, each of the
+node fi ∈ V is a CFG, and we denote it as fi = (B, A, φ)
+where the nodes in B represent the basic blocks and the edges
+in A denote their dependency relationships. φ is a mapping
+function that maps each basic block in the assembly form to
+its corresponding extracted attributes φ(vi) = Ck where C is a
+numeric value, and k is the number of attributes for the basic
+block (BB). Whereas the CFG structure is meant to provide
+more information at the lower BB level, the GoG structure
+is intended for recovering information at the overarching
+function level between the CFGs. Figure 3 is an example of
+a partial GoG structure with a closer inspection of one of its
+CFG nodes and another of a single CFG BB node, showing
+the set of features corresponding to that BB node. The goal is
+to design an efficient and effective graph embedding technique
+that can be used for reconstructing the function names for each
+function fi ∈ V .
+B. Ghidra Data ToolKit for Graph Extraction
+To extract the structured representation required for cfg2vec
+we leverage the state-of-the-art decompiler Ghidra [14] and
+the Ghidra Headless Analyzer1. The headless analyzer is a
+command-line version of Ghidra allowing users to perform
+many tasks (such as analyzing a binary file) supported by
+Ghidra via a command-line interface. For extracting GoG
+1Documentation of Ghidra Headless Analyzer: https://ghidra.re/ghidra
+docs/analyzeHeadlessREADME.html
+
+Fig. 3. An example of a Graph-of-Graph (GoG) of a binary compiled from a package Freecell with amd64 CPU architecture.
+from a binary, we developed our Ghidra Data Toolkit (GDT);
+GDT is a set of Java-based metadata extraction scripts used
+for instrumenting Ghidra Headless Analyzer. First, GDT pro-
+grammatically analyzes the given executable file and stores the
+extracted information in the internal Ghidra database. Ghidra
+provides a set of APIs to access the database and retrieve the
+information about the analyzed binary. GDT uses these APIs to
+export information such as Ghirda’s PCode and call graph for
+each function. Specifically, the FunctionManager API allows
+us to manipulate the information of each decompiled function
+in the binary and acquire the cross-calling dependencies be-
+tween functions. For each function, we utilized another Ghidra
+API DecompInterface2 to extract 12 attributes associated with
+each basic block in a function. These attributes precisely corre-
+spond to the total number of instructions, including arithmetic,
+logic, transfer, call, data transfer, SSA, compare, and pointer
+instructions, as well as other instructions not falling within
+those categories and the total number of constants and strings
+within that BB. Lastly, by integrating all of the information, we
+form a GoG representation G for each binary p. We repeat this
+process until all binaries are converted to the GoG structure.
+We feed the resulting GoG representations to our model in
+batches, with the batch size denoted as B.
+C. Hierarchical Graph Neural Network
+Once G is extracted from the GDT, we then feed it to
+our hierarchical network architecture (inspired from [46]) that
+contains both CFG Graph Embedding layer and GoG Graph
+Embedding Layer as Figure 4 shows. For each GoG structure,
+we denote it as G = (V , A) where V is a set of functions
+associated with G and A indicates the calling relationships
+between the functions in V . Each function in V is in the form
+of CFG fi = (B, A, φ) where each node b ∈ B is a BB
+represented in a fixed-length attributed vector b ∈ Rd, and d
+is the dimension that we have mentioned earlier. A encodes
+the pair-wise control-flow dependency relationships between
+these BBs.
+1) CFG Graph Embedding Layer: Our network architec-
+ture first feeds all functions in a batch of GoGs to the
+CFG Graph Embedding Layer consisting of multiple graph
+2Documentation of Ghidra API DecompInterface: https://ghidra.re/ghidra
+docs/api/ghidra/app/decompiler/DecompInterface.html
+convolutional layers and a graph readout operations. The input
+to this layer is a function fi = (B, A, φ) and the output is the
+fixed-dimensional vector representing a function. For each BB
+bk we let b0
+k = bk, and we update bt
+k to bt+1
+k
+with the graph
+convolution operation shown as follows:
+bt+1
+k
+= fG(Wbt
+k +
+�
+bm∈Ak
+Mbt
+m)
+where fG is a non-linear activation function such as ReLU,
+Ak is the list of adjacent BBs for bk, and W ∈ Rd×d and
+M ∈ Rd×d are the weights to be learned during the training.
+We run T iterations of such a convolution, which can be a
+tunable hyperparameter in our model. During the updates, each
+BB gradually aggregates the global information of the control-
+flow dependency relations into its representation, utilizing the
+representation of its neighbor. We obtain the final represen-
+tation for each BB as bT
+k . To acquire the representation for
+the function fi, we apply a graph readout operation such as
+sum-readout, described as follows,
+g(T ) =
+�
+bk∈B
+bT
+k
+(1)
+We assign the value of g(T ) (a.k.a. CFG embedding) to fi. The
+graph readout operation can be replaced with mean-readout or
+max-readout.
+2) GoG Graph Embedding Layer: Once all the functions
+have been converted to fixed-length graph embeddings, we
+then feed G to the second layer of cfg2vec, the GoG Embed-
+ding Layer. Here, for each function fi we apply another L
+iterations of graph convolution with F and C. The updates
+can be illustrated as follows,
+f (l+1)
+k
+= fGoG(Uf l
+k +
+�
+fm∈Ck
+V f (l)
+m )
+(2)
+where fGoG is a non-linear activation function and Ck is the
+list of adjacent functions (calling) for the function fk and U ∈
+Rd×d and V ∈ Rd×d are the weights to be learned during the
+training. Lastly, we take the f (L)
+k
+as the representation that
+considers both CFG and GoG graph structures. We use these
+updated representations to perform cross-architecture function
+similarity learning.
+
+1001011110001
+ALLSAR
+0101010101011
+GHIDRA
+山
+010001010010
+010101010000
+0101010101011
+18Fig. 4. The architecture of cfg2vec with a supervised hierarchical graph neural network approach.
+3) Siamese-based Cross-Architectural Function Similarity:
+Given a batch of GoGs B = {GoG1, GoG2, ..., GoGB},
+we apply the hierarchical graph neural network to acquire
+the set of updated function embeddings, denoted as BF =
+{f (T )
+1
+, f (T )
+2
+, ..., f (T )
+K }. We calculate the function similarity
+for each function pair with cosine similarity, denoted as
+ˆy ∈ [−1, 1]. The loss function J between ˆY and a ground-
+truth label y, which indicates whether a pair of functions have
+the same function or not, is calculated as follows,
+J(ˆy, y) =
+�
+1 − y,
+if y=1,
+MAX(0, ˆy − m),
+if y=-1,
+(3)
+the final loss L is then calculated as follows,
+L = H(Y, ˆY ) =
+�
+i
+(J( ˆyi, yi)),
+(4)
+where Y stands for ground-truth labels (either similarity or
+dissimilarity), and ˆY represents the corresponding predictions.
+More specifically, we denote a pair of functions as simi-
+lar if they are the same but compiled with different CPU
+architectures. The m is a constant to prevent the learned
+embeddings from becoming distorted (by default, 0.5). To
+maintain the balance between positive and negative training
+samples, we developed a custom batching algorithm. The
+function leverages the knowledge gained by adding a binary
+of some package to a given batch to find and add a binary
+for the same package, built for a different architecture, to the
+provided batch as a positive sample. It will also include a
+binary from another package as a negative sample. This will
+give any batch a balanced proportion of positive and negative
+samples. Finally, we use the loss L to update all the associated
+weights in our neural networks with an Adam optimizer. Once
+trained, we then use the model to perform function name
+reconstruction tasks.
+IV. EXPERIMENTAL RESULTS
+In this section, we evaluate cfg2vec’s capability in predicting
+function names. We first describe the dataset preparation and
+the training setup processes. Then, we present the comparison
+of cfg2vec against baseline in predicting function names.
+Although many baseline candidates tackle the same prob-
+lem [18], [20], [21], [37], some require purchasing a paid
+version of IDA Pro to preprocess datasets, and some even do
+not open source their implementations. Therefore, [18] was
+the only feasible choice, as running other models using our
+datasets was almost impossible. Next, we also show the result
+of the ablation study over cfg2vec. Besides, we exhibit that
+our cfg2vec can perform architecture-agnostic prediction better
+than the baseline. Lastly, we illustrate the real-world use case
+where our cfg2vec is integrated as a Ghidra plugin applica-
+tion for assisting in resolving challenging reverse engineering
+tasks. We conducted all experiments on a server equipped with
+Intel Core i7-7820X CPU @3.60GHz with 16GB RAM and
+two NVIDIA GeForce GTX Titan Xp GPUs.
+A. Dataset Preparation
+Our evaluating data source is the ALLSTAR (Assembled
+Labeled Library for Static Analysis Research) dataset, hosted
+by Applied Physics Laboratory (APL) [47]. It has over 30,000
+Debian Jessie packages pre-built from i386, amd64, ARM,
+MIPS, PPC, and s390x CPU architectures for software
+reverse engineering research. The authors used a modified
+Dockcross script in docker to build each package for each
+supported architecture. Then, they saved each resulting ELF
+with its symbols, the corresponding source code, header files,
+intermediate files (.o, .class, .gkd, .gimple), system headers,
+and system libraries altogether.
+To form our datasets, we selected the packages that have
+ELF binaries built for the amd64, armel, i386, and
+mipsel CPU architectures. i386 and amd64 are widely
+used by general computers, especially in the Intel and AMD
+products, respectively. MIPS and ARM are crucial in em-
+bedded systems, smartphones, and other portable electronic
+devices [48]. In practice, we excluded the packages with only
+one CPU architecture in the ALLSTAR dataset. Additionally,
+due to our limited local computing resources, we eliminated
+packages that were too large to handle. We checked each
+selected binary on whether the ground-truth symbol infor-
+mation exists using the Ghidra decompiler and Linux file
+command and removed the ones that do not have them. Lastly,
+we assembled our primary dataset, called the AS-4cpu-30k-bin
+dataset, that consists of 27572 pre-built binaries from 1117
+packages and 4 CPU architectures, as illustrated in Table I.
+Our preliminary experiment revealed that the evaluation
+had a data leakage issue when splitting the dataset randomly.
+Therefore, we performed a non-random variant train-test split
+with a 4-to-1 ratio on the AS-4cpu-30k-bin dataset, selecting
+roughly 80% of the binaries for the training dataset and
+leaving the rest for the testing dataset. We referenced [23]
+for their splitting methods, aiming to ensure that the binaries
+
+CFG
+GoG
+Embedding
+Embedding
+创金
+Layer
+Layerthat belong to the same packages stay in the same set, either
+the training or testing sets. Such a variant splitting method
+allows us to evaluate cfg2vec truly.
+Next, we converted binaries in the AS-4cpu-30k-bin dataset
+into their Graph-of-Graph (GoG) representations leveraging
+the GDT mentioned previously in Section III-B. Notably,
+we processed a batch of binaries related to one package
+at one time as developers might define user functions in
+different modules of the same package while putting prototype
+declarations in that package’s main module. For this case,
+Ghidra indeed recognizes two function instances while one
+only contains the function declaration and another has its
+actual function content. As these two instances correspond
+to the same function name and one contains only dummy
+instructions, they can thus create noise in our datasets, thus
+affecting our model’s learning. To cope with this, our GDT
+also searches from other binaries of the same package for the
+function bodies. If found, our GDT associates that user func-
+tion with the function graph node with the actual content data.
+Besides user functions, library function calls may exist, and
+searching their function bodies in the same package would fail
+for dynamically loaded binaries. Under such circumstances,
+Ghidra would recognize these functions as ThunkFunctions3
+which only contain one dummy instruction. As a workaround,
+we removed these ThunkFunctions from our data as they
+might mislead the model’s learning. Applying this workaround
+indicates that our model works in predicting function names
+for the user and statically linked functions.
+TABLE I
+THE STATISTICS OF DATASETS USED IN OUR EXPERIMENTS.
+Dataset / #
+pkg/bin
+func node/edge1
+bb node/edge2
+AS-4cpu-30k-bin
+1117/27,572
+51.17/97.14
+14.12/19.98
+AS-3cpu-9k-bin
+633/9,000
+44.01/79.06
+12.24/17.07
+AS-i386-3k-bin
+633/3,000
+45.31/87.70
+11.45/15.97
+AS-amd64-3k-bin
+633/3,000
+42.28/74.07
+12.28/17.18
+AS-armel-3k-bin
+633/3,000
+44.45/75.41
+13.00/18.07
+1 # of average functions and edges in each binary
+2 # of average bb blocks and edges from each function
+We experimented [18] with our datasets, referencing to
+their implementation4. As [18] used a dataset with 3,000
+binaries for experiments, we followed accordingly, prepar-
+ing datasets with smaller but similar sizes. We achieved
+this by downsampling from our primary AS-4cpu-30k-bin
+dataset, creating the AS-3cpu-9k-bin dataset which has 9,000
+binaries for i386, amd64, and armel CPU architectures.
+Furthermore, as [18] supports only one CPU architecture at
+a time, we then separated the AS-3cpu-9k-bin dataset into
+different CPU architectures, generating three training datasets
+for testing [18]: AS-i386-3k-bin, AS-amd64-3k-bin, and AS-
+armel-3k-bin. For training, we utilized the strip Linux
+command, converting our original data into three: the original
+binaries (debug), stripped binaries with debug information
+3ThunkFunction Manual: https://ghidra.re/ghidra docs/api/ghidra/program/
+model/listing/ThunkFunction.html
+4Debin’s [18] repository: https://github.com/eth-sri/debin
+(stripped), and stripped binaries without debug information
+(stripped wo symtab) to follow [18]’s required data format.
+For evaluation, we sampled 100 binaries from our primary
+dataset for each CPU architecture, labeled AS-amd-100-bin,
+AS-i386-100-bin, AS-armel-100-bin, and AS-mipsel-100-bin.
+We also have another evaluation dataset called AS-noMipsel-
+300-bin, which contains roughly 300 binaries produced for the
+amd64, i386, and armel platforms. Table I summarizes the
+data statistics for all these datasets, including the numbers of
+packages and binaries, the average number of function nodes,
+edges, and BB nodes. The following sections will detail how
+we utilized these datasets during our experiments.
+B. Evaluation: Function Name Prediction
+Table II demonstrates the results of cfg2vec in predicting
+function names. For the baseline, we followed [18]’s best
+setting where the feature dimension of register or stack offset
+are both 100 to train with our prepared datasets. For cfg2vec,
+we used three GCN layers and one GAT convolution layer
+in both graph embedding layers. For evaluation, we calculate
+the p@k (e.g., precision at k) metric, which refers to an
+average hit ratio over the top-k list of predicted function
+names. Specifically, we feed each binary represented in GoG
+into our trained model, converting each function f ∈ F and
+acquiring its function embedding hf. Then, we calculate pair-
+wise cosine similarities between hf and all the other function
+embeddings, forming a top-k list by selecting k names in
+which their embeddings are top-kth similar to hf. If the
+ground-truth function name is among the top-k list of function
+name predictions, we regard that as a hit; otherwise, it is a
+miss. During experiments, we set the top-k value to be 5, so
+our model can recommend the best five possible names for
+each function in a binary.
+As shown in Table II, cfg2vec, trained with the AS-3cpu-9k-
+bin dataset, can achieve a 69.75% prediction accuracy (e.g.,
+p@1) in inferring function names. For [18], we had to train
+their models for each CPU architecture separately as it cannot
+train in a cross-architectural manner. Even so, for amd64
+binaries, [18] only achieves 29.32% precision, while for i386
+and armel, it performs 52.64% and 53.65%, respectively.
+This result indicates that in any case, our cfg2vec outperforms
+[18]. Besides, while [18] only yields one prediction, our
+cfg2vec suggests five choices, making it flexible for our users
+(e.g., REs) to select what they believe best fits the function
+among the best k predicted names. The p@2 to p@5 in Table II
+demonstrate that our cfg2vec can provide enough hints of
+function names for users. For example, p@5 of cfg2vec trained
+with our AS-3cpu-9k-bin dataset can achieve 70.50% precision
+across all the CPU architecture binaries. We also experimented
+our cfg2vec with larger datasets. From Table II, we can observe
+that cfg2vec can have 5.04% performance gain in correctly
+predicting function names (e.g., p@1). Moreover, the gain
+increases to 28% when training cfg2vec with the AS-4cpu-30k-
+bin dataset. We believe training on a larger dataset implies
+training with a more diversified set of binaries. This allows
+our model to acquire more knowledge, thus being capable
+
+TABLE II
+THE PERFORMANCE EVALUATION OF cfg2vec FOR FUNCTION NAME PREDICTION AGAINST [18].
+Model
+Training dataset
+Testing dataset
+P@11
+P@21
+P@31
+P@41
+P@51
+cfg2vec
+AS-4cpu-30k-bin
+AS-noMipsel-300-bin
+97.05%
+99.47%
+99.47%
+99.47%
+99.47%
+cfg2vec
+AS-4cpu-20k-bin
+AS-noMipsel-300-bin
+74.22%
+75.76%
+75.78%
+75.78%
+78.78%
+AS-amd-100-bin
+69.18%
+69.98%
+69.98%
+69.98%
+69.98%
+cfg2vec
+AS-3cpu-9k-bin
+AS-i386-100-bin
+69.41%
+70.39%
+70.39%
+70.39%
+70.39%
+AS-armel-100-bin
+70.66%
+71.04%
+71.11%
+71.11%
+71.11%
+AS-noMipsel-300-bin
+69.75%
+70.47%
+70.50%
+70.50%
+70.50%
+[18]-amd642
+AS-amd64-3k-bin
+AS-amd-100-bin
+29.32%
+-
+-
+-
+-
+[18]-i3862
+AS-i386-3k-bin
+AS-i386-100-bin
+52.64%
+-
+-
+-
+-
+[18]-armel2
+AS-armel-3k-bin
+AS-armel-100-bin
+53.65%
+-
+-
+-
+-
+1 P@k measures if the actual function name is in the top k of the predicted function names.
+2 These models only provide the top 1 function name prediction; hence they only have P@1 value.
+of extracting more robust features for binary functions. In
+summary, this result indicates that compared to the baseline,
+our model can effectively provide contextually relevant names
+for functions in the decompiled code to our users.
+TABLE III
+THE COMPARISON BETWEEN cfg2vec AND ITS ABLATED VARIATIONS.
+Arch
+[18]
+GCN-GAT
+2GCN
+2GCN-GAT
+cfg2vec
+amd64
+29.32%1
+61.59%
+69.49%
+69.56%
+70.66%
+armel
+52.64%2
+66.40%
+68.59%
+68.92%
+69.19%
+mipsel
+53.65%3
+66.47%
+68.17%
+68.56%
+69.41%
+Overall
+45.20%
+64.82%
+68.75%
+69.01%
+69.75%
+1 Evaluation results for [18]-amd64 model.
+2 Evaluation results for [18]-i386 model.
+3 Evaluation results for [18]-armel model.
+We also experimented with various ablated network setups
+to study how each component of cfg2vec contributes to per-
+formance. First, we simplified our cfg2vec by stripping one
+GCN layer from the original experimental setup. As shown
+in Table III, we called this setup 2GCN-GAT which slightly
+decreased the performance by 0.75%. Then, from 2GCN-GAT
+setup, we further removed the GAT layer, calling it 2GCN.
+We again observed a marginal performance decrease (<1%).
+Next, we eliminated another GCN layer from 2GCN-GAT,
+constructing the GCN-GAT setup. For GCN-GAT, we saw
+a drastic drop (4.2%) which highlights that the number of
+GCN layers can be an essential factor in the performance.
+Specifically, we found that going from 1 to 2 GCN layers
+improves prediction accuracy by more than 4%. However, we
+do not observe a significant performance gain when increasing
+the number of GCN layers to more than three. Therefore,
+we retained the original cfg2vec model with its three GCN
+layers. All in all, as shown in Table III, all these ablated
+models, still outperform [18], which we attributed to the GoG
+representation we made for each binary in the dataset.
+C. Evaluation: Architectural-agnostic Prediction
+Table IV demonstrates our cfg2vec’s capability in terms
+of cross-architecture support. As [18] supports training one
+CPU architecture at a time, we had to train it multiple times
+during experiments. Specifically, we trained [18] on three
+datasets: AS-amd64-3k-bin, AS-i386-3k-bin, and AS-armel-3k-
+bin, calling resulting trained models, [18]-amd64, [18]-i386,
+and [18]-armel, respectively. For these baseline models, we
+observe that they perform well when tested with the binaries
+built on the same CPU architecture but poorly with the ones
+built on different CPU architectures. For instance, [18]-amd64
+achieves 29.3% accuracy for amd64 binaries, but performs
+worse for i386 and armel binaries (13.8% and 7.1%). Sim-
+ilarly, [18]-i386 achieves 52.6% accuracy for i386 binaries,
+but performs worse for amd64 and armel binaries (6.2%
+and 1.1%). Lastly, [18]-armel achieves 53.6% accuracy for
+armel binaries, but performs worse for amd64 and i386
+binaries (11.8% and 8.9%). We used the top-1 prediction
+generated from cfg2vec (a.k.a., p@1) as the comparing metric
+as [18] produces only one prediction per each function. From
+the results, we observe that cfg2vec outperforms [18] across all
+three tested CPU architectures. The fact that cfg2vec performs
+consistently well across all CPU architectures indicates that
+our cfg2vec supports cross-architecture prediction.
+To evaluate the capability of generalizing the learned knowl-
+edge, we tested all models with the AS-mipsel-100-bin dataset,
+which has binaries built from another famous CPU architec-
+ture, mipsel, that our cfg2vec does not train before. For [18],
+it has lower performance when testing on binaries built from
+the CPU architectures that it did not train before, exampled
+by the highest accuracy of [18] to be 13.84% when trained on
+amd64 binaries and evaluated on i386 binaries. In our work,
+as Table IV shows, our cfg2vec achieves 36.69% accuracy
+when trained with amd64, i386, and armel binaries but
+tested on mipsel binaries. For [18], it does not even support
+analyzing mipsel binaries. In short, these results demonstrate
+that our cfg2vec outperforms our baseline in the function name
+prediction task on cross-architectural binaries and generalizes
+better to the binaries built from unseen CPU architectures. To
+further investigate cfg2vec’s cross-architecture performance,
+we trained it on three datasets, each consisting of binaries
+built for two different architectures. We then gave the resulting
+trained models names that indicated the architectures from
+which the binaries were derived: cfg2vec-armel-i386, cfg2vec-
+amd64-i386, and cfg2vec-armel-amd64. These results show
+that our model performs well in the function name prediction
+job across all of these scenarios, including when tested on
+binaries compiled for unknown CPU architectures.
+
+TABLE IV
+THE CROSS-ARCHITECTURAL COMPARISON BETWEEN CFG2VEC AND [18]
+Model
+Testing dataset
+P@1
+AS-amd-100-bin
+69.18%
+cfg2vec-3-Arch
+AS-i386-100-bin
+70.66%
+AS-armel-100-bin
+69.41%
+AS-mipsel-100-bin∗
+36.69%
+AS-amd-100-bin
+68.53%
+cfg2vec-amd64-armel
+AS-i386-100-bin∗
+39.23%
+AS-armel-100-bin
+69.11%
+AS-mipsel-100-bin∗
+32.21%
+AS-amd-100-bin
+68.59%
+cfg2vec-amd64-i386
+AS-i386-100-bin
+69.09%
+AS-armel-100-bin∗
+34.20%
+AS-mipsel-100-bin∗
+38.26%
+AS-amd-100-bin∗
+42.96%
+cfg2vec-armel-i386
+AS-i386-100-bin
+67.45%
+AS-armel-100-bin
+63.86%
+AS-mipsel-100-bin∗
+36.61%
+AS-amd-100-bin
+29.32%
+[18]-amd64
+AS-i386-100-bin∗
+13.84%
+AS-armel-100-bin∗
+7.08%
+AS-mipsel-100-bin∗
+-
+AS-amd-100-bin∗
+6.23%
+[18]-i386
+AS-i386-100-bin
+52.64%
+AS-armel-100-bin∗
+1.05%
+AS-mipsel-100-bin∗
+-
+AS-amd-100-bin∗
+11.82%
+[18]-armel
+AS-i386-100-bin∗
+8.86%
+AS-armel-100-bin
+53.65%
+AS-mipsel-100-bin∗
+-
+∗ indicates that dataset was not used during the training.
+D. The Practical Usage of CFG2VEC
+In this section, we demonstrate how cfg2vec assists REs
+in dealing with Defense Advanced Research Projects Agency
+(DARPA) Assured MicroPatching (AMP) challenges binaries.
+The AMP program aims at enabling fast patching of legacy
+mission-critical system binaries, enhancing decompilation and
+guiding it toward a particular goal of a Reverse Engineer (RE)
+by integrating the existing source code samples, the original
+build process information, and historical software artifacts.
+1) The MINDSIGHT project: our multi-industry-academia
+initiative between Siemens, JHU/APL, BAE, and UCI jointly
+developed a project, Making Intelligible Decompiled Source
+by Imposing Homomorphic Transforms (MINDSIGHT). Our
+team focused on building an automated toolchain integrated
+with Ghidra, aiming to enable the decompilation process with
+(1) a less granular identification of modular units, (2) an
+accurate reconstruction of symbol names, (3) the lifting of
+binaries to stylized C code, (4) a principled and scalable
+approach to reason about code similarity, and (5) the bench-
+marking of new decompilation techniques using state-of-the-
+art embedded software binary datasets. To date, our team
+has developed an open-source tool, CodeCut5, to improve the
+accuracy and completeness of Ghidra’s module identification,
+providing an automated script-based decompilation analysis
+toolchain to ease the RE’s expert interpretation. Besides, we
+also developed a Homomorphic Transform Language (HTL) to
+describe transformations on Abstract Syntax Tree (AST) lan-
+guages and the rules of their composition. Our tool, integrated
+5CodeCut’s repository: https://github.com/DARPAMINDSIGHT/CodeCut
+with ghidra, allows developers to transform the decompiled
+code syntactically while keeping it semantically equivalent.
+The key idea is to use this HTL to morph a Ghidra AST
+into a GCC AST to lift the decompiled binary to a high-level
+C representation. This process can make it easier for REs to
+comprehend the binary code. cfg2vec is another tool developed
+in the MINDSIGHT project, enabling the reconstruction of
+function names, saving the manual guesswork from REs.
+Fig. 5. The plugin screenshot integrated into Ghidra.
+2) The cfg2vec plugin: In MINDSIGHT project, we incor-
+porated cfg2vec into Ghidra decompiler as a plugin applica-
+tion. Our cfg2vec plugin assists REs in comprehending the
+binaries by providing a list of potential function names for
+each function without its name. Technically, like all Ghidra
+plugins, our cfg2vec plugin bases on Java with its core
+inference modules implemented as a REST API in Python
+3.8. Once the metadata of a stripped binary is extracted from
+Ghidra decompiler, it is then sent to the cfg2vec end-point,
+which calculates and returns the inferred mappings for all
+the functions. Figure 5 demonstrates the user interface of
+our cfg2vec plugin. In this scenario, the user must provide
+the vulnerable and the reference binary with extra debug
+information, such as function names. The “Match Functions”
+button triggers cfg2vec functionality and displays the function
+mapping results in three tables:
+• Matched Table: displays the mapping of similar functions.
+• Mismatched Table: displays the mapping of dissimilar
+functions and, therefore, candidates for patching.
+• Orphan Table: displays the mapping of functions with a
+low confidence score.
+The groupings reduce REs’ workload. Rather than inspect-
+ing all functions, they can focus on patching candidate func-
+tions (mismatched functions) and the orphans. The “Explore
+Functions” button invokes Ghidra’s function explorer, where
+the two functions can be compared side-by-side, as shown
+in Figure 5. This utility allows the user to switch between
+C and assembly language, thus assisting in confirming or
+modifying the mappings from the three tables. Regarding
+
+MINDSIGHT Plugin
++
+x
+Select Vulnerable Program
+program_c
+Match Functions
+Select Patched Program
+program_c_patched
+Matched functions
+Vulnerable function
+Patched Function
+_DT_INIT
+_DT_INIT
+cxa_finalize
+cxa_finalize
+setsockopt
+setsockopt
+printf
+printf
+Mismatched functions
+Orphan functions
+Explore Functions
+Rename Function
+Program Diffcfg2vec’s function prediction, the “Rename Function” button
+takes the selected row from the tables and imposes the name
+from the patched binary in the vulnerable binary. When the
+“Match Functions” button fires, we invoke the FCG and CFG
+generators for the two programs (vulnerable and patched).
+3) The use-case for AMP challenge binaries: DARPA AMP
+challenges is about REs to patch a vulnerability regarding a
+weak encryption algorithm where the encryption of commu-
+nication traffic was accomplished with a deprecated cipher
+suite, Triple DES or 3DES [49]. For this challenge, REs
+have to analyze the vulnerable binary, identify functions and
+instructions to be patched, 3DES cipher suite in this case, and
+patch 3DES-related function calls and instructions with the
+ones for AES [50]. All these steps happen at the decompiled
+binary level, and the vulnerable binaries are optimized by
+a compiler and stripped of the debugging information and
+function names. Furthermore, these binaries are sometimes
+statically linked against libraries such as GNU C Library [51]
+or OpenSSL, which introduce many extra functions to the
+binary (some of which will never be called/used). Given these
+complications, it becomes a non-trivial task for an RE to make
+sense of all these functions, find the problem, and successfully
+patch the problem. The direct usage of our cfg2vec plugin was
+to pick a function of interest with stripped information and see
+predictions of potential function names or matching functions
+from the available reference binary to confirm that whether this
+function is in the critical path during RE’s problem solving.
+As Figure 5 shows, our plugin allows users to see possible
+matches between functions from a stripped vulnerable binary
+and functions from a patched (reference) binary with extra
+information. REs may then leverage such information and
+make appropriate notes for that function, allowing them to
+complete their jobs more efficiently. The main feedback we
+received from REs who used the tool was that this is the
+functionality REs would like to have. However, the accuracy
+and usability of the tool were not high enough to truly utilize
+the tool’s potential.
+V. CONCLUSION
+This paper presents cfg2vec, a Hierarchical Graph Neural
+Network-based approach for software reverse engineering.
+Building on top of Ghidra, our cfg2vec plugin can extract
+a Graph-of-Graph (GoG) representation for binary, combin-
+ing the information from Control-Flow Graphs (CFG) and
+Function-Call Graphs (FCG). Cfg2vec utilizes a hierarchical
+graph embedding framework to learn the representation for
+each function in binary code compiled into various archi-
+tectures. Lastly, our cfg2vec utilizes the learned function
+embeddings for function name prediction, outperforming the
+state-of-the-art [18] by an average of 24.54% across all tested
+binaries. By increasing the amount of data, our model achieved
+51.84% better. While [18] requires training once for each
+CPU architecture, our cfg2vec still can outperform consistently
+across all the architectures, only with one training. Besides,
+our model generalizes the learning better [18] to the binaries
+built from untrained CPU architectures. Lastly, we demonstrate
+that our cfg2vec can assist the real-world REs in resolving
+Darpa Assured MicroPatching (AMP) challenges.
+ACKNOWLEDGMENT
+This material is based upon work supported by the Defense
+Advanced Research Projects Agency (DARPA) and Naval
+Information Warfare Center Pacific (NIWC Pacific) under
+Contract Number N66001-20-C-4024. The views, opinions,
+and/or findings expressed are those of the author(s) and should
+not be interpreted as representing the official views or policies
+of the Department of Defense or the U.S. Government.
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+arXiv:2301.00158v1  [eess.SY]  31 Dec 2022
+1
+Robust Synergistic Hybrid Feedback
+(Extended Version)
+Pedro Casau, Ricardo G. Sanfelice, Fellow, IEEE, and Carlos Silvestre Senior Member, IEEE,
+Abstract—Synergistic hybrid feedback refers to a collection
+of feedback laws that allow for global asymptotic stabilization
+of a compact set through the following switching logic: given
+a collection of Lyapunov functions that are indexed by a logic
+variable, whenever the currently selected Lyapunov function
+exceeds the value of another function in the collection by a
+given margin, then a switch to the corresponding feedback law
+is triggered. This kind of feedback has been under development
+over the past decade and it has led to multiple solutions
+for global asymptotic stabilization on compact manifolds. The
+contributions of this paper include a synergistic controller
+design in which the logic variable is not necessarily constant
+between jumps, a synergistic hybrid feedback that is able to
+tackle the presence of parametric uncertainty, backstepping
+of adaptive synergistic hybrid feedbacks, and a demonstration
+of the proposed solutions to the problem of global obstacle
+avoidance.
+Index Terms—Hybrid Systems, Adaptive Control, Robotics,
+Uncertain Systems
+I. INTRODUCTION
+A. Background and Motivation
+In this paper, we consider the problem of globally asymp-
+totically stabilizing continuous-time plants of the form
+˙xp = Fp(xp, up, θ)
+(1)
+where xp ∈ Xp denotes the state of the plant, up ∈ Up is the
+input, and θ represents the parameters of the plant. To this
+end, we propose the following hybrid controller
+˙χc ∈ ˆFc(xp, χc, xc, uc)
+˙xc ∈ Fc(xp, χc, xc)
+�
+(xp, χc, xc) ∈ C, uc ∈ Uc
+χ+
+c = χc
+x+
+c ∈ Gc(xp, χc, xc)
+�
+(xp, χc, xc) ∈ D
+(2)
+P. Casau is with the Department of Electrical and Computer Engineering
+at Instituto Superior T´ecnico, Universidade de Lisboa, Lisboa, Portugal.
+E-mail address: pcasau@isr.tecnico.ulisboa.pt. C. Silvestre is
+with the Department of Electrical and Computer Engineering of the Faculty
+of Science and Technology of the University of Macau, Macau, China, and
+with Instituto Superior T´ecnico, Universidade de Lisboa, Lisboa, Portugal.
+E-mail address: csilvestre@um.edu.mo. R. G. Sanfelice is with the
+Department of Computer Engineering, University of California, Santa Cruz,
+CA 95064. Email address: ricardo@ucsc.edu. This work was partially
+supported by the Macao Science and Technology Development Fund under
+Grant FDCT/0146/2019/A3, by the University of Macau, Macao, China,
+under Project MYRG2020-00188-FST, by the Fundac¸˜ao para a Ciˆencia e
+a Tecnologia (FCT) through LARSyS - FCT Project UIDB/50009/2020
+and LAETA - FCT Project UIDB/50022/2020, and by FCT Scientific
+Employment Stimulus grant CEECIND/04652/2017. Research by R. G.
+Sanfelice has been partially supported by the National Science Foundation
+under Grant no. ECS-1710621, Grant no. CNS-1544396, and Grant no. CNS-
+2039054, by the Army Research Office under Grant no. W911NF-20-1-0253,
+and by the Air Force Office of Scientific Research under Grant no. FA9550-
+19-1-0053, Grant no. FA9550-19-1-0169, and Grant no. FA9550-20-1-0238.
+where χc ∈ ˆ
+Xc and xc ∈ Xc represent different components
+of the state of the controller, ˆFc and Fc are the flow maps
+associated with χc and xc, respectively, C denotes the flow
+set, Gc defines the update law for jumps of xc and D is the
+jump set. The key differences between χc and xc is the fact
+that χc does not change its value during jumps and also that
+the flows of χc depend on a virtual input variable uc ∈ Uc.
+More precisely, the goal in this paper is to design a controller
+that globally asymptotically stabilizes a compact set A for
+the closed-loop system resulting from the interconnection
+between (1) and (2) both when the parameter θ is known,
+but also when it is only known to belong to a given compact
+set Ω.
+In the presence of topological obstructions, this objective
+is not attainable via continuous feedback and, even though
+it might be attainable through discontinuous feedback, the
+resulting closed-loop system may not be robust to arbitrarily
+small noise (cf. [1] and [2]). To illustrate these limitations
+of continuous/discontinuous feedback, let us consider the
+problem of globally asymptotically stabilizing the point (1, 0)
+for the dynamical system
+˙x1 = −x2up,
+˙x2 = x1up,
+where xp := (x1, x2) ∈ X :=S1 := {xp ∈ R2 : |xp| = 1} is
+the state variable and up ∈ R denotes the input. In this direc-
+tion, let h(xp) = (1 − x1)/2 for each xp ∈ S1. The gradient-
+based feedback law is given by up =
+�x2
+−x1
+�
+∇h(xp)
+which represents the projection of the gradient of h onto the
+tangent space to S1 at xp. It follows from standard Lyapunov
+stability arguments that (1, 0) is asymptotically stable for
+the closed-loop system, but it is not globally asymptotically
+stable since x = (−1, 0) is also an equilibrium point.
+It can be argued that the discontinuous feedback law
+up = κp(xp) =
+
+
+
+
+
+
+
+−1
+if x1 = −1
+0
+if x1 = 1
+− x2
+|x2|
+otherwise
+(3)
+defined for each xp ∈ S1 globally asymptotically stabilizes
+(1, 0) if one considers Carath´eodory solutions to the discon-
+tinuous closed-loop system because, in this case, (−1, 0) is
+not an equilibrium point. However, due to the discontinuity
+of the feedback law (3), arbitrarily small noise can induce
+chattering which is a property that is ellucidated by consider-
+ing generalized solutions to discontinuous dynamical systems
+such as Krasovskii solutions (cf. [3]). These limitations of
+continuous and discontinuous feedbacks constitute the moti-
+vation for the development of synergistic hybrid feedback.
+
+Submitted for publication
+If ˆFc in (2) defining the dynamics of χc is given, then χc
+can become part of the state of (1) and the stated objective
+can be attained through the design of a hybrid controller
+Hc := (C, Fc, D, Gc) with state xc ∈ Xc and dynamics
+˙xc ∈ Fc(x, xc)
+(x, xc) ∈ C
+x+
+c ∈ Gc(x, xc)
+(x, xc) ∈ D
+assigning u := (up, uc) ∈ U := Up × Uc via a feedback law
+(x, xc) �→ κ(x, xc), where x := (xp, χc) ∈ X := Xp × ˆ
+Xc
+is the state of the system to control with dynamics described
+by the following differential inclusion
+˙x ∈ Fθ(x, xc, u) := Fp(xp, up, θ) × ˆFc(xp, χc, xc, uc), (5)
+where θ is a constant.
+This formulation enables the controller design for systems
+whose dynamics depend on the controller state. For example,
+given a plant with dynamics ˙xp = fp(xp) + Hp(xp)up +
+Wp(xp)θ where fp, Hp, Wp are functions with the appro-
+priate dimensions, suppose that the reference trajectory to
+be tracked is denoted by xd and that it is generated by the
+system ˙xd = fp(xd) + Hp(xd)ξd for some signal ξd. The
+tracking error x := xp − xd can be taken as the state of
+the system (5), in which case we have that Fθ(x, xc, u) =
+fp(xd+x)−fp(xd)−Hp(xd)ξd+Hp(xd+x)u+Wp(xd+x)θ
+by identifying up with u and by considering (xd, ξd) as
+components of the controller variable xc. More practically, x
+can be considered to be the part of the state of the closed-loop
+system that remains unchanged during jumps.
+In this paper, we present two novel synergistic hybrid
+controllers for global asymptotic stabilization of a compact
+set for a closed-loop system with the plant dynamics in (5).
+The first controller design considers that the parameter θ is
+known, while the second controller design considers that θ
+is unknown but belongs to a known compact set Ω.
+B. Literature Review
+Synergistic hybrid feedback is a hybrid control strategy
+that consists of a collection of potential functions that asymp-
+totically stabilize a given compact set by gradient descent
+feedback. If, for all equilibria that do not lie within the given
+compact set, there exists another function in the collection
+that has a lower value and does not share the same equilibria,
+then it is possible to achieve global asymptotic stabilization
+of the given compact set through hysteretic switching (see,
+e.g., [4]).
+Synergistic hybrid feedback came to prominence with the
+work [3] on quaternion-based feedback for global asymptotic
+stabilization attitude tracking, thereby solving the attitude
+control problem (cf. [5]). The framework of synergistic
+hybrid feedback provides not only a solution to the problem
+of attitude control but, more importantly, it provides a ro-
+bust solution for global asymptotic stabilization on compact
+manifolds. The works [6] and [7] leverage the concepts at
+the root of synergistic hybrid feedback and use them to
+design controllers that are applicable to a broad class of
+systems. However, most of the contributions on this class
+of hybrid controllers are on the control of robotic systems,
+such as pendulum stabilization [8], vector-based rigid body
+stabilization [9], [10], tracking for marine and aerial vehi-
+cle [11], [12], and rigid body tracking through rotation matrix
+feedback [13], [14]. Within the field of robotics, we single out
+the problem of obstacle avoidance, which is also addressed
+in this paper.
+Obstacle avoidance is an important and longstanding prob-
+lem that reflects the need to drive a the state of a system from
+one place to another while avoiding obstacles in its way.
+Several solutions to this problem have been proposed over
+the last few decades as highlighted in [15]. In particular, it is
+possible to find both stochastic [16] as well as deterministic
+approaches [17] to tackle the obstacle avoidance problem.
+However, it was shown in [18] that in a “sphere world,” there
+is at least one saddle equilibrium point for each obstacle
+within the state space, thus precluding global asymptotic
+stabilization of a setpoint by continuous feedback. To address
+this limitation, hybrid control solutions to the problem of
+obstacle avoidance were proposed in [19], [20], [7] and [21].
+Though not directly addressed in this paper, the concepts of
+synergistic hybrid feedback have also been used for observer
+design, optimization and control barrier function design in
+in [22], [23], and [21], respectively.
+C. Contributions
+The contributions in this paper are as follows: 1) We
+develop a dynamic synergistic hybrid feedback controller for
+global asymptotic stabilization of a broad class of dynamical
+systems. In particular, we consider that the distinguishing
+feature of synergistic hybrid feedback is the switching logic,
+thus we depart from earlier works which were limited to
+controller variables that were constant during flows; 2) We
+provide a modification to the dynamic synergistic controller
+that takes into account the presence of parametric uncertainty;
+3) We demonstrate how the proposed constructions can be
+used to develop an adaptive synergistic controller for the
+stabilization of compact sets for affine control systems under
+matched uncertainties; 4) We show that the proposed adaptive
+controller is amenable to hybrid backstepping; 5) We apply
+the proposed controller designs to the problem of global
+obstacle avoidance in the presence of parametric uncertainty
+and illustrate the behavior of the closed-loop system through
+simulations.
+The paper is organized as follows: in Section III we
+present the main assumptions on the plant dynamics. In
+Section IV-B we provide the conditions under which the
+closed-loop system is well-posed. In Section IV-C we provide
+sufficient conditions for global asymptotic stability of a
+compact set for the closed-loop system. In Section V, we
+develop the concept of robust synergistic hybrid feedback. In
+Section VI, we apply the synergistic approach to the devel-
+opment of an adaptive synergistic controller for stabilization
+of affine control systems subject to matched uncertainties. In
+Section VII, we apply the proposed controller to the problem
+of global obstacle avoidance. In Section VIII, we present
+some concluding remarks.
+2
+
+Submitted for publication
+A preliminary version of this paper was presented at the
+2019 ACC with a simpler synergistic controller design for
+global asymptotic stabilization of control affine systems and
+without the full proofs (cf. [24]). The original version of this
+paper has been submitted for publication.
+II. NOTATION & PRELIMINARIES
+A. Topology, Metric Spaces, Functions, and Set-Valued Maps
+Given a topological space X, a neighborhood of a set S is
+any open set that contains S. A topological space X is said to
+be Hausdorff if, given any pair of distinct points q1, q2 ∈ X,
+there exist neighborhoods U1 of q1 and U2 of q2 that do not
+intersect. Any metric space is Hausdorff, hence the Euclidean
+spaces are Hausdorff. Lemma 4.29 in [29] points out that
+any closed subspace of an locally compact Hausdorff space
+is itself locally compact Hausdorff. A set is said to be locally
+compact if for each point there is a neighborhood which is
+precompact, i.e., whose closure is a compact set.
+A topology on a set X is a collection T of subsets of X,
+called open sets, satisfying the following properties: X and ∅
+are elements of T ; T is closed under finite intersections; and
+T is closed under arbitrary unions. The subspace topology
+of a subset A of X is the collection of subsets of A that are
+obtained from the intersection of A with an open set of X.
+A subset A of a topological space X that is endowed with
+the subspace topology is said to be a subspace of X.
+A metric space is a set M together with a metric d. A set
+S ⊂ M is open in the metric space sense if for each x ∈ S
+there exists ǫ > 0 such that the set points y ∈ M satisfying
+d(x, y) < ǫ are contained in S. The metric topology on M is
+the collection of all subsets of M that are open in the metric
+space sense (cf. [29, Exercise 2.1]).
+The Cartesian Product Rn = R × . . . R of n copies of the
+real line together with scalar multiplication and component-
+wise addition of vectors is known as n-dimensional Euclidean
+space. The Euclidean metric topology is the one induced by
+the metric x �→ |x| :=
+√
+x⊤x. The n-dimensional Euclidean
+space has the topology generated by a countable basis of open
+balls of the form c + ǫB := {x ∈ Rn : |x − c| < ǫ}, where
+c ∈ Rn and ǫ > 0. More generally, given a set Ω ⊂ Rn,
+we define Ω + ǫB := �
+c∈Ω c + ǫB. The operators ∂S and S
+denote the boundary and the closure of a set S, respectively.
+Given a function f : Rm → Rn, the preimage of a set
+U ⊂ Rn through f is f −1(U) := {x ∈ Rm : f(x) ∈ U}.
+Similarly, the image of a set W through f is f(W) := {y ∈
+Rn : y = f(x) for some x ∈ W}.
+A set-valued map M from S ⊂ Rm to the power set of
+some Euclidean space Rn is represented by M : S ⇒ Rn.
+The domain of a set-valued map is given by dom M := {x ∈
+Rn : M(x) ̸= ∅}. Given a subset S of Rm, a set-valued map
+M : S ⇒ Rn is said to be outer semicontinuous (osc) relative
+to S if its graph, given by gph M := {(x, y) ∈ S × Rn : y ∈
+M(x)}, is closed relative to S × Rn. The set-valued map M
+is locally bounded at x ∈ Rm if there exists a neighborhood
+Ux of x such that M(Ux) ⊂ Rn is bounded. It is locally
+bounded relative to S if the set-valued mapping from Rm to
+Rn defined by M(x) for x ∈ S and ∅ for x ̸∈ S is locally
+bounded at each x ∈ S. It is convex-valued if M(x) is convex
+for each x ∈ S.
+A set-valued map M : S ⇒ Rn is upper semicontinuous
+(usc) at x if, for each open set V ⊂ Rn that contains M(x),
+there exists a neighborhood U of x such that x′ ∈ U ∩ S
+implies M(x′) ⊂ V . The map M is lower semicontinuous
+(lsc) at x if, for each open set V ⊂ Rn satisfying M(x)∩V ̸=
+∅, there exists a neighborhood U of x such that x′ ∈ U ∩ S
+implies M(x′) ∩ V ̸= ∅. The map M is continuous at x if it
+is both lsc and usc at x. The map M is usc, lsc, continuous
+on S if it is usc, lsc, continuous, respectively, at each x ∈ S.
+B. Differentiability
+The tangent cone to a set S ⊂ Rn at a point x ∈ Rn,
+denoted by TxS, is the set of all vectors w ∈ Rn for which
+there exists xi ∈ S, τi > 0 with xi → x, τi convergent to 0
+from above, and w = limi→∞
+xi−x
+τi .
+Given a differentiable function F : Rm×n → Rp×q, we
+define DF(X) :=
+∂ vec(F )
+∂ vec(X)⊤ (X) for each X ∈ Rm×n, where
+vec(A) := [e⊤
+1 A⊤ . . . e⊤
+mA⊤]⊤ for each A ∈ Rm×n and
+ei ∈ Rm is a vector of zeros, except for the i-th component,
+which is 1. If F has multiple arguments, say (X, Y ) ∈
+Rm×n × Rk×ℓ, we define DXF(X, Y ) :=
+∂ vec(F )
+∂ vec(X)⊤ (X, Y )
+for each (X, Y ) ∈ Rm×n × Rk×ℓ. If F : Rn → R, then
+∇F(x) := DF(x)⊤ for each x ∈ Rn. If F : Rn ×Rm → R,
+then ∇xF(x, y) := DxF(x, y)⊤ for each (x, y) ∈ Rn × Rm
+and ∇yF(x, y) := DyF(x, y)⊤ for each (x, y) ∈ Rn × Rm.
+Clarke’s generalized directional derivative of a function
+V : Rn → R in the direction v, is defined as follows (c.f. [26,
+Eq. (1)]): V ◦(x; v) := lim supy→x
+λց0
+V(y+λv)−V(y)
+λ
+.
+C. Stability of Hybrid Systems
+A hybrid system H with state space Rn is defined in [27]
+and [28] as
+˙ξ ∈ F(ξ)
+ξ ∈ C
+ξ+ ∈ G(ξ)
+ξ ∈ D
+(6)
+where ξ ∈ Rn is the state, C ⊂ Rn is the flow set,
+F : Rn ⇒ Rn is the flow map, D ⊂ Rn denotes the
+jump set, and G : Rn ⇒ Rn denotes the jump map. A
+solution ξ to H is parametrized by (t, j), where t denotes
+ordinary time and j denotes the jump time, and its domain
+dom ξ ⊂ R≥0×N is a hybrid time domain: for each (T, J) ∈
+dom ξ, dom ξ ∩ ([0, T ] × {0, 1, . . .J}) can be written in the
+form ∪J−1
+j=0 ([tj, tj+1], j) for some finite sequence of times
+0 = t0 ≤ t1 ≤ t2 ≤ · · · ≤ tJ, where Ij := [tj, tj+1] and the
+tj’s define the jump times. A solution ξ to a hybrid system
+is said to be maximal if it cannot be extended by flowing nor
+jumping and complete if its domain is unbounded.
+A set S is said to be forward pre-invariant for a hybrid
+system (6) if each maximal solution of (6) starting in S
+remains in S. It is said to be forward invariant if it is forward
+pre-invariant and each maximal solution from S is complete
+(see e.g. [28, Chapters 3 and 7]).
+The hybrid basic conditions provide a set of sufficient
+conditions for well-posedness and they are as follows (cf. [27,
+Assumption 6.5]):
+3
+
+Submitted for publication
+(A1) C and D are closed subsets of Rn;
+(A2) F : Rn ⇒ Rn is osc and locally bounded relative to C,
+C ⊂ dom F, and F(x) is convex for every x ∈ C;
+(A3) G : Rn ⇒ Rn is osc and locally bounded relative to D,
+and D ⊂ dom G.
+Given a function V : Rn → R≥0 that is Lipschitz continu-
+ous on a neighborhood of C in (6) and uc : Rn → R≥0, we
+say that the growth of V along flows of (6) is bounded by uc
+if the following holds:
+V ◦(ξ; f) ≤ uc(ξ)
+∀ξ ∈ C, ∀f ∈ F(ξ) ∩ TξC.
+(7)
+If, for some function uc : Rn → R≥0,
+V(ξ) − V(ξ) ≤ ud(ξ)
+∀ξ ∈ D, ∀ξ ∈ G(ξ),
+(8)
+then we say that the growth of V along jumps of (6) is
+bounded by ud. If both (7) and (8) hold, then we say that
+the growth of V along solutions to (6) is bounded by uc, ud.
+A compact set A is said to be stable for (6) if for
+every ǫ > 0 there exists δ > 0 such that every solution
+φ to (6) with |φ(0, 0)|A ≤ δ satisfies |φ(t, j)|A ≤ ǫ for
+all (t, j) ∈ dom φ;
+globally pre-attractive for (6) if every
+solution φ to (6) is bounded and, if it is complete, then
+also limt+j→+∞ |φ(t, j)|A = 0; globally pre-asymptotically
+stable for (6) if it is both stable and globally pre-attractive.
+If every maximal solution to (6) is complete then one may
+drop the prefix “pre.”
+III. PROBLEM SETUP
+Given sets X, Xc, and U, we consider a dynamical system
+with state x ∈ X that is governed by the dynamics (5) where
+xc ∈ Xc is a controller variable, u ∈ U is the input, θ is a
+constant parameter that belongs to a compact set Ω and Fθ
+is a set-valued map with the following properties.
+Assumption 1. Given sets X, Xc, U, and Fθ as in (5) the
+following properties hold:
+(S1) Each set X, Xc and U is a closed nonempty subset of
+some Euclidean space;
+(S2) The set-valued map Fθ is outer semicontinuous, locally
+bounded, and convex-valued.
+Assumption (S1) allows for the use of the analysis tools
+for hybrid dynamical systems that are provided in [27] which
+consider sets as subspaces of Euclidean spaces with the Eu-
+clidean metric topology. Assumptions (VC) and (S2) are used
+to prove that the resulting closed-loop system has nontrivial
+solutions and that it satisfies the hybrid basic conditions,
+respectively.
+Remark 1. Since the sets X, Xc and U are closed relative
+to their Euclidean ambient spaces, then any of their closed
+subsets are also closed in the ambient space and locally
+compact Hausdorff (cf. [29, Lemma 4.29]).
+In Section IV, we develop a dynamic synergistic controller
+with the objective of globally asymptotically stabilizing a
+compact set for the resulting closed-loop system under the
+assumption that θ is known. In Section V, we modify the
+dynamic synergistic controller to allow for θ ∈ Ω to be
+unknown, when Ω is known.
+IV. DYNAMIC SYNERGISTIC HYBRID FEEDBACK
+A. Controller Design
+Dynamic synergistic hybrid feedback (relative to the plant
+in Section III) is a hybrid control strategy that renders a
+compact set A ⊂ X × Xc globally asymptotically stable for
+the closed-loop system. It is comprised of a feedback law
+κ : dom κ → U
+(9)
+and of the hybrid dynamics that are described in the sequel.
+Given a function
+V : dom V → R≥0 ∪ {+∞},
+(10)
+satisfying X×Xc ⊂ dom V with dom V open in the Euclidean
+space containing X × Xc,1 and a set-valued map
+Dc : X × Xc ⇒ Xc
+(11)
+we define
+νV(x, xc) := min{V(x, g) : g ∈ Dc(x, xc)},
+(12a)
+̺V(x, xc) := arg min{V(x, g) : g ∈ Dc(x, xc)},
+(12b)
+µV(x, xc) := V(x, xc) − νV(x, xc)
+(12c)
+for each (x, xc) ∈ X × Xc, under the following assumption:
+(C1) The optimization problem in (12) is feasible for each
+(x, xc) ∈ X × Xc, i.e., for each (x, xc) ∈ X × Xc, there
+exists g ∈ Dc(x, xc) such that V(x, g) < +∞.
+Given a set-valued map Fc defined on X×Xc that satisfies
+the following assumption:
+(C2) Fc
+is outer semicontinuous, locally bounded, and
+convex-valued.
+we define the hybrid controller dynamics as follows:
+˙xc ∈ Fc(x, xc)
+(x, xc) ∈ C
+(13a)
+x+
+c ∈ ̺V(x, xc)
+(x, xc) ∈ D
+(13b)
+where
+C := {(x, xc) ∈ X × Xc : µV(x, xc) ≤ δ(x, xc)},
+D := {(x, xc) ∈ X × Xc : µV(x, xc) ≥ δ(x, xc)},
+(14)
+and δ : X × Xc → R is a continuous function. The switching
+logic in (13) implements the following functionality: if the
+solutions to the closed-loop system reach a state (x, xc)
+where µV(x, xc) is greater than or equal to the predefined
+value of δ(x, xc), then the variable xc is reset to some point
+g ∈ ̺V(x, xc) and the feedback law changes its value from
+κ(x, xc) to κ(x, g). Since the hybrid controller (13) is derived
+from κ, V, Dc, and Fc, we represent (13) using the 4-tuple
+(κ, V, Dc, Fc).
+The
+hybrid
+closed-loop
+system
+H
+:=
+(C, Fcl, D, Gcl)
+resulting
+from
+the
+interconnection
+1The function V maps values in X ×Xc to the one-point compactification
+of R≥0. More generally, given a topological space X that is noncompact
+locally compact Hausdorff space and an object ∞ not in X, the one point
+compactification of X is a topological space X∗ with the topology: T =
+{open subsets of X} ∪ {U ⊂ X∗ : X∗\U is a compact subset of X}.
+4
+
+Submitted for publication
+between
+(5)
+and
+(κ, V, Dc, Fc)
+is
+given
+by
+� ˙x
+˙xc
+�
+∈ Fcl(x, xc) :=
+�Fθ(x, xc, κ(x, xc))
+Fc(x, xc)
+�
+(x, xc) ∈ C
+(15a)
+�x+
+x+
+c
+�
+∈ Gcl(x, xc) :=
+�
+x
+̺V(x, xc)
+�
+(x, xc) ∈ D.
+(15b)
+Remark 2. Notice that if δ(x, xc) ≥ 0 for all (x, xc) ∈
+X × Xc then it follows from the construction of the hybrid
+controller (κ, V, Dc, Fc) that V (x, g)−V (x, xc) ≤ 0 for each
+(x, xc) ∈ D and each g ∈ ̺V(x, xc). In other words, if the
+function δ is nonnegative for all (x, xc) ∈ X × Xc, then the
+function V does not increase during jumps.
+The controller design presented in this section is informed
+by many preceding synergistic hybrid feedback controllers.
+As mentioned in Section I-C, we preserve the switching logic
+of the synergistic controllers in [28, Chapter 7], in the sense
+that controller switching is triggered when the difference
+between the current value of V and its lowest possible value
+exceeds a predefined threshold δ > 0. The main difference
+between the controller design presented in this paper and
+synergistic controllers in the literature is that, here, xc does
+not necessarily belong to a finite set. Instead, the flows of xc
+are described more generally by a differential inclusion and
+we constrain its jumps using a set-valued map Dc.
+In the sequel, we introduce the assumptions on the con-
+troller that allow for the global asymptotic stabilization of a
+compact subset of the state space.
+B. Basic Properties of the Closed-Loop System
+In this section, we provide some conditions on (9), (10)
+and (11) which ensure that the closed-loop system (15)
+satisfies the hybrid basic conditions of [27, Assumption 6.5]
+and that maximal solutions to (15) are complete. To this end,
+we introduce the following definitions.
+Definition 1. Given a compact subset A of X × Xc, κ, V,
+Dc and Fc we say that the hybrid controller (κ, V, Dc, Fc) is
+a synergistic candidate relative to A for (5) if (C1) and (C2)
+hold and:
+(C3) V is continuous, positive definite relative to A,2 and
+V −1([0, c]) is compact for each c ∈ R≥0;
+(C4) The set-valued map Dc is outer semicontinuous, lower
+semicontinuous, and locally bounded;
+(C5) The function κ is continuous and
+{(x, xc) ∈ X × Xc : V(x, xc) < +∞} ⊂ dom κ.
+Given a synergistic candidate relative to A, the prop-
+erty (C3) guarantees that sublevel sets of V are compact
+and the properties (C3) and (C4) guarantee that the synergy
+gap function µV in (12c) is continuous and that ̺V is outer
+semicontinuous, as proved in the next result.
+2A function V : X × Xc → R≥0 is positive definite relative to A ⊂
+X × Xc if V(x, xc) = 0 ⇐⇒ (x, xc) ∈ A.
+Lemma
+1. Given a compact subset A
+of X × Xc,
+if (κ, V, Dc, Fc) is a synergistic candidate relative to A
+for (5), then the following hold:
+1) The function νV in (12a) is continuous;
+2) The set-valued map ̺V in (12b) is outer semicontinuous
+and ̺V(x, xc) is compact for each (x, xc) ∈ X × Xc;
+3) The function µV in (12c) is continuous.
+Proof. It follows from (C4) that Dc is outer semicontinuous,
+hence Dc(x, xc) is closed for each (x, xc) ∈ X × Xc. Since
+Dc is also assumed to be locally bounded in (C4), we have
+that Dc(x, xc) is compact for each (x, xc) ∈ X × Xc. In
+addition, the outer semicontinuity and local boundedness
+of Dc imply that Dc is upper semicontinuous (cf. [27,
+Lemma 5.15]). Since Dc is assumed to be lower semicontinu-
+ous in (C4), we have that Dc is continuous. Since V is contin-
+uous by assumption (C3), it follows from [30, Theorem 9.14]
+that νV is continuous and that ̺V is compact-valued and
+upper semicontinuous. Since X is locally compact Hausdorff
+(cf. Remark 1), it follows from [29, Proposition 4.27] that
+each point (x, xc) ∈ X × Xc has a precompact neighborhood
+Ux. Since ̺V is compact-valued and upper semicontinuous, it
+follows from [30, Proposition 9.7] that ̺V(Ux) is compact.
+It follows from the fact that ̺V(Ux) is a subset of a the
+compact set ̺V(Ux) that ̺V is locally bounded. Since ̺V
+is compact-valued it is, in particular, closed-valued, hence
+it follows from [27, Lemma 5.15] that ̺V is outer semi-
+continuous. It follows from (C1) that νV(x, xc) < +∞ for
+each (x, xc) ∈ X × Xc, hence the function µV is continuous
+because it is the composition of continuous functions.
+The hybrid basic conditions in [27, Assumption 6.5] are
+very important to the synthesis of hybrid controllers, because
+they guarantee that the resulting hybrid closed-loop systems
+are endowed with nominal robustness to a wide range of
+perturbations/sensor noise and, in particular, they enable
+the application of invariance principles for hybrid systems
+(cf. [27, Chapter 8]). In the following result, we show that
+these conditions follow directly from the regularity of (12c)
+and (12) that was proved in Lemma 1.
+Corollary 1. Suppose that Assumption 1 holds. Given a
+compact subset A of X×Xc, if (κ, V, Dc, Fc) is a synergistic
+candidate relative to A for (5), then the hybrid closed-loop
+system (15) satisfies (A1), (A2), and (A3).
+Proof. Let h(x, xc)
+:=
+µV(x, xc) − δ(x, xc) for each
+(x, xc) ∈ X × Xc. Since δ is assumed to be continuous and
+µV is continuous under the given assumptions (cf. Lemma 1),
+it follows that h is a continuous function. Continuity of h
+implies that the flow and jump sets are closed, because they
+can be written as the preimage of closed sets through h,
+as follows: C = h−1((−∞, 0]) and D = h−1([0, +∞]),
+respectively (cf. [29, Lemma 2.7]).
+It follows from the construction of the hybrid con-
+troller (κ, V, Dc, Fc) that Fcl(x, xc) is defined for each
+(x, xc) ∈ C. From the continuity of κ in (C5) and the
+assumption that Fθ is outer semicontinuous, locally bounded
+and convex-valued (cf. (S2)), it follows that Fcl in (15)
+5
+
+Submitted for publication
+is outer semicontinuous and locally bounded relative to C
+and Fcl(x, xc) is convex for each (x, xc) ∈ C. The outer
+semicontinuity and local boundedness of Gcl in (15) relative
+to D follows from Lemma 1.
+C. Global Asymptotic Stability of A
+In this section, we present further assumptions on the
+hybrid controller (κ, V, Dc, Fc) that allow for the global
+asymptotic stabilization of a compact set A for (15).
+Definition 2. Given a compact subset A of X × Xc, we say
+that a synergistic candidate relative to A for (5) with data
+(κ, V, Dc, Fc), is synergistic relative to A for (5) if:
+(C6) The function V is Lipschitz continuous on a neighbor-
+hood of C and the growth of V along flows of (15) is
+bounded by uc with
+uc(x, xc) ≤ 0
+∀(x, xc) ∈ X × Xc;
+(C7) The largest weakly invariant subset of
+˙x ∈ Fθ(x, xc, κ(x, xc))
+˙xc ∈ Fc(x, xc)
+(16)
+in u−1
+c (0), denoted by Ψ, is such that
+δ1 := inf{µV(x, xc) : (x, xc) ∈ Ψ\A} > 0.
+(17)
+If one considers V as a Lyapunov function candidate, then
+Assumption (C6) implies that V is nonincreasing along flows
+to the closed-loop system (15), implying that there exists a
+choice of δ which renders A stable for (15).
+Lemma 2. Suppose that Assumption 1 holds. Given a com-
+pact subset A of X×Xc, if (κ, V, Dc, Fc) is a synergistic can-
+didate relative to A for (5) satisfying (C6) and δ(x, xc) ≥ 0
+for each (x, xc) ∈ X × Xc, then each sublevel set of V is
+forward pre-invariant for (15). If, for each (x, xc) ∈ C\D,
+(VC) there exists a neighborhood U of (x, xc) such that
+Fcl(ξ) ∩ TξC ̸= ∅, for every ξ ∈ U ∩ C
+then each maximal solution to (15) is complete and, conse-
+quently, each sublevel set of V is forward invariant.
+Proof. It follows from the discussion in Remark 2 that the
+growth of V along jumps of (15) is bounded by ud with
+ud(x, xc) ≤
+�
+−δ(x, xc)
+if (x, xc) ∈ D
+−∞
+otherwise
+(18)
+for each (x, xc) ∈ X × Xc. Together with assumption (C6)
+it follows that the growth of V along solutions to (15) is
+bounded by uc, ud satisfying
+uc(x, xc) ≤ 0,
+ud(x, xc) ≤ 0
+(19)
+for each (x, xc) ∈ X × Xc. It follows that each solution φ
+to (15) with initial condition ξ satisfies V(φ(t, j)) ≤ V(ξ) for
+all (t, j) ∈ dom φ, hence each sublevel set of V is forward
+pre-invariant for (15).
+It follows from Corollary 1 that (15) satisfies the hybrid
+basic conditions, hence we can use [27, Proposition 6.10]
+to prove the completeness of each maximal solution to (15).
+Since C ∪ D = X × Xc, then there are no solutions to (15)
+starting outside the union between the jump and flow sets.
+It follows from (VC) that (VC) in [27, Proposition 6.10]
+is satisfied, hence each maximal solution to (15) either
+“blows up,” leaves C ∪ D in finite time or is complete (cf.
+conditions (a),(b) and (c) of [27, Proposition 6.10]). Since
+Gcl(D) ⊂ C ∪ D, no solution can leave C ∪ D after a
+jump (hence, condition (c) in [27, Proposition 6.10] does
+not occur). Since each sublevel set of V is compact and
+forward pre-invariant, then solutions to (15) do not “blow
+up” (condition (b) in [27, Proposition 6.10] does not occur).
+It follows that each maximal solution to (15) is complete.
+Lemma 3. Suppose that Assumption 1 holds. Given a com-
+pact subset A of X × Xc, if (κ, V, Dc, Fc) is a synergistic
+candidate relative to A for (5) that satisfies (C6) and
+δ(x, xc) ≥ 0 for each (x, xc) ∈ X × Xc, then the set A
+is stable for (15).
+Proof. Since X × Xc ⊂ dom V, it follows that µV(x, xc)
+is defined for all (x, xc) ∈ X × Xc; hence, for any given
+continuous function δ : X × Xc → R, at least one of
+the following conditions holds: 1) µV(x, xc) ≥ δ(x, xc);
+2) µV(x, xc) ≤ δ(x, xc). It follows from (14) that C ∪ D =
+X × Xc ⊂ dom V. Since dom V is also assumed to be open
+in the Euclidean space containing X × Xc, it follows that
+dom V contains a neighborhood of A ∩ (C ∪ D ∪ G(D)).
+Positive definiteness of V with respect to A and continuity
+of V follows from (C3). From assumption (C6) and from (18),
+it follows that V is locally Lipschitz on a neighborhood of
+C and that the bounds [28, Eqs.(3.18), (3.19)] are satisfied.
+Since A is compact and the hybrid basic conditions are
+satisfied (cf. Corollary 1), it follows from [28, Theorem 3.19]
+that A is stable for (5).
+Assumption (C7) guarantees that there exists δ satisfying
+(D1) δ(x, xc) > 0 for each (x, xc) ∈ X × Xc;
+(D2) δ(x, xc) < µV(x, xc) for each (x, xc) ∈ Ψ\A with Ψ
+defined in (C7).
+We say that δ is positive if it satisfies (D1), and that a hybrid
+controller (κ, V, Dc, Fc) is synergistic relative to A for (5)
+with synergy gap exceeding δ if it is synergistic relative to
+A for (5) and satisfies (D2). When both conditions (D1)
+and (D2) are satisfied, all the points in the largest weakly
+invariant subset of (16) in u−1
+c (0) that are not in A lie in
+the jump set of (15), allowing us to prove that A is globally
+asymptotically stable for the closed-loop system (15).
+Theorem 1. Suppose that Assumption 1 holds. Given a
+compact subset A of X × Xc and a positive function
+δ : X × Xc → R, if (κ, V, Dc, Fc) is synergistic relative
+to A for (5) with synergy gap exceeding δ, then the set
+A is globally pre-asymptotically stable for (15). If, for
+each (x, xc) ∈ C\D, (VC) is satisfied, then A is globally
+asymptotically stable for (15).
+Proof. Stability of A is proved in Lemma 3 and complete-
+ness of solutions is demonstrated in Lemma 2. The global
+6
+
+Submitted for publication
+pre-asymptotic stability of A for (15) follows from pre-
+attractivity of A for (15), which is demonstrated next through
+an application of [27, Theorem 8.2].
+It follows from Lemmas 3 and 2 that each maximal solu-
+tion to (15) is precompact and the growth of V along solutions
+to (15) is bounded by uc, ud satisfying (19). Therefore, it
+follows from [27, Theorem 8.2] that every complete solution
+approaches the largest weakly invariant set
+V −1(r) ∩
+�
+u−1
+c (0) ∪ (u−1
+d (0) ∩ Gcl(u−1
+d (0))
+�
+(20)
+for some r in the image of V. From (18) and the assump-
+tion (D1), it follows that u−1
+d (0) = ∅, hence (20) can be
+rewritten as
+V −1(r) ∩ u−1
+c (0).
+(21)
+It follows from (C7), (D2) and the definition of D in (13)
+that the largest weakly invariant subset of (15) in (21) does
+not include points that are not in A and, consequently, A
+is globally pre-attractive for (15). Global asymptotic stability
+of A for (15) follows from global pre-asymptotic stability
+if each maximal solution to (15) is complete, which is
+guaranteed by Lemma 2 under assumption (VC).
+Note that, if there exists an accumulation point of Ψ\A
+in A, then δ1 in (17) is equal to 0. Therefore, the topology
+of Ψ and A may preclude global asymptotic stabilization of
+A for (15) since (C7) is not met. Conversely, if one is able
+to show that δ1 > 0, then Ψ\A does not have accumulation
+points in A. With additional conditions on δ, we are able to
+show that there exists a neighborhood of A contained in C.
+Proposition 1. Given a compact subset A of X × Xc, if
+the hybrid controller (κ, V, Dc, Fc) is synergistic relative to
+A and δ := inf{δ(x, xc) : (x, xc) ∈ X × Xc} satisfies
+δ ∈ (0, δ1), where δ1 is given in (C7), then there exists a
+neighborhood of A that is contained in C.
+Proof. Selecting ǫ ∈ (0, δ), we have that µ−1
+V ((−ǫ, ǫ)) is
+open because µV is continuous (cf. Lemma 1), contains A
+because µV(A) = 0 and it is a subset of C because ǫ < δ ≤
+δ(x, xc) for each (x, xc) ∈ X × Xc.
+Remark 3. Note that, if the hybrid controller (κ, V, Dc, Fc)
+is synergistic relative to A for (5), then δ can be chosen as
+a constant ∆ ∈ R as long as ∆ ∈ (0, δ1). In this case, the
+conditions of Proposition 1 hold, thus the fact that δ is state-
+dependent does not constrain the global asymptotic stability
+results and it provides more flexibility to the design of the
+hybrid controller.
+V. ROBUST SYNERGISTIC HYBRID FEEDBACK
+In this section, we propose a new kind of synergistic hybrid
+controller that, unlike the controller of Section IV, is able
+to handle the case where θ is unknown, but belongs to a
+known compact set Ω. In this direction, let A := {Aθ}θ∈Ω
+denote a collection of compact subsets of X × Xc and let
+V := {Vθ}θ∈Ω denote a collection of functions satisfying the
+following assumption:
+(C8) Given a compact set Ω and a collection V := {Vθ}θ∈Ω
+of functions Vθ : dom Vθ → R≥0 ∪ {+∞} satisfying
+X × Xc ⊂ dom Vθ for each θ ∈ Ω, we assume that
+(x, xc, θ) �→ V(x, xc, θ) := Vθ(x, xc)
+(22)
+is continuous.
+Remark 4. Note that Ω might be uncountable, thus the
+collections A := {Aθ}θ∈Ω and V := {Vθ}θ∈Ω are not
+necessarily finite nor countable.
+Using the previous definitions, we propose the following
+hybrid controller:
+˙xc ∈ Fc(x, xc)
+(x, xc) ∈ CΩ
+(23a)
+x+
+c ∈ Gc(x, xc)
+(x, xc) ∈ DΩ
+(23b)
+where
+CΩ :=
+�
+(x, xc) ∈ X × Xc : min
+θ∈Ω µVθ(x, xc) ≤ δ(x, xc)
+�
+DΩ :=
+�
+(x, xc) ∈ X × Xc : min
+θ∈Ω µVθ(x, xc) ≥ δ(x, xc)
+�
+(24)
+with δ : X × Xc → R continuous, and
+min
+θ∈Ω µVθ(x, xc) = min
+θ∈Ω {Vθ(x, xc) − νVθ(x, xc)}
+= min
+θ∈Ω
+�
+V(x, xc, θ) −
+min
+g∈Dc(x,xc) V(x, g, θ)
+�
+for each (x, xc) ∈ X × Xc, in accordance with the defini-
+tions (12c), (12a) and (22), and Gc : X × Xc ⇒ X satisfies
+the following assumptions:
+(C9) The set-valued map Gc is outer semicontinuous and
+locally bounded;
+(C10) For each θ ∈ Ω, we assume that
+Vθ(x, xc) − Vθ(x, g) ≥ min
+θ∈Ω µVθ(x, xc)
+(25)
+for each (x, xc) ∈ DΩ and each g ∈ Gc(x, xc)
+Remark 5. Under assumption (C10), we guarantee by con-
+struction that
+Vθ(x, xc) − Vθ(x, g) ≥ δ(x, xc)
+for each (x, xc) ∈ DΩ and each g ∈ Gc(x, xc), which
+implies that the function Vθ does not increase during jumps
+if δ(x, xc) ≥ 0 for all (x, xc) ∈ X × Xc (cf. Remark 2).
+Note that the jump map of the hybrid controller (13) is
+constructed from the data V and Dc as shown in (12b). On the
+other hand, the jump map Gc in (23) is left undefined for the
+sake of generality. It is possible to construct Gc in (23) from
+the data V and Dc, but this requires additional assumptions,
+as shown in the following remark. Owing to the fact that (23)
+is derived from κ, V, Dc, Fc, and Gc, we refer to (23) using
+the 5-tuple (κ, V, Dc, Fc, Gc).
+7
+
+Submitted for publication
+Remark 6. Suppose that Ω is compact and convex and that
+Dc in (11) is convex and compact for each (x, xc) ∈ X ×Xc.
+Given V := {Vθ}θ∈Ω, suppose that Gc defined as
+Gc(x, xc) := arg max
+g∈Dc(x,xc)
+min
+θ∈Ω {V(x, xc, θ) − V(x, g, θ)}
+∀(x, xc) ∈ X × Xc
+(26)
+is outer semicontinuous with V(x, xc, θ) = Vθ(x, xc) for each
+(x, xc, θ) ∈ X × Xc × Ω. Furthermore, suppose that, for
+each (x, xc) ∈ X × Xc, the function h(g, θ) := Vθ(x, xc) −
+Vθ(x, g) is continuous, quasi-concave as a function of g,
+and quasi-convex as a function of θ.3 Then, it follows
+from [31, Theorem 3.4] that the min and max operations
+in maxg∈Dc(x,xc) minθ∈Ω h(g, θ) commute, yielding
+max
+g∈Dc(x,xc) min
+θ∈Ω h(g, θ) = min
+θ∈Ω
+max
+g∈Dc(x,xc) h(g, θ)
+= min
+θ∈Ω
+�
+Vθ(x, xc) −
+min
+g∈Dc(x,xc) Vθ(x, g)
+�
+.
+(27)
+It
+follows
+from
+(27)
+and
+(12c)
+that
+maxg∈Dc(x,xc) minθ∈Ω h(g, θ)
+=
+minθ∈Ω µVθ(x, xc).
+We conclude that, for each θ ∈ Ω, the following holds
+Vθ(x, xc) − Vθ(x, g)
+≥
+minθ∈Ω µVθ(x, xc),
+for each
+(x, xc) ∈ X × Xc and each g belonging to (26), hence
+condition (25) is verified.
+The following definition extends the notion of a synergistic
+controller in order to address the case where θ ∈ Ω is not
+known.
+Definition 3. Given a compact set Ω, a continuous func-
+tion δ : X × Xc → R, a collection of compact subsets
+A := {Aθ}θ∈Ω of X × Xc, and a collection of continuous
+functions V
+:= {Vθ}θ∈Ω, we say that the hybrid con-
+troller (κ, V, Dc, Fc, Gc) is synergistic relative to A for (5)
+with robustness margin Ω if (C8), (C9), (C10) hold, and
+if, for each θ ∈ Ω, the hybrid controller (κ, Vθ, Dc, Fc) is
+synergistic relative to Aθ for (5).
+The
+assumption
+that
+the
+hybrid
+con-
+troller
+(κ, V, Dc, Fc, Gc)
+is
+synergistic
+relative
+to
+A
+for (5) with robustness margin Ω ensures that the hybrid
+closed-loop system
+�
+˙x
+˙xc
+�
+∈ Fcl(x, xc) :=
+�
+Fθ(x, xc, κ(x, xc))
+Fc(x, xc)
+�
+(x, xc) ∈ CΩ
+(28a)
+�x+
+x+
+c
+�
+∈ GΩ(x, xc) :=
+�
+x
+Gc(x, xc)
+�
+(x, xc) ∈ DΩ
+(28b)
+satisfies the hybrid basic conditions as proved next.
+Lemma 4. Suppose that Assumption 1 holds. Given a
+compact set Ω and a collection of compact subsets A :=
+{Aθ}θ∈Ω of X × Xc if (κ, V, Dc, Fc, Gc) is synergistic
+3A function (x, y) �→ h(x, y) on X × Y is quasi-concave as a function
+of x if the set {x ∈ X : h(x, y) ≥ c} is convex for each y ∈ Y and
+each c ∈ R. The function h is quasi-convex as a function of y if the set
+{y ∈ X : h(x, y) ≤ c} is convex for each x ∈ X and each c ∈ R.
+relative to A for (5) with robustness margin Ω, then the
+hybrid closed-loop system (28) satisfies (A1), (A2), and (A3).
+Proof. The continuity of µVθ (for a fixed θ ∈ Ω) is es-
+tablished in Lemma 1. It follows from the continuity of
+(x, xc, θ) �→ V(x, xc, θ) = Vθ(x, xc) that is assumed in (C8),
+compactness of Ω and from [30, Theorem 9.14] that the
+function
+(x, xc) �→ min
+θ∈Ω µVθ(x, xc)
+(29)
+is continuous on X×Xc. It follows from the continuity of (29)
+and of δ that CΩ and DΩ are closed, because they are the
+preimage of the closed sets (−∞, 0] and [0, +∞], respec-
+tively. It follows from Assumption 1, (C2), and (C5) that
+the flow map FΩ is outer semicontinuous, locally bounded
+and convex-valued. It follows from (C9) that GΩ is outer
+semicontinuous, and locally bounded relative to DΩ.
+In the sequel, we demonstrate that, for each θ ∈ Ω,
+the set Aθ ∈ A is globally asymptotically stable under
+appropriate assumptions on δ. The next result asserts forward
+pre-invariance of sublevel sets of Vθ ∈ V for the closed-loop
+system (28) when δ is a continuous and nonnegative function.
+Lemma 5. Suppose that Assumption 1 holds. Given a
+compact set Ω and a collection of compact subsets A :=
+{Aθ}θ∈Ω of X × Xc, if (κ, V, Dc, Fc, Gc) is synergistic
+relative to A for (5) with robustness margin Ω and if
+δ(x, xc) ≥ 0 for each (x, xc) ∈ X×Xc, then, for each θ ∈ Ω,
+each sublevel set of Vθ is forward pre-invariant for (28). If,
+for each (x, xc) ∈ CΩ\DΩ,
+(VC’) there exists a neighborhood U of (x, xc) such that
+Fcl(ξ) ∩ TξCΩ ̸= ∅, for every ξ ∈ U ∩ CΩ
+then each maximal solution to (28) is complete and, conse-
+quently, each sublevel set of Vθ is forward invariant for (28).
+Proof. As explained in Remark 5, it follows from (C10)
+and (24) that Vθ(x, xc) − Vθ(x, g) ≥ δ(x, xc) for each
+g ∈ Gc(x, xc) and each (x, xc) ∈ DΩ. Hence, the growth
+of Vθ during jumps of (28) is bounded by
+ud,θ(x, xc) :=
+�
+−δ(x, xc)
+if (x, xc) ∈ DΩ
+−∞
+otherwise
+(30)
+for each (x, xc) ∈ X × Xc. Since (κ, V, Dc, Fc, Gc) is
+synergistic relative to A for (5) with robustness margin
+Ω, it follows that (κ, Vθ, Dc, Fc) is synergistic relative to
+Aθ for (5) and, due to this assumption, the remainder of
+the proof follows closely that of Lemma 2. From Assump-
+tion (C6) it follows that the growth of Vθ along solutions
+to (28) is bounded by uc,θ, ud,θ, with uc,θ(x, xc) ≤ 0 and
+ud,θ(x, xc) ≤ 0 for each (x, xc) ∈ X × Xc. This implies that
+sublevel sets of Vθ are forward pre-invariant for (28). The
+completeness of solutions under (VC’) follows closely the
+proof in Lemma 2, thus it is omitted here.
+Let Ψθ denote the largest weakly invariant subset of
+( ˙x, ˙xc) ∈ Fcl(x, xc)
+(x, xc) ∈ u−1
+c,θ(0)
+8
+
+Submitted for publication
+where uc,θ is the upper bound on the growth of Vθ during
+flows of (28) as defined in (C6). Given a function δ : X ×
+Xc → R and a hybrid controller (κ, V, Dc, Fc, Gc) that is
+synergistic relative to A for (5) with robustness margin Ω,
+we say that it has synergy gap exceeding δ if, for each θ ∈ Ω
+and each (x, xc) ∈ Ψθ\Aθ, δ(x, xc) < µVθ(x, xc).
+Theorem 2. Suppose that Assumption 1 holds. Given a
+compact set Ω, a positive function δ : X × Xc → R, and
+a collection of compact subsets A := {Aθ}θ∈Ω of X × Xc,
+if (κ, V, Dc, Fc, Gc) is synergistic relative to A for (5) with
+robustness margin Ω and synergy gap exceeding δ, then, for
+each θ ∈ Ω, the set Aθ is globally pre-asymptotically stable
+for (28). If, for each (x, xc) ∈ CΩ\DΩ, (VC’) is satisfied,
+then Aθ is globally asymptotically stable for (28).
+Proof. For each θ ∈ Ω, it follows from (C3) that each
+sublevel set of Vθ is compact and, since it is also forward pre-
+invariant as shown in Lemma 5, we have that each solution
+to (28) is bounded. In addition, it follows from the proof
+of Lemma 5 that the growth of Vθ along jumps of (28) is
+bounded by (30) and, since δ(x, xc) > 0 by assumption, it
+follows from [27, Theorem 8.2] that each complete solution
+to (28) approaches the largest weakly invariant subset of
+V −1
+θ
+(r) ∩ u−1
+c,θ(0) for some r in the image of Vθ, which is to
+say that each complete solution to (28) approaches Ψθ ∩CΩ.
+Since each point (x, xc) ∈ Ψθ\Aθ belongs to DΩ\CΩ by
+Assumption (C7), it follows that each complete solution
+to (28) converges to Aθ, which concludes the proof of global
+pre-attractivity of Aθ for (28). The proof of stability of Aθ
+for (15) follows closely the proof of Lemma 3. We conclude
+that Aθ is globally pre-asymptotically stable for (28). Global
+asymptotic stability of Aθ for (28) under assumption (VC’)
+follows directly from global pre-asymptotic stability and
+completeness of solutions, as shown in Lemma 5.
+In the next section, we apply the proposed controller to
+the design of adaptive synergistic feedback control laws for
+a class of affine systems with matched uncertainties.
+VI. ADAPTIVE BACKSTEPPING OF SYNERGISTIC HYBRID
+FEEDBACK FOR AFFINE CONTROL SYSTEMS
+A. Nominal Synergistic Hybrid Feedback
+In this section, we apply the controller design of Section V
+to the problem of global asymptotic stabilization of a compact
+set A ⊂ X × Xc for a control affine system subject to
+parametric uncertainty, where X and Xc denote the spaces
+of the state and controller variables, respectively. In this
+direction, let Fθ in (5) be given by
+Fθ(x, xc, u) := f(x, xc) + H(x, xc)u + W(x, xc)θ
+(31)
+for each (x, xc, u) ∈ X × Xc × U, where u denotes an input
+variable subject to the constraint u ∈ U, and
+θ ∈ Ω := {θ ∈ Rℓ : |θ| ≤ θ0}
+(32)
+represents the parametric uncertainty of the model whose
+norm is assumed to be bounded by a known parameter
+θ0 ∈ R≥0. The controller design in this section is applicable
+under the assumption of matched uncertainties stated next.
+Assumption 2. There exists a continuously differentiable
+function �
+W such that W(x, xc) = H(x, xc)�
+W(x, xc) for
+each (x, xc) ∈ X × Xc.
+In addition, we assume that we are given a synergistic
+hybrid controller for the nominal (unperturbed) system as
+defined next.
+Definition 4. Given a compact set A ⊂ X × Xc and a
+continuous function δ : X × Xc → R, the hybrid con-
+troller (κ0, V0, Dc, Fc) is said to be nominally synergistic
+relative to A for (31) with synergy gap exceeding δ if it
+is synergistic relative to A for
+˙x = F0(x, xc, u) := f(x, xc) + H(x, xc)u
+(33)
+with synergy gap exceeding δ, and V0 is continuously differ-
+entiable on {(x, xc) ∈ X × Xc : V0(x, xc) < +∞.}.
+The dynamical system (33) is obtained from (31) by
+considering that there are no perturbations, i.e., θ = 0. It
+follows from Theorem 1 that A is globally asymptotically
+stable for the closed-loop system H in (15) resulting from the
+interconnection of (33) and a nominally synergistic controller
+relative to A for (31) when θ = 0. In the next section, we
+present variations of the nominal synergistic controller so as
+to deal with nonzero disturbances.
+B. Adaptive Synergistic Hybrid Feedback
+In this section, we modify the nominal synergistic con-
+troller given in Section VI-A to globally asymptotically
+stabilize
+A1,θ := A × {θ}
+(34)
+for the closed-loop system when θ in (31) is nonzero.4 In
+this direction, let ˆθ ∈ Rℓ denote an estimate of the parameter
+θ that is generated via
+˙ˆθ = Γ1 Proj(W(x, xc)⊤∇xV0(x, xc), ˆθ),
+(35)
+where Γ1 ∈ Rℓ×ℓ is a positive definite matrix and Proj :
+Rℓ × Rℓ → Rℓ is given by
+Proj(η, ˆθ) :=
+�
+η
+if p(ˆθ) ≤ 0 or ∇ p(ˆθ)⊤η ≤ 0
+�
+Iℓ − p(ˆθ)∇ p(ˆθ)∇ p(ˆθ)⊤
+∇ p(ˆθ)⊤∇ p(ˆθ)
+�
+η
+otherwise
+(36)
+for each (η, ˆθ) ∈ Rℓ × Rℓ,
+p(ˆθ) :=
+ˆθ⊤ˆθ − θ2
+0
+ǫ2 + 2ǫθ0
+(37)
+for each ˆθ ∈ Rℓ, with ǫ > 0 and θ0 > 0 given in (32), and
+W as in (31). The function Proj in (36) has the following
+properties (cf. [33]):
+(P1) Proj is Lipschitz continuous;
+4As the controller design exploits ideas in the literature of adaptive control,
+we refer the reader to [32] for an overview of adaptive controller design and
+backstepping under the influence of model uncertainty.
+9
+
+Submitted for publication
+(P2) Each solution t �→ ˆθ(t) to ˙ˆθ = Γ1 Proj(η(t), ˆθ), from
+ˆθ ∈ Ω+ǫB with input t �→ η(t) satisfies rge ˆθ ⊂ Ω+ǫB;
+(P3) Given θ ∈ Ω, (θ − ˆθ)⊤ Proj(η, ˆθ) ≥ (θ − ˆθ)⊤η for each
+(η, ˆθ) ∈ Rℓ × Rℓ;
+with
+ǫ
+>
+0
+as
+in
+(37).
+Given
+a
+hybrid
+con-
+troller (κ0, V0, Dc, Fc) that is nominally synergistic relative
+to A for (31) with synergy gap exceeding δ, and the controller
+variable xc,1 := (xc, ˆθ) ∈ Xc,1 := Xc × (Ω + ǫB), we define
+κ1(x, xc,1) := κ0(x, xc) − �
+W(x, xc)ˆθ
+(38a)
+V1,θ(x, xc,1) := V0(x, xc) + 1
+2(θ − ˆθ)⊤Γ−1
+1 (θ − ˆθ) (38b)
+Dc,1(x, xc,1) := Dc(x, xc) × (Ω + ǫB)
+(38c)
+Fc,1(x, xc,1) =
+�
+Fc(x, xc)
+Γ1 Proj(W(x, xc)⊤∇xV0(x, xc), ˆθ)
+�
+(38d)
+for each (x, xc,1) ∈ X × Xc,1, where �
+W comes from
+Assumption 2. The hybrid closed-loop system resulting
+from the interconnection between (31) and the hybrid con-
+troller (κ1, V1,θ, Dc,1, Fc,1), is given by
+( ˙x, ˙xc,1) ∈ Fcl,1(x, xc,1)
+(x, xc,1) ∈ C1
+(39a)
+(x+, x+
+c,1) ∈ Gcl,1(x, xc,1)
+(x, xc,1) ∈ D1
+(39b)
+where
+C1 := {(x, xc,1) ∈ X × Xc,1 : µV1,θ(x, xc,1) ≤ δ(x, xc)}
+D1 := {(x, xc,1) ∈ X × Xc,1 : µV1,θ(x, xc,1) ≥ δ(x, xc)}
+and
+Fcl,1(x, xc,1) :=
+�Fθ(x, xc, κ1(x, xc,1))
+Fc,1(x, xc,1)
+�
+∀(x, xc,1) ∈ C1
+(40a)
+Gcl,1(x, xc,1) :=
+�
+x
+̺V1,θ(x, xc,1)
+�
+∀(x, xc,1) ∈ D1.
+(40b)
+where, for each (x, xc,1) ∈ X × Xc,1,
+νV1,θ(x, xc,1) = νV0(x, xc)
+(41a)
+̺V1,θ(x, xc,1) = ̺V0(x, xc) × {θ}
+(41b)
+µV1,θ(x, xc,1) = µV0(x, xc) + 1
+2(θ − ˆθ)⊤Γ−1
+1 (θ − ˆθ) (41c)
+are directly computed from (12a), (12b) and (12c), respec-
+tively.
+Remark 7. For the hybrid controller (κ1, V1,θ, Dc,1, Fc,1),
+the functions (41) are not realizable, because µV1,θ and
+̺V1,θ in (41) depend on the unknown constant θ. This
+dependence will be removed when we show that there
+exists Gc,1
+: X × Xc,1
+⇒ Xc,1 such that the hybrid
+controller (κ1, V1, Dc,1, Fc,1, Gc,1) with V := {V1,θ}θ∈Ω
+is synergistic relative to A1 := {A1,θ}θ∈Ω for (31) with
+robustness margin Ω.
+To design (23), we start by showing that the hybrid
+controller (κ1, V1,θ, Dc,1, Fc,1) is synergistic relative to A1,θ
+for (31).
+Proposition 2. Suppose that the sets X, Xc, U, and the set-
+valued map Fθ in (31) satisfy Assumption 1, and that Assump-
+tion 2 holds. Given θ ∈ Ω, a compact set A ⊂ X ×Xc, and a
+hybrid controller (κ0, V0, Dc, Fc) that is nominally synergis-
+tic relative to A for (31), the controller (κ1, V1,θ, Dc,1, Fc,1)
+given in (38) is a synergistic candidate relative to A1,θ
+for (31).
+Proof. The optimization problems in (12) are feasible for
+each x ∈ X, because they are feasible for V0, hence (C1) is
+satisfied.
+Since V1,θ corresponds to the sum of V0 with (θ −
+ˆθ)⊤Γ−1
+1 (θ − ˆθ)/2 and both terms are continuous, it follows
+that V1,θ is continuous. Since V0 is positive definite with
+respect to A and ˆθ �→ (θ− ˆθ)⊤Γ−1
+1 (θ− ˆθ) is positive definite
+relative to θ, it follows that V1,θ is positive definite relative
+to A1,θ. It follows from the assumption that V −1
+0
+([0, c])
+is compact for each c ≥ 0 and radial unboundedness of
+ˆθ �→ (θ − ˆθ)⊤Γ−1
+1 (θ − ˆθ) relative to θ that V −1
+1,θ ([0, c]) is
+compact for each c ≥ 0, thus proving that V1,θ satisfies (C3).
+From (38c), we have that Dc,1(x, xc,1) is the Cartesian
+product between Dc(x, xc) and Ω + ǫB for each (x, xc) ∈
+X × Xc. Since Dc satisfies (C4) by assumption, we have that
+Dc,1 also satisfies (C4). Since κ0 satisfies (C5), then κ1 also
+satisfies (C5).
+Proposition 3. Suppose that the sets X, Xc, U, and the set-
+valued map Fθ in (31) satisfy Assumption 1, and that Assump-
+tion 2 holds. Given θ ∈ Ω, a compact set A ⊂ X ×Xc, and a
+hybrid controller (κ0, V0, Dc, Fc) that is nominally synergis-
+tic relative to A for (31), the controller (κ1, V1,θ, Dc,1, Fc,1)
+given in (38) satisfies (C6).
+Proof. It follows from (38b), (40a) and (P3) that, for each
+(x, xc,1) ∈ X × Xc,1 and each fcl,1 ∈ Fcl,1(x, xc,1)
+∇V1,θ(x, xc,1)⊤fcl,1 ≤∇V0(x, xc)⊤
+�Fθ(x, xc, κ1(x, xc,1))
+fc
+�
+− (θ − ˆθ)⊤W(x, xc)⊤∇xV0(x, xc)
+where fc ∈ Fc(x, xc) is the component of fcl,1 that deter-
+mines the dynamics of xc, i.e., ˙xc = fc. Replacing (31)
+and (38a) in (VI-B), we obtain
+∇V1,θ(x, xc,1)⊤fcl,1 ≤ ∇V0(x, xc)⊤
+�
+F0(x, xc, κ0(x, xc))
+fc
+�
++ ∇xV0(x, xc)⊤(−H(x, xc)�
+W(x, xc)ˆθ + W(x, xc)θ)
+− (θ − ˆθ)⊤W(x, xc)⊤∇V0(x, xc)
+(X1)
+for each (x, xc,1) ∈ X × Xc,1 with F0 given in (33). Hence,
+it follows from Assumption 2 that
+∇V1,θ(x, xc,1)⊤fcl,1 ≤ ∇V0(x, xc)⊤
+�F0(x, xc, κ0(x, xc))
+fc
+�
+for each (x, xc,1) ∈ X × Xc,1. From the assumption that the
+hybrid controller (13) with data (κ0, V0, Dc, Fc) is synergistic
+relative to A for (33), we have that ∇V1,θ(x, xc,1)⊤fcl,1 ≤ 0
+for each (x, xc,1) ∈ X × Xc,1 and each fcl,1 ∈ Fcl,1(x, xc,1),
+which proves (C6).
+10
+
+Submitted for publication
+Since the hybrid controller (κ1, V1,θ, Dc,1, Fc,1) satis-
+fies (C6), we have that V1,θ is nonincreasing along solutions
+to the closed-loop system (39), but satisfying (C7) requires
+further assumptions on the data, as shown next.
+Proposition 4. Suppose that the sets X, Xc, U, and the
+set-valued map Fθ in (31) satisfy Assumption 1, and that
+Assumption 2 holds. Given θ ∈ Ω, a compact set A ⊂ X×Xc,
+and a hybrid controller (κ0, V0, Dc, Fc) that is nominally
+synergistic relative to A for (31) with synergy gap exceeding
+δ, let Ψ denote the largest weakly invariant subset of
+( ˙x, ˙xc) ∈ Fcl,0(x, xc) =
+�
+F0(x, xc, κ0(x, xc))
+Fc(x, xc)
+�
+on (x, xc) ∈ E := {(x, xc) ∈ X ×Xc : ∇V0(x, xc)⊤fcl,0 = 0
+for some fcl,0 ∈ Fcl,0(x, xc)} and let Ψ1,θ denote the
+largest weakly invariant subset of
+( ˙x, ˙xc,1) ∈ Fcl,1(x, xc,1)
+(x, xc,1) ∈ E1
+with E1 := {(x, xc,1) ∈ X × Xc,1 : ∇V1,θ(x, xc,1)⊤fcl,1 =
+0
+for some fcl,1
+∈ Fcl,1(x, xc,1)}. If the projection of
+Ψ1,θ\A1,θ onto X × Xc is a subset of Ψ\A, i.e., 5
+πX×Xc(Ψ1,θ\A1,θ) ⊂ Ψ\A,
+(42)
+then the hybrid controller (κ1, V1,θ, Dc,1, Fc,1) in (38) is syn-
+ergistic relative to A1,θ for (31) with synergy gap exceeding
+δ.
+Proof. It follows from the definition of µV1,θ in (41c) that
+µV1,θ(x, xc,1) is the sum of µV0(x, xc) with a quadratic
+nonnegative term, hence
+µV1,θ(x, xc,1) ≥ µV0(x, xc)
+(43)
+for each (x, xc,1) ∈ Ψ1,θ\A1,θ and, consequently, we have
+that
+δ2 := inf{µV1,θ(x, xc, θ) : (x, xc,1) ∈ Ψ1,θ\A1,θ}
+≥ inf {µV0(x, xc) : (x, xc,1) ∈ Ψ1,θ\A1,θ} .
+(44)
+The
+fact
+that
+(x, xc,1)
+∈
+Ψ1,θ\A1,θ
+implies
+(x, xc)
+∈
+πX×Xc(Ψ1,θ\A1,θ)
+together
+with
+(44)
+allow
+us
+to
+derive
+the
+following
+inequality:
+δ2
+≥
+inf {µV0(x, xc) : (x, xc) ∈ πX×Xc(Ψ1,θ\A1,θ)} . It follows
+from (42) that δ2 ≥ inf{µV0(x, xc) : (x, xc) ∈ Ψ\A}
+which is greater than zero by the assumption that the
+controller (κ0, V0, Dc, Fc)
+is
+synergistic relative to
+A
+for (33) with synergy gap exceeding δ. In addition, we
+have that µV1,θ(x, xc,1)
+≥
+µV0(x, xc)
+>
+δ(x, xc) for
+each (x, xc,1) ∈ Ψ1,θ\A1,θ, which proves that the hybrid
+controller (κ1, V1,θ, Dc,1, Fc,1) in (38) is synergistic relative
+to A1,θ for (31) with synergy gap exceeding δ.
+In the next result, we complete the construction of the
+robust synergistic controller (23) from the data of a nominally
+5Given a subset S of X := X1 × X2, the projection of S onto X1
+is represented by πX1(S) := {x1 ∈ X1 : (x1, x2) ∈ S for some x2 ∈
+X2}. Similarly, the projection of S onto X2 is denoted by πX2(S) :=
+{x2 ∈ X2 : (x1, x2) ∈ S for some x1 ∈ X1}.
+synergistic controller (κ0, V0, Dc, Fc), by designing a set-
+valued map Gc,1 : X × Xc,1 ⇒ X that is outer semicon-
+tinuous, locally bounded and satisfies (25).
+Proposition 5. Suppose that the sets X, Xc, U, and the
+set-valued map Fθ in (31) satisfy Assumption 1, and that
+Assumption 2 holds. Given Ω in (32), a compact set A ⊂
+X × Xc, and a hybrid controller (κ0, V0, Dc, Fc) that is
+nominally synergistic relative to A for (31) with synergy
+gap exceeding δ, A1 := {A1,θ}θ∈Ω with A1,θ in (34),
+V1 := {V1,θ}θ∈Ω with V1,θ in (38b), then the hybrid con-
+troller (κ1, V1, Dc,1, Fc,1, Gc,1) where
+Gc,1(x, xc,1) := ̺V0(x, xc) × ˆG(ˆθ)
+(45)
+for each (x, xc,1) ∈ X × Xc,1, and
+ˆG(ˆθ) := arg max
+g∈Ω+ǫB
+min
+θ∈Ω (θ − ˆθ)⊤Γ−1
+1 (θ − ˆθ)
+− (θ − g)⊤Γ−1
+1 (θ − g)
+for each ˆθ ∈ Ω + ǫB, is synergistic relative to A1 for (31)
+with robustness margin Ω and synergy gap exceeding δ.
+Proof. In Proposition 4 we demonstrate that the hybrid
+controller (κ1, V1,θ, Dc,1, Fc,1) is synergistic relative to A1,θ
+as required by Definition 3. It remains to be shown that
+the hybrid controller (κ1, V1, Dc,1, Fc,1, Gc,1) satisfies as-
+sumptions (C8), (C9) and (C10). To prove (C8), one must
+show that X × Xc,1 ⊂ dom V1,θ. From the definition of
+V1,θ in (38b), we have that dom V1,θ = dom V0 × (Ω +
+ǫB). It follows from the assumption that the hybrid con-
+troller (κ0, V0, Dc, Fc) is nominally synergistic relative to A
+for (31) that X × Xc ⊂ dom V0, hence X × Xc,1 ⊂ dom V1,θ.
+The function (x, xc,1, θ) �→ V1(x, xc,1, θ) := V1,θ(x, xc,1)
+is continuous because it results from the composition of
+continuous functions, hence (C8) holds.
+To prove (C9) and (C10), one must show that Gc,1 is outer
+semicontinuous, locally bounded and that it satisfies (25).
+Since Gc,1(x, xc,1) is the Cartesian product of ̺V0(x, xc)
+and ˆG(ˆθ) for each (x, xc,1) ∈ X × Xc,1 and ̺V0 is outer
+semicontinuous and locally bounded as proved in Lemma 1,
+to demonstrate that (C9) is satisfied it only remains to be
+shown that ˆG is outer semicontinuous and locally bounded.
+Let h(g, θ) := (θ − ˆθ)⊤Γ−1
+1 (θ − ˆθ)− (θ − g)⊤Γ−1
+1 (θ − g) for
+each (g, θ) ∈ (Ω+ ǫB)× Ω. Since h results from the compo-
+sition of continuous functions it is also continuous. It follows
+from the compactness of Ω and from [30, Theorem 9.14] that
+h(g) := min{h(g, θ) : θ ∈ Ω}
+∀g ∈ Ω + ǫB
+(46)
+is continuous. Since Dc,1 is continuous and compact-valued,
+it follows from the continuity of (46) and [30, Theorem 9.14]
+that ˆG is compact-valued and upper semicontinuous. The
+remainder of the proof of outer semicontinuity and local
+boundedness of ˆG follows closely that of Lemma 1, thus
+it will be omitted.
+The fact that Gc,1 satisfies (25) follows from the obser-
+vations in Remark 6 by noticing that Ω and Ω + ǫB are
+convex and compact spaces and the function h, which can be
+11
+
+Submitted for publication
+rewritten as h(g, θ) = 2θ⊤Γ−1
+1 (g − ˆθ) − g⊤Γ−1
+1 g + ˆθ⊤Γ−1
+1 ˆθ
+for each (g, θ) ∈ (Ω+ǫB)×Ω, is quasi-concave as a function
+of g and quasi-convex as a function of θ.
+The hybrid closed-loop system resulting from the intercon-
+nection between (κ1, V1, Dc,1, Fc,1, Gc,1) and (31) is given
+by:
+( ˙x, ˙xc,1) ∈ Fcl,1(x, xc,1)
+(x, xc,1) ∈ CΩ,1
+(47a)
+(x+, x+
+c,1) ∈ GΩ,1(x, xc,1)
+(x, xc,1) ∈ DΩ,1
+(47b)
+where
+CΩ,1 :=
+�
+(x, xc,1) ∈ X × Xc,1 : min
+θ∈Ω µV1,θ(x, xc,1) ≤ δ(x, xc)
+�
+DΩ,1 :=
+�
+(x, xc,1) ∈ X × Xc,1 : min
+θ∈Ω µV1,θ(x, xc,1) ≥ δ(x, xc)
+�
+and
+GΩ,1(x, xc,1) :=
+�
+x
+Gc,1(x, xc,1)
+�
+∀(x, xc,1) ∈ DΩ,1.
+(48)
+Global asymptotic stability of A1,θ for (47) follows from
+the application of Theorem 2 and it is summarized in the
+next corollary.
+Corollary 2. Suppose that the sets X, Xc, U, and the
+set-valued map Fθ in (31) satisfy Assumption 1, and that
+Assumption 2 holds. Given Ω in (32), a positive function
+δ : X × Xc �→ R, a compact set A ⊂ X × Xc, and a hybrid
+controller (κ0, V0, Dc, Fc) that is nominally synergistic rel-
+ative to A for (31) with synergy gap exceeding δ, for each
+θ ∈ Ω, the set A1,θ is globally asymptotically stable for (47).
+Proof. It follows from (43) that min{µV1,θ(x, xc,1) : θ ∈
+Ω} ≥ µV0(x, xc) for each (x, xc,1) ∈ Ψ1,θ\A1,θ. Since
+µV0(x, xc) > δ(x, xc) for each (x, xc,1) ∈ Ψ1,θ\A1,θ as
+shown in the proof of Proposition 4, and δ satisfies (D1), the
+conditions of Theorem 2 apply and we are able to conclude
+that A1,θ is globally asymptotically stable for (47).
+C. Backstepping
+Given a nominally synergistic controller (κ0, V0, Dc, Fc),
+we extend the dynamics of the controller in Section VI-B to
+include the input u as a controller state:6
+˙xc,2 ∈ Fc,2(x, xc,2)
+:=
+
+
+
+
+
+fc
+Γ1 Proj(υ(x, xc,2), ˆθ)
+fu(x, xc,2) + Dxc(κ1(x, xc,1))fc
+
+ : fc ∈ Fc(x, xc)
+
+
+
+(49)
+with xc,2 := (xc,1, u) ∈ Xc,2 := Xc × (Ω + ǫB) × Rm,
+Γ2 ∈ Rm×m positive definite, ku > 0,
+υ(x, xc,2) := W(x, xc)⊤∇xV0(x, xc)
+− W(x, xc)⊤Dx(κ1(x, xc,1))⊤Γ−1
+2 (u − κ1(x, xc,1))
+(50)
+6Alternatively, one may consider u as a plant state rather than a controller
+state, in which case u would remain constant during jumps. We have
+included u as a controller variable because it is an approach less often
+found in the literature.
+for each (x, xc,2) ∈ X × Xc,2, and
+fu(x,xc,2) := −�
+W(x, xc)Γ1 Proj(υ(x, xc,2), ˆθ)
+− ku(u − κ1(x, xc,1)) − Γ2H(x, xc)⊤∇xV0(x, xc)
++ Dx(κ1(x, xc,1))F(x, xc, u, ˆθ)
+(51)
+which is defined for each (x, xc,2) ∈ X × Xc,2 assuming that
+κ0 is continuously differentiable and that F(x, xc, u, ˆθ) =
+Fˆθ(x, xc, u) denotes the dynamics (31) with θ is equal to the
+estimated value ˆθ.
+Given the compact set Ω of possible (unknown) values
+of θ in (32), a compact set A ⊂ X × Xc, and a nominal
+synergistic controller (κ0, V0, Dc, Fc) relative to A for (31)
+with synergy gap exceeding δ, the main goal of this section
+is to design a controller of the form (23) that is synergistic
+relative to A2 := {A2,θ}θ∈Ω for (31) with robustness margin
+Ω and synergy gap exceeding δ, where
+A2,θ := {(x, xc,2) ∈ X × Xc,2 : (x, xc,1) ∈ A1,θ,
+u = κ1(x, xc,1)}.
+(52)
+In this direction, we define the Lyapunov function
+V2,θ(x, xc,2) := V1,θ(x, xc,1)
++ 1
+2(u − κ1(x, xc,1))⊤Γ−1
+2 (u − κ1(x, xc,1))
+(53)
+for each (x, xc,2) ∈ X × Xc,2 and the set-valued map
+Dc,2(x, xc,2) := {(gc,1, gu) ∈ Xc,2 : gc,1 ∈ Dc,1(x, xc,1),
+gu = κ1(x, gc,1)}
+(54)
+for each (x, xc,2) ∈ X × Xc,2. The choice u = κ1(x, gc,1)
+in (54) may seem peculiar, but it turns out that this value
+minimizes (53) with respect to u, hence it is suitable for the
+jump logic.
+From the interconnection between (31) and the hybrid
+controller (κ2, V2,θ, Dc,2, Fc,2) with κ2(x, xc,2) = u for each
+(x, xc,2) ∈ X×Xc,2, we obtain the hybrid closed-loop system
+( ˙x, ˙xc,2) ∈ Fcl,2(x, xc,2)
+(x, xc,2) ∈ C2
+:= {(x, xc,2) ∈ X × Xc,2 : µV2,θ(x, xc,2) ≤ δ(x, xc)}
+(55a)
+(x+, x+
+c,2) ∈ Gcl,2(x, xc,2)
+(x, xc,2) ∈ D2
+:= {(x, xc,2) ∈ X × Xc,2 : µV2,θ(x, xc,2) ≥ δ(x, xc)}
+(55b)
+where
+Fcl,2(x, xc,2) :=
+�Fθ(x, xc, u)
+Fc,2(x, xc,2)
+�
+∀(x, xc,2) ∈ C2
+(56a)
+Gcl,2(x, xc,2) :=
+�
+x
+̺V2,θ(x, xc,2)
+�
+∀(x, xc,2) ∈ D2. (56b)
+12
+
+Submitted for publication
+Note
+that,
+from
+the
+definitions
+(12b)
+and
+(12c),
+we
+have
+the
+following
+identities
+for
+the
+hybrid
+controller (κ2, V2,θ, Dc,2, Fc,2):
+̺V2,θ(x, xc,2) = {(gc,1, gu) ∈ Xc,2 : gc,1 ∈ ̺V1,θ(x, xc,1),
+gu = κ1(x, gc,1)},
+µV2,θ(x, xc,2) = µV1,θ(x, xc,1) + 1
+2
+���Γ
+− 1
+2
+2
+(u − κ1(x, xc,1))
+���
+2
+for each (x, xc,2) ∈ X × Xc,2,7; hence, similarly to (39), the
+closed-loop system (55) is impossible to implement due to
+dependence on θ in C2, D2, and Gcl,2, but, similarly to the
+controller of Section VI-B, this dependence will be removed
+with the design of a hybrid controller that is synergistic
+relative to A2 := {A2,θ}θ∈Ω for (31) with robustness margin
+Ω (cf. Remark 7).
+We are able to prove the following result using arguments
+similar to those of Proposition 4.
+Proposition 6. Suppose that the sets X, Xc, U, and the set-
+valued map Fθ in (31) satisfy Assumption 1, and that Assump-
+tion 2 holds. Given θ ∈ Ω, a compact set A ⊂ X ×Xc, and a
+hybrid controller (κ0, V0, Dc, Fc) that is nominally synergis-
+tic relative to A for (31) with synergy gap exceeding δ, if (42)
+is satisfied then the hybrid controller (κ2, V2,θ, Dc,2, Fc,2)
+is synergistic relative to A2,θ for (31) with synergy gap
+exceeding δ.
+Proof. Similarly to the proof of Proposition 4, it is possible
+to show that properties (C1), (C3) and (C5) follow directly
+from the fact that A2,θ is compact and from the assumption
+that (κ0, V0, Dc, Fc) is synergistic relative to A for (33).
+It follows from the continuity of Dc,1 and κ1 that Dc,2
+is continuous. That Dc,2 is compact-valued follows from
+compactness of Dc,1 and continuity of κ1, hence (C4) is
+satisfied. It remains to be shown that properties (C6) and (C7)
+also hold. It follows from (53) and (56a) that
+∇V2,θ(x, xc,2)⊤fcl,2 = ∇V0(x, xc)⊤
+�Fθ(x, xc, u)
+fc
+�
+− (θ − ˆθ)⊤ Proj(υ(x, xc,2), ˆθ)
++ (u − κ1(x, xc,1))⊤Γ−1
+2
+�
+fu(x, xc,2)
+− D(κ1(x, xc,1))
+
+
+Fθ(x, xc, u)
+fc
+Γ1 Proj(υ(x, xc,2), ˆθ)
+
+
+�
+(X2)
+for each (x, xc,2) ∈ X×Xc,2 and each fcl,2 ∈ Fcl,2(x, xc,2),
+where V2,θ is continuously differentiable and fc ∈ Fc(x, xc)
+is the component of fcl,2 that describes the dynamics of xc.
+7Since Γ2 ∈ Rm×m is assumed to be positive definite, Γ
+− 1
+2
+2
+exists and
+is unique (cf. [34, Section 8.5]).
+Replacing (51) in (X2), we obtain
+∇V2,θ(x, xc,2)⊤fcl,2 = ∇V0(x, xc)⊤
+�Fθ(x, xc, u)
+fc
+�
+− (θ − ˆθ)⊤ Proj(υ(x, xc,2), ˆθ)
+− ku(u − κ1(x, xc,1))⊤Γ−1
+2 (u − κ1(x, xc,1))
+− (u − κ1(x, xc,1))⊤H(x, xc)⊤∇xV0(x, xc)
+− (u − κ1(x, xc,1))⊤Γ−1
+2 Dx(κ1(x, xc,1))W(x, xc)(θ − ˆθ).
+(X3)
+It follows from (P3) and (X3) that
+∇V2,θ(x, xc,2)⊤fcl,2 ≤ ∇V0(x, xc)⊤
+�Fθ(x, xc, u)
+fc
+�
+− (θ − ˆθ)⊤υ(x, xc,2)
+− ku(u − κ1(x, xc,1))⊤Γ−1
+2 (u − κ1(x, xc,1))
+− (u − κ1(x, xc,1))⊤H(x, xc)⊤∇xV0(x, xc)
+− (u − κ1(x, xc,1))⊤Γ−1
+2 Dx(κ1(x, xc,1))W(x, xc)(θ − ˆθ).
+(X4)
+Replacing (50) in (X4), we obtain
+∇V2,θ(x, xc,2)⊤fcl,2 ≤ ∇V0(x, xc)⊤
+�Fθ(x, xc, u)
+fc
+�
+− (θ − ˆθ)⊤W(x, xc)⊤∇xV0(x, xc)
+− ku(u − κ1(x, xc,1))⊤Γ−1
+2 (u − κ1(x, xc,1))
+− (u − κ1(x, xc,1))⊤H(x, xc)⊤∇xV0(x, xc).
+(X5)
+The control affine structure of (31) allows us to derive the
+following inequality from (X5):
+∇V2,θ(x, xc,2)⊤fcl,2
+≤ ∇V0(x, xc)⊤
+�Fθ(x, xc, κ1(x, xc,1))
+fc
+�
+− (θ − ˆθ)⊤W(x, xc)⊤∇xV0(x, xc)
+− ku(u − κ1(x, xc,1))⊤Γ−1
+2 (u − κ1(x, xc,1)).
+(X6)
+Note that it was proved in Proposition 3 that
+∇V0(x, xc)⊤(Fθ(x, xc, κ1(x, xc,1)) − W(x, xc)(θ − ˆθ))
+≤ ∇V0(x, xc)⊤F0(x, xc, κ0(x, xc)),
+(X7)
+thus, from the assumption that (κ0, V0, Dc, Fc) is synergistic
+relative to A for (33), we have that
+∇V2,θ(x, xc,2)⊤fcl,2 ≤ ∇V0(x, xc)⊤F0(x, xc, κ0(x, xc))
+− ku(u − κ1(x, xc,1))⊤Γ−1
+2 (u − κ1(x, xc,1)) ≤ 0
+(57)
+for each (x, xc,2) ∈ X × Xc,2 satisfying V2,θ(x, xc,2) < +∞
+and each fcl,2 ∈ Fcl,2(x, xc,2), hence property (C6) is
+satisfied. Let Ψ2,θ denote the largest weakly invariant subset
+of
+( ˙x, ˙xc,2) ∈ Fcl,2(x, xc,2)
+(x, xc) ∈ E2
+(58)
+with E2 := {(x, xc,2) ∈ X × Xc,2 : ∇V2,θ(x, xc,2)⊤fcl,2 = 0
+for some fcl,2
+∈
+Fcl,2(x, xc,2)}.
+To
+verify
+that (κ2, V2,θ, Dc,2, Fc,2) is synergistic relative to A2,θ
+for (49), we need to check that δ2 := inf{µV2,θ(x, xc,2) :
+13
+
+Submitted for publication
+(x, xc,2) ∈ Ψ2,θ\A2,θ} > 0. It follows from (57) that Ψ2,θ ⊂
+{(x, xc,2) ∈ X × Xc,2 : (x, xc,1) ∈ Ψ1,θ, u = κ1(x, xc,1)}
+where Ψ1,θ is defined in Proposition 4. It follows from (52)
+that
+δ2 ≥ inf{µV2,θ(x, xc,2) : (x, xc,1) ∈ Ψ1,θ\A1,θ,
+u = κ1(x, xc,1)}
+= inf{µV1,θ(x, xc,1) : (x, xc,1) ∈ Ψ1,θ\A1,θ}
+(59)
+which we have shown in Proposition 4 to satisfy δ2 >
+0, under assumption (42). In addition, µV2,θ(x, xc,2)
+=
+µV1,θ(x, xc,1) ≥ µV0(x, xc) > δ(x, xc) for each (x, xc,2) ∈
+Ψ2,θ\A2,θ, hence the hybrid controller (κ2, V2,θ, Dc,2, Fc,2)
+is synergistic relative to A2,θ for (31) with synergy gap
+exceeding δ.
+To finalize the design of a robust synergistic controller, we
+provide the construction of the jump map Gc,2 in the next
+proposition.
+Proposition 7. Suppose that the sets X, Xc, U, and the set-
+valued map Fθ in (31) satisfy Assumption 1, and that Assump-
+tion 2 holds. Given Ω in (32), a compact set A ⊂ X × Xc,
+and a hybrid controller (κ0, V0, Dc, Fc) that is nominally
+synergistic relative to A for (31) with synergy gap exceeding
+δ, A2 := {A2,θ}θ∈Ω with A2,θ in (52), V2 := {V2,θ}θ∈Ω with
+V2,θ in (53), the hybrid controller (κ2, V2, Dc,2, Fc,2, Gc,2)
+where
+Gc,2(x, xc,2) := {(gc,1, gu) ∈ Dc,2(x, xc,2) :
+gc,1 ∈ Gc,1(x, xc,1)}
+(60)
+for each (x, xc,2) ∈ X × Xc,2 is synergistic relative to A2
+for (31) with robustness margin Ω and synergy gap exceeding
+δ.
+Proof. In Proposition 6 we demonstrate that the hybrid
+controller (κ2, V2,θ, Dc,2, Fc,2) is synergistic relative to A2,θ
+with synergy gap exceeding δ as required by Definition 3.
+The proof that (C8) is satisfied follows closely the proof of
+Proposition 5, hence it is omitted here. The outer semicon-
+tinuity and local boundedness of Gc,2 follows from outer
+semicontinuity and local boundedness of Gc,1 in addition
+to the continuity of κ1, thus (C9) is verified. For each
+(x, xc,2) ∈ X × Xc,2 and for each gc,2 ∈ X2, we have that
+V2,θ(x, xc,2) − V2,θ(x, gc,2)
+≥ min
+θ∈Ω V2,θ(x, xc,2) − V2,θ(x, gc,2).
+(61)
+From (60), it follows that gc,2
+:= (gc,1, gu) with gc,1
+belonging to (45) and gu = κ1(x, gc,1). Replacing (53)
+in (61) and plugging in the aforementioned values of gc,1
+and gu, we have that
+V2,θ(x, xc,2) − V2,θ(x, gc,2)
+≥
+max
+gc,1∈Dc,1(x,xc,1) min
+θ∈Ω V1,θ(x, xc,1) − V1,θ(x, gc,1)
++ 1
+2(u − κ1(x, xc,1))⊤Γ−1
+2 (u − κ1(x, xc,1))
+=
+max
+gc,2∈Dc,2(x,xc,2) min
+θ∈Ω V2,θ(x, xc,2) − V2,θ(x, gc,2)
+(62)
+for each (x, xc,2) ∈ X × Xc,2 and each gc,2 := (gc,1, gu) ∈
+Gc,2(x, xc,2). Since the max and min operators in (62)
+commute as shown in the proof of Proposition 5, it follows
+that
+V2,θ(x, xc,2) − V2,θ(x, gc,2) ≥ min
+θ∈Ω µV2,θ(x, xc,2)
+for each (x, xc,2) ∈ X × Xc,2 and each gc,2 := (gc,1, gu) ∈
+Gc,2(x, xc,2), thus verifying (C10).
+The hybrid closed-loop system resulting from the intercon-
+nection between (κ2, V2, Dc,2, Fc,2, Gc,2) and (31) is given
+by:
+( ˙x, ˙xc,2) ∈ Fcl,2(x, xc,2)
+(x, xc,2) ∈ CΩ,2
+(63a)
+(x+, x+
+c,2) ∈ GΩ,2(x, xc,2)
+(x, xc,2) ∈ DΩ,2
+(63b)
+where
+CΩ,2 :=
+�
+(x, xc,2) ∈ X × Xc,2 : min
+θ∈Ω µV2,θ(x, xc,2) ≤ δ(x, xc)
+�
+DΩ,2 :=
+�
+(x, xc,2) ∈ X × Xc,2 : min
+θ∈Ω µV2,θ(x, xc,2) ≥ δ(x, xc)
+�
+and
+GΩ,2(x, xc,2) :=
+�
+x
+Gc,2(x, xc,2)
+�
+∀(x, xc,2) ∈ DΩ,2.
+The global asymptotic stability of A2,θ for (63) follows
+from Theorem 2 and it is stated in the next corollary for
+the sake of completeness. The proof is omitted because it is
+identical to the proof of Corollary 2
+Corollary 3. Suppose that the sets X, Xc, U, and the
+set-valued map Fθ in (31) satisfy Assumption 1, and that
+Assumption 2 holds. Given Ω in (32), a positive function
+δ : X × Xc �→ R, a compact set A ⊂ X × Xc, and a hybrid
+controller (κ0, V0, Dc, Fc) that is nominally synergistic rel-
+ative to A for (31) with synergy gap exceeding δ, for each
+θ ∈ Ω, the set A2,θ is globally asymptotically stable for (63).
+In the next section, we apply the controllers proposed in
+Sections VI-B and VI-C to global asymptotic stabilization of
+a setpoint for a two-dimensional system in the presence of
+an obstacle.
+VII. SYNERGISTIC HYBRID FEEDBACK FOR ROBUST
+GLOBAL OBSTACLE AVOIDANCE
+To demonstrate the applicability of the synergistic adaptive
+controller of Section VI, we consider the problem of globally
+asymptotically stabilizing the origin for a vehicle moving on
+a plane with an obstacle N := z0 + rB with z0 ∈ R2 and
+r > 0 such that the origin is not contained in N. We consider
+that the evolution in time of the position z ∈ R2\N of the
+vehicle is described by
+˙z = u + θ
+(64)
+where u ∈ R2 is the input and θ ∈ R2 is an unknown
+constant. We have shown in [7, Section IV] that ψ(z) :=
+�
+log(|z − z0| − r)
+z−z0
+|z−z0|
+�⊤
+is a diffeomorphism between
+14
+
+Submitted for publication
+R2\N and R × S1, hence global asymptotic stabilization
+of the origin for (64) is equivalent to the global asymptotic
+stabilization of ψ(0). for
+˙x = Dψ(ψ−1(x))u + Dψ(ψ−1(x))θ.
+(65)
+with x ∈ X := R × S1. Before moving to the controller
+design, we show that Assumption 1 is verified for the
+particular problem at hand.
+Proposition 8. The sets X := R × S1, Xc := {−1, 1}, U :=
+R2 and the set-valued map
+Fθ(x, xc, u) := Dψ(ψ−1(x))u + Dψ(ψ−1(x))θ
+(66)
+defined for each (x, xc, u) ∈ X × Xc × U satisfies Assump-
+tion 1 and
+(⋆) The intersection between Fθ(x, xc, u) and the tan-
+gent space to X at (x, xc, u) is nonempty for each
+(x, xc, u) ∈ X × Xc × U.
+Proof. To check that the condition (S1) holds, note that the
+sets X, Xc and U are closed subsets of R3, R and R2,
+respectively. It follows from the fact that ψ is a diffeomor-
+phism between R2\N and R × S1 that Dψ(ψ−1(x)) is an
+isomorphism between the tangent space to R2\N at ψ−1(x)
+and the tangent space to R × S1 at x for each x ∈ X
+(cf. [29, Proposition 3.6]), thus (⋆) is verified. Since ψ is
+a diffeomorphism it also follows that x �→ Dψ(ψ−1(x))
+is continuous, thus Fθ is also continuous and single-valued,
+hence it verifies (S2).
+Remark 8. The condition (⋆) is pivotal in the verification of
+the conditions (VC) and (VC’) for this particular example,
+which, in turn, allows us to check the completeness of max-
+imal solutions as shown in Theorems 1 and 2, respectively.
+The controller design of Section VI requires the existence
+of a hybrid controller of the form (κ0, V0, Dc, Fc) that is
+nominally synergistic relative to
+A := {(x, q) ∈ X × Xc : x = ψ(0)}.
+(67)
+for (66), thus we start by showing that the controller provided
+in [7, Section IV] satisfies the requirements (C1)-(C7). In
+this direction, let the controller variable xc in (31) be a logic
+variable q which is either 1 or −1 and whose values does
+not change during flows, i.e., xc = q ∈ Xc := {−1, 1} and
+˙q = Fc(x, q) := 0 for all (x, q) ∈ X × Xc which veri-
+fies (C2). Following the controller design of [7, Section IV],
+let φq(x) :=
+�x1
+x2
+1−qx3
+�⊤ for each x := (x1, x2, x3) ∈
+Uq := {x ∈ X : qx3 ̸= 1} with q ∈ Xc := {−1, 1}.
+Furthermore, we define
+V0(x, q) :=
+
+
+
+1
+2 |φq(x) − φq(ψ(0))|2
+if x ∈ Uq
++∞
+otherwise
+(68)
+for each (x, q) ∈ X × Xc. Defining Dc(x, q) = Xc for
+each (x, q) ∈ X × Xc and noting that {(Uq, φq)}q∈Xc covers
+R × S1 we have that the optimization problem in (12) is
+feasible, hence assumption (C1) is verified. Since each chart
+φq : Uq → R2 is a diffeomorphism, φq(x) = φq(ψ(0)) if
+and only if x = ψ(0), hence V in (68) is positive definite
+relative to (67). Moreover, V0 is continuous and V −1
+0
+([0, c])
+is compact for each c ∈ R≥0, thus (C3) is verified. Since
+Dc is constant and equal to the finite set Xc for each
+(x, xc) ∈ X × Xc, we have that Dc is outer semicontinuous,
+lower semicontinuous and locally bounded, hence (C4) is
+verified. Condition (C5) is verified for
+κ0(x, q) = −
+�
+Dψ(ψ−1(x))
+�⊤ Dφq(x)⊤(φq(x) − φq(ψ(0)))
+for each (x, q) ∈ dom κ0 = {(x, q) ∈ X ×Xc : x ∈ Uq}. The
+previous arguments allow us to make the following assertion.
+Proposition
+9.
+Given
+A
+in
+(67),
+the
+hybrid
+con-
+troller (κ0, V0, Dc, Fc) is a synergistic candidate relative to
+A in (67) for (65).
+Even though (68) is continuous, it is not Lipschitz contin-
+uous everywhere, hence the proof that (C6) holds might not
+be immediately obvious. From (12c) and using the fact that
+Dc(x, q) := Xc := {−1, 1} for each (x, q) ∈ X × Xc, we
+have that µV0(x, q) = max{0, V0(x, q)−V0(x, −q)} for each
+(x, q) ∈ X × Xc and, in particular, we have that µV0(x, q) =
++∞ for each (x, q) ̸∈ Uq, hence, for any function δ, it
+follows that each (x, q) ∈ X×Xc satisfying (x, q) ∈ Uq does
+not belong to C. Since Uq is open relative to X := R × S1
+for each q ∈ Xc, {(x, q) ∈ X × Xc : x ∈ X\Uq} and C are
+disjoint closed sets, and there exists a neighborhood of C
+where V0 is Lipschitz continuous. The generalized derivative
+of V0 at (x, q) is the direction Fcl,0(x, q) is given by
+V ◦
+0 (x, q; Fcl,0(x, q)) =
+−
+��Dψ(ψ−1(x))⊤Dφq(x)⊤(φq(x) − φq(ψ(0)))
+��2 ,
+(69)
+for each (x, q) ∈ C, where Fcl,0 is the flow map for
+the closed-loop system resulting from the interconnection
+between (κ0, V0, Dc, Fc) and (65) with θ = 0 (cf. (15)). It
+follows from (69) that the growth of V0 along the flows of
+the closed-loop system is upper bounded by 0, hence (C6)
+is verified. It follows from the fact that ψ and {φq}q∈Xc
+are diffeomorphisms that Assumption (C7) is satisfied with
+Ψ = A (cf. [7]), thus the following holds.
+Proposition 10. Given A in (67) and a continuous function
+δ : X × Xc → R, the hybrid controller (κ0, V0, Dc, Fc)
+is nominally synergistic relative to A in (67) for (65) with
+synergy gap exceeding δ.
+An additional property of the kind of synergistic feedback
+presented above is that any function δ satisfies (D2) since
+Ψ\A = ∅. Therefore, any choice of δ satisfying (D1) yields
+global asymptotic stability for the hybrid closed-loop system
+as proved in Theorem 1. Since Assumption 2 is satisfied,
+we meet all the requirements for the controller design of
+Section VI, thus it is only a matter of applying the procedures
+described therein to obtain an adaptive synergistic hybrid
+feedback controller that is able to deal with parametric
+uncertainty. In the next section, we present some numerical
+results that illustrate the behaviour of the closed-loop system.
+15
+
+Submitted for publication
+PSfrag replacements
+N
+z1
+z2
+−0.5
+0
+0.5
+1
+1.5
+2
+2.5
+−1
+−0.5
+0
+0.5
+1
+1.5
+2
+Section VI-B with q(0, 0) = −1
+Section VI-B with q(0, 0) = 1
+Section VI-C with q(0, 0) = −1
+Section VI-C with q(0, 0) = 1
+Fig. 1.
+Trajectories t �→ z(t) of (64) for the closed-loop system with
+parameters given in Section VII-A.
+A. Simulation Results
+In this section, we present simulation results of the closed-
+loop system resulting from the interconnection between (65)
+and the hybrid controllers that are presented in Section VI
+considering that there is an obstacle N := z0 + rB with
+z0 =
+�
+1
+0
+�⊤ and r = 0.5. Furthermore, we consider that
+θ =
+�√
+2/2
+√
+2/2
+�⊤ and that the controller parameters are
+ku = 1, Γ1 = Γ2 = I2, ǫ = 1, θ0 = 1, and δ(x, q) = 1 for
+each (x, q) ∈ X × Xc. For this particular choice of Γ1, we
+have that
+ˆG(ˆθ) =
+
+
+
+ˆθ
+if
+���ˆθ
+��� ≤ θ0
+θ0
+ˆθ
+|ˆθ|
+otherwise
+for each ˆθ ∈ Ω + ǫB, which is outer semicontinuous and
+locally bounded.
+Figure 1 represents the trajectory of the vehicle starting
+from rest at z(0) =
+�2
+0�⊤ for each of the controllers
+presented in Section VI. It can be verified both through
+Figure 1 as well as Figure 2 that the trajectories before and
+after backstepping are comparable, since the evolution of
+the distance of the vehicle to the desired setpoint is fairly
+similar in both cases. The bottom half of Figure 2 depicts
+the evolution of the estimation error, which has a smaller
+settling time for the closed-loop system with the controller
+of Section VI-C than the controller of Section VI-B for this
+particular simulation. To find out more about the simulation
+and its implementation, you may explore the source code at
+https://github.com/pcasau/synergistic.
+PSfrag replacements
+t
+���ˆθ(t) − θ
+���
+|z(t)|
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+0
+0.5
+1
+1.5
+2
+2.5
+Section VI-B with q(0, 0) = −1
+Section VI-B with q(0, 0) = 1
+Section VI-C with q(0, 0) = −1
+Section VI-C with q(0, 0) = 1
+Fig. 2. Evolution in time of the distance to the origin (top) and the parametric
+estimation error (bottom) for the closed-loop system with parameters given
+in Section VII-A.
+VIII. CONCLUSION
+Synergistic hybrid feedback has taken many forms over
+the years, depending on the particular dynamical system
+being studied. The unifying framework for synergistic hybrid
+feedback that we presented in this paper captures the most
+salient features of existing synergistic hybrid feedbacks in
+order to help others distinguish between the particular and the
+general in different instances of synergistic hybrid feedback
+across the literature. In addition, we provided a controller
+design that starts from an existing synergistic controller
+and modified it in order to yield an adaptive controller
+that is able to compensate for the presence of bounded
+matched uncertainties in affine control systems. Furthermore,
+we demonstrated that the proposed controller is amenable to
+backstepping and can be applied to the problem of global
+obstacle avoidance.
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+Automatica, vol. 58, pp. 90–98, 2015.
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+[34] D. S. Bernstein, Matrix Mathematics. Princeton, NJ: Princeton Uni-
+versity Press, 2009.
+Pedro Casau is a Research Fellow at the Institute
+for Systems and Robotics, Lisbon, Portugal. He re-
+ceived the B.Sc. in Aerospace Engineering in 2008
+from Instituto Superior T´ecnico (IST), Lisbon, Por-
+tugal. In 2010, he received the M.Sc. in Aerospace
+Engineering from IST and enrolled in the Elec-
+trical and Computer Engineering Ph.D. program
+at the same institution which he completed with
+distinction and honours in 2016. He participated on
+several national and international research projects
+on guidance, navigation and control of unmanned
+air vehicles (UAVs) and satellites. His current research interests include
+nonlinear control, hybrid control systems, vision-based control systems,
+controller design for autonomous air-vehicles.
+Ricardo G. Sanfelice received the B.S. degree
+in Electronics Engineering from the Universidad
+de Mar del Plata, Buenos Aires, Argentina, in
+2001, and the M.S. and Ph.D. degrees in Electrical
+and Computer Engineering from the University
+of California, Santa Barbara, CA, USA, in 2004
+and 2007, respectively. In 2007 and 2008, he held
+postdoctoral positions at the Laboratory for Infor-
+mation and Decision Systems at the Massachusetts
+Institute of Technology and at the Centre Automa-
+tique et Syst`emes at the ´Ecole de Mines de Paris. In
+2009, he joined the faculty of the Department of Aerospace and Mechanical
+Engineering at the University of Arizona, Tucson, AZ, USA, where he
+was an Assistant Professor. In 2014, he joined the University of California,
+Santa Cruz, CA, USA, where he is currently Professor in the Department
+of Electrical and Computer Engineering. Prof. Sanfelice is the recipient of
+the 2013 SIAM Control and Systems Theory Prize, the National Science
+Foundation CAREER award, the Air Force Young Investigator Research
+Award, the 2010 IEEE Control Systems Magazine Outstanding Paper Award,
+and the 2020 Test-of-Time Award from the Hybrid Systems: Computation
+and Control Conference. He is Associate Editor for Automatica and a
+Fellow of the IEEE. His research interests are in modeling, stability, robust
+control, observer design, and simulation of nonlinear and hybrid systems
+with applications to power systems, aerospace, and biology.
+Carlos Silvestre received the Licenciatura degree
+in Electrical Engineering from the Instituto Supe-
+rior Tecnico (IST) of Lisbon, Portugal, in 1987 and
+the M.Sc. degree in Electrical Engineering and the
+Ph.D. degree in Control Science from the same
+school in 1991 and 2000, respectively. In 2011 he
+received the Habilitation in Electrical Engineering
+and Computers also from IST. Since 2000, he
+is with the Department of Electrical Engineering
+of the Instituto Superior Tecnico, where he is
+currently an Associate Professor of Control and
+Robotics on leave. Since 2015 he is a Professor of the Department of Electri-
+cal and Computers Engineering of the Faculty of Science and Technology of
+the University of Macau. Over the past years, he has conducted research on
+the subjects of navigation, guidance and control of air and ocean robots. His
+research interests include linear and nonlinear control theory, hybrid control,
+sensor based control, coordinated control of multiple vehicles, networked
+control systems, fault detection and isolation, and fault tolerant control.
+17
+

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+arXiv:2301.03184v1  [math.RT]  9 Jan 2023
+G-SPECTRA OF CYCLIC DEFECT
+TONY FENG, DAVID TREUMANN, AND ALLEN YUAN
+Abstract. Brou´e’s Abelian Defect Conjecture predicts interesting derived
+equivalences between derived categories of modular representations of finite
+groups. We investigate a generalization of Brou´e’s Conjecture to ring spectrum
+coefficients and prove this generalization in the cyclic defect case, following an
+argument of Rouquier.
+Contents
+1.
+Introduction
+1
+2.
+Brou´e’s Conjecture over ring spectra
+2
+3.
+Associative algebras, modules, and bimodules in spectra
+9
+4.
+Permutation modules and Rouquier’s equivalence over S
+24
+References
+33
+1. Introduction
+Representation theory behaves significantly differently depending on whether the
+ground field has characteristic 0 or characteristic p. Since more general rings, like
+the p-adic integers Zp, in some sense interpolate between characteristics 0 and p,
+the study of representations on modules over such rings plays an important role in
+modular representation theory.
+In this paper, we investigate representations over an even more general class of
+rings that arises in algebraic topology, called ring spectra. Ordinary commutative
+rings are ring spectra, but ring spectra also include differential graded algebras, as
+well as non-algebraic objects such as (to name two of many) the one that repre-
+sents complex K-theory and the one that represents the stable homotopy groups
+of spheres. In a way, enlarging the class of rings makes the “interpolation between
+characteristics 0 and p” more finely resolved. There is a rich theory and we hope
+that the tools and concepts of that theory could be applied to make progress on
+representation-theoretic questions, even on classical problems that have nothing
+to do with algebraic topology. We review some of this theory, and some of its
+precedent applications to other areas, in §2.4.
+Conversely, some of those representation-theoretic questions suggest very natural
+questions in algebraic topology that have seemingly not been considered before. In
+this paper, we attend to one such question, Brou´e’s Abelian Defect Conjecture
+[Bro90]. It predicts that the derived categories of mod p or p-adic representations
+of a finite group G and of a normalizer subgroup NG(D) have direct factors in
+common when D is (1) abelian of p-power order and (2) a defect group of one of
+the mod p blocks of G. We review it in greater detail in §2.1–2.3.
+1
+
+2
+TONY FENG, DAVID TREUMANN, AND ALLEN YUAN
+For thirty years Brou´e’s Conjecture has been a guiding problem in modular
+representation theory.
+The purpose of this paper is to formulate the analogous
+problem with p-adic rings replaced by p-complete ring spectra: do the categories
+of p-complete G-spectra and of p-complete NG(D)-spectra have direct factors in
+common?
+In §4.6 we show that the original Brou´e conjecture is a consequence of our spec-
+trum version. We hope that the original conjecture could be attacked by apply-
+ing some of the tools and concepts of stable homotopy theory: the Segal Con-
+jecture/Carlsson’s Theorem, Tate fixed points/Frobenius, organizing by chromatic
+level, etc.; we discuss these further in §2.4, §3.10–3.12, §4.4. However, this hope is
+not realized in this paper. Before pursuing it very seriously, it is natural to first
+ask if there is any evidence that the spectrum version is true.
+In §4, we prove our spectrum version when D is cyclic. The result does not
+illuminate the original form of the Brou´e conjecture, which has been known for a
+long time in the cyclic defect case. Different proofs have been given by Rickard
+and by Rouquier. Both arguments end in the construction of “tilting complexes”
+of (G, NG(D))-bimodules that implement the derived equivalence. In general there
+are many obstructions to lifting a tilting complex to a G × NG(D)op-spectrum.
+We deduce the desired equivalence of categories by showing that Rouquier’s tilting
+complex does lift.
+Acknowledgements. The authors would like to thank Robert Burklund and Jay
+Taylor for helpful conversations. The first author was supported by NSF grant DMS
+1902927. The third author was supported in part by NSF grant DMS-2002029.
+2. Brou´e’s Conjecture over ring spectra
+2.1. Review of modular representation theory. This is a paper about finite
+group actions on p-complete spectra, but we will start this extended introduction
+by reviewing an example and some of the theory of finite group actions on p-adic
+abelian groups. Three basic questions in modular representation theory are:
+(1) What are the simple representations over a p-adic field of characteristic 0?
+(2) What are the simple representations over a finite field of characteristic p?
+(3) What are the projective representations in characteristic p, or over a p-adic
+ring of integers?
+The answer to these questions, and some information about blocks, defects, and
+Brauer trees, are given below in the case G = PSL2(F7) and p = 7. We will use
+this example to illustrate some of the concepts of modular representation theory.
+Conventionally, one chooses the coefficient rings and fields to have sufficiently
+many roots of unity. We wish to avoid adding pth roots of unity (or 4th roots when
+p = 2) to our coefficients for algebraic topology reasons we’ll come to later §2.4.
+We describe the representation theory of PSL2(F7) over Z7 and Q7.
+2.1.1. Irreducible Q7[PSL2(F7)]-modules. Let G = PSL2(F7) be the simple group
+of order 168 and let p = 7. There are 5 irreducible representations of G on Q7-vector
+spaces that we will call
+1, 7, 8, 6 and 33
+of dimensions 1, 7, 8, 6, and 6 respectively.
+
+G-SPECTRA OF CYCLIC DEFECT
+3
+The trivial representation is 1; the other representations have names coming from
+the theory of SL2: 7 is the Steinberg representation, 8 is in the principal series, the
+two 6-dimensional representations are in the discrete series.
+The representations 1, 7, 8, and 6 are all absolutely irreducible but if we adjoin
+√−7 or
+7√
+1 to Q7, the representation 33 splits as a sum of two 3-dimensional
+representations.
+2.1.2. Simple and indecomposable projective Z7[PSL2(F7)]-modules. The theory of
+projective covers gives a bijection between the simple and the indecomposable pro-
+jective Zp[G]-modules. Over Z7, there are four indecomposable projective PSL2(F7)-
+representations: a lattice in 7, a lattice in 1 ⊕ 6, a lattice in 6 ⊕ 8, and a lattice in
+8 ⊕ 33.
+The simple quotients of these modules are F7-vector spaces of dimensions 7, 1, 5,
+and 3, respectively; they don’t play a prominent role in this section, and in general
+simple modules don’t play a prominent role in this paper.
+2.1.3. Blocks. If G is any finite group and p is any prime, then the group algebra
+Zp[G] is a product of associative rings called block algebras:
+(2.1.1)
+Zp[G] = Zp[G]b1 × Zp[G]b2 × · · ·
+The way direct products of rings work, every Zp[G]-module M is canonically a
+direct sum of modules for the block algebras
+M = Mb1 ⊕ Mb2 ⊕ · · ·
+The notation Mbi means M is a module for Zp[G]bi. We say that “M belongs to
+the block bi” if M = Mbi.
+Any indecomposable Zp[G]-module belongs to a single block.
+All the Zp[G]-
+lattices in an irreducible Qp[G]-module belong to the same block. Thus, the blocks
+partition the set of irreducible Qp[G]-modules. One way to name or classify the
+blocks of G is by describing this partition. For example, there are two blocks of
+Z7[PSL2(F7)] and they are labeled by
+(2.1.2)
+{1, 8, 6, 33} and {7}
+In general the block containing the trivial representation is called the principal
+block. When G = PSL2(F7), we can call the other block the Steinberg block, since
+the only finitely generated indecomposable module that belongs to it is a lattice in
+the Steinberg representation 7.
+2.1.4. Defect groups and a Brauer tree. “Defects” are invariants of blocks.
+A block b has defect zero if there’s only one irreducible Qp[G]-module that be-
+longs to b and every lattice in that irreducible module is projective. For instance,
+the Steinberg block has defect zero. A block of defect zero is categorically and
+homologically boring — its category of modules is the same as the category of
+modules over Zp, or of a division ring over Zp.
+Groups whose p-Sylow is of order p furnish the basic examples of blocks of defect
+one. The principal block of such a group has defect one. For instance the principal
+block of PSL2(F7) (labeled {1, 8, 6, 33} in (2.1.2)) is a block of defect one.
+Brauer observed a “tree” structure in the decomposition matrix of a block of
+defect one.
+The decomposition matrix records how the indecomposable projec-
+tive Zp[G]-modules break up into irreducible Qp[G]-modules. Brauer showed that,
+
+4
+TONY FENG, DAVID TREUMANN, AND ALLEN YUAN
+sticking to modules in a block of defect one, this matrix is the incidence matrix of
+a tree.
+For example, the Brauer tree of the principal block of PSL2(F7) is
+(2.1.3)
+1
+6
+8
+33
+Its vertices are labeled by the irreducible Q7[PSL2(F7)]-modules in the principal
+block. It is natural to label its edges by the indecomposable projective Z7[PSL2(F7)]-
+modules: if P ⊗ Q7 splits up as M ⊕ N, draw an edge between M and N.
+In general the “defect” of a block is more detailed information than a num-
+ber n; it is a conjugacy class of subgroups of G of order pn.
+A simple way to
+characterize the defect group is as follows: the defect group of b is the smallest
+p-subgroup Q ⊂ G such that every module M belonging to b is a direct summand
+of IndG
+Q(a Zp[Q]-module).
+2.1.5. Coefficients. In our running example of PSL2(F7), Brou´e’s conjecture (to be
+introduced below) is easy to check over Z7. But in general we will need to consider
+a larger coefficient system.
+Let O be a finite extension of Zp whose fraction field K has characteristic 0
+and whose residue field k has characteristic p. Sometimes modular representation
+theorists call the triple (K, O, k) a “p-modular system.” We will take a triple of
+the following form:
+• k = Fq contains a primitive nth root of unity whenever G has an element
+of order n which is prime to p.
+• O = Zq is the ring of Witt vectors of k. (This is the minimal extension of
+Zp with residue field Fq.)
+• K = Qq is the fraction field of O.
+Remark 2.1.1. The reduction map O[G] → k[G] always induces a bijection on
+blocks. The hypothesis on Fq has the following additional consequence for blocks:
+that the map Fq[G] → k′[G] induces a bijection on blocks for any extension field
+k′/Fq, and the map Zq[G] → O′[G] induces a bijection on blocks for any extension
+O′ of Zq.
+Remark 2.1.2. A standard additional hypothesis on p-modular systems is that
+O and K contain exp(G)th roots of unity, where exp(G) is the exponent of G —
+sometimes (K, O, k) is called a “splitting p-modular system” when this hypothesis
+holds. We warn that (Qq, Zq, Fq) is usually not a splitting p-modular system in
+this sense, for example not if p is odd and divides the order of |G|, or if p = 2 and
+the 2-Sylow subgroup of G is not elementary abelian. Our reasons for avoiding pth
+roots of unity are explained in §2.4.
+Brou´e’s conjecture, introduced below, concerns blocks of O[G] and k[G] where
+(K, O, k) is a “sufficiently large” p-modular system. Usually, “sufficiently large” at
+least implicitly means “splitting” but we will work with (K, O, k) = (Qq, Zq, Fq),
+where Fq obeys the condition above.
+Though not splitting, the conjecture for
+(Qq, Zq, Fq) implies the conjecture for any larger (K, O, k), and conversely Rickard’s
+refined form of the Brou´e conjecture for Fq [Ric96] implies it for Zq [Ric96, §5].
+2.2. Brou´e’s conjecture. Write Db(Zq[G]b)fg for the bounded derived category
+of finitely generated Zq[G]b-modules. Whenever the defect group D of b is abelian,
+
+G-SPECTRA OF CYCLIC DEFECT
+5
+Brou´e’s conjecture predicts an equivalence of derived categories
+(2.2.1)
+Db(Zq[G]b)fg ∼= Db(Zq[NG(D)]b′)fg
+for some block b′ of the normalizer NG(D) of D. Given b, there is an explicit recipe
+to determine b′, which is called the “Brauer correspondent” of b. In particular, if b
+is the principal block of Zq[G] then b′ is the principal block of Zq[NG(D)].
+The conjecture has been known for a long time for blocks of cyclic defect —
+proved first by Rickard [Ric89] and later by Rouquier [Rou98].
+In case G =
+PSL2(F7), the ring Zq is Z7 itself; the example is pretty typical and Rouquier’s
+equivalence can be described quickly, as follows.
+The normalizer of the defect group of the principal block of G = PSL2(F7) is
+the “Borel” subgroup B ⊂ G, of order 21. The group algebra Z7[B] has only one
+block. This block has defect one and its Brauer tree is
+(2.2.2)
+1′
+B
+1B
+33B
+❈
+❈
+❈
+❈
+❈
+❈
+❈
+❈
+④
+④
+④
+④
+④
+④
+④
+1′′
+B
+The central vertex is another module that would split over a field with √−7, deco-
+rated with a subscript B to distinguish it from the irreducible Q7[G]-module with
+the same property and similar name. The other vertices are 1-dimensional mod-
+ules, two of them nontrivial. The Rouquier equivalence sends a G-module M in the
+principal block to the two-term complex
+(2.2.3)
+ResG
+B(M) → Q ⊗ HomG(P, M)
+In the formula (2.2.3), P and Q are indecomposable projectives, which we can
+specify by indicating their corresponding edges in the Brauer trees:
+1
+6
+P
+8
+33
+1′
+1B
+33B
+❈
+❈
+❈
+❈
+❈
+❈
+❈
+❈
+Q
+④
+④
+④
+④
+④
+④
+④
+④
+1′′
+The differential in (2.2.3) is a natural transformation ResG
+B(−) → Q⊗HomG(P, −).
+A Yoneda/Morita argument identifies the set of such natural transformations with
+a ball in a p-adic vector space, specifically with HomB(ResG
+B P, Q) ∼= Z5
+7. A little
+care is necessary in choosing the differential in this ball: the corresponding homo-
+morphism ResG
+B P → Q must be surjective. This condition is both closed and open
+in the usual 7-adic metric on Z5
+7.
+
+6
+TONY FENG, DAVID TREUMANN, AND ALLEN YUAN
+2.3. Triangulated categories of G-modules. The block decomposition (2.1.1)
+of Zq[G] induces a decomposition of the derived category Db(Zq[G])fg as a direct
+product of triangulated categories
+Db(Zq[G])fg ∼= Db(Zq[G]b1)fg × Db(Zq[G]b2)fg × · · ·
+Each of these categories has a natural dg enrichment. In other words, each of them
+is the homotopy category of a Zq-linear stable ∞-category. Our notation for these
+∞-categories is not the usual one in representation theory: we write LMod(Zq[G]b)ft
+for the Zq-linear stable ∞-category whose homotopy category is Db(Zq[G]b)fg. Let
+us make some comments about this notation, which is developed in §3.2–3.3.
+• LMod stands for “left module spectra” — LMod(R) is the ∞-category of
+left module spectra over the ring spectrum R.
+Every ring in the usual
+sense — i.e., every “discrete ring” — determines a ring spectrum, namely
+its Eilenberg-MacLane spectrum. In this paper we abuse notation and use
+the same symbol for a discrete ring as for its Eilenberg-MacLane spectrum.
+Thus, LMod(Zq[G]b) is the ∞-category of left module spectra over the
+Eilenberg-MacLane spectrum of Zq[G]b.
+• The homotopy category of LMod(Zq[G]b) is equivalent not to Db(Zq[G]b)fg
+but to D(Zq[G]b), the unbounded derived category of Zq[G]b. Its objects
+are complexes that are not required to be bounded, or to obey any other
+finiteness condition.
+• The superscript in LMod(Zq[G]b)ft stands for “finite type.” The notation
+is based on the following characterization of Db(Zq[G]b)fg as a subcategory
+of D(Zq[G]b): it is the full subcategory spanned by those complexes whose
+underlying complex of Zq-modules is bounded and finitely generated in each
+degree.
+2.4. Spectra. Extraordinary cohomology theories (cobordism, K-theory, . . . ) are
+represented by spectra. The category of spectra can be be viewed as a homotopy
+theoretic refinement of the derived category of abelian groups. For instance, it
+is triangulated, and the category of abelian groups embeds inside spectra by the
+Eilenberg-MacLane spectrum construction A �→ H∗(−; A). Any spectrum X has
+homotopy groups, which are the graded abelian group
+π∗X = H−∗(pt; X).
+Moreover, the category of spectra has a symmetric monoidal structure, and so one
+can consider ring spectra and commutative ring spectra, analogously to DGA’s and
+CDGA’s in D(Z).
+In this “higher algebra” of ring spectra, the initial ring spectrum is S, the sphere
+spectrum, whose homotopy groups are the stable homotopy groups of spheres. The
+∞-category of spectra is equivalent LMod(S) — for a commutative ring spectrum
+such as S, we usually drop the L and write Mod(S) := LMod(S). In the body of
+the paper we will assume a greater familiarity with spectra and with ∞-categories;
+here in the introduction we will make some basic comments:
+(1) The map g : S → Z induces an isomorphism in homotopy groups in non-
+positive degrees, and the positive degree homotopy groups of S are the
+well-studied stable homotopy groups of spheres. These groups form a graded
+commutative ring under composition, elements in positive degree are known
+to be torsion [Ser53] and nilpotent [Nis73].
+
+G-SPECTRA OF CYCLIC DEFECT
+7
+(2) For each prime p there is a natural p-completion of S, another commutative
+ring spectrum denoted Sp. Its homotopy groups are the homotopy groups
+of S, tensored with Zp.
+(3) When n is prime to p, there is a natural construction that “adjoins nth
+roots of unity to Sp”, e.g. [Lur18, Ex. 5.2.7]. We will denote the result by
+Sq, where q is the cardinality of the field obtained by adjoining nth roots of
+unity to Fp. Its homotopy groups are the homotopy groups of S, tensored
+with Zp(
+n√
+1). On the other hand, when p is odd, it is not possible to adjoin
+pth roots of unity to Sp [LN14, §A.6.iii], and for p = 2, it is not possible to
+adjoin 4th roots of unity to S2 [SVW99].
+2.4.1. Relation to classical algebra. Regarding Z as a ring spectrum, Mod(Z) is a
+stable ∞-category whose homotopy category is D(Z). Algebra over S and over Z
+is related by the unique homotopy class of ring spectrum maps g : S → Z and the
+induced extension and restriction of scalar functors
+g∗ : Mod(S) → Mod(Z)
+g∗ : Mod(Z) → Mod(S)
+The map g : S → Z can be thought of as a nilpotent thickening: it is an isomorphism
+in non-positive degrees, and its kernel on homotopy groups consists of nilpotent
+torsion elements. But these extra elements in S yield some striking consequences:
+(1) Commutative algebras over Sp admit a natural Frobenius [NS18, §IV.1].
+(2) There is a natural chromatic filtration on Sp:
+Sp = lim
+←−
+n
+LnSp
+which interpolates between mixed characteristic phenomena over Sp and
+characteristic zero phenomena over L0Sp = Qp.
+The above Frobenius
+“moves filtration by one” in an appropriate sense.
+These features have seen recent application in the study of mixed characteristic
+phenomena. For instance, the Frobenius of (1) underlies Bhatt–Morrow–Scholze’s
+work on p-adic Hodge theory [BMS19], and (2) has led to the construction of some
+new quantum groups by Yang–Zhao [YZ21], realizing earlier character formulas of
+Lusztig [Lus89, Lus15]. We hope that studying analogues of Brou´e’s conjecture
+over Sp may shed light on the original form of the conjecture.
+2.5. G-spectra. Morally, a G-spectrum is a spectrum with an action of the group
+G.
+G-spectra ought to represent G-equivariant cohomology theories.
+Algebraic
+topologists have a few inequivalent ways of modeling them. In this paper we deal
+with “Borel equivariant” G-spectra, which in ∞-categorical language have a simple
+definition: a Borel G-spectrum is a functor to Mod(S) from the classifying space
+BG of G. The ∞-category of Borel G-spectra is Fun(BG, Mod(S)).
+The ∞-category of Borel G-spectra is an ∞-category of module spectra:
+Fun(BG, Mod(S)) ∼= LMod(S[G])
+Here, S[G] is the “group algebra of G over S” whose underlying spectrum is �
+g∈G S.
+We similarly have the variants
+Fun(BG, Mod(Sp)) ∼= LMod(Sp[G])
+(2.5.1)
+Fun(BG, Mod(Sq)) ∼= LMod(Sq[G])
+(2.5.2)
+
+8
+TONY FENG, DAVID TREUMANN, AND ALLEN YUAN
+where we first p-complete and then adjoint appropriate roots. The group algebra
+Sq[G], like Zq[G], splits into block algebras:
+Sq[G] = Sq[G]b1 × Sq[G]b2 × · · ·
+The blocks of Sq[G] naturally correspond to those of Zq[G], since π0(Sq[G]b) ∼=
+Zq[G]b. The category LMod(Sq[G]) — and LMod(Sq[G])ft, see below — splits up
+in the same way:
+(2.5.3)
+LMod(Sq[G])
+∼= LMod(Sq[G]b1)
+× LMod(Sq[G]b2)
+× · · ·
+LMod(Sq[G])ft ∼= LMod(Sq[G]b1)ft × LMod(Sq[G]b2)ft × · · ·
+Let’s discuss LMod(Sq[G])ft, which we propose as a spectral analog of Db(Zq[G])fg.
+Let Mod(Sq)ω denote the full subcategory of Mod(Sq) spanned by compact objects
+§3.2. Then LMod(Sq[G])ft is the full subcategory of LMod(Sq[G]) spanned by mod-
+ules whose underlying Sq-module is compact. Similar to (2.5.1) we have
+Fun(BG, Mod(Sq)ω) ∼= LMod(Sq[G])ft
+In a way LMod(Sq[G])ft is a difficult category to work with, because it is not
+known whether it has a finite set of generators. (An alternative, LMod(Sq[G])ω,
+is by definition generated by Sq[G]). Nevertheless we are able to prove some cases
+(the cyclic defect case) of the obvious LModft-analog of Brou´e’s conjecture:
+2.6. Brou´e’s Conjecture for G-spectra. Since Sq[G] and Zq[G] have the same
+blocks, each block of defect D of Sq[G] has a Brauer corresponding block of Sq[NG(D)]:
+if Zq[NG(D)]b′ is the Brauer correspondent of Zq[G]b, then Sq[NG(D)]b′ is the
+Brauer correspondent of Sq[G]b. The natural analog of Brou´e’s conjecture for G-
+spectra would be a positive answer to the following question:
+Question. Suppose b is a block of Sq[G] with abelian defect group D ⊂ G, and
+that b′ is the corresponding block of Sq[NG(D)]. Then there is an equivalence of
+stable ∞-categories
+LMod(Sq[G]b) ∼= LMod(Sq[NG(D)]b′)
+which restricts to an equivalence between the full subcategories LMod(Sq[G]b)ft
+and LMod(Sq[NG(D)]b′)ft. (This last statement about finite type subcategories is
+in fact automatic, by Proposition 4.6.1.)
+We are not quite bold enough to state this as a conjecture, but we prove the
+following as Theorem 4.7.2, answering the Question in the case where the defect
+group is cyclic.
+Theorem. Suppose b is a block of Sq[G] with defect D ⊂ G, and that b′ is the
+corresponding block of Sq[NG(D)]. If D is cyclic, then there is an equivalence of
+stable ∞-categories
+LMod(Sq[G]b) ∼= LMod(Sq[NG(D)]b′)
+which restricts to an equivalence between the full subcategories LMod(Sq[G]b)ft
+and LMod(Sq[NG(D)]b′)ft.
+
+G-SPECTRA OF CYCLIC DEFECT
+9
+2.7. Rickard vs. Rouquier. There are two old proofs of Brou´e’s conjecture for
+blocks of cyclic defect, one by Rickard [Ric89] and one by Rouquier [Rou98]. Our
+proof is an adaptation of Rouquier’s argument.
+Rickard’s proof. The Brauer tree of a block of cyclic defect has a little bit of
+extra structure: a ribbon structure and a distinguished vertex labeled by an integer
+called its “multiplicity.”
+In the case of PSL2(F7) (2.1.3) or its Borel subgroup
+(2.2.2), the ribbon structure is the embedding in the page, the distinguished vertex
+is 33, and the multiplicity is 2.
+Very few trees come from blocks. But given any tree with these decorations,
+Rickard defined by generators and relations an associative algebra that is Morita
+equivalent to a block when the tree is the Brauer tree of that block. Then, he
+proved that any two of these tree algebras have the same derived category, as long
+as the trees have the same number of edges and the integer called “multiplicity”
+is the same. An elementary argument shows that these numbers match for Brauer
+corresponding blocks of G and NG(D).
+In higher algebra, it takes more work than “generators and relations” to define
+an associative ring spectrum. For this reason we have not generalized Rickard’s
+proof, but it might be possible and interesting to do so.
+Rouquier’s proof. Morita theory tells us that functors between module categories
+can be given by bimodules.
+When D is cyclic, Rouquier found a very explicit
+(G, NG(D))-bimodule that gives Brou´e’s equivalence.
+This bimodule and its inverse bimodule have a simple structure: they are two-
+term complexes of summands of permutation G × NG(D)op-modules (the inverse
+is a two-term complex of summands of permutation NG(D) × Gop-modules), one
+term of which is projective. Because of this simple structure, it is easy to find
+Sp-versions of these bimodules.
+More generally, Brou´e’s conjecture is known for many noncyclic defect groups,
+usually by constructing bimodules. But at present we do not know how to lift these
+known bimodules to Sp; it seems to be a difficult problem.
+3. Associative algebras, modules, and bimodules in spectra
+In this section, we review some basic Morita theory over the spectrum analog
+of a finite-dimensional algebra.
+The relevant material is developed in detail in
+[Lur16, Ch. 4], which we draw heavily from and we refer the reader to for further
+details. In §3.1, we review some basic notations from the theory of ∞-categories.
+In §3.2–§3.6, we discuss the theory of bimodules and its interactions with two
+finiteness conditions: “compact” modules and “finite type” modules. In §3.7 we
+prove the main theorem of this section, which will be used in §4 to lift certain
+Z-linear equivalences of ∞-categories to S-linear equivalences.
+In §3.10–§3.12, we discuss K(n)-local algebras. When A and B are group algebra
+over a K(n)-local algebra k, the theory of ambidexterity provides a larger class a
+functors LMod(A)ft → LMod(B)ft. This material is not used in our proof of §2.6.
+3.1. ∞-categorical notation. We write S for the ∞-category of spaces, S for the
+sphere spectrum, and Mod(S) for the ∞-category of spectra. If x and y are objects
+of the ∞-category C, we write Maps(x, y) ∈ S for the space of maps between them,
+and [x, y] for π0Maps(x, y). If C is stable, then we write Maps(x, y) for the cor-
+responding mapping spectrum, whose associated infinite loop space is Maps(x, y).
+We write Σ for the suspension functor in a stable ∞-category.
+
+10
+TONY FENG, DAVID TREUMANN, AND ALLEN YUAN
+We write Fun(C, D) for the ∞-category of functors between ∞-categories C and
+D. The full subcategory spanned by functors that have right adjoints is FunL(C, D).
+If C and D are stable, the full subcategory spanned by functors that preserve finite
+limits and colimits is Funex(C, D).
+We write BG for the classifying space of a finite group G, which we regard as
+equipped with a natural basepoint pt → BG. If G is a finite group and c is an
+object of the ∞-category C, then an action of G on c is by definition a functor
+BG → C whose value at the basepoint is c. We write
+chG := lim
+←−
+BG
+c
+chG := lim
+−→
+BG
+c
+for the homotopy fixed points and homotopy quotient of a G-object c, when these
+limits and colimits exist in C.
+If C = Mod(S), there is a more sophisticated notion of a G-object in C: the
+notion of a “genuine” G-spectrum. This paper mostly does not touch the genuine
+theory except in §4.4.
+3.2. Finiteness conditions for modules over a commutative ring spec-
+trum. In this paper, a commutative ring spectrum is an E∞-algebra object in
+Mod(S). Let k be a commutative ring spectrum. Write (Mod(k), ⊗k) for the sym-
+metric monoidal stable ∞-category of k-module spectra. The mapping spectrum
+Maps(M, N) attached to two objects of Mod(k) has a natural k-module structure.
+The following conditions are equivalent in Mod(k) [Lur16, Proposition 7.2.4.4]:
+(1) M is compact, i.e., the functor Maps(M, −) : Mod(k) → S commutes with
+filtered colimits.
+(2) Maps(M, −), regarded either as a functor Mod(k) → Mod(S) or Mod(k) →
+Mod(k), commutes with direct sums. [Lur09, Prop. 15.1]
+(3) M is perfect i.e., it is a summand of an object which carries a finite filtration
+whose subquotients have the form Σnk.
+(4) M is dualizable, i.e., there is a second object M ∗ and a pair of maps k →
+M ∗ ⊗k M and M ⊗k M ∗ → k such that the composite
+M = M ⊗k k → M ⊗k (M ∗ ⊗k M) = (M ⊗k M ∗) ⊗k M → k ⊗k M = M
+is an isomorphism.
+We will write Mod(k)ω ⊂ Mod(k) for the full subcategory spanned by the com-
+pact objects.
+For example, when k = Z, the homotopy category of Mod(Z) is
+the “traditional” unbounded derived category of abelian groups, and the homotopy
+category of Mod(Z)ω is the full subcategory spanned by bounded complexes with
+finitely generated homology groups.
+Let Catex
+∞ denote the ∞-category of small stable ∞-categories and exact func-
+tors. Then Catex
+∞ admits a canonical symmetric monoidal structure ⊗, which comes
+equipped with a universal map C × D → C ⊗ D which commutes with finite colimits
+separately in each variable. We also have reason to consider the “large” setting of
+presentable ∞-categories. The ∞-category PrL of presentable ∞-categories also ad-
+mits a symmetric monoidal structure ⊗, which commutes with (arbitrary) colimits
+separately in each variable [Lur16, Corollary 4.8.1.4].
+Example 3.2.1. We have Mod(k)ω ∈ Catex
+∞ (as well as LMod(A)ω, LMod(A)ft to
+be introduced in the following section) and
+Mod(k) ∼= Ind(Mod(k)ω) ∈ PrL.
+
+G-SPECTRA OF CYCLIC DEFECT
+11
+More generally, the Ind-category construction determines a symmetric monoidal
+functor Ind : Catex
+∞ → PrL.
+We will call a stable ∞-category C ∈ Catex
+∞ (resp.
+C ∈ PrL) k-linear when
+it is endowed with an action of the algebra Mod(k)ω ∈ Catex
+∞ (resp. Mod(k) ∈
+PrL) (cf. [Lur11, Def. 6.2]). We denote the bifunctor Mod(k)ω ⊗ C → C (resp.
+Mod(k) ⊗ C → C) by ⊗k. The mapping spectrum Maps(x, y) between objects of a
+k-linear category has a k-module structure, which is not always perfect but which
+represents the functor
+Maps(− ⊗k x, y) : Mod(k)ω,op → S.
+If C and D are k-linear categories, we write Funex
+k (C, D) for the ∞-category of exact
+functors that preserve this action.
+3.3. Finiteness conditions on associative algebra spectra and their mod-
+ules. For a commutative ring spectrum k, a k-algebra spectrum will mean an as-
+sociative k-algebra spectrum — that is, an E1-algebra over k. We write LMod(A)
+for the ∞-category of left A-module spectra. It is presentable, stable, and has a
+k-linear structure.
+We write RMod(A) and Bimod(A, B) for the ∞-categories of right A-modules
+and of (A, B)-bimodules respectively. The theory of right modules and bimodules
+is related to the theory of left modules via the natural equivalences RMod(A) ∼=
+LMod(Aop) and Bimod(A, B) ∼= LMod(A ⊗k Bop) [Lur16, Proposition 4.3.2.7,
+Proposition 4.6.3.11].
+For an object M ∈ LMod(A), the following are equivalent [Lur16, Proposition
+7.2.4.4]:
+(1) Maps(M, −) : LMod(A) → S commutes with filtered colimits.
+(2) Maps(M, −), regarded either as a functor LMod(A) → Mod(k) or LMod(A) →
+Mod(S), commutes with direct sums [Lur09, Prop. 15.1].
+(3) M is a summand of an object that has a finite filtration whose associated
+graded pieces have the form ΣnA.
+A module obeying these conditions is called perfect. We denote the full subcategory
+spanned by perfect left A-modules by LMod(A)ω.
+If A is perfect as a k-module, then it is natural to consider a weaker finiteness
+condition: we say that M ∈ LMod(A) has finite type if its underlying k-module
+belongs to Mod(k)ω. Write LMod(A)ft for the full subcategory spanned by finite
+type A-modules. We have a containment
+LMod(A)ω ⊂ LMod(A)ft if and only if A ∈ Mod(k)ω.
+Remark 3.3.1. If k = Z and A is an associative ring, the homotopy category of
+LMod(A)ω coincides with the category of bounded complexes of projective modules
+and chain homotopy classes of maps between them. For such an A, the condition
+that the underlying additive group of A is finitely generated is equivalent to the
+condition that A ∈ Mod(k)ω, in which case the homotopy category of LMod(A)ft
+is the same as Db(f.g. left A-modules).
+3.4. Projective A-module spectra. We call an object of LMod(A) a free module
+if it is isomorphic to a direct sum of objects of the form ΣdA. Write Free(A) ⊂
+
+12
+TONY FENG, DAVID TREUMANN, AND ALLEN YUAN
+LMod(A) and Free(A)ω ⊂ LMod(A)ω for the full subcategories spanned by free
+modules.
+We call an object of LMod(A) a projective module if it is a direct summand of
+a free module. Write Proj(A) ⊂ LMod(A) and Proj(A)ω ⊂ LMod(A)ω for the full
+subcategories spanned by projective modules.
+The free modules ΣdA represent the functor πd : hLMod(A) → Ab where the
+domain is homotopy category of LMod(A) and the codomain is the usual 1-category
+of abelian groups. Furthermore, for any M ∈ LMod(A), the graded abelian group
+π∗(M) := �
+i∈Z πi(M) has the structure of a graded π∗(A)-module, and the natural
+map
+(3.4.1)
+[ΣdA, M] → Homπ∗(A)(Σdπ∗(A), π∗(M))
+is an isomorphism. The codomain in (3.4.1) denotes the set of homomorphisms of
+graded abelian groups that are compatible with the grading and the π∗(A)-module
+structure, and Σdπ∗(A) denotes a grading shift. We record two consequences of
+this observation:
+Proposition 3.4.1. Let ¯P be a graded abelian group equipped with a left π∗(A)-
+module structure. If ¯P is projective, then there is an object of Proj(A) such that
+π∗(P) ∼= ¯P as graded π∗(A)-modules.
+Proof. First suppose that ¯P = ¯F is free: ¯F ∼=
+�
+i∈I Σdiπ∗(A). Then we may take
+P = F = �
+i∈I ΣdiA.
+In general, ¯P is the image of an idempotent endomorphism e : ¯F → ¯F. The
+isomorphism (3.4.1) shows this lifts to an idempotent endomorphism of F, and
+idempotents in LMod(A) split by [Lur16, Lemma 1.2.4.6].
+□
+Proposition 3.4.2. Let P ∈ Proj(A) and M ∈ LMod(A). Then the map
+(3.4.2)
+[P, M] → Homπ∗(A)(π∗(P), π∗(M))
+is an isomorphism.
+Note one consequence of Proposition 3.4.2 is that the lift in Proposition 3.4.1 of
+¯P to P is unique up to isomorphism.
+Proof. This is a weaker version of [Lur16, Corollary 7.2.2.19]; it can be proved
+as follows.
+Since P is a summand of a free module F = �
+i ΣdiA, the do-
+main of (3.4.2) is a retract of �
+i[ΣdiA, M] and the codomain is a retract of
+�
+i Homπ∗(A)(Σdiπ∗(A), π∗(M)).
+The retractions that are induced by an idem-
+potent endomorphism of F commute with the maps (3.4.2) for P and for F. Since
+the retract of an isomorphism is an isomorphism, we are reduced to proving it for
+free modules, and then further reduced to the case P = ΣdA, which is (3.4.1).
+□
+3.5. Finiteness conditions for functors. Let A and B be k-algebra spectra.
+Then LMod(A) and LMod(B) are k-linear presentable stable ∞-categories, and we
+can consider k-linear exact functors between them. For such a functor, the following
+are equivalent:
+(1) F : LMod(A) → LMod(B) preserves filtered colimits.
+(2) F : LMod(A) → LMod(B) preserves all colimits.
+(3) F : LMod(A) → LMod(B) preserves (arbitrary) direct sums.
+(4) F has a right adjoint.
+
+G-SPECTRA OF CYCLIC DEFECT
+13
+We write FunL
+k (LMod(A), LMod(B)) ⊂ Funex
+k (LMod(A), LMod(B)) for the full
+subcategory spanned by colimit-preserving functors.
+If F is any such functor, then F(A) supports a (B, A)-bimodule structure, F is
+isomorphic to F(A) ⊗A −, and
+(3.5.1)
+FunL
+k (LMod(A), LMod(B)) ∼= LMod(B ⊗k Aop)
+[Lur16, Proposition 4.8.4.1]. Furthermore, the restriction to LMod(A)ω induces an
+equivalence:
+(3.5.2)
+FunL
+k (LMod(A), LMod(B))
+∼
+−→ Funex
+k (LMod(A)ω, LMod(B)).
+Proposition 3.5.1. Let A and B be k-algebra spectra and let F be a colimit-
+preserving k-linear functor LMod(A) → LMod(B). The following are equivalent:
+(1) F(A) is perfect as a left B-module
+(2) F carries LMod(A)ω into LMod(B)ω
+In other words after (3.5.1) we have
+(3.5.3)
+Funex
+k (LMod(A)ω, LMod(B)ω) ∼=
+
+
+
+
+full subcategory of
+Bimod(B, A) spanned
+by bimodules that are
+perfect as left B-modules
+
+
+
+
+Proof. Condition (2) implies condition (1) because A is an object of LMod(A)ω. If
+M ∈ LMod(A)ω has a filtration
+0 → M≤0 → M≤1 → · · · → M≤n = M
+with M≤i/M≤i−1 ∼= ΣdiA, then F(M) has a filtration whose subquotients are iso-
+morphic to ΣdiF(A). If M ′ is a summand of such an M then F(M ′) is a summand
+of F(M). Since F(A) is perfect so is ΣdiF(A), therefore so is F(M), and therefore
+so is F(M ′) — this shows that (1) implies (2).
+□
+Proposition 3.5.2. Let A and B be k-algebra spectra and let F be a k-linear
+colimit-preserving functor LMod(A) → LMod(B). Suppose that F(A) is perfect as
+a right A-module and as a k-module. Then F carries LMod(A)ft into LMod(B)ft.
+Remark 3.5.3. When we consider LMod(A)ft, it will typically be the case that A
+is perfect as a k-module — in that case, when F(A) is perfect as a right A-module
+it is automatically perfect as a k-module.
+Proof. Since F(A) ⊗A M belongs to LMod(B)ft exactly when the underlying k-
+module belongs to Mod(k)ω, it suffices to show that the bifunctor (−) ⊗A (−) :
+RMod(A) × LMod(A) → Mod(k) carries RMod(A)ω × LMod(A)ft into Mod(k)ω.
+Let N ∈ RMod(A)ω and let M ∈ LMod(A)ft. Fix a filtration
+0 → N≤0 → · · · → N≤n = N
+such that N≤i/N≤i−1 is a suspension of A (as a right A-module). Since ΣdiA ⊗A
+M = ΣdiM, and M is perfect as a k-module, it follows that N ⊗A M is perfect as
+a k-module. If N ′ is a summand of N then N ′ ⊗A M is a summand of N ⊗A M so
+it is also perfect as a k-module.
+□
+
+14
+TONY FENG, DAVID TREUMANN, AND ALLEN YUAN
+Remark 3.5.4. It is tempting to guess that there are weaker hypotheses than
+those of Prop. 3.5.2 that would guarantee that F carries LMod(A)ft to LMod(B)ft.
+The following example shows some limitations — at least, that it can fail even
+when F(A) is perfect or finite type as a left B-module. Let A = k[G], B = k, and
+let M = k be the trivial (B, A)-bimodule. Tensoring with M is isomorphic to the
+colimit-preserving functor
+(−)hG : LMod(k[G]) → LMod(k),
+i.e. to homotopy G-coinvariants. The functor carries the trivial module to the k-
+homology of BG, which often does not lie in LMod(k)ω = LMod(k)ft. For example,
+the k-homology of BG is not perfect when G is finite and nontrivial and k = S or
+Z, and if p divides the order of G then it is not perfect when k = Sp, Zp, or Fp.
+Thus, it does not carry LMod(k[G])ft into LMod(k)ft, even though M is perfect
+and finite type as a left B-module. (Meanwhile, note that since M is not perfect
+as a right A-module, it also does not carry LMod(k[G])ω into LMod(k)ω.)
+For some values of k, the Greenlees-Sadofsky “Tate vanishing” or Hopkins-Lurie
+“ambidexterity” can repair this issue in a significant way §3.12.
+3.6. Finiteness conditions for the right adjoint functor. Let F : LMod(A) →
+LMod(B) be a colimit-preserving k-linear functor, so that it has a k-linear right
+adjoint G. Just as F(M) ∼= F(A) ⊗A M for F(A) endowed with its natural (B, A)-
+bimodule structure (3.5.1), G(M) is given by the formula
+(3.6.1)
+G(M) = MapsLMod(B)(F(A), M)
+The right A-module structure of F(A) induces a left A-module structure on (3.6.1).
+Note in particular that G(B) ∼= MapsLMod(B)(F(A), B).
+Proposition 3.6.1. Let A and B be k-algebra spectra, and let F : LMod(A) →
+LMod(B) be a colimit-preserving k-linear functor. Let G : LMod(B) → LMod(A)
+be its right adjoint. The following are equivalent:
+(1) F(A) is perfect as a left B-module
+(2) F carries LMod(A)ω into LMod(B)ω
+(3) G preserves colimits
+(4) There is a natural isomorphism G(M) ∼= G(B) ⊗B M
+Proof. (1) and (2) are equivalent by Prop. 3.5.1. (1) and (3) are equivalent by
+(3.6.1), the fact that colimits in LMod(B) are computed in Mod(k), and §3.3. (3)
+and (4) are equivalent by §3.5.
+□
+As a corollary we have the following criterion for a colimit-preserving k-linear
+functor to induce a pair of adjoint functors between LMod(A)ft and LMod(B)ft:
+Proposition 3.6.2. Suppose that A and B are perfect over k, let F : LMod(A) →
+LMod(B) be a colimit-preserving k-linear functor, and let G be its right adjoint.
+Suppose that F(A) is perfect as a left B-module and as a right A-module. Then F
+carries LMod(A)ft into LMod(B)ft and G carries LMod(B)ft into LMod(A)ft.
+Proof. Since F(A) is perfect as a right A-module (and since A is perfect over k), F
+carries LMod(A)ft into LMod(B)ft by Prop. 3.5.2. Since F(A) is perfect as a left
+B-module, the adjoint G is colimit-preserving by Proposition 3.6.1, and we may
+detect whether G carries LMod(B)ft into LMod(A)ft by applying Prop. 3.5.2 to
+G(B) = MapsLMod(B)(F(A), B)
+
+G-SPECTRA OF CYCLIC DEFECT
+15
+But this is F(A)∨, which is perfect as a right B-module by Prop. 3.8.1.
+□
+Remark 3.6.3. Not every pair of adjoint functors LMod(A)ft ⇆ LMod(B)ft ex-
+tends to a colimit-preserving functor LMod(A) → LMod(B). For example when G
+is a commutative p-group, [Tre15, §3.6] constructs self-equivalences of LMod(KUp[G])ft
+that exchange the trivial module KUp and the free module KUp[G]
+KUp ↔ KUp[G]
+Since the trivial module is not perfect, the equivalence does not preserve the subcat-
+egory LMod(KUp[G])ω and does not extend to a self-equivalence of LMod(KUp[G]).
+Another consequence of Prop. 3.6.1 is that any k-linear functor LMod(A)ω →
+LMod(B)ω extends to a colimit-preserving functor LMod(A) → LMod(B) whose
+right adjoint is also colimit-preserving. We will want a criterion for this right adjoint
+to carry LMod(B)ω back into LMod(A)ω. According to Prop. 3.5.1, a necessary
+and sufficient condition is that
+(3.6.2)
+G(B) = MapsLMod(B)(F(A), B) is perfect as a left A-module
+It is perhaps hard to tell at a glance whether this is the case—for instance to give
+a criterion in terms of the right A-module structure on F(A). We can give a useful
+criterion like that when A and B are both “symmetric”; this will be discussed in
+§3.9.
+3.7. Change of rings. Let k continue to denote a commutative ring spectrum
+(i.e. what it has been denoting since §3.2). Suppose we have a second commutative
+ring spectrum k′, and a map u : k → k′. Then u induces a symmetric monoidal,
+colimit-preserving functor
+(3.7.1)
+u∗ : Mod(k) → Mod(k′)
+u∗(M) := k′ ⊗k M
+that carries Mod(k)ω into Mod(k′)ω.
+We also denote by u∗ the functor induced by (3.7.1) from algebras in Mod(k) to
+algebras in Mod(k′). If A is a k-algebra spectrum and A′ := u∗(A) is the induced
+k′-algebra spectrum, LMod(A) and LMod(A′) are related via the formula in PrL
+(3.7.2)
+LMod(A′) ∼= Mod(k′) ⊗Mod(k) LMod(A).
+It follows that a k-linear equivalence between LMod(A) and LMod(B) induces
+a k′-linear equivalence between LMod(A′) and LMod(B′), where B is a second
+k-algebra spectrum and B′ := u∗(B).
+Remark 3.7.1. We can also deduce an equivalence LMod(A′)ω ∼= LMod(B′)ω
+whenever LMod(A)ω ∼= LMod(B)ω. But an equivalence LMod(A)ft ∼= LMod(B)ft
+may not induce an equivalence LMod(A′)ft ≇ LMod(B′)ft. The self-equivalences
+in Remark 3.6.3 provide an example with k = KUp and k′ = Qp[β, β−1], and
+the map k → k′ being the Chern character. If G is a commutative p-group then
+LMod(k′[G])ft = LModω(k′[G]) is semisimple and no self-equivalence of it can ex-
+change the trivial representation for the regular representation.
+We would like to study the converse problem: given a k′-linear equivalence
+LMod(A′) ∼= LMod(B′), can we conclude that LMod(A) ∼= LMod(B)?
+
+16
+TONY FENG, DAVID TREUMANN, AND ALLEN YUAN
+Proposition 3.7.2. Let A and B be k-algebras which are perfect over k. Let F(A)
+be a (B, A)-bimodule which is perfect separately as a left B-module and as a right
+A-module. Let u : k → k′ be a commutative k-algebra and put A′ := u∗(A) and
+B′ = u∗(B). If
+(3.7.3)
+u∗ : Mod(k)ω → Mod(k′)ω is conservative
+and the induced functor
+(3.7.4)
+F ′ : LMod(A′) → LMod(B′)
+is an equivalence of k′-linear ∞-categories, then LMod(A) → LMod(B) is an equiv-
+alence of k-linear ∞-categories.
+Remark 3.7.3. The hypothesis (3.7.3) applies when k is a discrete local ring and
+u : k → k′ is the quotient by the maximal ideal. More generally if k and k′ are
+discrete rings, then (3.7.3) holds if and only if the image of Spec (k′) → Spec (k)
+contains all the closed points.
+Hypothesis (3.7.3) also applies to the truncation map S → Z: this is one formu-
+lation of the Whitehead theorem for homology groups.
+Proof. After Prop. 3.6.1 and (3.6.2) the right adjoint functor to F(A) ⊗A − is
+Maps(F(A), B) ⊗B −.
+The counit for the adjunction is induced by a map
+(3.7.5)
+F(A) ⊗A Maps(F(A), B) → B
+of (B, B)-bimodules, and the unit is induced by a map of (A, A)-modules
+(3.7.6)
+A → Maps(F(A), B) ⊗B F(A)
+The induced k′-linear functors between LMod(A′) and LMod(B′) (F ′ (3.7.4) and
+its adjoint G′) are isomorphic to
+u∗F(A) ⊗A′ −
+and
+u∗(Maps(F(A), B)) ⊗B′ −
+By assumption, F ′ is an equivalence. Its inverse equivalence must be G′ and the
+maps
+u∗F(A) ⊗A′ u∗(Maps(F(A), B)) → B′
+A′ → u∗Maps(F(A), B) ⊗B′ u∗F(A)
+are isomorphisms of bimodules, and since u∗ is a symmetric monoidal functor so
+are
+u∗(F(A) ⊗A Maps(F(A), B)) → B′
+A′ → u∗(Maps(F(A), B) ⊗B F(A))
+Then (3.7.5) and (3.7.6) are isomorphisms by (3.7.3).
+□
+3.8. Duality. There are two dualities between Bimod(A, B) and Bimod(B, A),
+“left” and “right” [Lur16, §4.6.2, §4.6.4]. In this section we review these concepts in
+the special case B = k (dualities between LMod(A) ∼= Bimod(A, k) and RMod(A) ∼=
+Bimod(k, A)) and make some remarks.
+
+G-SPECTRA OF CYCLIC DEFECT
+17
+3.8.1. Duality of perfect A-modules. For any k-algebra spectrum A and any left A-
+module M, the (A, A)-bimodule structure on A endows MapsLMod(A)(M, A) with
+the structure of a right A-module.
+We denote it by M ∨; the construction is a
+contravariant functor
+LMod(A)op → RMod(A) : M �→ M ∨
+Proposition 3.8.1. For any k-algebra spectrum A, the functor LMod(A)op →
+RMod(A) : M �→ M ∨ restricts to an equivalence
+(LMod(A)ω)op ∼= RMod(A)ω
+Proof. If M ∈ LMod(A) has a filtration
+0 → M≤0 → M≤1 → · · · → M≤n = M
+by left A-modules, such that M≤i/M≤i−1 ∼= ΣdiA for each i, then M ∨ has a
+filtration by right A-modules:
+(M/M≤n)∨ → · · · (M/M≤1)∨ → (M/M≤0)∨ → M ∨
+whose graded pieces have the form Σ−diA∨. Since A∨ ∼= A, it follows that M ∨ is
+perfect as a right A-module, and so is any summand of M ∨. Thus (−)∨ carries
+(LMod(A)ω)op into RMod(A)ω. The composite
+RMod(A)ω = LMod(Aop)ω
+(−)∨
+−−−→ (RMod(Aop)ω)op = (LMod(A)ω)op)
+is the inverse functor.
+□
+3.8.2. Duality of k-modules. If M ∈ LMod(A)ft, then M ∗ (the monoidal dual of M
+regarded as a k-module, notation as in §3.2) has the structure of a right A-module
+spectrum. If N is a second A-module spectrum of finite type, then [Lur16, Prop.
+4.6.2.1] the natural map
+(3.8.1)
+MapsLMod(A)(M, N) → MapsRMod(A)(N ∗, M ∗)
+is an equivalence of k-module spectra. In fact we have a commutative square:
+(3.8.2)
+LMod(A)ftop
+∼
+=
+�
+forget
+�
+RModft(A)
+forget
+�
+(Mod(k)ω)op
+∼
+=
+� Mod(k)ω
+where the horizontal maps are given by (−)∗.
+Proposition 3.8.2. Suppose A is perfect over k and that A∗ is perfect as a
+right A-module. Then (3.8.2) is a fully faithful embedding of (LMod(A)ω)op into
+RMod(A)ω. If A∗ is also perfect as a left A-module, then (3.8.2) restricts to an
+equivalence (LMod(A)ω)op ∼= RMod(A)ω.
+Remark 3.8.3. The example in §3.8.3 shows the hypothesis on A∗ cannot be
+removed. The hypothesis is easy to verify for symmetric algebras.
+Proof. If M ∈ LMod(A)ω has a filtration
+0 → M≤0 → M≤1 → · · · → M≤n = M
+
+18
+TONY FENG, DAVID TREUMANN, AND ALLEN YUAN
+by left A-modules, such that M≤i/M≤i−1 ∼= ΣdiA for each i, then M ∗ has a filtra-
+tion by right A-modules:
+(M/M≤n)∗ → · · · (M/M≤1)∗ → (M/M≤0)∗ → M ∗
+whose graded pieces have the form Σ−diA∗. By assumption, A∗ is perfect as a right
+A-module and therefore M ∗ is as well, and so is any summand of M ∗.
+The functor is fully faithful by (3.8.1). If A∗ is perfect as a left A-module, the
+(Aop)∗ is perfect as a right (Aop)-module and the same argument, together with
+M ∼= (M ∗)∗ when M is a perfect k-module, shows that RModω(A)op → LMod(A)ω
+is the inverse equivalence.
+□
+3.8.3. Example. Suppose A ∈ Mod(k)ω, i.e. that LMod(A)ω ⊂ LMod(A)ft. The
+equivalence
+(LMod(A)ft)op ∼= RModft(A)
+does not always carry (LMod(A)ω)op into RMod(A)ω. For instance, consider the
+case where k is a field and A is the four-dimensional algebra with basis e1, e2, f, ǫ,
+where e1 and e2 are orthogonal idempotents and
+e1fe2 = f
+e2ǫe2 = 0
+ǫf = 0
+ǫ2 = 0
+Left modules over this ring are representations of the quiver
+•
+f
+� •
+ǫ
+�
+subject to ǫ2 = 0 and ǫf = 0
+while right modules are representation of the quiver
+•
+•
+f T
+�
+ǫT
+�
+subject to (ǫT )2 = 0 and f T ǫT = 0
+The representation
+k
+=
+� k
+0
+�
+is projective but its dual
+k
+k
+=
+�
+0
+�
+is not, nor does it have a finite projective resolution.
+3.9. Symmetric structures. In the literature on derived equivalences between
+group algebras and their block algebras, the natural “symmetric structures” on
+these algebras is sometimes used, e.g. [Rou01]. The symmetric structure has some
+pleasant consequences for the right adjoint of a functor Db(A) → Db(B) that is
+given by a complex of (B, A)-bimodules. In this section we explain the analog of
+these consequences for algebra spectra.
+Definition 3.9.1. Let B be a k-algebra spectrum that is perfect as a k-module.
+A symmetric structure on B is an isomorphism of (B, B)-bimodules B ∼= B∗.
+Let F : LMod(A) → LMod(B) be a colimit-preserving k-linear functor, i.e.
+F(M) = F(A) ⊗A M.
+We have already seen that F carries LMod(A)ω into
+LMod(B)ω if and only if the (B, A)-bimodule F(A) is perfect as a left B-module.
+When B is perfect over k, we can also conclude that F(A) is perfect over k and
+consider F(A)∗ with its (A, B)-bimodule structure.
+
+G-SPECTRA OF CYCLIC DEFECT
+19
+Proposition 3.9.2. Suppose that B is perfect over k and is endowed with a sym-
+metric structure. Let F : LMod(A) → LMod(B) be a colimit-preserving functor
+that carries LMod(A)ω into LMod(B)ω. Then the right adjoint G : LMod(B) →
+LMod(A) is given by
+G(M) = F(A)∗ ⊗B M
+Proof. By Prop. 3.6.1, G is colimit-preserving and is given by the formula G(M) =
+G(B) ⊗B M, and (3.6.2) gives a formula for G(B):
+G(B) = MapsLMod(B)(F(A), B)
+Since both F(A) and B are perfect over k, we furthermore have an isomorphism of
+left A-modules (3.8.1)
+(3.9.1)
+MapsLMod(B)(F(A), B) ∼= MapsRMod(B)(B∗, F(A)∗)
+where on the domain the left A-module structure is induced by the right A-module
+structure on F(A), and on the codomain by the left A-module structure on F(A)∗.
+The composite is
+G(B) ∼= MapsRMod(B)(B∗, F(A)∗)
+The bimodule isomorphism B∗ ∼= B is also a right B-module isomorphism and it
+induces a further isomorphism
+G(B) ∼= MapsRMod(B)(B, F(A)∗) ∼= F(A)∗
+where the second isomorphism MapsRMod(B)(B, N) ∼= N holds for any right B-
+module N.
+□
+When both A and B are perfect over k, and both are endowed with symmetric
+structures, we can make some further conclusions:
+Corollary 3.9.3. Suppose that both A and B are perfect over k and can be
+endowed with symmetric structures. Let F : LMod(A) → LMod(B) be a colimit-
+preserving functor that carries LMod(A)ω into LMod(B)ω. Then the following are
+equivalent:
+(1) F(A) is perfect as a right A-module
+(2) F(A)∗ is perfect as a left A-module
+(3) The right adjoint G(M) = F(A)∗ ⊗B M carries LMod(B)ω into LMod(A)ω
+When these equivalent conditions hold, F carries LMod(A)ft into LMod(B)ft and
+G carries LMod(B)ft into LMod(A)ft.
+Proof. (1) and (2) are equivalent by Prop. 3.8.2, and the equivalence of (2) and (3)
+is immediate from the formula G(M) = F(A)∗ ⊗B M.
+That F carries LMod(A)ft into LMod(B)ft is immediate from Prop. 3.5.2. That
+G carries LMod(B)ft into LMod(A)ft follows from the same Proposition applied to
+G(B).
+□
+3.10. K(n)-local algebras. While Proposition 3.5.1 is a complete description of
+functors LMod(A)ω → LMod(B)ω in terms of (B, A)-bimodules, Proposition 3.5.2
+only gives a sufficient condition for a bimodule to determine a functor LMod(A)ft →
+LMod(B)ft. In this and the next section, we discuss how we can do better when
+A and B are group algebras for finite groups and k obeys a technical condition
+
+20
+TONY FENG, DAVID TREUMANN, AND ALLEN YUAN
+(Equation (3.11.1)) related to the theory of K(n)-local spectra; the main result is
+Corollary 3.12.7. This material is not used in the proof of §2.6.
+For n ≥ 1 let K(n) = K(n)p be the nth Morava K-theory spectrum at the prime
+p, with
+πi(K(n)) =
+�
+Z/p
+if i ∈ (2pn − 2)Z
+0
+otherwise.
+An object c of (any) presentable stable ∞-category C is called K(n)-acyclic if
+K(n)⊗S c = 0, and d ∈ C is called K(n)-local if Maps(c, d) is contractible whenever
+c is K(n)-acyclic. The full subcategory of K(n)-local objects of C is the essential
+image of an idempotent functor LK(n) : C → C and is denoted LK(n)C.
+We have two useful formulas for the limit and for the colimit of a diagram
+D : I → LK(n)C:
+lim
+←− D = lim
+←−(LK(n)C ֒→ C) ◦ D
+lim
+−→ D = LK(n) lim
+−→(LK(n)C ֒→ C) ◦ D
+In other words, the inclusion of LK(n)C → C preserves limits and the localization
+functor C → LK(n)C preserves colimits.
+We are interested in the categories LK(n)LMod(A), where A is an associative
+k-algebra spectrum and k is a commutative ring spectrum. We will also gener-
+ally assume k and A are themselves K(n)-local (as otherwise, we get equivalent
+categories by replacing them with their K(n)-localizations).
+Proposition 3.10.1. For M ∈ LMod(A), the following are equivalent:
+(1) M ∈ LK(n)LMod(A)
+(2) The k-module underlying M is in LK(n)Mod(k).
+Proof. The actions of K(n)⊗S on LMod(A) and on Mod(k) are intertwined by the
+forgetful functor, so if N is K(n)-acyclic as an A-module, it is also K(n)-acyclic as
+a k-module. Similarly A⊗k N, A⊗k A⊗k N,. . . are K(n)-acyclic k-modules if N is.
+Thus, for all n, Mapsk(A⊗kn ⊗k N, M) = 0 when M is K(n)-local and N is K(n)-
+acyclic. We conclude that (2) implies (1) from the bar model for MapsA(N, M),
+i.e.
+MapsA(N, M)
+∼=
+lim
+←−n Mapsk(A⊗n ⊗ N, M)
+=
+lim
+←−(Mapsk(A ⊗k N, M) ⇒ Mapsk(A ⊗k A ⊗k N, M) · · · )
+which vanishes as long as all the terms in the limit do, i.e. as long as M is K(n)-local
+as a k-module.
+To show that (1) implies (2), use again that A ⊗k N is K(n)-acyclic if N is, and
+that Mapsk(N, M) ∼= MapsA(A ⊗k N, M).
+□
+3.11. Finiteness conditions for K(n)-local modules.
+Remark 3.11.1. There are analogous finiteness conditions to §3.2 in LK(n)Mod(k),
+but they do not necessarily coincide with the corresponding notions in the larger
+category Mod(k), and they no longer all agree.
+(1) The notion of perfect is unchanged (any perfect k-module is automatically
+K(n)-local).
+(2) Compact (or perfect) objects in Mod(k) are not in general compact in
+LK(n)Mod(k); for example k itself is usually not compact in LK(n)Mod(k).
+This is analogous to the fact that Zp is not compact in p-complete Zp-
+modules.
+
+G-SPECTRA OF CYCLIC DEFECT
+21
+(3) The natural tensor product on LK(n)Mod(k) is not (M, N) �→ M ⊗k N but
+(M, N) �→ LK(n)(M ⊗k N). We will call objects which are dualizable with
+respect to this tensor structure K(n)-locally dualizable. Perfect modules are
+always K(n)-locally dualizable, but there are generally more K(n)-locally
+dualizable objects than this [HS99, §15].
+While the categories LMod(A)ft are defined in terms of the underlying k-modules
+being perfect, it turns out that the notion of K(n)-local dualizability has more useful
+formal properties (cf. also Proposition 3.12.2):
+Lemma 3.11.2. Let k be a K(n)-local commutative ring spectrum and M ∈
+LK(n)Mod(k). Then M is K(n)-locally dualizable if and only if the functor
+LK(n)(− ⊗k M) : LK(n)Mod(k) → LK(n)Mod(k)
+preserves limits.
+Proof. The only if direction follows from LK(n)Mod(k) being a closed symmetric
+monoidal category.
+For the if direction, note that the adjoint functor theorem
+[Lur17, Cor. 5.5.2.9] and Proposition 3.12.1 imply that − ⊗k M has a left adjoint
+given by LK(n)(− ⊗k N), and N is easily seen to be the K(n)-local dual of M.
+□
+In light of Remark 3.11.1(3), we highlight a condition where this distinction
+disappears:
+(3.11.1)
+A K(n)-local commutative ring spectrum k satisfies condition
+(3.11.1) if every K(n)-locally dualizable k-module M is perfect.
+Since perfect k-modules are always dualizable, the classes of dualizable and per-
+fect K(n)-local k-modules coincide when k obeys this condition. Hence, LMod(A)ft
+coincides with the subcategory of LK(n)LMod(A) whose underlying k-module is
+K(n)-locally dualizable.
+Example 3.11.3. Condition (3.11.1) is satisfied when K(n) = K(1) and k = KUp.
+In fact, for any n, any Lubin-Tate theory (equivalently, Morava E-theory) obeys
+the condition. To see this, let E be a Lubin-Tate theory, let I = (p, v1, · · · , vn−1) ⊂
+π0(E) denote a Landweber ideal in π0(E), and let K = E/(p, v1, · · · , vn−1) de-
+note the corresponding Morava K-theory. If M is a dualizable E-module, then
+π∗(M/(p, v1, · · · , vn−1)) is a dualizable π∗(K)-module, and therefore it is a finitely
+generated π∗(K)-module. By the proof of [HS99, Prop. 2.4], this means that π∗(M)
+is finitely generated over π∗(E), and hence by [HS99, Lem. 8.11] that M is perfect
+as an E-module.
+3.12. Functors and bimodules in LK(n)-local categories.
+Proposition 3.12.1. Let k be a K(n)-local commutative ring spectrum and let
+A and B be K(n)-local associative k-algebra spectra. If F is a colimit-preserving
+k-linear functor
+F : LK(n)LMod(A) → LK(n)LMod(B),
+then F is isomorphic to LK(n)(F(A) ⊗A −).
+Here F(A) gets a (B, A)-bimodule structure as in §3.5. The analog of (3.5.1) is
+FunL
+k(LK(n)LMod(A), LK(n)LMod(B)) ∼= LK(n)Bimod(B, A)
+
+22
+TONY FENG, DAVID TREUMANN, AND ALLEN YUAN
+The functor from the right category to the left category sends a K(n)-local bimodule
+N to the functor LK(n)(N ⊗A −).
+Proof. Suppose that F is a colimit-preserving k-linear functor LK(n)LMod(A) →
+LK(n)LMod(B).
+For M ∈ LK(n)LMod(A) the natural map
+(3.12.1)
+F(A) ⊗A M → F(M)
+factors through a natural map
+(3.12.2)
+LK(n)(F(A) ⊗A M) → F(M).
+Indeed, since F(M) is K(n)-local, (3.12.2) is LK(n) applied to (3.12.1). The map
+(3.12.2) is an isomorphism when M = A, and since F preserves colimits is therefore
+an isomorphism for all M ∈ LK(n)LMod(A).
+□
+When A is perfect over k, we have containments
+LModω(A) ⊂ LMod(A)ft ⊂ LK(n)LMod(A).
+As in Remark 3.11.1, we warn that neither category agrees with compact objects
+in LK(n)LMod(A). If A and B are both perfect over k, it is natural to ask for a
+criterion for LK(n)(F(A) ⊗A −) to carry LMod(A)ft into LMod(B)ft. The criterion
+of Prop. 3.5.2 still applies, but is strictly stronger than necessary. When A and B
+are group algebras, a phenomenon called K(n)-local Tate vanishing or ambidexterity
+gives a less restrictive criterion.
+The study of these phenomena originates in work of Greenlees–Sadofsky and
+Hovey–Sadofsky [GS96, HS96].
+Generalizing their work, Kuhn [Kuh04] showed
+that the Tate cohomology of finite groups vanishes in the K(n)-local setting1. The
+n = 0 case of Kuhn’s theorem is the familiar fact that the additive transfer map from
+orbits to fixed points is an isomorphism when working over Q. These results were
+further generalized and reinterpreted by the Hopkins–Lurie theory of ambidexterity
+[HL13], which has been generalized and developed extensively by Carmeli–Schlank–
+Yanovski and Barthel–Carmeli–Schlank–Yanovski [CSY22], [CSY21a], [CSY21b],
+[BCSY22]. For the convenience of the reader, the following proposition collects
+some basic features of the theory.
+Proposition 3.12.2. Let G be a finite group and let k be a K(n)-local commuta-
+tive ring spectrum. Then:
+(1) The functors Fun(BG, LK(n)Mod(k)) → LK(n)Mod(k) given by (−)hG and
+LK(n)(−)hG are identified.
+(2) The k-modules kBG and LK(n)k[BG] (which are isomorphic by (1)) are
+K(n)-locally dualizable over k.
+If G is a p-group, then we also have
+(3) There is an equivalence of categories
+Fun(BG, LK(n)Mod(k)) ∼= LK(n)Mod(kBG)
+given by M �→ M hG.
+(4) Under the equivalence in (3), both functors in (1) are identified with re-
+striction of scalars along k → kBG.
+1In fact, he showed this in the (conjecturally) more general setting of T(n)-local homotopy
+theory.
+
+G-SPECTRA OF CYCLIC DEFECT
+23
+Proof. Statement (1) is [HS96, Thm 1.1] (or [HL13, Thm. 5.2.1]) and (3) and (4)
+are [HL13, Thm. 5.4.3]. To see (2), note first that if P ⊂ G is a p-Sylow, then a
+standard transfer argument shows that LK(n)k[BG] is a summand of LK(n)k[BP]
+and so it suffices to consider the case when G is a p-group. The transitivity of
+homotopy orbits then reduces to the case G = Cp, and by base-change, it suffices
+to consider the case k = LK(n)S.
+Finally, by [HS99, Thm 8.6], the K(n)-local
+dualizability of LK(n)S[Cp] amounts to seeing K(n)∗(BCp) is finite, which follows
+from [RW80].
+□
+Proposition 3.12.3. Let k be a K(n)-local commutative ring spectrum, let A =
+k[G] be the group algebra (§4.2) of a finite group G, and let F(A) be a right
+A-module whose underlying k-module is K(n)-locally dualizable. Then the functor
+F : LK(n)LMod(A) → LK(n)Mod(k)
+M �→ LK(n)(F(A) ⊗A M)
+preserves the property of having K(n)-locally dualizable underlying k-module.
+Proof. In light of Proposition 3.12.2(2-4), this is a consequence of the following
+lemma 3.12.4 applied to the map k → kBG.
+□
+Lemma 3.12.4. Let f : k → k′ be a map of K(n)-local commutative ring spectra
+such that k′ is K(n)-locally dualizable as a k-module. Then restriction of scalars
+along f sends K(n)-locally dualizable k′-modules to K(n)-locally dualizable k-
+modules.
+Proof. Let M be a K(n)-locally dualizable k′-module. Then note that the functor2
+(3.12.3)
+LK(n)(− ⊗k M) : LK(n)Mod(k) → LK(n)Mod(k)
+can be written as the composite
+LK(n)Mod(k)
+−⊗kk′
+−−−−→ LK(n)Mod(k′)
+−⊗k′ M
+−−−−−→ LK(n)Mod(k′)
+forget
+−−−→ LK(n)Mod(k).
+By the dualizability hypotheses and Lemma 3.11.2, the first two arrows preserve
+limits, and the third arrow preserves limits as it is a right adjoint. Thus, (3.12.3)
+preserves limits as well and the claim follows from Lemma 3.11.2.
+□
+Remark 3.12.5. Lemma 3.12.4 is not specific to K(n)-local spectra — the analo-
+gous statement holds more generally in any presentable symmetric monoidal stable
+∞-category.
+Remark 3.12.6. The conclusion of Proposition 3.12.3 can fail when A is not the
+group algebra of a finite group. For example, let A = k[S1] be the k-homology
+spectrum of the circle. Then we have an isomorphism
+LK(n)(k ⊗A k) ∼= k[BS1],
+which is not generally K(n)-locally dualizable.
+When k satisfies the condition (3.11.1), then perfect modules and K(n)-locally
+dualizable modules coincide, and therefore we immediately deduce the following
+consequence of Proposition 3.12.3 for finite type modules:
+2Here and in the following equation, we need not K(n)-localize the tensor product due to the
+assumed dualizability.
+
+24
+TONY FENG, DAVID TREUMANN, AND ALLEN YUAN
+Corollary 3.12.7. Let k be a K(n)-local commutative ring spectrum satisfying
+condition (3.11.1). Let A = k[G] and B = k[H] be the group algebras (§4.2) of
+finite groups G and H. Let F(A) be a (B, A)-bimodule whose underlying k-module
+is perfect. Then the functor
+M �→ LK(n)(F(A) ⊗A M)
+carries LMod(A)ft to LMod(B)ft.
+4. Permutation modules and Rouquier’s equivalence over S
+We now turn to lifting Rouquier’s equivalence to S. In §4.1, we start by reviewing
+the form of Rouquier’s original equivalence. Then in §4.2-4.4, we discuss the theory
+of permutation G-modules over S. In §4.5, we show that summands of permutation
+modules lift from Zq to Sq. Finally, in §4.7, we put these ingredients together with
+the results of §3 to prove our main theorem, Theorem 4.7.2.
+4.1. Rouquier’s two-term complexes. Let G be a finite group and let D ⊂ G
+be a cyclic group of p-power order.
+The left multiplication of G and the right
+multiplication of NG(D) on G endow Zq[G] with a (G, NG(D))-bimodule structure,
+whose associated functor
+LMod(Zq[NG(D)]) → LMod(Zq[G])
+is equivalent to induction along the inclusion NG(D) ⊂ G. The adjoint is restriction
+along the same inclusion, represented by Zq[G] regarded as a (NG(D), G)-bimodule.
+Now let Fq[G]b and Fq[NG(D)]b′ be blocks whose defect group is D and which
+are Brauer correspondents of each other. Then (abusing notation and writing b
+and b′ also for the unique lifts of these idempotents to Zq[G] and Zq[NG(D)]) the
+(Zq[G]b, Zq[NG(D)]b′)-summand of the bimodule Zq[G] is bZq[G]b′, which repre-
+sents the composite functor
+(4.1.1)
+LMod(Zq[NG(D)]b′) → LMod(Zq[NG(D)]) → LMod(Zq[G]) → LMod(Zq[G]b)
+This functor carries LMod(Zq[NG(D)]b′)ft into LMod(Zq[G]b)ft and LMod(Zq[NG(D)]b′)ω
+into LMod(Zq[G]b)ω. Its right adjoint is represented by b′Zq[G]b and carries LMod(Zq[G]b)ft
+into LMod(Zq[NG(D)]b′)ft and LMod(Zq[G]b)ω into LMod(Zq[NG(D)]b′)ω.
+The composition (4.1.1) is not an equivalence. For many finite groups, it does
+induce a “stable equivalence” (for instance, for groups whose p-Sylows have trivial
+intersection [Bro94, §5, 6.4]; in general one has to pass to a summand) — one
+formulation of this is that it induces an equivalence of quotient categories
+LMod(Zq[NG(D)]b′)ft
+LMod(Zq[NG(D)]b′)ω ∼= LMod(Zq[G]b)ft
+LMod(Zq[G]b)ω
+In particular the composite of (4.1.1) with its adjoint differs from the identity
+functor by a bounded complex of projective (Zq[NG(D)]b′, Zq[NG(D)]b′)-bimodules
+or projective (Zq[G]b, Zq[G]b)-bimodules.
+Rouquier showed that, when D is cyclic, one can introduce a relatively simple
+correction to the bimodule representing (4.1.1) to obtain an equivalence predicted
+by Brou´e’s Conjecture.
+Theorem 4.1.1 (Rouquier [Rou98]). Let G be a finite group and let D ⊂ G be
+a cyclic subgroup of p-power order. Let Fq[G]b and Fq[NG(D)]b′ be blocks whose
+
+G-SPECTRA OF CYCLIC DEFECT
+25
+defect group is D and which are Brauer correspondents of each other. Then there
+is a two-term complex of (Zq[G]b, Zq[NG(D)]b′)-bimodules
+M0 = {· · · → 0 → N ′
+0 → N0 → 0 → · · · }
+with the following properties:
+(1) N0 is a direct summand of Zq[G], with its (Zq[G], Zq[NG(D)])-bimodule
+structure coming from the left multiplication action of G and the right
+multiplication action of NG(D).
+(2) N ′
+0 is a projective (Zq[G]b, Zq[NG(D)]b′)-bimodule.
+(3) The resulting functor
+M0 ⊗Zq[NG(D)]b′ − : LMod(Zq[NG(D)]b′) → LMod(Zq[G]b)
+is an equivalence, and restricts to an equivalence on LModft and LModω.
+4.2. Group rings and permutation modules. If k is a commutative ring spec-
+trum, denote by X �→ k[X] ∈ Mod(k) the functor that carries a space to its k-
+homology spectrum. For example, if k = S, then S[X] is the suspension spectrum
+of X with a disjoint basepoint and in general k[X] = S[X] ⊗S k. A useful formula
+is
+(4.2.1)
+[k[X], Σik] ∼= Hi(X; k)
+where the right-hand side denotes the ith (extraordinary) cohomology of X with
+coefficients in k.
+If G acts on X, then we may regard k[X] as a G-object in Mod(k) (i.e., as an
+object of Fun(BG, Mod(k))) by composing
+BG → S
+k[−]
+−−−→ Mod(k).
+The analog of (4.2.1) is
+(4.2.2)
+[k[X], Σik]Fun(BG,Mod(k)) ∼= Hi
+G(X; k) := Hi(XhG; k)
+where XhG := lim
+−→BG X ∼= (X × EG)/G is the Borel construction.
+If X is a finite G-set, then k[X] = �
+x∈X k is called a permutation module.
+The associative k-algebra structure on k[G] induced by the multiplication on G
+coincides with the endomorphism algebra of k[G/{1}] regarded as a permutation
+module, which generates Fun(BG, Mod(k)), so that we have
+Fun(BG, Mod(k)) ∼= LMod(k[G]).
+The symmetric monoidal structure on Mod(k) induces a symmetric monoidal struc-
+ture on Fun(BG, Mod(k)), which we denote by ⊗k.
+The unit of the monoidal
+structure is the trivial module k.
+Lemma 4.2.1. The permutation modules k[X] are self-dual with respect to the
+monoidal structure in Mod(k) and in Fun(BG, Mod(k)).
+Proof. We produce evaluation and coevaluation maps satisfying the necessary re-
+lations.
+The coevaluation map comes from the composition
+(4.2.3)
+k → k[X]
+∆X
+−−→ k[X × X] ∼= k[X] ⊗k k[X]
+
+26
+TONY FENG, DAVID TREUMANN, AND ALLEN YUAN
+where the first map is the diagonal k → �
+x∈X k. The composite
+[k[X] ⊗ k[X], k]
+⊗k[X]
+−−−−→ [k[X] ⊗ k[X] ⊗ k[X], k[X]]
+idk[X]⊗(4.2.3)∗
+−−−−−−−−−−→ [k[X], k[X]]
+is a bijection. The homotopy class of the evaluation map k[X] ⊗ k[X] → k is the
+image of idk[X] under the inverse bijection.
+□
+Remark 4.2.2. The k-homology of manifolds furnish a more general class of du-
+alizable objects of Mod(k), and the k-homology of G-manifolds furnish dualizable
+objects of Fun(BG, Mod(k)). For a closed n-manifold M, the homology spectrum
+k[M] is self-dual up to a shift so long as M is k-orientable — indeed one defini-
+tion of a k-orientation of M is a homotopy class of maps k → Σ−nk[M] with the
+property that the composite map
+k → Σ−nk[M] → Σ−nk[M × M] → Σ−nk[M] ⊗k k[M]
+exhibits Σ−nk[M] as the dual of k[M]. Lemma 4.2.1 is the case n = 0.
+Under the equivalence
+Bimod(k[G], k[G]) ∼= LMod(k[G] ⊗ k[G]op) ∼= LMod(k[G × Gop]),
+the diagonal bimodule k[G] is a permutation G× Gop-module, with G× Gop acting
+on G by left and right multiplication. It follows from (4.2.3) that each k[G] has a
+natural symmetric structure in the sense of §3.9. Furthermore any block of k[G] has
+a symmetric structure: indeed, if A = A1 ×A2 ×· · · , then a symmetric structure on
+A induces a symmetric structure on each Ai by the composite Ai → A ∼= A∗ → A∗
+i .
+The self-duality of permutation modules gives an identification between homo-
+topy classes of maps k[X] → k[Y ] and equivariant k-cohomology classes in X × Y :
+[k[X], k[Y ]]k[G] ∼= [k[X] ⊗ k[Y ], k]k[G] ∼= [k[X × Y ], k]k[G] ∼= H0
+G(X × Y ; k).
+4.3. The Burnside ring. For a finite group G and a finite G-set X, let BurnG(X)
+denote the Grothendieck group of the commutative monoid whose objects are iso-
+morphism classes of finite G-sets Y equipped with a G-equivariant map to X, and
+whose addition is given by disjoint union. We will refer to BurnG(X) as the Burn-
+side ring of virtual finite G-sets over X, as it acquires a commutative multiplication
+given by fiber product over X.
+Remark 4.3.1. In general, BurnG(X) = �
+x∈G\X BurnGx(pt), where the sum is
+over G-orbit representatives and Gx denotes the stabilizer. In particular, BurnG(G/H)
+is isomorphic to BurnH(pt) and BurnG(G/{1}) = Z.
+A map of G-sets X → X′ induces by fiber-product over X′ a ring homomor-
+phism BurnG(X′) → BurnG(X). In particular the map X → pt gives BurnG(X) a
+BurnG(pt)-module structure, and the map G → pt induces the augmentation map
+BurnG(pt) → BurnG(G) = Z, carrying a G-set to the cardinality of its underlying
+set. Let I ⊂ BurnG(pt) denote the kernel of the augmentation map, called the
+augmentation ideal.
+Theorem 4.3.2 (Carlsson [Car84], weak form of Segal’s conjecture). Let G be a
+finite group, which we regard as acting trivially on S, and let ShG be the homotopy
+fixed-point spectrum of this action §3.1. Then there is a natural map of rings
+(4.3.1)
+BurnG(pt) → π0(ShG)
+which exhibits the target as the I-adic completion of the source.
+
+G-SPECTRA OF CYCLIC DEFECT
+27
+Remark 4.3.3. As the G-action on S is trivial, the homotopy fixed points ShG
+are naturally identified with MapsMod(S)(S[BG], S) and with MapsLMod(S[G])(S, S).
+(Similarly, ShG is naturally identified with S[BG].)
+We denote the completion at I of BurnG(pt) by BurnG(pt)∧
+I . If X is a finite
+G-set, we also define BurnG(X)∧
+I using the natural BurnG(pt)-module structure on
+BurnG(X).
+Proposition 4.3.4. Let G be a finite group. Then for any two finite G-sets X and
+Y , we have an identification
+π0MapsLMod(S[G])(S[X], S[Y ]) ∼= BurnG(X × Y )∧
+I .
+In particular, by taking Y = pt, we have
+(4.3.2)
+π0MapsLMod(S[G])(S[X], S) ∼= BurnG(X)∧
+I .
+Proof. In fact, it suffices to prove the special case (4.3.2) noted in the statement,
+as S[Y ] is canonically self-dual and therefore
+MapsG(S[X], S[Y ]) ∼= MapsG(S[X] ⊗ S[Y ], S) ∼= MapsG(S[X × Y ], S).
+For this, note that both sides of (4.3.2) convert disjoint union into products, so it’s
+enough to consider the case where X has a single orbit, say X = G/H. Then
+MapsG(S[G/H], S) ∼= MapsH(S, S),
+whose π0 is identified by Theorem 4.3.2 with BurnH(pt)∧
+IH ∼= BurnG(G/H)∧
+IG.
+□
+If we further complete BurnG(X)∧
+I at p, or equivalently if we study BurnG(X)∧
+(I,p)
+where (I, p) is the kernel of BurnG(X)
+aug
+−−→ Z → Z/p, we have the following simple
+description:
+Lemma 4.3.5. The Zp-module BurnG(pt)∧
+(I,p) is free of finite rank, generated by
+isomorphism classes of G-sets of the form G/P for P ⊂ G a p-group.
+Because of this and Remark 4.3.1, BurnG(X)∧
+(I,p) is a free Zp-module generated
+by the set of isomorphism classes of G-maps G/P → X, where P ⊂ G is a p-group.
+Proof. Recall Burnside’s marking homomorphism BurnG(pt) → �
+H Z, also called
+the “table of marks” [Bur11]. The factors in the direct product are indexed by
+conjugacy class representatives of subgroups of G, and the projection onto the factor
+H sends the G-set X to the cardinality of its H-fixed points. If s ∈ BurnG(pt)
+we will write #sH for the composition of the marking homomorphism with the
+projection onto the Z factor indexed by H.
+The marking homomorphism is a ring homomorphism, and it is easy to see that it
+is injective (a reference is [tD79, Proposition 1.2.2]). Since its domain and codomain
+are free abelian groups of the same rank, this means it becomes an isomorphism
+after applying − ⊗ Q or − ⊗ Qp. Write eH ∈ BurnG(pt) ⊗ Q for the virtual G-set
+with one H-fixed point and no H′-fixed points when H′ is not conjugate to H:
+#eH
+H = 1
+#eH′
+H = 0 when H′ is not conjugate to H
+The eH are the primitive idempotents in BurnG(pt) ⊗ Q and in BurnG(pt) ⊗ Qp.
+We remark that eH only involves G-sets corresponding to subconjugates of H: to
+see this, consider the variant of the marking homomorphism where on the source,
+one takes the subring generated by G-sets subconjugate to H, and on the target,
+
+28
+TONY FENG, DAVID TREUMANN, AND ALLEN YUAN
+one considers only factors corresponding to subconjugates of H. This variant is
+easily seen to be injective again, and is therefore an isomorphism after tensoring
+with Q.
+The primitive idempotents in BurnG(pt) ⊗ Zp ∼= BurnG(pt)∧
+p were identified by
+Dress [Dre69]. They are in one-to-one correspondence with conjugacy classes of
+p-perfect subgroups ̟ ⊂ G (a group is called p-perfect when it has no normal
+subgroups of p-power index). The idempotent corresponding to ̟ is ε̟ = �
+H eH,
+where the sum runs through class representatives of subgroups H that contain a
+conjugate of ̟ as a normal subgroup of p-power index.
+The trivial subgroup is p-perfect; let us denote its corresponding idempotent by
+ε := ε1. Thus ε := ε1 = �
+Q eQ where the sum runs through class representatives
+of p-subgroups Q ⊂ G. The natural map
+(4.3.3)
+BurnG(pt)∧
+p → BurnG(pt)∧
+(I,p)
+kills all the other ε̟ and has target a local ring (the target is the completion
+of BurnG(pt) at a maximal ideal, namely the kernel of augmentation modulo p).
+Therefore, (4.3.3) factors through
+(4.3.4)
+ε BurnG(pt)∧
+p → BurnG(pt)∧
+(I,p)
+Since the source ring ε BurnG(pt)∧
+p is local and (by the injectivity of the marking
+homomorphism) finitely generated and free over Zp, the (I, p)-adic topology on it
+coincides with the p-adic topology: ε BurnG(pt)∧
+p /(p) is a finite local ring so the
+image of the ideal (I, p) is nilpotent. Therefore, ε BurnG(pt)∧
+p is isomorphic to its
+own completion with respect to the ideal generated by (I, p), so that (4.3.4) is an
+isomorphism.
+Furthermore, note that for any p-subgroup Q ⊂ G, we have εG/Q = G/Q.
+This can be seen from the fact that they have the same image under the marking
+homomorphism, as (G/Q)H is nonempty if and only if H is conjugate to a subgroup
+of Q, and ε is the indicator function on the factors indexed by p-groups H.
+It therefore suffices to show that the G-sets of the form εG/Q span ε BurnG(pt)∧
+p .
+But the G-sets of the form G/H span BurnG(pt) and inside ε BurnG(pt)∧
+p , we
+have G/H = εG/H. As ε = �
+Q eQ only involves G-sets of the form G/Q for
+p-subgroups Q and G/H × G/Q involves only G-sets with isotropy subconjugate to
+Q, we conclude.
+□
+4.4. Digression on the strong Segal conjecture and Tate cohomology. This
+subsection is not used in the proof of §2.6.
+Segal formulated (4.3.1) by analogy with Atiyah’s theorem [Ati61], which iden-
+tifies the complex K-theory KU0(BG) with a completion of the representation ring
+R(G) of G, i.e. the Grothendieck ring of finite-dimensional representations of C[G]:
+(4.4.1)
+R(G) → KU0(BG).
+There is a more sophisticated version of both Atiyah’s theorem and of the Segal
+conjecture, which in a way is the start of “genuine” equivariant stable homotopy
+theory.
+(1) There is a commutative ring spectrum KUG, the (genuine) G-equivariant
+K-theory of a point, together with a map
+KUG → KUhG
+
+G-SPECTRA OF CYCLIC DEFECT
+29
+which recovers (4.4.1) on π0. It is constructed out of the category of com-
+plex representations of G in about the same way that KU is constructed
+out of the category of complex vector spaces. As an object of Mod(S) or
+Mod(KU), KUG has a simple structure: it is a free KU-module spanned by
+the irreducible complex representations:
+KUG ∼=
+�
+Irr(C[G])
+KU .
+(2) Let SG denote the group completion K-theory spectrum of the category of
+finite G-sets, with its symmetric monoidal structure given by disjoint union
+of G-sets. The Cartesian product of G-sets endows SG with the structure
+of a commutative ring spectrum. As an object of Mod(S) one has
+SG ∼=
+�
+H
+S[B(NG(H)/H)]
+where the sum ranges over conjugacy class representatives of subgroups
+H ⊂ G.
+Carlsson proved more than (4.3.1):
+Theorem 4.4.1 ([Car84], strong form of Segal’s conjecture). Let G be a finite
+group. There is a natural map of commutative ring spectra
+SG → ShG
+which exhibits π∗(ShG) as the I-adic completion of π∗(SG).
+This has a surprising consequence: ShG is connective. (To see that it is sur-
+prising, note that replacing S with the Eilenberg-MacLane spectrum of Z yields
+an object concentrated in non-positive degrees, as πi(ZhG) is the (−i)th group
+cohomology of G).
+A second consequence of the Segal conjecture is that one can compute the Tate
+cohomology of some finite groups with coefficients in the sphere. For a finite group
+G and a G-spectrum M, the G-Tate cohomology M tG ∈ Mod(S) of M is defined
+via a cofiber sequence
+MhG
+Nm
+−−→ M hG → M tG
+When M is discrete, the map Nm is given on π0 by the formula m �→ �
+g∈G gm,
+and the homotopy groups π∗M tG = ˆH−∗(G; M) coincide with the classical Tate
+cohomology groups of G with coefficients in M.
+Example 4.4.2. When G = Cp, the cyclic group of order p, the nature of the
+augmentation ideal is that the strong Segal conjecture identifies ShCp as Sp ⊕
+S[BCp]. The norm map is the inclusion of the second summand, S[BCp] = ShCp.
+It follows that StCp ∼= Sp. It is in this form that the Segal conjecture for Cp was
+originally proved by Lin (p = 2, [Lin80]) and Gunawardena (p odd, [Gun80]).
+4.5. p-permutation Sq-modules. We say that a Zq[G]-module M0 lifts to Sq if
+there exists M ∈ LMod(Sq[G]) such that there is an equivalence
+M ⊗Sq Zq ∼= M0
+of Zq[G]-modules. The question of whether a Zq[G]-module lifts to Sq is a close
+relative of the “equivariant Moore space problem,” and negative examples were first
+given by Carlsson in the cases G = Cp × Cp [Car81]. In this section we discuss a
+
+30
+TONY FENG, DAVID TREUMANN, AND ALLEN YUAN
+class of modules for which a natural lift to the sphere does exist: for summands of
+permutation modules, by virtue of the following proposition:
+Proposition 4.5.1. Let G be a finite group, let X be a finite G-set, and suppose
+that Zq[X] ∼= M0 ⊕ N0 is a Zq[G]-module splitting. Then, there exists a Sq[G]-
+module splitting Sq[X] ∼= M ⊕ N which recovers the Zq[G]-module splitting above
+after applying ⊗SqZq.
+The map of ring spectra Sq → Zq, given by truncation, induces a map of endo-
+morphism rings
+(4.5.1)
+π0MapsSq[G](Sq[X], Sq[X]) → π0MapsZq[G](Zq[X], Zq[X])
+and the content of the Proposition is that any primitive idempotent in the target
+lifts to a primitive idempotent in the source. By §4.3, this map can be identified
+with the natural map
+BurnG(X × X)∧
+(p,I) ⊗Zp Zq → π0MapsZq[G](Zq[X × X], Zq)
+which sends a G-set over X × X to its rank function. In particular, the source
+(Lemma 4.3.5) and target are free Zq-modules of finite rank, and the map is a
+surjection of (not necessarily commutative) Zq-algebras, so this is a consequence of
+the following purely algebraic fact:
+Proposition 4.5.2. Let ¯A and ¯B be associative Zq-algebras that are finitely gen-
+erated as Zq-modules. Let f : ¯A → ¯B be a surjective algebra homomorphism. Then
+any primitive idempotent of ¯B lifts to a primitive idempotent of ¯A.
+We put bars over these Zq-algebras to emphasize that they are ordinary abelian
+groups, not ring spectra. It is likely that this is an old result, but we didn’t find
+any place that it was stated very plainly, so we supply a proof in the rest of this
+section.
+Lemma 4.5.3. Let ¯A and ¯B be finitely generated associative algebras over a field,
+and let f : ¯A → ¯B be a surjective algebra homomorphism. Then every invertible
+element of ¯B has an invertible preimage under f.
+Proof. Let I ⊂ ¯A be the kernel of f, so that f is isomorphic to the quotient map
+¯A → ¯A/I. We will show that every invertible element of ¯A/I lifts to an invertible
+element of ¯A.
+Let J be the Jacobson radical of ¯A. Let us first prove the Lemma in the case
+that I ∩ J = 0. If I ∩ J = 0, then the natural map ¯A → ¯A/I × ¯
+A/I+J ¯A/J is an
+isomorphism of algebras, thus for every unit u+I ∈ ¯A/I, we seek a unit v+J ∈ ¯A/J
+such that u + I + J = v + I + J. Such a v + J exists by Wedderburn’s theorem:
+¯A/J is semisimple thus a product of a finite set of simple algebras, and the image
+of I in ¯A/J is a product of a subset of the same simple algebras.
+In case I ∩J ̸= 0, the above argument shows we can lift units along A/(I ∩J) →
+A/I. Since ¯A is finite-dimensional over a field, it is an Artinian ring so J (and
+therefore I ∩ J) is nilpotent [Her94, Theorem 1.3.1]. It follows that units lift along
+A → A/I ∩ J.
+□
+Lemma 4.5.4. Let ¯A and ¯B be associative Zp-algebras that are finitely generated
+as Zp-modules. Let f : ¯A ։ ¯B be a surjective algebra homomorphism. Then every
+invertible element of ¯B has an invertible preimage under f.
+
+G-SPECTRA OF CYCLIC DEFECT
+31
+Proof. Suppose b ∈ ¯B is invertible. Let us first show there is an a ∈ ¯A such that
+(1) a + px is invertible for all x ∈ A
+(2) f(a) = b mod p ¯B
+Indeed b + p ¯B is invertible in the Fp-algebra ¯B/p ¯B, and by applying the previous
+Lemma to ¯A/p ¯A → ¯B/p ¯B we conclude there is an a ∈ ¯A such that a + p ¯A is
+invertible in A/p ¯A and f(a) + p ¯B = b + p ¯B. Since ¯A is finitely generated over
+Zp, it is p-adically complete and such an a is invertible in ¯A: if aa′ = 1 + pǫ then
+a′(1 − pǫ + p2ǫ2 − · · · ) is the inverse to a.
+It remains to find x such that f(a + px) = b. By (2) above, f(a) − b = py for
+some y ∈ ¯B. Let x be such that f(x) = −y. Then
+f(a + px) − b = f(a) + pf(x) − b = f(a) − b + pf(x) = py + pf(x) = 0
+□
+Lemma 4.5.5. Let ¯A be a Zp-algebra that is finitely generated as a Zp-module.
+Let i, j ∈ ¯A be two idempotent elements. Then ¯Ai ∼= ¯Aj as left ¯A-modules if and
+only if i is conjugate to j.
+Proof. If i = uju−1, then right multiplication by u induces an isomorphism ¯Ai =
+¯Auju−1 ∼
+→ ¯Aj. Let us show the converse. Suppose φ : ¯Ai
+∼
+→ ¯Aj is an isomorphism
+of left ¯A-modules. Since φ(ai) = aφ(i), φ is determined by φ(i) ∈ ¯Aj. Furthermore,
+since i2 = i, φ(i) = φ(i2) = iφ(i), so φ(i) ∈ i ¯Aj. Similarly, φ−1 is determined by
+φ−1(j) ∈ j ¯Ai. The computations
+φ−1(j)φ(i) = φ(φ−1(j)i) = φ(φ−1(j)) = j
+φ(i)φ−1(j) = φ−1(φ(i)j) = φ−1(φ(i)) = i
+show that φ−1(j)φ(i) = j and φ(i)φ−1(j) = i
+Recall that Zp-algebras that are finitely generated as Zp-modules are “semiper-
+fect” in the sense of [Bas60, §2.1]. This means that finitely generated ¯A-modules
+— such as ¯A itself — have the Krull-Schmidt property, a unique direct sum decom-
+position into indecomposable summands. Since ¯A = ¯Ai⊕ ¯A(1 − i) = ¯Aj ⊕ ¯A(1 − j),
+the hypotheses of the Lemma also show that ¯A(1 − i) ∼= ¯A(1 − j), via an iso-
+morphism ψ. Thus by the same argument as above, ψ(1 − i) ∈ (1 − i)A(1 − j),
+ψ−1(1 − j) ∈ (1 − j)A(1 − i), and
+ψ−1(1 − j)ψ(1 − i) = 1 − i
+ψ(1 − i)ψ−1(1 − j) = 1 − j.
+Let u = φ(i) + ψ(1 − i) and v = φ−1(j) + ψ−1(1 − j). Then uv = 1 = vu, so
+v = u−1, and u conjugates j to i:
+uju−1
+=
+(φ(i) + ψ(1 − i))j(φ−1(j) + ψ−1(1 − j))
+=
+(φ(i)j + ψ(1 − i)j)(φ−1(j) + ψ−1(1 − j))
+=
+(φ(i) + 0)(φ−1(j) + ψ−1(1 − j))
+=
+φ(i)φ−1(j) + φ(i)ψ−1(1 − j)
+=
+φ(i)φ−1(j) + 0
+=
+i
+□
+Proof of Proposition 4.5.1. It suffices to prove Proposition 4.5.2. Choose a decom-
+position 1 ¯
+A = � ei, where the ei are orthogonal idempotents. Then 1 ¯
+B = f(1 ¯
+A) =
+� f(ei). The set of nonzero f(ei) is a set of orthogonal idempotents in ¯B that sum
+to 1.
+
+32
+TONY FENG, DAVID TREUMANN, AND ALLEN YUAN
+Now suppose the ei are all primitive, and let us show that the nonzero f(ei) are
+also primitive. Each ¯Aei surjects onto ¯Bf(ei); since this is a map of ¯A-modules
+and ¯Aei is an indecomposable projective ¯A-module, it follows that ¯Bf(ei) is inde-
+composable as an ¯A-module.
+Since the action of ¯A factors through the surjection to ¯B, ¯Bf(ei) is also inde-
+composable as a ¯B-module. Thus each f(ei) is primitive.
+If e′ is some other primitive idempotent of ¯B, then ¯B ∼= ¯Be′ ⊕ ¯B(1 − e′) ∼=
+� ¯Bf(e)i, so Be′ ∼= Bf(ei) for some i by the Krull-Schmidt property. By Lemma
+4.5.5, there is an invertible u ∈ B such that uf(ei)u−1 = e′, and by Lemma 4.5.4
+there is a unit u′ ∈ ¯A such that f(u′) = u. Then u′ei(u′)−1 is a primitive idempotent
+of ¯A that lifts e′.
+□
+4.6. Blocks of Sq[G]. Let b1, b2, . . . be the block idempotents of Zq[G], so that
+(4.6.1)
+Zq[G] = Zq[G]b1 × Zq[G]b2 × · · ·
+By Proposition 3.4.1, there is a corresponding splitting of Sq[G/{1}] as a left G-
+module — write Sq[G/{1}]bi for the summand matching Zq[G]bi. Let Sq[G]bi de-
+note the endomorphism Sq-algebra spectrum of Sq[G/{1}]bi. By Proposition 3.4.2,
+[Sq[G/{1}]bi, Sq[G/{1}]bj] = 0 when i ̸= j, and therefore there is an isomorphism
+of Sq-algebra spectra
+(4.6.2)
+Sq[G] = Sq[G]b1 × Sq[G]b2 × · · ·
+that induces (4.6.1) on taking π0 or on applying ⊗SqZq.
+Proposition 4.6.1. Let G and G′ be finite groups, let b be a block of Zq[G] and let
+b′ be a block of Zq[G′]. Suppose that there is an equivalence of stable ∞-categories
+(4.6.3)
+LMod(Sq[G]b) ∼= LMod(Sq[G′]b′).
+Then for any Sq-algebra spectrum k, there are equivalences
+(1) LMod(k[G]b) ∼= LMod(k[G′]b′)
+(2) LMod(k[G]bi)ft ∼= LMod(k[G′]b′
+i)ft
+(3) LMod(k[G]bi)ω ∼= LMod(k[G]bi)ω
+Proof. That (4.6.3) implies (1) and (3) is a consequence of (3.7.2); that it implies
+(2) is a consequence of Corollary 3.9.3.
+□
+4.7. The Rouquier equivalence over Sq.
+Proposition 4.7.1. Suppose we are in the situation of Theorem 4.1.1, and let
+M0 denote the complex of bimodules of that Theorem regarded as an object in
+Bimod(Zq[G]b, Zq[NG(D)]b′). Then there is an M ∈ Bimod(Sq[G]b, Sq[NG(D)]b′)
+such that M ⊗Sq Zq is isomorphic to M0.
+Proof. By Proposition 4.5.1, N0 lifts to a (Sq[G]b, Sq[NG(D)]b′)-bimodule N such
+that N ⊗Sq Zq ∼= N0. Since N ′
+0 is projective, it is the image of an idempotent
+endomorphism of (Zq[G]b ⊗Zq Zq[NG(D)]b′op)⊕n, which §3.4 lifts to an idempo-
+tent endomorphism of (Sq[G]b ⊗ Sq[NG(D)]b′op)⊕n.
+This idempotent splits in
+Bimod(Sq[G]b, Sq[NG(D)]b′), so N ′
+0 lifts to a projective bimodule as well.
+By
+Proposition 3.4.2 it follows that the map N ′
+0 → N0 lifts to a map N ′ → N, and
+M := Cone(N ′ → N) lifts M0.
+□
+
+G-SPECTRA OF CYCLIC DEFECT
+33
+Theorem 4.7.2. Let b be a block of Sq[G], let D be its defect group, let b′ be the
+Brauer correspondent block of Sq[NG(D)]. Then there is an equivalence of stable
+∞-categories LMod(Sq[G]b) ∼= LMod(Sq[NG(D)]b′) that carries LMod(Sq[G]b)ft
+onto LMod(Sq[NG(D)]b′)ft and LMod(Sq[G]b)ω onto LMod(Sq[NG(D)]b′)ω.
+Proof. This is a consequence of Proposition 3.7.2. We apply it in the case where
+k → k′ is the truncation map Sq → Zq, A = Sq[NG(D)]b′, B = Sq[G]b, and
+F(A) is the bimodule M of Proposition 4.7.1.
+By Rouquier’s Theorem 4.1.1,
+M ⊗Sq Zq induces an equivalence LMod(Zq[NG(D)]b′) ∼= LMod(Zq[G]b) — the
+hypotheses of Proposition 3.7.2 are satisfied and so the functor M ⊗Sq[NG(D)]b′ −
+induces an equivalence LMod(Sq[NG(D)]b′) ∼= LMod(Sq[G]b).
+Like any equiv-
+alence, it preserves the subcategories of compact objects and so restricts to an
+equivalence LMod(Sq[NG(D)]b′)ω ∼= LMod(Sq[G]b)ω. Since each term of the com-
+plex M of Proposition 4.7.1 carries LMod(Sq[NG(D)]b′)ft to LMod(Sq[G]b)ft, the
+equivalence carries LMod(Sq[NG(D)]b′)ft into LMod(Sq[G]b)ft. Finally, this func-
+tor LMod(Sq[NG(D)]b′)ft → LMod(Sq[G]b)ft is an equivalence by Proposition 4.6.1.
+□
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+David Treumann. Representations of finite groups on modules over K-theory (with an
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+arXiv preprint arXiv:2105.14681, 2021.
+

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+arXiv:2301.13805v1  [math.PR]  31 Jan 2023
+PARABOLIC EQUATIONS AND SDES WITH TIME-INHOMOGENEOUS
+MORREY DRIFT
+D. KINZEBULATOV
+Abstract. We prove the unique weak solvability of stochastic differential equations with time-
+inhomogeneous drift in essentially the largest (scaling-invariant) Morrey class, i.e. with integra-
+bility parameter q > 1 close to 1. The constructed weak solutions constitute a Feller evolution
+family.
+The proofs are based on a detailed Sobolev regularity theory of the corresponding
+parabolic equation.
+1. Introduction
+1. We consider the problem of weak well-posedness of stochastic differential equation
+Xt = x −
+� t
+0
+b(r, Xr)dr +
+√
+2Bt,
+x ∈ Rd,
+(1)
+where Bt is a Brownian motion in Rd, under minimal assumptions on the time-inhomogeneous
+vector field b : R × Rd → Rd (drift), d ≥ 3.
+This equation or, more generally, stochastic
+equations additionally having variable, possibly discontinuous diffusion coefficients, arise e.g. in
+the problems of stochastic optimization and serve as a basis for many physical models. This
+requires, generally speaking, dealing with irregular, locally unbounded drifts. An illustrative
+example is equation (1) with velocity field b obtained by solving 3D Navier-Stokes equations,
+which models the motion of a small particle in a turbulent flow [26]. One is thus led to the
+problem of establishing weak and strong well-posedness of (1) under minimal assumptions on b.
+The latter can also be stated as the problem of finding the most general integral characteristics
+of b that determine whether (1) is weakly/strongly well-posed.
+Let us give a brief outline of the literature on stochastic differential equations (SDEs) with
+singular drift. We will try to keep the chronological order, but will be somewhat loose with the
+terminology by including in “well-posedness” the uniqueness results of different strength.
+The “sub-critical” Ladyzhenskaya-Prodi-Serrin class
+|b| ∈ Ll(R, Lp(Rd)),
+p ≥ d, l ≥ 2,
+d
+p + 2
+l < 1
+(2)
+was attained by Portenko [28] (weak solutions) and Krylov-Röckner [25] (strong solutions). See
+also Zhang [34, 35, 36]. Between [28] and [25], Bass-Chen [3] proved existence and uniqueness
+2010 Mathematics Subject Classification. 60H10, 47D07 (primary), 35J75 (secondary).
+Key words and phrases. Stochastic differential equations, weak solutions, singular drifts, Morrey class, Feller
+property.
+The research of the author is supported by the NSERC (grant RGPIN-2017-05567).
+1
+
+2
+D. KINZEBULATOV
+in law of weak solutions of (1) for b = b(x) in the Kato class of vector fields. The Kato class
+contains {|b| ∈ Lp(Rd), p > d} as well as some vector fields with entries not even in L1+ε
+loc (Rd),
+ε > 0, however, it does not contain {|b| ∈ Ld(Rd)}. Speaking of time-homogeneous drifts, of
+course, the fact that p = d is the optimal exponent on the scale of Lebesgue spaces can be seen
+from rescaling the parabolic equation.
+In [4], Beck-Flandoli-Gubinelli-Maurelli developed an approach to proving strong well-posedness
+of (1) with b in the critical Ladyzhenskaya-Prodi-Serrin class
+|b| ∈ Ll(R, Lp(Rd)),
+p ≥ d, l ≥ 2,
+d
+p + 2
+l ≤ 1.
+(3)
+for a.e. starting point x ∈ Rd via stochastic transport and stochastic continuity equations. They
+also considered the following example. Let
+b(x) =
+√
+δ d − 2
+2
+1|x|<1|x|−2x,
+(4)
+so |b| just misses to be Ld(Rd). If δ > 4(
+d
+d−2)2, i.e. the attraction to the origin by the drift is
+large enough, then SDE (1) with the starting point x = 0 does not have a weak solution. In [15],
+Semënov and the author showed that the Feller generator ∆−b·∇ with “weakly form-bounded”
+b = b(x), see (23) below, determines for every starting point x ∈ Rd a weak solution to (1)
+that is, moreover, unique among weak solutions that can be constructed via approximation. (To
+the best of the author’s knowledge, this was the first result on weak well-posedness of (1) that
+included both |b| ∈ Ld(Rd) and the model vector field (4) with δ small; it also included the
+elliptic Morrey class with q > 1 and the Kato class considered by Bass-Chen.) Returning to
+time-inhomogeneous drifts, we note that almost at the same time Wei-Lv-Wu [32], and later Nam
+[27], obtained results on weak well-posedness of (1) for every x ∈ Rd for time-inhomogeneous
+vector fields b that can be more singular than the ones in (2). Nevertheless, their results excluded
+b = b(x) with |b| ∈ Ld(Rd). In [21], Krylov proved strong well-posedness of (1), for every initial
+point and |b| ∈ Ld(Rd), by developing an approach based on his and Veretennikov’s old criterion
+for a weak solution to be a strong solution [31]. In [33], Xia-Xie-Zhang-Zhao established, among
+other results, weak well-posedness of (1) for every initial point and b ∈ Cb(R, Ld(Rd)). Röckner-
+Zhao [29] furthermore established weak well-posedness of (1), with any x ∈ Rd, for drifts in
+L∞(R, Ld,w(Rd)), plus the drifts in the critical LPS class. Here Ld,w(Rd) denotes the weak Ld
+class that contains vector fields in Ld(Rd), as well as more singular vector fields, such as (4). In
+[30], they obtained strong well-posedness of (1), for any starting point and b in the critical LPS
+class. In [13], the author and Madou established weak well-posedness of (1), for every starting
+point and form-bounded drifts, see example 5) below. This class contains L∞(R, Ld,w(Rd)) as
+well as some drifts that are not even in L∞(R, L2+ε(Rd)) for a given ε > 0. By the way, in [18]
+Semënov, Song and the author showed that the approach of [4] to the strong well-posedness of
+(1) via the stochastic transport/continuity equations also works for form-bounded vector fields
+b = b(x), although, again, one obtains strong well-posedness of (1) only for a.e. starting point.
+In a recent major advancement, Krylov [23] further verified the Veretennikov-Krylov criterion
+
+PARABOLIC EQUATIONS AND SDES WITH TIME-INHOMOGENEOUS MORREY DRIFT
+3
+for drifts in a large parabolic Morrey class:
+b ∈ Eq,
+q > d
+2 + 1,
+(5)
+see definition below (ignoring here some differences with [23] in the definition of parabolic Morrey
+class), thus establishing, for every starting point, strong well-posedness of (1) with drift in class
+(5). Class (5) contains L∞(R, Ld,w) as well as some drifts that are not in L∞(R, Lq+ε) for a
+given ε > 0 (see e.g. example in [22, Sect. 2]). Let us note in passing that some steps in his proof,
+such as gradient estimates, are, in fact, carried out for a larger class of form-bounded vector
+fields.
+The above outline does not discuss many interesting results on distribution-valued drifts and
+on the drifts satisfying additional assumptions on their structure, such as div b ≤ 0. We also did
+not discuss results on partial well-posedness of (1) with |b| ∈ Lp(Rd) in the supercritical regime
+d
+2 < p < d (in the sense of scaling). In this regard, see Zhao [37].
+In the present paper we consider drifts in the Morrey class Eq with integrability parameter
+q > 1 that can be chosen arbitrarily close to 1. Denote
+Cr(t, x) := {(s, y) ∈ Rd+1 | t ≤ s ≤ t + r2, |x − y| ≤ r}
+and, given a vector field b : Rd+1 → Rd with components in Lq
+loc(Rd+1), q ∈ [1, d + 2], set
+∥b∥E+
+q :=
+sup
+r>0,z∈Rd+1 r
+� 1
+|Cr|
+�
+Cr(z)
+|b(t, x)|qdtdx
+� 1
+q
+and, reversing the direction of time in b,
+∥b∥E−
+q :=
+sup
+r>0,z∈Rd+1 r
+� 1
+|Cr|
+�
+Cr(z)
+|b(−t, x)|qdtdx
+� 1
+q
+.
+Definition. We say that a vector field b belongs to the parabolic Campanato-Morrey (or, for
+brevity, Morrey) class Eq if ∥b∥Eq := ∥b∥E+
+q ∨ ∥b∥E−
+q
+< ∞.
+One has
+∥b∥Eq ≤ ∥b∥Eq1 if q < q1.
+If above b = b(x), then one obtains the usual elliptic Morrey class Mq, that is, |b| ∈ Lq
+loc(Rd)
+and
+∥b∥Mq :=
+sup
+r>0,y∈Rd r
+� 1
+|Br|
+�
+Br(y)
+|b(x)|qdx
+� 1
+q
+< ∞,
+where Br(y) is the closed ball of radius r centered at y.
+Our result, stated briefly, is as follows (see Theorems 1-3 for details). We will be using some
+notations defined in the end of this section.
+Theorem. Let d ≥ 3, let b : Rd+1 → Rd be a vector field in the Morrey class Eq with q > 1 close
+to 1. Let p ∈]1, ∞[. There exists a constant cd,p,q such that if ∥b∥Eq < cd,p,q, then the following
+are true:
+
+4
+D. KINZEBULATOV
+(i) There exists a unique weak solution to
+(λ − ∂t − ∆ + b(t, x) · ∇)u = 0,
+t < r,
+u(r, ·) = g(·) ∈ Lp(Rd) ∩ L2(Rd).
+The difference
+u(t, ·) −
+e−λ(r−t)
+(4π(r − t))
+d
+2
+�
+Rd e− |·−y|2
+4(r−t)g(y)dy
+(t < r),
+extended by 0 to t > r,
+is in the parabolic Bessel potential space W1+ 1
+p ,p(Rd+1).
+(ii) For p > d + 1, operators {P t,r}t<r defined by
+P t,rg := u(t),
+g ∈ C∞(Rd) ∩ L2(Rd)
+are extended to a backward Feller evolution family on C∞(Rd) that determines, for every x ∈ Rd,
+a weak solution to SDE (1) that is, moreover, unique in an appropriate class.
+Here are some examples of vector fields in Eq, q > 1:
+1) The critical Ladyzhenskaya-Prodi-Serrin class
+|b| ∈ Ll(R, Lp(Rd)),
+p ≥ d, l ≥ 2,
+d
+p + 2
+l ≤ 1.
+To prove the inclusion of this class into Eq it suffices to consider, by an elementary interpolation
+argument, only the cases l = 2, p = ∞ and l = ∞, p = d. In the former case the inclusion is
+trivial, in the latter case the inclusion follows using Hölder’s inequality.
+This example is strengthened in the next two examples.
+2) The vector fields b with |b| ∈ L2,w(R, L∞(Rd)) are in Eq for 1 < q < 2. Indeed, by a well
+known characterization of weak Lebesgue spaces, we have
+r
+� 1
+|Cr|
+�
+Cr
+|b|qdz
+� 1
+q
+≤ Cr
+� 1
+r2
+� t+r2
+t
+|˜b|qds
+� 1
+q
+˜b(t) := ∥b(t, ·)∥L∞(Rd)
+≤ C∥˜b∥L2,w(R).
+Hence, for example, a vector field b that satisfies
+∥b(t, ·)∥L∞(Rd) ≤ C
+√
+t,
+t > 0
+(and defined to be zero for t ≤ 0) is in Eq with 1 < q < 2.
+3) Moreover, by the well known inclusion of the weak Lebesgue space Ld,w(Rd) in the elliptic
+Morrey class,
+|b| ∈ L∞(R, Ld,w(Rd))
+⇒
+b ∈ Eq with 1 < q ≤ d.
+4) For every ε > 0, one can find a vector field b ∈ Eq such that |b| is not in Lq+ε
+loc (Rd+1).
+(In particular, selecting q > 1 sufficiently close to 1, we obtain vector fields b satisfying the
+assumptions of the theorem, and that are not in L1+ε
+loc (Rd+1) for a given ε > 0.)
+
+PARABOLIC EQUATIONS AND SDES WITH TIME-INHOMOGENEOUS MORREY DRIFT
+5
+5) A vector field b is said to be form-bounded if |b| ∈ L2
+loc(Rd+1) and for a.e. t ∈ R
+∥b(t, ·)ϕ∥2
+L2(Rd) ≤ δ∥∇ϕ∥2
+L2(Rd) + g(t)∥ϕ∥2
+L2(Rd)
+for all ϕ = ϕ(·) ∈ C∞
+c (Rd), for some constant δ > 0 and a function g ∈ L1
+loc(R) (written as
+b ∈ Fδ). The constant δ is called a form-bound of b.
+This class itself contains 1) and 3). In particular, by Hardy’s inequality, the vector field (4)
+is in Fδ with g = 0 (but not in any Fδ′ with δ′ < δ).
+Note that if a form-bounded vector field b depends only on time, then b ∈ L2
+loc(R). One can
+compare this with example 2).
+2. Let us say a few more words about the form-bounded vector fields. In the time-homogeneous
+case b = b(x) the form-boundedness of b ensures, by the Lax-Milgram Theorem, that, whenever
+δ < 1, the formal operator ∆ − b · ∇ has a realization as the generator of a quasi-contraction
+semigroup in L2(Rd) which, moreover, is bounded as an operator W 1,2(Rd) → W −1,2(Rd). The
+form-boundedness of b with δ < 1 is essentially the broadest condition on |b| that provides the
+minimal “classical” theory of operator −∆ + b · ∇ in L2.
+By considering dilates and translates of a bump function, one can show that the class of
+form-bounded vector fields Fδ is contained in E2. (Let us note, on the other hand, that the
+time-homogeneous vector fields in Eq, q > 2 are form-bounded [7], see also [6]. This remark
+extends right away e.g. to time-inhomogeneous vector fields b = b(t, ·) having uniformly bounded
+in t elliptic Morrey norm ∥ · ∥Mq, q > 2; they are, obviously, contained in Eq.) Note also that if
+b is form-bounded with g = 0, then ∥b∥E2 = c
+√
+δ for appropriate constant c = cd.
+The class of form-bounded vector fields is well known in the literature on regularity theory of
+parabolic equations [19] (although, perhaps, less so than the Morrey class). As we mentioned
+earlier, in the context of SDEs, condition b ∈ Fδ with δ < d−2 was used to develop a Sobolev
+regularity theory of the corresponding parabolic equation and construct a Feller evolution family
+[11] that determines, for every x ∈ Rd, a weak solution to (1) which is, moreover, unique in a
+broad class of weak solutions [13].
+3. In the present paper we deal with time-inhomogeneous vector fields that can be more
+singular than the ones covered by the form-boundedness condition. These new vector fields are
+situated in Eq − Es, 1 < q ≤ 2, s > 2. Let us note that the distinction between Eq, 1 < q < 2
+and smaller class Es, s > 2 is not only quantitative. Namely, consider a vector field b = b(x) in
+Es, s > 2. Since it is form-bounded, the term b · ∇ in the parabolic equation can be handled
+using quadratic inequality:
+�
+Rd(b · ∇u)udx ≤ γ
+�
+Rd |b|2u2dx + 1
+4γ
+�
+Rd |∇u|2dx,
+γ > 0,
+(6)
+where one estimates
+�
+Rd |b|2u2dx from above, using form-boundedness of b, by
+�
+Rd |∇u|2dx. This
+elementary argument plays a key role e.g. in the proof of gradient estimates in [4, 23, 11], needed
+to prove well-posedness of the corresponding stochastic equations. However, quadratic inequality
+(6) is unavailable under our assumption b ∈ Eq, 1 < q < 2.
+
+6
+D. KINZEBULATOV
+4. The proofs in [13] use a Feller evolution family constructed in [11] using a parabolic variant
+of the iteration procedure of [19]. In this paper we pursue a simpler operator-theoretic approach,
+replacing the iterations with the Duhamel series (written in “resolvent form”).
+Namely, we
+construct solution to inhomogeneous equation
+(λ + ∂t − ∆ + b(t, x) · ∇)u = f ∈ Lp(Rd+1)
+(which is essential to the rest of the paper) as
+u := (λ + ∂t − ∆)−1f − (λ + ∂t − ∆)− 1
+2− 1
+2p Qp(1 + Tp)−1Rp(λ + ∂t − ∆)− 1
+2p′ f,
+(7)
+where p′ :=
+p
+p−1 and, if b ∈ Eq, q > 1, the operators
+Rp = b
+1
+p · ∇(λ + ∂t − ∆)− 1
+2 − 1
+2p ,
+Qp = (λ + ∂t − ∆)− 1
+2p′ |b|
+1
+p′
+are bounded on Lp(Rd+1) and, provided ∥b∥Eq is sufficiently small and λ is large, the operator
+Tp = RpQp has norm ∥Tp∥p→p < 1, so the Duhamel series converges. The regularizing factor
+(λ+∂t−∆)−1/2−1/2p in (7) now yields the sought regularity of solution u provided that p is large
+(> d + 1). A similar argument was used in [12] in the elliptic setting, where an even larger than
+{b = b(x)} ∩ Eq, q > 1 class of time-homogeneous vector fields was treated (see (23) below); the
+weak well-posedness of SDEs with such drifts was addressed in [15]. The proof of the gradient
+estimates in [12], however, depends on the symmetry of the resolvents, and the transition from
+elliptic estimates to the results for the parabolic equations required development of some new
+approaches to constructing semigroups, using old ideas of Hille (pseudoresolvents) and Trotter,
+see also [14]. In the present paper the proofs are much shorter since we work directly with the
+parabolic operator. See also discussion after Theorem 3.
+The necessity of the assumption “∥b∥Eq cannot be too large” follows from the aforementioned
+counterexample to solvability of (1) with drift (4) when x = 0 and δ > 4(
+d
+d−2)2 (note that there
+∥b∥Eq = c
+√
+δ). It was recently proved in [17] that, given an arbitrary b ∈ Fδ with δ < 4, SDE
+(1) has a weak solution for every starting point x ∈ Rd, so the above example and this result
+become essentially sharp in high dimensions. (The fact that δ = 4 is critical can be seen from
+multiplying the parabolic equation (11) corresponding to (1) by u|u|p−2, integrating by parts
+and using quadratic inequality and form-boundedness as above.
+The admissible p that give
+e.g. an energy inequality turn out to be p >
+2
+2−
+√
+δ.)
+Notations. Set for 0 < α ≤ 2
+(λ − ∂t − ∆)− α
+2 h(t, x) :=
+� ∞
+t
+�
+Rd e−λ(s−t)
+1
+(4π(s − t))
+d
+2
+1
+(s − t)
+2−α
+2
+e− |x−y|2
+4(s−t) h(s, y)dsdy,
+(8)
+(λ + ∂t − ∆)− α
+2 h(t, x) :=
+� t
+−∞
+�
+Rd e−λ(t−s)
+1
+(4π(t − s))
+d
+2
+1
+(t − s)
+2−α
+2
+e− |x−y|2
+4(t−s) h(s, y)dsdy,
+(9)
+where λ ≥ 0. By a standard result, if λ > 0, then these operators are bounded on Lp(Rd+1),
+1 ≤ p ≤ ∞, with operator norm λ− α
+2 . If λ > 0, then (λ ± ∂t − ∆)−1 is the resolvent of a Markov
+generator on Lp(Rd+1), 1 ≤ p < ∞, which we will denote by λ ± ∂t − ∆, respectively. (The
+abuse of notation resulting from not indicating p should not cause any confusion.) In particular,
+
+PARABOLIC EQUATIONS AND SDES WITH TIME-INHOMOGENEOUS MORREY DRIFT
+7
+one has well defined fractional powers (λ ± ∂t − ∆)
+α
+2 . We refer to articles [2, 9], among others,
+regarding the properties of these operators.
+Denote by ⟨ , ⟩ the integration in d + 1 variables, i.e.
+⟨h⟩ :=
+�
+Rd+1 hdz,
+⟨h, g⟩ := ⟨hg⟩
+(all functions considered in this paper are real-valued).
+We denote by B(X, Y ) the space of bounded linear operators between Banach spaces X → Y ,
+endowed with the operator norm ∥ · ∥X→Y . Set B(X) := B(X, X).
+Put
+∥h∥p := ⟨|h|p⟩
+1
+p .
+Denote by ∥ · ∥p→q the (Lp(Rd+1), ∥ · ∥p) → (Lq(Rd+1), ∥ · ∥q) operator norm.
+We write T = s-X- limn Tn for T, Tn ∈ B(X) if Tf = limn Tnf in X for every f ∈ X.
+Let λ > 0 be fixed. Set
+Wα,p(Rd+1) := (λ + ∂t − ∆)− α
+2 Lp(Rd+1)
+endowed with the norm ∥h∥Wα,p := ∥(λ + ∂t − ∆)− α
+2 h∥p.
+Denote by C∞(Rd) the space of continuous functions on Rd vanishing at infinity, endowed
+with the sup-norm. Define in the same way C∞(Rd+1).
+We fix T > 0 and put
+DT := {(s, t) ∈ R2 | 0 ≤ s ≤ t ≤ T}.
+Recall that a family of positivity preserving L∞ contractions {Ut,s}(s,t)∈DT ⊂ B(C∞(Rd)) is
+called a Feller evolution family (on C∞(Rd)) if Ut,rUr,s = Ut,s for all r ∈ [s, t], Us,s = Id and
+Ur,s = s-C∞(Rd)- lim
+t↓r Ut,s
+for all s ≤ r < T.
+Define the following regularization of b : Rd+1 → Rd, b ∈ Eq, q > 1:
+bn := 1nb,
+where 1n is the indicator of {|b| ≤ n} ⊂ Rd+1.
+(10)
+We can additionally mollify bn to obtain a C∞ smooth approximation of b such that the Morrey
+norm of the approximating vector field does not exceed (1 + ε)∥b∥Eq for any fixed ε > 0, as is
+needed to turn a priori Sobolev regularity estimates for (11) into a posteriori estimates. However,
+a regularization of b given by (10) will suffice (in particular, we will be able to apply Itô’s formula
+to solutions of parabolic equations with drift bn).
+Put b
+1
+p := b|b|−1+ 1
+p .
+Let E = Ep := ∪ε>0e−ε|b|Lp(Rd+1), a dense subspace of Lp(Rd+1).
+Acknowledgements. The author is grateful to Renming Song for some useful comments.
+
+8
+D. KINZEBULATOV
+2. Main results
+1. We first develop a Sobolev regularity theory of the inhomogeneous parabolic equation
+(λ + ∂t − ∆ + b(t, x) · ∇)u = f
+on Rd+1.
+(11)
+The next theorem is essential for the rest of the paper.
+Theorem 1 (Sobolev regularity theory). Let b = bs + bb, where
+|bs| ∈ Eq for some q > 1 close to 1, and |bb| ∈ L∞(Rd+1)
+(12)
+(indices s and b stand for “singular” and “bounded”, respectively).
+The following are true:
+(i) For every p ∈]1, ∞[ there exist constants cd,p,q and λd,p,q such that if
+∥bs∥Eq < cd,p,q,
+then, for every λ ≥ λd,p,q, solutions un ∈ Lp(Rd+1) to the approximating parabolic equations
+(λ + ∂t − ∆ + bn · ∇)un = f,
+f ∈ Lp(Rd+1)
+converge in W1+ 1
+p ,p(Rd+1) to
+u := (λ + ∂t − ∆)−1f − (λ + ∂t − ∆)− 1
+2− 1
+2p Qp(1 + Tp)−1Rp(λ + ∂t − ∆)− 1
+2p′ f,
+(13)
+where the operators
+Rp = Rp(b) := b
+1
+p · ∇(λ + ∂t − ∆)− 1
+2− 1
+2p ,
+Qp = Qp(b) :=
+�(λ + ∂t − ���)− 1
+2p′ |b|
+1
+p′ ↾ E
+�clos
+p→p
+(14)
+are bounded on Lp(Rd+1), and the operator Tp := RpQp has norm
+∥Tp∥p→p < 1.
+(15)
+(ii) If above p > d+1, then, by (13) and by the parabolic Sobolev embedding, the convergence
+is uniform on Rd+1 and u ∈ C∞(Rd+1).
+Remarks. 1. If b is bounded, then Qp = (λ+∂t −∆)− 1
+2p′ |b|
+1
+p′ , and so the RHS of (13) is simply
+the Duhamel series representation for the solution to (11) in Lp(Rd+1) provided by the standard
+theory.
+2. The constraint ∥bs∥Eq < cd,p,q is needed to ensure that ∥Tp∥p→p < 1, see Proposition 2.
+3. If e.g. b ∈ C∞(R, Ld(Rd)) or b = b(x) is in Ld(Rd), then one can represent b = bs + bb with
+∥bs∥Eq arbitrarily small (by defining bb to be a cutoff of b such that the remaining part bs has
+sufficiently small L∞(R, Ld(Rd)) norm).
+Given a general b satisfying (12), it is natural to ask, in what sense u defined by (13) solves
+the parabolic equation (11)? Let us consider the case p = 2 and f ∈ L2(Rd+1).
+
+PARABOLIC EQUATIONS AND SDES WITH TIME-INHOMOGENEOUS MORREY DRIFT
+9
+Definition. We say that a function u ∈ W
+3
+2 ,2 is a weak solution to (11) if the following identity
+is satisfied:
+⟨(λ + ∂t − ∆)
+3
+4u, (λ + ∂t − ∆)
+3
+4 η⟩ + ⟨R2(b)(λ + ∂t − ∆)
+3
+4u, Q∗
+2(b)(λ + ∂t − ∆)
+3
+4 η⟩
+= ⟨f, (λ − ∂t − ∆)− 1
+4 (λ + ∂t − ∆)
+3
+4 η⟩
+(16)
+for all η ∈ C∞
+c (Rd+1).
+(Identity (16) is obtained by formally multiplying equation (11) by (λ−∂t−∆)− 1
+4(λ+∂t−∆)
+3
+4 η
+and integrating over Rd+1. Note that Q∗
+2(b) = |b|
+1
+2 (λ − ∂t − ∆)− 1
+4 is in B(L2) by Proposition 1.)
+Then u defined by (13) is the unique in W
+3
+2,2 weak solution to (11). See Remark 4 for the
+proof.
+Remark 1. Thus, in order to have weak well-posedness of (11) for drifts satisfying (12) for
+q > 1 we have to shift the standard scale of Hilbert spaces W1,2 ⊂ L2(Rd+1) ⊂ W−1,2 to
+W
+3
+2,2 ⊂ W
+1
+2 ,2 ⊂ W− 1
+2,2. If we were to consider instead of (11) a more general equation (λ + ∂t −
+∇ · a · ∇ + b · ∇)u = f with a uniformly elliptic discontinuous matrix a, then the second order
+term would force us to work in the standard scale, and hence would require more restrictive
+assumptions on b: the form-boundedness, see the beginning of the paper (on the scale of Morrey
+spaces this will be (12) with q > 2).
+An analogous result can be obtained for f ∈ Lp(Rd+1) with (16) modified according to (13).
+2. Fix T > 0. For given n = 1, 2, . . . and 0 ≤ r < T, let vn ∈ Cb([r, T], C∞(Rd)) denote the
+solution to the Cauchy problem
+
+
+
+(λ + ∂t − ∆ + bn(t, x) · ∇)vn = 0
+(t, x) ∈]r, T] × Rd,
+vn(r, ·) = g(·) ∈ C∞(Rd),
+(17)
+where bn’s are defined by (10). By a standard result, for every n, the operators
+Ut,r
+n g := vn(t),
+0 ≤ r ≤ t ≤ T
+constitute a Feller evolution family on C∞(Rd).
+Let δs=r denote the delta-function in the time variable s. Put
+(λ + ∂t − ∆)−1δs=rg(t, x) := 1t≥re−λ(t−r)(4π(t − r))− d
+2
+�
+Rd e− |x−y|2
+4(t−r) g(y)dy,
+∇(λ + ∂t − ∆)− 1
+2 − 1
+2p′ δs=rg := 1t≥re−λ(t−r)(t − r)− 1
+2+ 1
+2p′ (4π(t − r))− d
+2
+�
+Rd ∇xe− |x−y|2
+4(t−r) g(y)dy.
+Recall DT = {0 ≤ r ≤ t ≤ T}.
+Theorem 2 (C∞ theory). Under the assumptions of Theorem 1, let ∥b∥Eq < cd,p,q for a p > d+1.
+Then the following are true:
+(i) The limit
+Ut,r := s-C∞(Rd)- lim
+n Ut,r
+n
+uniformly in (r, t) ∈ DT
+exists and determines a Feller evolution family on C∞(Rd).
+
+10
+D. KINZEBULATOV
+(ii) For every initial function g ∈ C∞(Rd) ∩ W 1,p(Rd), v(t) := Ut,rg, where (r, t) ∈ DT , has
+representation
+v = (λ + ∂t − ∆)−1δs=rg − (λ + ∂t − ∆)− 1
+2− 1
+2p Qp(1 + Tp)−1GpSpg,
+(18)
+where Gp = Gp(b) := b
+1
+p (λ + ∂t − ∆)− 1
+2p ∈ B(Lp(Rd+1)) and Spg := ∇(λ + ∂t − ∆)− 1
+2− 1
+2p′ δs=rg
+satisfies ∥Spg∥Lp(Rd+1) ≤ Cp,d∥∇g∥Lp(Rd).
+(iii) As a consequence of (18) and the parabolic Sobolev embedding, we obtain
+sup
+(r,t)∈DT ,x∈Rd |v(t, x; r)| ≤ C∥g∥W 1,p(Rd).
+3. Recall that a probability measure Px, x ∈ Rd on (C([0, T], Rd), Bt = σ(ωr | 0 ≤ r ≤ t)),
+where ωt is the coordinate process, is said to be a martingale solution to SDE
+Xt = x −
+� t
+0
+b(r, Xr)dr +
+√
+2Bt.
+(19)
+if
+1) Px[ω0 = x] = 1;
+2) Ex
+� r
+0 |b(t, ωt)|dt < ∞, 0 < r ≤ T;
+3) for every f ∈ C2
+c (Rd) the process
+r �→ f(ωr) − f(x) +
+� r
+0
+(−∆f + b · ∇f)(t, ωt)dt
+is a Br-martingale under Px.
+A martingale solution Px of (19) is said to be a weak solution if, upon completing Bt with
+respect to Px (to, say, ˆBt), there exists a Brownian motion Bt on
+�C([0, T], Rd), ˆBt, Px
+� such that
+ωr = x −
+� r
+0
+b(t, ωt)dt +
+√
+2Br,
+r ≥ 0
+Px-a.s.
+Put
+P t,r(b) := UT−t,T−r(˜b),
+˜b(t, x) = b(T − t, x)
+where 0 ≤ t ≤ r ≤ T.
+Theorem 3. Under the assumptions of Theorem 2 the following are true:
+(i) The backward Feller evolution family {P t,r}0≤t≤r≤T is conservative, i.e. the density P t,r(x, ·)
+satisfies
+⟨P t,r(x, ·)⟩ = 1
+for all x ∈ Rd,
+and determines probability measures Px, x ∈ Rd on (C([0, T], Rd), Bt), such that
+Ex[f(ωr)] = P 0,rf(x),
+0 ≤ r ≤ T,
+f ∈ C∞(Rd).
+(ii) For every x ∈ Rd, the probability measure Px is a weak solution to (19).
+
+PARABOLIC EQUATIONS AND SDES WITH TIME-INHOMOGENEOUS MORREY DRIFT
+11
+(iii) For every x ∈ Rd and f satisfying (12), given a p > d + 1 as in Theorem 2, there exists
+a constant c such that for all h ∈ Cc(Rd+1)
+Ex
+� T
+0
+|f(r, ωr)h(r, ωr)|dr ≤ c∥1[0,T]|f|
+1
+p h∥p.
+(In particular, one can take f = b.)
+(iv) Any martingale solution Qx to (19) that satisfies
+EQx
+� T
+0
+|b(r, ωr)h(r, ωr)|dr ≤ c∥1[0,T]|b|
+1
+p h∥p,
+h ∈ Cc(Rd+1),
+for some p > d + 1,
+(20)
+coincides with Px.
+2.1. Key bounds. Here is the key bound used in the proofs of Theorems 1 and 2.
+Proposition 1. Let |b| ∈ Eq for some q > 1 close to 1. Then for every p ∈]1, ∞[ there exists a
+constant cp,q such that
+∥|b|
+1
+p (± ∂t − ∆)− 1
+2p ∥p→p ≤ cp,q∥b∥
+1
+p
+Eq
+In the time homogeneous case b = b(x) the estimate on ∥|b|
+1
+p (λ−∆)− 1
+2p ∥Lp(Rd)→Lp(Rd) in terms
+of the elliptic Morrey norm of |b| is due to [1, Theorem 7.3]. Similar estimates in the parabolic
+case were obtained in [24, proof of Prop. 4.1]. There the proof is carried out for a different set
+of parameters than the one needed in this paper, so we included the details in Section 4.
+As an immediate consequence of Proposition 1 we obtain the following
+Proposition 2. Let b = bs + bb, satisfy (12). Then, for all λ > 0,
+∥Rp∥p→p ≤ Cd,p,q∥bs∥
+1
+p
+Eq + cλ− 1
+2p ∥bb∥
+1
+p
+L∞(Rd+1)
+(21)
+∥Qp∥p→p ≤ C′
+p,q∥bs∥
+1
+p′
+Eq + c′λ− 1
+2p′ ∥bb∥
+1
+p′
+L∞(Rd+1).
+(22)
+The first estimate (21) follows from Proposition 1 using the boundedness of parabolic Riesz
+transforms (see [9]). The second estimate (22) follows from Proposition 1 by duality.
+2.2. Some remarks.
+Remark 2. Theorems 1, 2 are time inhomogeneous counterparts of the results in [11], Theorem
+3 is a time inhomogeneous counterpart of the result in [15], where the authors treated vector
+fields b : Rd → Rd that can be more singular than the ones considered in this paper, namely,
+|b| ∈ L1
+loc(Rd) and there exists δ > 0 such that, for some λ > 0,
+��|b|
+1
+2ϕ
+��
+L2(Rd) ≤ δ
+��(λ − ∆)
+1
+4 ϕ
+��
+L2(Rd),
+∀ ϕ ∈ C∞
+c (Rd).
+(23)
+These vector fields are called weakly form-bounded. The fact that the time-homogeneous vector
+fields in Eq, 1 < q ≤ 2 are weakly form-bounded follows from [1, Theorem 7.3]. A key difference
+between [11] and the present paper is in the elliptic analogue of estimate (15), i.e.
+∥ ˜Tp∥Lp(Rd)→Lp(Rd) < 1,
+where ˜Tp := b
+1
+p · ∇(µ − ∆)−1|b|
+1
+p′ ,
+(24)
+
+12
+D. KINZEBULATOV
+needed to ensure the convergence of the Neumann series. This estimate is proved in [11] directly,
+without splitting ˜Tp into a product of operators b
+1
+p ·∇(µ−∆)− 1
+2+ 1
+2p and (µ−∆)− 1
+2p′ |b|
+1
+p′ as we do
+for Tp in Theorem 1. In fact, the previous two operators are not even bounded on Lp(Rd) under
+the assumption (23); to have the Lp(Rd) boundedness one has to replace the exponents − 1
+2 + 1
+2p,
+− 1
+2p′ by − 1
+2 + 1
+2p − ε, − 1
+2p′ − ε for a ε > 0 (this shows, by the way, that the difference between
+the elliptic Morrey class Mq with q > 1 and the larger class (23) is quite significant). However,
+the proof of (24) in [11] requires inequalities for symmetric Markov generators, not valid for the
+parabolic operator ∂t − ∆ (although, it seems, one can address a parabolic counterpart of (23)
+via an appropriate symmetrization of ∂t − ∆, which we plan to do in a subsequent paper).
+Remark 3. Despite what was said in the introduction, [13, 11] and the present paper are not
+directly comparable. Namely, the results in [13, 11] with b ∈ Fδ can be extended more or less
+directly to the non-divergence form equation
+� − a(t, x) · ∇2 + b(t, x) · ∇
+�u = f,
+u(s, ·) = g(·),
+t > s,
+(25)
+and the corresponding Itô’s SDE having “form-bounded diffusion coefficients”.
+Namely, the
+matrix a is assumed to be bounded, uniformly elliptic and have
+∇aij ∈ Fδij,
+1 ≤ i, j ≤ d.
+(26)
+See [16] where this scheme was carried out in the time-homogeneous case b = b(x), a = a(x).
+(Let us note that since ∇aij(x) are form-bounded, they are in the elliptic Morrey class with
+q = 2, see the introduction. Hence by Poincaré’s inequality such aij belong to the VMO class,
+see details in [22, Sect.3].) However, a similar extension of the results of the present paper for
+(1) with b ∈ Eq, 1 < q ≤ 2 requires more regular a, see e.g. Remark 1.
+3. Some corollaries of Theorem 1 and Theorem 2(ii)
+The following results will be needed in the proof of Theorem 3.
+Corollary 1. Let vector fields b, f satisfy (12). Let bn be given by (10) and let us define fn
+similarly. Then, under the assumptions on ∥bs∥Eq of Theorem 1, for every λ ≥ λd,p,q solutions
+un ∈ Lp(Rd+1) to the approximating parabolic equations
+(λ + ∂t − ∆ + bn · ∇)un = |fn|h,
+h ∈ Cc(Rd+1)
+converge in W1+ 1
+p ,p(Rd+1) to
+u := (λ + ∂t − ∆)−1|f|h − (λ + ∂t − ∆)− 1
+2− 1
+2p Qp(b)(1 + Tp(b))−1Rp(b)Qp(f)|f|
+1
+p h,
+In particular, if p > d + 1, then the convergence is uniform on Rd+1 and
+sup
+Rd+1 |u| ≤ C∥f
+1
+p h∥p.
+
+PARABOLIC EQUATIONS AND SDES WITH TIME-INHOMOGENEOUS MORREY DRIFT
+13
+The reason we include h in Corollary 1 is because in general |f| is only in L1
+loc(Rd+1), not in
+L1(Rd+1). In the proof of Theorem 3 we will be applying Corollary 1 and Corollaries 2, 3 below
+to fn equal to either bn or (with some abuse of notation) to bn − bk.
+Corollary 2. Under the assumptions and notation of Corollary 1, if p > d + 1, then, for
+every λ ≥ λd,p,q, solutions vn ∈ Cb([r, T], C∞(Rd)) to the approximating inhomogeneous Cauchy
+problems
+(λ + ∂t − ∆ + bn · ∇)vn = 1[r,T]|fn|h
+on ]r, T] × Rd,
+vn(r, ·) = g(·) ∈ C∞(Rd) ∩ W 1,p(Rd),
+where 0 ≤ r < T, converge uniformly on DT × Rd to
+v := (λ + ∂t − ∆)−1�1|f|h + δs=rg
+�
+− (λ + ∂t − ∆)− 1
+2− 1
+2p Qp(b)(1 + Tp(b))−1�Rp(b)Qp(1f)|1f|
+1
+p h + Gp(b)Spg
+�,
+1 := 1[r,T],
+sup
+(r,t)∈DT ,x∈Rd |v(t, x; r)| ≤ C1∥1f
+1
+p h∥p + C2∥g∥W 1,p(Rd).
+We also have the following weighted variant of Corollary 2, which we will record for bounded
+vector fields bn and fn and λ = 0, as will be needed in the proof of Theorem 3 in Section 8.
+Let q > 1 (close to 1) be from the hypothesis on b in Theorem 1. Set
+ρ(x) := (1 + l|x|2)−ν,
+x ∈ Rd,
+where ν > d
+2p +
+1
+pq′ is fixed (so that ρ ∈ Lp(Rd) and (42) below holds) and l > 0 is to be chosen.
+We have
+|∇ρ| ≤ ν
+√
+lρ =: c1
+√
+lρ,
+|∆ρ| ≤ 2ν(2ν + d + 2)lρ =: c2lρ.
+(⋆)
+Corollary 3. Under the assumptions of Corollary 1, if p > d + 1, then, provided that the
+constant l in the definition of ρ is chosen sufficiently small, solutions vn ∈ Cb([r, T], C∞(Rd))
+to the approximating inhomogeneous Cauchy problems
+(∂t − ∆ + bn · ∇)vn = ±1[r,T]|fn|
+on [r, T] × Rd,
+vn(r, ·) = g ∈ C∞(Rd) ∩ W 1,p(Rd),
+where 0 ≤ r < T, satisfy, for all t ∈]r, T],
+sup
+[r,t]×Rd |ρvn| ≤ C1∥ρ1[r,t]|fn|
+1
+p ∥p + C2∥ρg∥W 1,p(Rd)
+(27)
+and, putting ρy(x) := ρ(x − y),
+sup
+[r,t]×Rd |vn| ≤ sup
+y∈Zd(C1∥ρy1[r,t]|fn|
+1
+p ∥p + C2∥ρyg∥W 1,p(Rd))
+(28)
+≤ ˜C1(t − r)γ∥f∥
+1
+p
+Eq + ˜C2∥g∥W 1,p
+(29)
+with constants C1, C2, ˜C1, ˜C2 and γ > 0 independent of n and t.
+
+14
+D. KINZEBULATOV
+We prove Corollary 3 in Section 7.
+4. Proof of Proposition 1
+It suffices to carry out the proof for bn defined by (10) and then use the Dominated Conver-
+gence Theorem. So, without loss of generality, everywhere below b is bounded. Below we follow
+[24].
+Set
+Mβh(t, x) := sup
+ρ>0
+ρβ 1
+|Cρ|
+�
+Cρ(t,x)
+|h|dz,
+0 ≤ β ≤ d − 2,
+and define the maximal function
+Mh(t, x) := sup
+ρ>0
+1
+|Cρ|
+�
+Cρ(t,x)
+|h|dz.
+Also, define
+ˆ
+Mh(t, x) :=
+sup
+(t,x)∈C
+1
+|C|
+�
+C
+|h|dz,
+where the supremum is taken over all cylinders C = Cρ(z) ∋ (t, x), z ∈ Rd+1, ρ > 0.
+Put Pα := (−∂t − ∆)− α
+2 . We will need
+Lemma 1 (see [24, Lemma 2.2]). For every β ∈]0, d + 2], 0 < α < β there exists C > 0 such
+that, for all f ≥ 0,
+Pαf ≤ C(Mβf)
+α
+β (Mf)1− α
+β .
+Let us prove the first inequality:
+|⟨|b|(P 1
+p f)p⟩| ≤ cp
+p,q∥b∥Eq∥f∥p
+p,
+f ∈ Lp(Rd+1)
+(30)
+(the proof of the second one is similar). Put u := P 1
+p f. Then we estimate the LHS of (30) as
+|⟨|b|(P 1
+p f)p⟩| = |⟨|b||u|p−1, P 1
+p f⟩|
+≤ |⟨P ∗
+1
+p (|b||u|p−1), f⟩| ≤ ∥P ∗
+1
+p (|b||u|p−1)∥p′∥f∥p.
+(31)
+Here P ∗
+α = (∂t − ∆)− α
+2 is the adjoint of the operator Pα in L2(Rd+1). To obtain (30), we need
+to bound the coefficient ∥P ∗
+1
+p (|b|up−1)∥p′. To this end, we estimate pointwise
+P ∗
+1
+p (|b||u|p−1) = P ∗
+1
+p (|b|
+1
+p +γ|b|
+1
+p′ −γ|u|p−1) ≤ P ∗
+1
+p (|b|1+γp)
+1
+p (P ∗
+1
+p (|b|1−γp′|u|p))
+1
+p′
+for a small γ > 0 such that 1 + γp < q0 for some fixed q0 < q. Hence
+∥P ∗
+1
+p (|b||u|p−1)∥p′
+p′ ≤ ⟨|b|1−γp′|u|p, P 1
+p [P ∗
+1
+p (|b|1+γp)]
+1
+p−1⟩.
+(32)
+
+PARABOLIC EQUATIONS AND SDES WITH TIME-INHOMOGENEOUS MORREY DRIFT
+15
+1) By Lemma 1 with α = 1
+p, β = 1 + γp (or rather its straightforward variant for P ∗
+α),
+P ∗
+1
+p (|b|1+γp) ≤ C∥b∥
+1
+p
+E1+γp( ˆ
+M|b|1+γp)1− 1
+p
+1
+1+γp
+≤ C∥b∥
+1
+p
+Eq0( ˆ
+M|b|1+γp)1− 1
+p
+1
+1+γp
+At this point let us assume that
+ˆ
+M|b|1+γp ≤ C0|b|1+γp,
+(33)
+but will get rid of this assumption later. Then
+P ∗
+1
+p (|b|1+γp) ≤ C2∥b∥
+1
+p
+Eq0|b|1+γp− 1
+p .
+2) After applying the last estimate in (32), it remains to estimate P 1
+p (|b|
+1
+p +γp′). Selecting γ
+even smaller, if needed, one may assume that 1
+p + γp′ < q0. By Lemma 1,
+P 1
+p (|b|
+1
+p +γp′) ≤ C(M 1
+p +γp′|b|
+1
+p +γp′)
+1
+p
+1
+1
+p +γp′ (M|b|
+1
+p +γp′)
+1− 1
+p
+1
+1
+p +γp′
+≤ C∥b∥
+1
+p
+Eq0( ˆ
+M|b|
+1
+p +γp′)
+1− 1
+p
+1
+1
+p +γp′
+In addition to (33), let us temporarily assume that
+ˆ
+M|b|
+1
+p +γp′ ≤ C0|b|
+1
+p +γp′.
+(34)
+Then
+P 1
+p (|b|
+1
+p +γp′) ≤ C2∥b∥
+1
+p
+Eq0|b|γp′.
+3) Applying the results from 1), 2) in (32), we obtain
+∥P ∗
+1
+p (|b||u|p−1)∥p′ ≤ C3∥b∥
+1
+p
+Eq0⟨|b||u|p⟩
+1
+p′ .
+Therefore, (31) yields
+⟨|b||u|p⟩ ≤ C3∥b∥
+1
+p
+Eq0⟨|b||u|p⟩
+1
+p′ ∥f∥p
+so
+⟨|b||u|p⟩
+1
+p ≤ C3∥b∥
+1
+p
+Eq0∥f∥p.
+(35)
+4) Now one gets rid of the assumptions (33) and (34) at expense of replacing ∥b∥Eq0 in (35)
+by ∥b∥Eq, where, recall, q0 < q. This, in turn, will give (30). Fix q0 < q1 < q and define
+˜b := ( ˆ
+M|b|q1)
+1
+q1 . Then ˜b ≥ |b| and ˜b satisfies
+ˆ
+M˜bq0 ≤ C0˜bq0
+(36)
+(see [8, p.158]). Since 1+ γp < q0, 1
+p + γp′ < q0, both inequalities (33) and (34) for ˜b follow from
+(36), and so we have
+⟨|b||u|p⟩
+1
+p ≤ C3∥˜b∥
+1
+p
+Eq0∥f∥p.
+
+16
+D. KINZEBULATOV
+It remains to apply inequality ∥˜b∥Eq0 ≤ C∥b∥Eq, which was established in [24, proof of Prop. 4.1].
+□
+5. Proof of Theorem 1
+The fact that Qp, Rp are bounded on Lp(Rd+1), and the operator Tp = RpQp has norm
+∥Tp∥p→p < 1 provided that ∥bs∥Eq is sufficiently small and λ is sufficiently large, is an immediate
+consequence of Proposition 2. Thus,
+Θp(b) := (λ + ∂t − ∆)−1 − (λ + ∂t − ∆)− 1
+2− 1
+2p Qp(1 + Tp)−1Rp(λ + ∂t − ∆)− 1
+2p′
+is in B(Lp).
+Recall: bn := 1nb, where 1n is the indicator of {(t, x) ∈ Rd+1 | |b(t, x)| ≤ n}. If un ∈ Lp(Rd+1)
+denotes the solution to (∂t − ∆ + bn · ∇)un = f, which exists by the classical theory, then
+un = Θp(bn)f,
+where Θp(bn)f coincides with the Duhamel series representation for un.
+Next, let us note that
+Rp(bn) → Rp(b), Qp(bn) → Qp(b)
+strongly in Lp(Rd+1)
+(37)
+as follows from (21), (22) and the Dominated Convergence Theorem. Hence
+un := Θp(bn) → u = Θp(b) in W1+ 1
+p ,p(Rd+1),
+(38)
+as needed.
+□
+Remark 4. In the comment after Theorem 1 we promised to prove the existence and uniqueness
+of weak solution to (11). The argument is standard and goes as follows. In the proof of Theorem
+1 above take p = 2, so u ∈ W
+3
+2 ,2. Multiplying (λ + ∂t − ∆ + bn · ∇)un = f, n = 1, 2, . . . , by
+ϕ = (λ − ∂t − ∆)− 1
+4 (λ + ∂t − ∆)
+3
+4 η, η ∈ C∞
+c (Rd+1) and integrating over Rd+1, we have
+⟨(λ + ∂t − ∆)
+3
+4un, (λ + ∂t − ∆)
+3
+4 η⟩+⟨R2(bn)(λ + ∂t − ∆)
+3
+4 un, Q∗
+2(bn)(λ + ∂t − ∆)
+3
+4 η⟩
+= ⟨f, (λ − ∂t − ∆)− 1
+4 (λ + ∂t − ∆)
+3
+4 η⟩,
+where, recall, Q∗
+2(b) = |b|
+1
+2(λ − ∂t − ∆)− 1
+4 ∈ B(L2). In view of (38),
+⟨(λ + ∂t − ∆)
+3
+4un, (λ + ∂t − ∆)
+3
+4 η⟩ → ⟨(λ + ∂t − ∆)
+3
+4u, (λ + ∂t − ∆)
+3
+4 η⟩
+(n → ∞).
+Next,
+⟨R2(bn)(λ + ∂t − ∆)
+3
+4 un, Q∗
+2(bn)(λ + ∂t − ∆)
+3
+4 η⟩
+= ⟨R2(bn)(λ + ∂t − ∆)
+3
+4 (un − u), Q∗
+2(bn)(λ + ∂t − ∆)
+3
+4η⟩
++ ⟨R2(bn)(λ + ∂t − ∆)
+3
+4 u, (Q∗
+2(bn) − Q∗
+2(b))(λ + ∂t − ∆)
+3
+4η⟩
++ ⟨R2(bn)(λ + ∂t − ∆)
+3
+4 u, Q∗
+2(b)(λ + ∂t − ∆)
+3
+4η⟩.
+
+PARABOLIC EQUATIONS AND SDES WITH TIME-INHOMOGENEOUS MORREY DRIFT
+17
+By (38), Q∗
+2(bn) → Q∗
+2(b) strongly in L2(Rd+1) (by the same argument as in (37)) we get
+that the first two terms in the RHS tend to 0 as n → ∞. By (37), the last term tends to
+⟨R2(b)(λ + ∂t − ∆)
+3
+4u, Q∗
+2(b)(λ + ∂t − ∆)
+3
+4η⟩. Hence u is a weak solution to (11) in the sense of
+definition (16).
+Let v ∈ W
+3
+2,2 be another weak solution. Put
+τ[v, η] := ⟨(λ + ∂t − ∆)
+3
+4 v, (λ + ∂t − ∆)
+3
+4 η⟩ + ⟨R2(b)(λ + ∂t − ∆)
+3
+4v, Q∗
+2(b)(λ + ∂t − ∆)
+3
+4η⟩,
+where η ∈ C∞
+c (Rd+1). We have
+|⟨R2(b)(λ + ∂t − ∆)
+3
+4 v, Q∗
+2(b)(λ + ∂t − ∆)
+3
+4 η⟩| ≤ c∥v∥W
+3
+2 ,2∥η∥W
+3
+2 ,2
+where c < 1 by our assumption on b. We extend τ[v, η] to η ∈ W
+3
+2,2 by continuity. Now we have
+τ[v − u, η] = 0, where u is the weak solution constructed above, so it suffices to choose η = v − u
+to arrive at 0 = τ[v − u, v − u] ≥ (1 − c)∥v∥2
+W
+3
+2 ,2, hence v = u.
+6. Proof of Theorem 2
+Let us define Ut,rg := v(t) (t ≥ r), g ∈ C∞(Rd) ∩ W 1,p(Rd) where v(t) is given by (18). Since
+Ut,r
+n
+are L∞ contractions, it suffices to prove (we consider convergence in (r, t) ∈ DT )
+U = Cb(DT , C∞(Rd))- lim
+n Ung,
+(39)
+and then extend operators Ut,r by continuity to g ∈ C∞(Rd). The reproduction property of Ut,r
+and the preservation of positivity will follow from the corresponding properties of Ut,r
+n .
+Proof of (39). Put vn := Ut,r
+n g. We have
+vn = (λ + ∂t − ∆)−1δs=rg − (λ + ∂t − ∆)− 1
+2− 1
+2p Qp(bn)(1 + Tp(bn))−1Gp(bn)Spg.
+This is the usual Duhamel series representation for vn. We know from the proof of Theorem 1
+that operators Qp(bn), Tp(bn), Gp(bn) are bounded on Lp(Rd+1) with operator norms indepen-
+dent of n. In turn, operator Sp satisfies
+∥Spg∥Lp(Rd+1) ≤ Cp,d∥∇g∥Lp(Rd).
+(Indeed, taking for brevity r = 0, we have by definition
+Spg(t, x) = 1t≥0e−λtt− 1
+2+ 1
+2p′ (4πt)− d
+2
+�
+Rd ∇xe− |x−y|2
+4t
+g(y)dy.
+Hence
+∥Spg∥p
+Lp(Rd+1) =
+�
+R
+1t≥0∥Sp(t)∥p
+Lp(Rd)dt ≤
+� ∞
+0
+e−λptt(− 1
+2+ 1
+2p′ )pdt∥∇g∥p
+Lp(Rd),
+where (− 1
+2 +
+1
+2p′ )p = − 1
+2 (> −1 so the integral in time converges).)
+
+18
+D. KINZEBULATOV
+Clearly, (λ + ∂t − ∆)−1δs=rg ∈ Cb([r, ∞[, C∞(Rd)). Thus, to prove (39), it remains to note
+that Qp(bn) → Qp(b), Tp(bn) → Tp(b) and Gp(bn) → Gp(b) strongly in Lp(Rd+1) (see proof of
+Theorem 1), so that by the parabolic Sobolev embedding, since p > d + 1,
+(λ + ∂t − ∆)− 1
+2 − 1
+2p Qp(bn)(1 + Tp(bn))−1Gp(bn)Spg
+→ (λ + ∂t − ∆)− 1
+2− 1
+2p Qp(b)(1 + Tp(b))−1Gp(b)Spg
+in C∞(Rd+1) as n → ∞.
+The required convergence (39) follows.
+□
+7. Proof of Corollary 3 (weighted bounds)
+It will be convenient to carry out the proof for solutions vn to
+(λ + ∂t − ∆ + bn · ∇)vn = ±1[r,T]|fn|,
+on [r, T] × Rd,
+vn(r, ·) = g ∈ C∞(Rd) ∩ W 1,p(Rd)
+where λ ≥ λd,p,q > 0. Since we are working on a fixed finite time interval [r, T], this change will
+amount to multiplying solution vn by a bounded function eλ(t−r) which, clearly, does not affect
+the sought estimates.
+Put for brevity v = vn. We will carry out the proof on the maximal interval [r, t], i.e. for
+t = T. Also, without loss of generality, the RHS of the equation is 1[r,T]|fn| and the initial
+function g ≥ 0, so v ≥ 0. We have for the weight ρ defined before the corollary,
+λρv + ∂t(ρv) − ∆(ρv) + bn · ∇(ρv)
+= ρ(λv + ∂tv − ∆v + bn · ∇v) − 2∇ρ · ∇v + (−∆ρ)v + bnv · ∇ρ
+(v solves the parabolic equation above)
+= ρ1[r,T]|fn| + K,
+K := −2∇ρ · ∇v + (−∆ρ)v + bnv · ∇ρ.
+Let us rewrite the term K as follows:
+K = −2
+�∇ρ
+ρ · ∇(ρv) − (∇ρ)2
+ρ
+v
+�
++ (−∆ρ)v + bnv · ∇ρ.
+Hence
+λρv + ∂t(ρv) − ∆(ρv) + ˜bn · ∇(ρv) = ρ1[r,T]|fn| + ˜K,
+(40)
+where ˜bn := bn + 2∇ρ
+ρ and
+˜K = 2(∇ρ)2
+ρ
+v + (−∆ρ)v + bnv · ∇ρ.
+Note that by (⋆) the term 2∇ρ
+ρ
+in ˜bn is a bounded vector field whose sup norm can be made
+as small as needed by selecting l in the definition of ρ sufficiently small; we will be selecting l
+sufficiently small.
+
+PARABOLIC EQUATIONS AND SDES WITH TIME-INHOMOGENEOUS MORREY DRIFT
+19
+By (⋆), the first two terms in ˜K are smaller than (c2
+1 + c2)lρv, so they can be handled by
+selecting λ ≥ λd,p,q + (c2
+1 + c2)l. Corollary 2 applied to (40) gives us
+sup
+[r,T]×Rd |ρv| ≤ C1∥ρ1[r,T]|fn|
+1
+p ∥p + C′
+1
+√
+l∥|bn|
+1
+p ρv∥p + C2∥ρg∥W 1,p(Rd),
+where, when we were treating the last term in ˜K, we used again (⋆). However, now we have
+to deal with ∥|bn|
+1
+p ρv∥p. So, we will have to use instead a finer consequence of the solution
+representation of Corollary 2:
+1
+2
+sup
+[r,T]×Rd |ρv| + C0
+2 ∥(λ + ∂t − ∆)
+1
+2p ρv∥p
+≤ C1∥ρ1[r,T]|fn|
+1
+p ∥p + C′
+1
+√
+l∥|bn|
+1
+p ρv∥p + C2∥ρg∥W 1,p(Rd)
+(41)
+for appropriate constant C0 > 0. (Here we only need to justify that (λ + ∂t − ∆)− 1
+2 − 1
+2p′ δs=rρg is
+in Lp(Rd+1) and its norm is bounded by ∥ρg∥Lp(Rd). The proof of this fact repeats the argument
+for operator Sp from the proof of Theorem 2.) We have
+∥|bn|
+1
+p ρv∥p ≤ ∥|bn|
+1
+p (λ + ∂t − ∆)− 1
+2p ∥p→p∥(λ + ∂t − ∆)
+1
+2p ρv∥p
+(we are applying Proposition 2)
+≤ Cλ∥(λ + ∂t − ∆)
+1
+2p ρv∥p.
+Applying the last bound in (41) with l chosen sufficiently small so that C0
+2 − C′
+1Cλ
+√
+l > 0, we
+obtain (27).
+Assertion (28) follows from (27) using translations.
+Armed with (28), we now prove (29). We obtain from (⋆) that supy∈Zd ∥ρyg∥W 1,p ≤ c0∥g∥W 1,p
+for appropriate c0 > 0. It remains to show that if |f| ∈ Eq, q > 1, then
+sup
+y∈Zd ∥ρy1[r,T]|f|
+1
+p ∥p ≤ c(T − r)γ∥f∥
+1
+p
+Eq
+(42)
+for constants c = c(d, p, l) and γ = γ(q, p) > 0.
+(Actually, f in Corollary 3 satisfies (12),
+i.e. f = f1 + f2 where |f1| ∈ Eq, q > 1 and f2 is bounded on Rd+1. The term f2 is dealt with using
+the fact that ρ ∈ Lq(Rd+1), so we only need to apply (42) to |f1|.)
+
+20
+D. KINZEBULATOV
+To prove (42), we estimate
+∥ρy1[r,T]|f|
+1
+p ∥p
+p = ⟨ρp
+y1[r,T]|f|⟩ ≤
+∞
+�
+k=0
+(1 + lk2)−νp⟨1[r,T]|f|1Ck+1(r,y)−Ck(r,y)⟩
+C0(r, y) := ∅
+≤
+∞
+�
+k=1
+(1 + lk2)−νp��1[r,T]|f|1Ck+1(r,y)⟩
+≤
+∞
+�
+k=1
+(1 + lk2)−νp|Ck+1(r, y) ∩ ([r, T] × Rd)|
+1
+q′ |Ck+1|
+1
+q
+k + 1 (k + 1)
+�
+1
+|Ck+1|⟨|f|q1Ck+1(r,y)⟩
+� 1
+q
+≤ c0(d)
+∞
+�
+k=1
+(1 + lk2)−νp|T − r|
+1
+q′ (k + 1)
+d
+q′ |Ck+1|
+1
+q
+k + 1 ∥f∥Eq
+≤ c1(d, l)|T − r|
+1
+q′
+∞
+�
+k=1
+k−2νpk
+d
+q′ k
+d+2
+q −1∥f∥Eq =: cp|T − r|
+1
+q′ ∥f∥Eq,
+where c = c(d, p, l) < ∞ since, by our choice of ν in (⋆), −2νp + d
+q′ + d+2
+q
+− 1 < −1.
+□
+8. Proof of Theorem 3
+Below we follow an argument from [13] but use different embeddings, i.e. the ones established
+in Corollaries 1-3.
+The first assertion (i), i.e. for all x ∈ Rd, 0 ≤ t ≤ r ≤ T,
+⟨P t,r(x, ·)⟩ = 1,
+follows from from (27) with f = 0 and Theorem 2(i). Namely, we first show for a fixed x, using
+weight ρ, that for every ε > 0 there exists R > 0 such that ⟨P t,r
+m (x, ·)1Rd−B(0,R)(·)⟩ < ε for all
+m = 1, 2, . . . , and so ⟨P t,r
+m (x, ·)1B(0,R)(·)⟩ ≥ 1−ε. Passing to the limit in m and then in R → ∞,
+we obtain ⟨P t,r(x, ·)⟩ ≥ 1 − ε, which yields the required.
+(ii) For every n = 1, 2, . . . , let Xn
+t = Xn
+t,x denote the strong solution to the approximating
+SDE
+Xn
+t = x −
+� t
+0
+bn(s, Xn
+s )ds +
+√
+2Bt,
+x ∈ Rd,
+on a complete probability space (Ω, Ft, P), where {bn} are given by (10).
+Step 1: There exists a constant C > 0 independent of n, k such that
+sup
+n
+sup
+x∈Rd E
+� r
+s
+|bk(t, Xn
+t,x)|dt ≤ CF(r − s)
+(43)
+for 0 ≤ s ≤ r ≤ T, where
+F(h) := hγ,
+constants C and γ > 0 (from Corollary 3) are independent of n and k. Indeed, let v = vn,k be
+the solution to the terminal-value problem
+(∂t + ∆ − bn · ∇)v = −|bk|,
+v(r, ·) = 0,
+t ≤ r.
+
+PARABOLIC EQUATIONS AND SDES WITH TIME-INHOMOGENEOUS MORREY DRIFT
+21
+By Itô’s formula,
+v(r, Xn
+r ) = v(s, Xn
+s ) +
+� r
+s
+(∂tv + ∆v − bn · ∇v)(t, Xn
+t )dt +
+√
+2
+� r
+s
+∇v(t, Xn
+t )dBt,
+hence
+0 = v(s, Xn
+s ) −
+� r
+s
+|bk(t, Xn
+t )|dt +
+√
+2
+� r
+s
+∇v(t, Xn
+t )dBt.
+Taking expectation, we obtain
+E
+� r
+s
+|bn(t, Xn
+t )|dt = Ev(s, Xn
+s ).
+Since Ev(s, Xn
+s ) ≤ ∥v(s, ·)∥L∞(Rd), we obtain from Corollary 3 with fk = bk and initial data
+g = 0
+E
+� r
+s
+|bk(t, Xn
+t )|dt ≤ C(r − s)γ
+with constants C and γ > 0 independent of k, n ⇒ (43).
+By a standard result (see e.g. [10, Ch. 2]), given a conservative backward Feller evolution
+family, there exist probability measures Px (x ∈ Rd) on (D([0, T], Rd), B′
+t = σ(ωr | 0 ≤ r ≤ t)),
+where D([0, T], Rd) is the space of right-continuous functions having left limits, and ωt is the
+coordinate process, such that
+Ex[f(ωr)] = P 0,rf(x),
+0 ≤ r ≤ T.
+Here and below, Ex := EPx. Also, put {Pn
+x := (PXn)−1}∞
+n=1 and set En
+x := EPnx.
+Step 2: Ex[
+� r
+0 |b(t, ωt)|dt] < ∞.
+Indeed, by Step 1, supn supx∈Rd En
+x
+� r
+s |bk(t, ωt)|dt ≤ CF(r − s). Hence by the convergence
+result in Corollary 2 (with f := 1B(0,k)bk)
+Ex[
+� r
+s
+|1B(0,k)bk(t, ωt)|dt] ≤ CF(r − s) < ∞.
+It remains to apply Fatou’s Lemma in k.
+Step 3: For every f ∈ C2
+c (Rd), the process
+Mf
+r := f(ωr) − f(x) +
+� r
+0
+(−∆f + b · ∇f)(t, ωt)dt
+(44)
+is a B′
+r-martingale under Px.
+Indeed, let us note first that
+Em
+x [f(ωr)] → Ex[f(ωr)],
+Em
+x [
+� r
+0
+(−∆f)(ωt)dt] → Ex[
+� r
+0
+(−∆f)(ωt)dt]
+(m → ∞),
+(⋆)
+as follows from the convergence result in Theorem 2(i). Next, we note that
+Em
+x
+� r
+0
+(bm · ∇f)(t, ωt)dt → Ex
+� r
+0
+(b · ∇f)(t, ωt)dt
+(m → ∞).
+(⋆⋆)
+
+22
+D. KINZEBULATOV
+The latter follows from
+Em
+x
+����
+� r
+0
+�(bm − bn) · ∇f
+�(t, ωt)dt η(ω)
+���� ≤ C∥η∥∞∥1[0,r]|bm − bn|
+1
+p |∇f|∥p
+(a)
+as m, n → ∞;
+Em
+x
+� � r
+0
+(bn · ∇f)(t, ωt)dt · η(ω)
+�
+→ Ex
+� � r
+0
+(bn · ∇f)(t, ωt)dt · η(ω)
+�
+(b)
+as m → ∞;
+Ex
+����
+� r
+0
+�(b − bn) · ∇f
+�(t, ωt)dt η(ω)
+���� ≤ C∥η∥∞∥1[0,r]|b − bn|
+1
+p |∇f|∥p → 0
+(c)
+as n → ∞. The proof of the inequality in (a) follows the proof of (43) but uses Corollary 2
+with f = bn − bm and g = 0. The convergence in (a) follows from the fact that bn − bm → 0 in
+[L1
+loc(Rd+1)]d. Assertion (b) follows from Corollary 2. The proof of (c) is similar to the proof of
+(a) except that we pass to the limit in m and then in k using Fatou’s Lemma.
+Now, since
+Mf
+r,m := f(ωr) − f(x) +
+� r
+0
+(−∆f + bm · ∇f)(t, ωt)dt
+is a B′
+r-martingale under Pm
+x ,
+x �→ Em
+x [f(ωr)] − f(x) + Em
+x
+� r
+0
+(−∆f + bm · ∇f)(t, ωt)dt
+is identically zero on Rd,
+and so by (⋆), (⋆⋆)
+x �→ Ex[f(ωr)] − f(x) + Ex
+� r
+0
+(−∆f + b · ∇f)(t, ωt)dt
+is identically zero in Rd.
+Since {Px}x∈Rd are determined by a Feller evolution family, and thus constitute a Markov
+process, we can conclude (see e.g. the proof of [20, Lemma 2.2]) that Mf
+r is a B′
+r-martingale
+under Px.
+Step 4: {Px}x∈Rd are concentrated on (C([0, T], Rd), Bt).
+By Step 3, ωt is a semimartingale under Px, so Itô’s formula yields, for every g ∈ C∞
+c (Rd),
+that
+g(ωt) − g(x) =
+�
+s≤t
+�g(ωs) − g(ωs−)
+� + St,
+(45)
+where St is defined in terms of some integrals and sums of (∂xig)(ωs−) and (∂xi∂xjg)(ωs−) in s,
+see [5, Sect. 2] for details. Now, let A, B be arbitrary compact sets in Rd such that dist(A, B) > 0.
+Fix g ∈ C∞
+c (Rd) that separates A, B, say, g = 0 on A, g = 1 on B. Set
+Mg
+t := g(ωt) − g(x) +
+� t
+0
+(−∆g + b · ∇g)(ωs)ds,
+Kg
+t :=
+� t
+0
+1A(ωs−)dMs.
+
+PARABOLIC EQUATIONS AND SDES WITH TIME-INHOMOGENEOUS MORREY DRIFT
+23
+In view of (44) and (45), when evaluating Kg
+t one needs to integrate 1A(ωs−) with respect to
+St, however, one obtains zero since (∂xig)(ωs−) = (∂xi∂xjg)(ωs−) = 0 if ωs− ∈ A. Thus,
+Kg
+t =
+�
+s≤t
+1A (ωs−) g(ωs) +
+� t
+0
+1A(ωs−)
+�−∆g + b · ∇g
+�(ωs)ds
+=
+�
+s≤t
+1A (ωs−) g(ωs).
+Since Mg
+t is a martingale, so is Kg
+t . Thus, Ex
+��
+s≤t 1A(ωs−)g(ωs)
+� = 0. Using the Dominated
+Convergence Theorem, we further obtain Ex
+��
+s≤t 1A(ωs−)1B(ωs)
+� = 0, which yields the re-
+quired. (By the way, this construction, in a more general form, can be used to control the jumps
+of stable process perturbed by a drift, see [5].)
+We denote the restriction of Px from (D([0, T], Rd), B′
+t) to (C([0, T], Rd), Bt) again by Px, and
+thus obtain that for every x ∈ Rd and all f ∈ C2
+c (Rd)
+Mf
+r = f(ωr) − f(x) +
+� r
+0
+(−∆f + b · ∇f)(t, ωt)dt,
+ω ∈ C([0, T], Rd),
+is a Br-martingale under Px.
+Thus, Px is a Br-martingale solution to (19). (Alternatively, we could have used a tightness
+argument, cf. [13].)
+To show that Px is a weak solution it suffices to show that Mf
+r is also a martingale for
+f(x) = xi and f(x) = xixj, which is done by following closely [15, proof of Lemma 6] and
+employing weight ρ and (27) in Corollary 3.
+(iii) This follows from Corollary 2, cf. proof of (ii) above.
+(iv) Suppose that there exist P1
+x, P2
+x, two martingale solutions to (19), that satisfy
+Ei
+x
+� T
+0
+|b(r, ωt)h(t, ωt)|dt ≤ c∥1[0,T]b
+1
+p h∥p,
+h ∈ Cc(Rd+1)
+(46)
+with constant c independent of h (i = 1, 2). Here and below, E1
+x := EP1x, E2
+x := EP2x. We will
+show that for every F ∈ Cc(Rd+1) we have
+E1
+x[
+� T
+0
+F(t, ωt)dt] = E2
+x[
+� T
+0
+F(t, ωt)dt],
+(47)
+which implies P1
+x = P2
+x.
+Proof of (47). Let un ∈ C([0, T], C∞(Rd)) be the solution to
+(∂t − ∆ + bn · ∇)un = F,
+un(T, ·) = 0,
+(48)
+where, recall, bn = 1nb, and 1n is the indicator of {|b| ≤ n}. Set τR := inf{t ≥ 0 | |ωt| ≥ R},
+R > 0. By Itô’s formula
+Ei
+xun(T ∧ τR, ωT∧τR) = un(0, x) + Ei
+x
+� T∧τR
+0
+F(t, ωt)dt
++ Ei
+x
+� T∧τR
+0
+�(b − bn) · ∇un
+�(t, ωt)dt
+(49)
+
+24
+D. KINZEBULATOV
+(i = 1, 2). We have
+����Ei
+x
+� T∧τR
+0
+�(b − bn) · ∇un
+�(t, ωt)dt
+���� ≤ Ei
+x
+� T∧τR
+0
+�|b|(1 − 1n)|∇un|
+�(t, ωt)dt
+(we are applying (46))
+≤ c∥1[0,T]×B(0,R)|b|
+1
+p (1 − 1n)|∇un|∥p.
+At this point we note that ˜un(t) := eλ(T−t)un(t) satisfies
+(λ + ∂t + ∆ + bn · ∇)un = 1[0,T]eλ(T−t)F.
+Hence we can apply to |b|
+1
+p |∇un| the solution representation of Corollary 2 (after reversing time
+and taking there g = 0).
+Using |b| ≥ |bn|, we then obtain an independent on n Lp(Rd+1)
+majorant on |b|
+1
+p |∇un|. Therefore, since 1 − 1n → 0 a.e. on Rd+1 as n → ∞, we have
+Ei
+x
+� T∧τR
+0
+�(b − bn) · ∇un
+�(t, ωt)dt → 0
+(n → ∞).
+We are left to note, using again Corollary 2, that solutions un converge to a function u ∈
+C([0, T], C∞(Rd)). Therefore, we can pass to the limit in (49), first in n and then in R → ∞, to
+obtain
+0 = u(0, x) + Ei
+x
+� T
+0
+F(t, ωt)dt
+i = 1, 2,
+which gives (47).
+□
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+26
+D. KINZEBULATOV
+Université Laval, Département de mathématiques et de statistique, Québec, QC, Canada
+

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